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Gray, C. G.
Theory of molecular fluids. — (The international
series of monographs on chemistry; 9)
Vol 1: Fundamentals
1. Fluid dynamics
I. Title II. Gubbins, K. E. III. Series
532'.05 QC151
ISBN 0-19-855602-0
Library of Congress Cataloging in Publication Data
Gray, C. G.
Theory of molecular fluids.
(The International series of monographs on chemistry; 9)
Includes bibliographies and index.
Contents: v. 1. Fundamentals.
1. Molecular theory. 2. Fluids. I. Gubbins,
Keith E. II. Title. III. Series.
OD461.G7178 1984 541'.0422 83-23767
ISBN 0-19-855602-0 (t>. 1)
Filmset and printed in Northern Ireland by The Universities Press (Belfast) Ltd.


. PREFACE The theory of molecular fluids (e.g. N2, H2O) must account for qualita- qualitatively new effects that are not present for atomic fluids (e.g. Ar); for example, the orientational correlations between molecules affect diffrac- diffraction patterns, give rise to new types of phase changes and dielectric effects, and cause alignment of molecules at liquid-gas and liquid-solid interfaces. Statistical mechanical theories of molecular fluids have de- developed rapidly in the last ten years. However, existing books cover the theory of atomic, rather than molecular fluids. The present volumes were written to fill this gap in the literature. We restrict ourselves to equilibrium properties, both to keep the book within reasonable bounds and because progress has been less rapid for dynamic properties. We deal with gases and liquids, but the emphasis is on liquids since some of the topics on gases are dealt with in other books. An underlying theme is the role played by the various sorts of inter- molecular forces in determining the different fluid properties. The scope is restricted to small nearly rigid molecules, for example N2, HC1, CO2, CH4, and H2O. The treatment is mainly classical, but we include lowest- order quantum corrections where appropriate. The books are aimed at beginning graduate students and research workers in chemistry, physics, and engineering. We assume an under- undergraduate knowledge of statistical mechanics, thermodynamics, elementary electromagnetic theory, and vector analysis. A few sections require some knowledge of quantum mechanics. Parts of the books have been used in graduate courses at Cornell, Guelph, Berkeley, and Florida. The books should be useful to experimentalists, as well as theorists, as we have included detailed derivations, tables of numerical results, and (in Volume 2) comparisons of theory and experiment. Volume 1 is devoted to the basic principles. In the introductory chapter we discuss critically the basic assumptions underlying many of the later calculations. In Chapter 2 we describe the various kinds of isotropic and anisotropic intermolecular forces that are relevant for molecular fluids. Our discussion of the isotropic forces is brief, since they have been discussed in recent books on atomic fluids; we give a detailed discussion of the anisotropic forces, however. In Chapter 3 we describe and relate the various distribution functions and correlation functions which arise in the statistical mechanics of molecular fluids. Chapters 4 and 5 contain discussions of the perturbation and integral equation theories for the pair correlation functions and thermodynamic properties used in later chapters on applications. A number of technical points are relegated to appendices following particular chapters. There are also some general appendices on
vi PREFACE useful mathematical results, spherical tensor methods, multipole moments and polarizabilities, and virial and hypervirial theorems. Volume 1 is nearly self-contained; only in a few places do we utilize results derived in Volume 2. Volume 2 is devoted to applications such as thermodynamics of pure (Chapter 6) and mixed (Chapter 7) fluids, surface properties (Chapter 8), X-ray and neutron diffraction structure factors (Chapter 9), dielectric properties (Chapter 10), and spectroscopic properties (Chapter 11). Symmetry and invariance arguments play a key role throughout the book. The reason is as follows. Intermolecular pair potentials and iso- tropic fluid pair correlation functions depend on the mutual orientation of two molecules, and are therefore invariant under coordinate rotations. The description of molecular and fluid properties in terms of invariants and other simply transforming quantities (tensors, e.g. dipole moments, polarizabilities, etc.) thus arises naturally; in addition, calculations are greatly simplified using tensor methods. Appendices A and B provide the background information on Cartesian and spherical tensors needed for these descriptions. Most of the existing books on these matters are written largely from the point of view of the quantum theory of angular momentum, so that in Appendix A we have recast some of the needed results into the more generally useful language of tensors, focussing particularly on their rotational transformation properties. ^ Guelph and Cornell C.G.G. October 1982 K.E.G.
ACKNOWLEDGMENTS Many friends and colleagues have read sections of the manuscript and offered helpful advice; we are grateful to J. A. Barker, A.D. Buckingham, P.T. Cummings, D. Henderson, C.G. Joslin, W.F. Murphy, B.G. Nickel, B.J. Orr, M. Rigby, G. Stell, D.E. Sullivan, and B. Widom. We are particularly indebted to J.S. Rowlinson for his help in planning the project, for a critical reading of the entire manuscript, and for meticulous editorial work. P.M. Gubbins exercised great care in drawing all of the figures. We thank Demetra Dentes, Kathleen Harris, and Connie York for their efficient and expert typing of the final manuscript. The financial support of the Natural Sciences and Engineering Research Council of Canada and the National Science Foundation is gratefully acknowledged. Finally, it is a pleasure to thank Oxford University Press for their patience and courtesy during the eight years the volumes were being written.
TO VIRGINIA AND PAULINE
CONTENTS VOLUME 1: FUNDAMENTALS COMMONLY OCCURRING ABBREVIATIONS xiii NOTE ON UNITS xiv 1. INTRODUCTION 1 1.1 Theory of fluids in equilibrium 1 1.2 Molecular fluids 4 1.3 Scope of this book 16 References and notes 18 2. INTERMOLECULAR FORCES 27 2.1 Model potentials 30 2.2 Isotropic and anisotropic parts of the potential 38 2.3 Spherical harmonic expansion for the potential 39 2.4 Multipole expansions 45 2.5 Induction interactions 85 2.6 Dispersion interactions 91 2.7 Overlap interactions 100 2.8 Spherical harmonic expansion of the atom-atom potential 106 2.9 Convergence of the inverse distance and spherical harmonic expansions 114 2.10 Three-body interactions 123 2.11 Generalization to nonrigid molecules, etc. 130 References and notes 131 3. BASIC STATISTICAL MECHANICS 143 3.1 Distribution functions in the canonical ensemble 145 3.2 Spherical harmonic expansion of the pair correlation function 180 3.3 Distribution functions in the grand canonical ensemble 187 3.4 Hierarchies of equations for the distribution functions 192 3.5 Distribution functions for mixtures 204 3.6 Density expansions of the correlation functions and thermo- dynamic properties 206 Appendix 3A Relation between the canonical and angular momenta 210 Appendix 3B Spherical harmonic form of the OZ equation 212 Appendix 3C Volume derivatives and the intensivity condition 219 Appendix 3D The classical limit and quantum corrections 224 Appendix 3E Direct correlation function expressions for some thermodynamic and structural properties 237 References and notes 240 4. PERTURBATION THEORY 248 4.1 Brief historical outline , 248 4.2 A simple example: the u-expansion for free energy 254 4.3 General expansion for the angular pair correlation function 256 4.4 General expansion for the free energy 257 4.5 Further development of the u -expansion 258
x CONTENTS 4.6 The /-expansion 274 4.7 Expansions for non-spherical repulsive potentials 284 4.8 Non-spherical reference potentials 289 4.9 Effective central potentials 300 4.10 Perturbation theory for non-additive potentials 304 4.11 Conclusions 315 Appendix 4A Perturbation expansion for a general property (B) 317 References and notes 328 5. INTEGRAL EQUATION METHODS 341 5.1 The PY approximation for hard spheres 344 5.2 Spherical harmonic form of the OZ equation 348 5.3 Separation of the OZ equation into reference and perturbation parts 350 5.4 Theories for the angular pair correlation function g(ra>io>2) 354 5.5 Theories for the site-site pair correlation functions g^gCr) 397 5.6 Conclusions 414 Appendix 5A Factorization method of solution of the OZ equation 415 Appendix 5B Thermodynamics of the HNC approximation 423 References and notes 426 APPENDIX A. SPHERICAL HARMONICS AND RELATED QUANTITIES 441 A.I Spherical harmonics 442 Definition 442. List of P, and Plm 444. Other notations 444. Eigenfunctions 445. Active and passive rotations 446. Rotation operator 446. Commutation relations 447. Angular gradient operator 448. Potential functions 448. Angular Laplacian 449. Examples of rlYlm 449. Recurrence relation 450. Orthogonality 450. Completeness 450. Addition theorem 451. Product rule 452. Integrals 452. Transformation under rotations 452. Inversion 454. Gradient formula 454. Example, proof of special case of gradient formula 455. Special cases of Y,m 456. List of Y,m 457. A.2 Rotation matrices 458 Definition 458. Euler angles 459. Formula for Dlmn 460. Sym- Symmetry properties of D'mn 460. Eigenfunctions 460. Dlmn as generalized (four-dimensional) spherical harmonics 462. Angular gradient and Laplacian 462. Rotation operator 462. Recurrence relations 463. Orthogonality 463. Completeness 464. Unitarity 464. Product rule 464. Integrals 465. Group representation prop- property 465. Transformation properties under rotations 466. Special cases of DL 467. d'(8) and d\8) 467. Example, rotations about Z 468. Example, D\a(}y) 469. Direction cosine matrix 470. A.3 Clebsch-Gordan coefficients 471 Definition 471. Selection rules 472. Symmetry properties 472. 3/ symbols 473. Orthogonality relations 473. Sum rules 473. Ab- Abbreviated notation 475. Special values 475. Tables and formulae 476. A.4 Tensors 476 Irreducible tensors 471. Irreducible spherical tensors 477. Exam- Example, Y,,,, as irreducible tensors 478. Example, rank-one tensors (vectors) 478. Example, rank-two tensor 478. Example, DLOT)* as
CONTENTS xi irreducible tensop 478. Reduction and contraction 479. Example, product of two vectors 479. Example, polarizability tensor 480. Example, spherical harmonic addition theorem 481. Reduction of a product of three irreducible spherical tensors 481. Example; An = 1, B12-2, C|3«i 482. Rotational invariants 483. Example, rotational invariants formed from products of three Dlmn and three Ylm 483. Generalization to noninvariants 484. Cartesian tensor analogues 485. Integrals over products of direction cosine matrices 487. Example, (QaeQys)n 487. Short-cut for last example 488. Transfor- Transformation between Cartesian and spherical components 489. Definition of Cartesian tensor 489. Reduction of second-rank Cartesian tensor 490. Reduction of third-rank Cartesian tensors, direct and spherical methods 490 and 493. T,m in terms of the TaS 492. Example, differential light scattering 493. Example, dipole-dipole interaction 496. Example, dispersion interaction 498. A.5 The / symbols 500 The 6/ symbol 500. Definition 501. Racah coefficient 501. Selec- Selection rules 502. Symmetry properties 502. Tables and formulae 503. Further sum rules 503. The 9/ symbol 504. Definition 504. Selection rules 504. Symmetry properties 505. Further sum rules 505. Example, PY approximation 506. Example, A terms in induc- induction interaction 509. Higher / symbols 510. References and notes 511 APPENDIX B. SOME USEFUL MATHEMATICAL RESULTS 517 B.I Vector and tensor notation 517 B.2 Quantum notation 522 B.3 The delta function 524 B.4 The generalized Parseval theorem 526 B.5 The Rayleigh expansion 527 B.6 Pade approximants 530 B.7 Matrices 531 References and notes 536 APPENDIX C. MOLECULAR POLARIZABILITIES 538 C.I Introduction 538 C.2 Quantum calculation of polarizabilities 541 C.3 Dispersion and induction intermolecular forces 550 C.4 Glossary of polarizability symbols 560 References and notes 561 APPENDIX D. TABLE OF MULTIPOLE MOMENTS AND <-64 POLARIZABILITIES 564 D.I The multipole moments ^E9 D.2 The polarizabilities ^74 D.3 Units rg? References and notes APPENDIX E. VIRIAL AND HYPERVIRIAL THEOREMS 602 E.I The virial theorem 603 E.2 Hypervirial theorems 607 References and notes 611 INDEX 617
VOLUME 2: APPLICATIONS Chapter 6 Thermodynamic properties of pure fluids Appendix 6A Geometry of convex bodies Appendix 6B Evaluation of perturbation theory averages Chapter 7 Thermodynamic properties of mixtures Chapter 8 Surface properties Appendix 8A Functionals Chapter 9 Chapter 10 Chapter 11 Structure factor Dielectric properties Appendix 10A Expressions for the dielectric constant Appendix 10B Perturbation expansion for the angular cor- correlation parameters G, Appendix IOC Evaluation of G, and e in RISM Appendix 10D DID approximation for collision-induced polarizability Appendix 10E Expressions for G[ in terms of g^eM Spectroscopic properties Appendix 11A Macroscopic relations for absorption Appendix 11B Classical calculation of A(&>) Appendix 11C Quantum calculation of A(&>) Appendix 11D Kramers—Kronig relations Appendix HE Quantum corrections to allowed moments
COMMONLY OCCURRING ABBREVIATIONS Abbreviation DID BBGKY CG CIA GMF (=LHNC = SSC) GMSA HNC HS LHNC (=GMF = SSC) LJ MC MD MSA oz PT PY QQ RISM SSC (= LHNC = GMF) /x/x Meaning Dipole-induced-dipole Bogoliubov, Born, Green, Kirkwood, Yvon hierarchy of equations Clebsch-Gordan coefficient Collision-induced absorption Generalized mean field theory Generalized mean spherical approximation Hypernetted chain theory Hard sphere Linearized hypernetted chain theory Lennard-Jones Monte Carlo Molecular dynamics Mean spherical approximation Ornstein-Zernike equation Perturbation theory Percus-Yevick theory Quadrupole-quadrupole interaction Reference interaction site model Single super chain theory Dipole-dipole interaction Introduced p. 311 p. 203 p. 39 p. 567 p. 379 p. 355 p. 342 Fig. 3.5 p. 380 p. 30 Table 4.1 Table 4.4 p. 355 p. 184 Fig. 4.6 p. 342 ¦ Fig. 3.5 p. 398 p. 379 p. 375
NOTE ON UNITS In dealing with molecular properties and intermolecular forces and their effect on fluid properties we shall be concerned almost entirely with electrostatics; electromagnetic forces and macroscopic circuit problems play only a very minor role. We have therefore found it convenient to use the electrostatic system of units rather than the rationalized Systeme International (SI)t in our treatment of these topics (particularly in Chap- Chapters 2 and 10, and in Appendices C and D). In Chapter 11 we use the Gaussian system (which derives naturally from the esu system) when dealing with light absorption and scattering. This makes for a more compact notation, since the factors of Atre0 that occur in the SI system are avoided. In Appendix D.3 the relations among the electrostatic (esu), SI, and atomic units (au) systems are discussed in some detail. t For a detailed discussion of various systems of units see: Jackson, J. D. A975), Classical electrodynamics, 2nd edn Appendix, Wiley, New York; Marion, J. B. and Heald, M. A. A980), Classical electromagnetic radiation, 2nd edn p. 2 and Appendix D, Academic, New York; Danloux-Dumesnils, M A969), The metric system, University of London Athlone Press.
1 INTRODUCTION The first problem of molecular science is to derive from the observed properties of bodies as accurate a notion as possible of their molecular constitution. The knowledge we may gain of their molecular constitution may then be utilized in the search for formulas to represent their observable properties. A most notable achievement in this direction is that of van der Waals, in his celebrated memoir, . . . J. Willard Gibbs, Proc. Am. Acad. 16, 458 A889). 1.1 Theory of fluids in equilibrium The application of statistical mechanics to the study of fluids over the past fifty yearst or so has progressed through a series of problems of gradually increasing difficulty. The first and most elementary calculations were for the thermodynamic functions (heat capacities, entropies, free energies, etc.) of perfect gases. These properties are related to the molecular energy levels, which for perfect gases can be determined theoretically (by quantum calculations) or experimentally (by spectroscopic methods, for example). For simple molecules (CO2, CH4, etc.) the energy levels, and hence the thermodynamic properties, can be determined with great accuracy, and even for quite complex organic molecules it is now possible to obtain thermodynamic properties with satisfactory accuracy.1 With the advent of digital computers it became possible to calculate ther- thermodynamic properties for a wide variety of substances and temperatures, and several useful tabulations of perfect gas properties now exist.2'3 Having successfully treated the perfect gas, it was natural to consider gases of moderate density, where intermolecular forces begin to have an effect, by expanding the thermodynamic functions in a power series (or virial series) in density. Although the mathematical basis for a theoretical treatment of this series was laid by Ursell4 in 1927, it was not exploited until ten years later, when Mayer5 re-examined the problem.6 Since that time a great deal of effort has been put into evaluating the virial coefficients that appear in the series for a variety of intermolecular force models.7 As the expressions for the virial coefficients are exact, they provide a very useful means of checking such force models by comparison of calculated and experimental coefficients. tThe earlier history, including the classic work of van der Waals, is reviewed in the historical references given at the end of the chapter.
2 INTRODUCTION 1.1 While the theory of dilute gases at equilibrium is essentially complete,11 this is far from being the case for all dense gases and liquids. The virial series cannot be applied directly13 to liquids.20'3 As an alternative to the 'dense gas' approach to liquids, there were early attempts to treat liquids as disordered solids by using cell or lattice theories;24^ these were popular from the mid-1930s until the early 1960s. The problem of calculating the partition function was simplified by assuming that the liquid was made up of a series of cells. In the simplest of these models, due to Lennard-Jones and Devonshire,27 the liquid volume was assumed to be spanned by a lattice, and each molecule was confined to its cell by the repulsive forces of its neighbours, Such models suffer from a number of defects, including a large number of adjustable parameters and a failure to predict a sufficiently large entropy. A variety of modifications have been intro- introduced in an attempt to improve these models; none is entirely satisfac- satisfactory, although they may provide a useful theory for special situa- tions20-28-29 (e.g. melting). An alternative and more fundamental approach to liquids is provided by the integral equation methods20'30'31 (sometimes called distribution function methods), initiated by Kirkwood and Yvon in the 1930s. One starts by writing down an exact equation for the molecular distribution function of interest, usually the pair function, and then introduces one or more approximations to effect a solution. These approximations are often motivated by considerations of mathematical simplicity, so that their validity depends on a posteriori agreement with computer simulation or experiment. The result is an integral equation which must be solved numerically in most cases. The theories in question include those of YBG (Yvon, Born, and Green), PY (Percus and Yevick), and the HNC (hyper- netted chain) approximation. They provide the distribution functions directly, and are thus applicable to a wide variety of properties. A particularly successful approach to liquids has been provided by the thermodynamic perturbation theories.20-31~3 In this approach the proper- properties of the fluid of interest, in which the intermolecular potential energy is °U say, are related to those for a reference fluid34 with potential °U0 through a suitable expansion. One attempts to choose a reference system that is in some sense close to the real system, and whose properties are well known (e.g. through computer simulation studies or an integral equation theory). Although the statistical mechanical basis for these theories was laid in papers published in the early 1950s,35~7 successful forms of the theory were not developed until the late 1960s.38'39 They are capable of giving excellent quantitative results for liquids and dense gases, except in the vicinity of the critical point. For molecular liquids a number of problems still remain to be worked out in detail, e.g. the calculation of the liquid pair correlation function for strong anisotropic potentials (H2O, etc.); see Chapters 4 and 5.
1.1 THEORY OF FLUIDS IN EQUILIBRIUM 3 Finally we mention the scaled-particle theory ,40~3 which was developed about the same time as the Percus-Yevick theory. It gives good results for the thermodynamic properties of hard molecules (spheres4041 or convex molecules42-43). It is not a complete theory (in contrast to the integral equation and perturbation theories) since it does not yield the molecular distribution functions (although they can be obtained for some finite range of intermolecular separations20-44). However, it does provide a useful and quite accurate equation of state for hard convex molecules (see Chapter 6). This brief discussion of the theory of fluids would not be complete with- without some mention of computer simulation studies,20'3 lA5~9 which played an important role in the development of the theory. Simulations50 can be carried out by either the Monte Carlo or molecular dynamics methods.5' In the Monte Carlo method one employs random numbers to generate a sequence of molecular configurations distributed canonically (for example), and then obtains property estimates by calculating the arithmetic mean over these configurations. The Monte Carlo method is suitable for treating static properties only, but has the advantage that a variety of different ensembles can be used, e.g. the canonical (T, N, V fixed), the grand canonical (T, jx, V fixed), or the isobaric (T, N, p fixed). In the molecular dynamics method the Newton equations for the transla- tional and rotational motions are solved for each molecule. Physical properties are obtained by time-averaging the appropriate function of the molecular positions, orientations, velocities, and angular velocities, Molecular dynamics has the advantage that both static and time- dependent properties are obtained, but its use is restricted (see, however, ref. 51) to the microcanonical ensemble (E, N, V fixed). In certain situations, e.g. the study of gas-liquid or liquid-liquid equilibria, where one needs to calculate the free energy, this is a disadvantage, and it may be easier to use the Monte Carlo method with the isobaric or grand canonical ensembles. In both the Monte Carlo and molecular dynamics methods the number of molecules used is usually in the range 100-1000. However, this seems sufficient to provide an accurate estimate of fluid properties, except perhaps for surface properties52 and properties in the critical region. Computer simulations play a role that is intermediate between theory and experiment (see Fig. 1.1) By comparing theory with simulation results it is possible to eliminate any uncertainties as to the form of the intermolecular potential energy, which is the same in both theory and simulation. In this role, the simulation provides 'experimental data' on a precisely denned model fluid, against which the theory can be tested. (In contrast, comparisons between theory and experiment test jointly the theory and the intermolecular potential model, and therefore fail to provide a conclusive test of the theory.) We shall frequently make use of
INTRODUCTION 1.2 Experiment Theory and potential Potential Theory Computer simulation Fig. 1.1. Methods for testing theoretical approximations and intermolecular potential models. simulation data for tests of theories throughout this book. A second use of computer simulation data is in comparisons with experimental data for real fluids, in order to test the intermolecular potential model.53 In this role the simulation provides the 'theory' which is tested against experi- experiment. By continually refining the potential and comparing the simulation cesults with experiment, one hopes to end up with a realistic model for the potential. In addition to the above, computer simulations can be used to obtain results which are difficult or impossible to obtain experimentally; exam- examples54 are the static angular pair correlation function for molecular liquids,47 and the density (and orientation) profile at liquid-vapour527172 and liquid-solid73 interfaces. Similarly, difficult theoretical questions can be studied to advantage with simulation methods, e.g. ergodicity45 (by comparing Monte Carlo and molecular dynamic results), the approach to* equilibrium,5560 the N-dependence (where N is the number of molecules) and ensemble dependence of thermal averages, and the dependence of structural and thermodynamic properties on the long- and short-range parts of the intermolecular potential. 1.2 Molecular fluids Early work (before about 1970) on the theory of dense fluids dealt almost exclusively with simple atomic fluids, in which the intermolecular forces are between the centres of spherical molecules and depend only on the separation distance r (Fig. 1.2(a)). Examples of such simple fluids are the inert gases, the alkali metals, and certain molten salts. Throughout this book we shall deal with fluids composed of relatively small but non- spherical molecules. In such fluids the intermolecular forces depend on the molecular orientations, vibrational coordinates, etc., in addition to r.
1.2 MOLECULAR FLUIDS (a) X x Fig. 1.2. (a) The intermolecular potential u(r) between two spherical molecules depends only on their separation r. (b) The intermolecular potential u(raIaJ) for two non spherical molecules depends on their separation r and on their orientations ti, and u>2. Here o>f represents the set of angles which give the orientation of the body-fixed axes (x^Zj) relative to the space-fixed axes {XYZ). In the 'intermolecular frame' Z is chosen to point along r. The dependence of the intermolecular forces on the molecular orienta- orientations leads to qualitatively new features in the liquid properties, when compared to atomic liquids. The angular correlations between the molecules are manifested in both the structural and thermodynamic properties. For example, additional peaks occur in the X-ray and neutron structure factors (Chapter 9), and the orientational energy and entropy can influence strongly the phase diagrams of pure fluids and mixtures (Chapters 6 and 7). For example, some binary mixtures exhibit a 'lower critical mixing temperature', as well as the usual upper critical mixing temperature. As further examples we mention the mean-squared torque on a molecule, which vanishes for purely central forces (see Chapter 11), and orientational polarization of molecules at surfaces (Chapter 8). In most of the subsequent development we shall make use of three basic approximations, and we discuss these briefly in what follows. These approximations are (a) the rigid molecule assumption, (b) classical treat- treatment of the translational and rotational motions, and (c) pairwise additiv- ity of the intermolecular forces. 1.2.1 The rigid molecular approximation In the 'rigid molecule approximation' it is assumed that the intermolecu- intermolecular potential energy °U(rN<t>N) depends only on the positions of the centres of mass rN=r,r2...rN for the N molecules and on their orientations a>N = a>1aJ •• <^n- This implies that the vibrational coordinates of the molecules are dynamically and statistically independent74 of the centre of mass and orientation coordinates, and, that internal rotations75 are either absent, or are independent of the rN and a)N coordinates. The molecules are also assumed to be in their ground electronic states. The rigid
INTRODUCTION 1.2 molecule approximation eliminates from consideration long-chain molecules having relatively free internal rotations, particularly polymers. These assumptions should be quite realistic for many fluids composed of small molecules such as N2, CO, CO2, Br2, CH4, C6H6, etc. The assumption that vibrational states are independent of the rotational and translational states should not lead to serious error for molecules in which the separation between vibrational energy levels greatly exceeds both kT and the separation between rotational levels; this is usually the case for simple molecules.7677 The characteristic vibrational temperature 6V = ha>/k = hv/k where v is the vibrational frequency in s, is given in Table 1.1 for some cases of interest; here h = h/2-ir, h is Planck's constant, and k is the Boltzmann constant. When TS0v the population of the excited vibrational energy levels will be appreciable. 0V is seen to be large in most Table 1.1 Relevant quantities for estimating the importance of quantum effects in \ liquids at their normal boiling points t Liquid He Ne Ar Xe H2 HF HCI HBr N2 CO o2 Cl2 h co2 H2O CH4 CF4 NH, N2O CC14 SF6 QHf, T/K 4.2 27.1 87.2 165.0 20.4 292.5 188.1 206.7 77.3 81.6 90.1 239 457.5 216.6§ 373.1 111.6 145.1 239.7 184.7 349.9 222.3§ 353.2 ejKt — — — 6215 5960 4227 3814 3374 3100 2274 810 310 954 2290 1890 629 1360 850 310 523 582 A*t 2.67 0.593 0.186 0.063 1.73 0.199 0.144 0.081 0.226 0.220 0.201 0.067 0.024 0.011 0.021 0.239 0.080 0.186 0.105 0.033 0.046 0.045 A*/BttT*)' 1.708 0.271 0.087 0.029 0.930 0.074 0.058 0.035 0.100 0.097 0.092 0.033 0.010 0.004 0.006 0.110 0.033 0.093 0.042 0.013 0.017 0.020 0r/KH — 85.3 30.2 15.02 12.02 2.88 2.77 2.07 0.351 0.054 0.561 40.1 7.54 0.275 13.6 0.603 0.0823 0.131 0.27 er/T — 4.18 0.1032 0.0799 0.0582 0.0373 0.0339 0.0230 0.00147 0.00012 0.00259 0.108 0.0676 0.00189 0.0567 0.00326 0.00024 0.00059 0.00076 i Here T* = kT/e, A* = hl<j(me)\ 0V= hv/k where v is the vibrational frequency in s ', and 0, = hcBJk where Be is the usual rotational constant in cm. + fl^ values from refs. 26, 76-78. For the polyatomic molecules, which have more than one vibrational mode, only the lowest 0V is listed. § Triple-point temperature. H From refs. 8, 26, 76-78. For the symmetric and asymmetric top molecules, which have two and three principal moments of inertia respectively, we have listed only the highest 8r.
1.2 MOLECULAR FLUIDS '' Table 1.2 Shift in vibration frequency for various substances dissolved in liquid argon at 90-100 K and in carbon tetrachloride at room temperature (from ref. 91)t Frequency shift Substance Mode In At In CC14 H2 CO HC1 N7O co2 cs2 CH4 CF4 C2H4 (CH3JO vb(OCH) 4155.1 2143.3 2886.0 1284.9 588.8 2223.8 667.4 2349.2 1535.4 730.3 3288.4 3019.5 1306.2 1260.9 1282.6 949.3 947.9 614.5 1102 1179 -10 -5.3 -15 -1.4 -1.2 -3.3 -2.3 -6.2 -6.9 2.7 -4.4 -5.5 0 -2.9 -7.6 -0.7 -8.3 -0.8 -1.5 -6.0 -23 -8 -56 -3.6 -2.8 -7.8 -13 -14 -26 -12 -5 t Here v = v/c = 1/A = frequency in cm is the wavelength in cm. , where v is the frequency in s ', and A cases; for these cases almost all molecules will be in their ground vibrational states. Exceptions are I2, CC14, and SF6. That the vibrational states are not completely unaffected by the transla- tional or rotational states is shown by experimental studies of bond vibrational frequencies,79'80 relaxation times,81 bond lengths,84 and con- formational changes8 in dense fluids. However, these effects are usually small for the molecules to be considered here. Tables 1.2 and 1.3 show typical results of spectroscopic studies of the effect of density on bond vibrational frequencies; vibrational frequencies in the liquid are at most only a few per cent different from those in the gas. Although small, these vibrational perturbations can sometimes give rise to observable thermodynamic effects, for example an 'inverse isotope effect' (see § 1.2.2). Bond lengths in the liquid or solid phase can be determined by neutron or X-ray diffraction measurements carried out at large values of momentum transfer. These liquid-phase bond lengths84 appear to be at most only two or three per cent different from the vapour-phase values
8 INTRODUCTION 1.2 Table 1.3 Shift in vibration frequency for the a{ C-H vibration for pure substances on going from the gas to the liquid phase (from ref. 92) Substance Frequency shiftt (j/h() —i>sas)/cm ' CHC13 -13.5 CH2C12 -12.0 CH2Br2 -18.0 CH3C1 -10.0 CH3I -22.0 cis C2H2C12 -10.0 cyclo C5HIC, -11.5 tThe value of v^ for the C-H bond is about 2980 cm, the exact value depending on the molecule considered. determined by electron diffraction. Thus, for liquid bromine neutron diffraction84 yields bond lengths ranging from 2.26 to 2.32 A, while the gas value is 2.28 A. Effects due to non-rigidity of the molecules can also be studied theoretically808893~5 and by computer simulation. Such studies have been made for H2O, NH3, HF, CH4, N2, CO, Br2 and alkanes94-9^100 (the case of H2O is discussed in § 5.5.4). The rigid molecule approximation implies that the intermolecular po- potential energy u(ro>1a>2) for a pair of molecules depends on the vector r = (rQ4>) from the centre of molecule 1 to the centre of molecule 2, and on the orientations o>i and <o2 of the molecules relative to some space- fixed set of axes (Fig. 1.2(b)).un Here o>; =6^ for linear molecules (e.g. CO2, N2, HC1) or <k0iXi for non-linear molecules (e.g. H2O, CC14). For linear molecules 0( and 4>i are simply the polar angles (Figs 2.6 and A.I), while for non-linear molecules the Euler angles (see Figs 2.5 and A.6) are usually used. It is often convenient to adopt the 'intermolecular frame' or 'r-frame', in which the space-fixed Z-axis is chosen to point along r; thus 0 -0. We shall write the molecular orientations in this frame as co[ to distinguish them from the orientations o>j for an arbitrary space-fixed frame. In Fig. 2.9 the intermolecular frame is illustrated for the particular case of two linear molecules. 1.2.2 The classical approximation A second basic assumption made is that the fluids behave classically. For rigid molecules one expects quantum effects in the translational and rotational motions of the molecules to become important for light molecules and/or at low temperatures. Thus some molecules containing hydrogen (e.g. HF, HC1, HD, CH4, H2O, NH3, and particularly H2) should exhibit quantum effects at sufficiently low temperatures.
1.2 MOLECULAR FLUIDS 9 Translatiohal quantum effects are of two types,8 (i) diffraction effects102 which result from the wave nature of the molecules, and (ii) symmetry effects which result from the boson or fermion character of the molecules. The symmetry effects arise for identical molecules, and are negligible for all applications that we shall consider. In a perfect gas such symmetry effects are negligible provided that the molecular thermal de Broglie wavelength103 At is much smaller than the mean separation between neighbouring molecules. The latter is of order p~\ where p is the fluid number density, so that we have A,«p-i A.1) for a perfect gas when symmetry effects are negligible. Here A, is given by where m is the molecular mass. However, the molecules of real gases and liquids have hard cores which suppress104 the symmetry effects in almost all cases of practical interest except108 for He. The hard cores prevent the molecules from getting sufficiently close to 'notice' their ferinion or boson character. For example,105 even in (gaseous) H2 the effect on the pair correlation function is negligible until T=s5K. Rotational quantum effects are also of two types, (i) kinetic energy effects due to the discrete nature of the rotational energy levels of a molecule; closely related are the intramolecular exchange effects (e.g. the ortho and para hydrogen species), (ii) intermolecular potential energy effects which tend to quan- quantize the motion of a hindered rotator. A quantitative treatment of the quantum effects for equilibrium109 properties can be obtained by expanding the molecular distribution functions and the partition function (and hence the thermodynamic properties) in powers of ft (the Wigner-Kirkwood expansion8-31'78112). If fermion-boson effects can be neglected, and if the repulsive core of the intermolecular potential is not infinitely hard, the series contains only even powers of ft.113 The partition function Q, for example, is given by112 Q = Qd[l + O(ft2)] A.3) where Qd is the classical partition function and O(h2) is the order-ft2 fractional correction. For N linear molecules this is given by °(ft)~ 24(kTKm where I is the moment of inertia (with respect to the centre of mass), and (F2) and (t2) are the classical mean-squared force and torque (about the
10 INTRODUCTION 1.2 centre of mass) on a molecule. Similar expressions can be derived for spherical top, symmetric top and asymmetric top molecules (see Appen- Appendix 3D and Chapter 6). The series A.3) is useful at not too low temperatures; at low temperatures a full quantum treatment,781146 a non-perturbative approximation method, or an ^-series acceleration technique (see Chapter 6 for references) must be used. The three terms on the right-hand side of A.4) correspond to the translational diffraction effect, the rotational potential energy effect, and the rotational kinetic energy effect, respectively. It is possible to give a qualitative explanation of these three terms as follows. We first consider the translational diffraction effect. For simplicity, we view the complicated translational motion of the molecule in a dense liquid as having a large vibrational component. The vibrational motion can be treated classically if the vibrational energy level spacing heo (or the zero-point energy for molecules in their ground state) is small compared to kT, where <o ~ («/mM is the vibrational frequency and k~(V2<%) is the mean force 'constant'.117 Here °U is the total intermolecular potential energy, Vj the gradient operator with respect to the position of a particular molecule, and {. . .) denotes a classical thermal average. Using the classical hyper- virial relation118 <V^) = (kT)-1((V1%J> A.5) to eliminate the mean Laplacian of the potential, one can write the classical condition ha>/kT« 1, or (ha>lkTJ« 1, as A.6) (kTfm where (F2) = ((V^J) is the mean squared force on a typical molecule (e.g. number 1). This condition corresponds119 to the first term on the right-hand side of A.4), apart from numerical factors. An alternative form of the condition A.6) is obtained by noting that diffraction effects will be negligible if the de Broglie wavelength At is much less than the molecular diameter <x, where A, is given by A.2). In terms of the dimensionless de Broglie wavelength A* = /t/<x(me)' (the so-called de Boer parameter8), where e is the well-depth of the isotropic part of the intermolecular potential, the classical condition is A* A.7) B7TT*)i where T* = kT/e is the reduced temperature. Using the estimate (F2)~ (s/(jJ in A.6), together with the fact that T*~l in liquids, we see that A.6) reduces to A.7) apart from numerical factors. The ratio A*/BTrT*)i
1.2 MOLECULAR FLUIDS 11 is given in Table 1.1 for some liquids at their normal boiling points, and is seen to be small (though not always negligible) in all cases except for He, H2, and Ne. Similar qualitative arguments can be used for the rotational potential energy effect. For simplicity we view the complicated rotational motion of the molecule in a dense liquid as having a large librational component. The quantum effects in the librational motion should be small if the librational energy level spacing h<o (alternatively, the zero-point energy) is small compared to kT, where <o ~ (k/7I is the frequency, with I the moment of inertia and k ~(Sl,fll) the mean torque constant. Here V<O] is the angular gradient operator120 about the centre of mass of a typical molecule. Using the classical hypervirial relation118 <V2v%) = (fcT)-1<(V<1)l%J) A.8) we can eliminate the mean angular Laplacian of °U and write the classical condition ha>lkT«l, or (ha>lkTJ«l, as (kTKI where (t2) = ((V^fllJ) is the mean squared torque on a molecule (about the centre of mass). This condition corresponds to the second term119 on the right-hand side of A.4). Using the estimate (see Chapter 11) (V^fli) — ea, where ea is the strength of the anisotropic intermolecular potential for r~~a, we can approximate A.9) as (h2/IkT)(eJkT)«l. A.10) The factor (h2/IkT) is usually small (see below), so that rotational potential energy effects become important only for very strong aniso- anisotropic forces.121 A case in point28122123 may be liquid H2O at room temperature. The librational frequencies are of the order 500 cm, so that hh)lkT« 1 is clearly not satisfied. Rotational quantum corrections are also non-negligible for gaseous H2O and NH3 (see below). Finally, rotational kinetic energy effects will become important when kT is of the order of the spacing between low-lying rotational energy levels, which are given by Ej =(h2/2I)J(J+l) for a linear or spherical top molecule. The level spacing is thus of order h2/2I, so that the classical condition ft2/2I« fcT can be written as t,2 A.11) 2IkT which corresponds to the third term on the right in A.4). The quantity124 6r=h2/2Ik is thus a characteristic rotational temperature and is listed in
12 INTRODUCTION 1.2 Table 1.1. The condition A.11) can be written as 0r/T«l, A.12) and 6r/T is also listed in Table 1.1. The condition A.12) is satisfied at the boiling point except for H2. From Table 1.1 we have seen that quantum effects are usually expected to be small for molecular liquids, but can be significant at low tempera- temperatures. As a simple example,125 the h2 quantum correction to the Helm- holtz free energy calculated from A.3) is 10 per cent for liquid Ne near the triple point. The quantum correction given by A.3) is at constant density. The corresponding corrections to the pressure, and the liquid- vapour coexistence curves,125 can of course be larger than 10 per cent. The traditional method of studying quantum effects in the thermodynamic properties of liquids is by isotope substitution experiments,12*^8 e.g. for vapour pressures, liquid volumes, heats of mixing, solubilities, and surface tension. According to classical statistical mechanics, different isotopes, which have the same intermolecular forces (neglecting (i) the difference produced by averaging over slightly different vibrational motions, and (ii) the extremely small non-adiabatic corrections to the electronic ground- state energy), should have the same configurational properties, since the mass and moment of inertia of the molecules do not appear in the configurational partition function. Quantum mechanically, on the other hand, the partition function does not factorize into kinetic and configura- configurational parts, so that the mass and moment of inertia enter in a non-trivial way. The results A.3) and A.4) show explicitly the dependence on m and I in the 'high temperature approximation'. The quantum effects on vapour pressure are the most studied.126128 Classically the composition of a vapour of a binary isotope mixture would be the same as that of the liquid with which it is in equilibrium. The configurational contribution to the vapour/liquid equilibrium is unaffected by isotope substitution for the reason discussed above; the kinetic contribution is also unaffected, since the velocities of the molecules are affected equally on the vapour and liquid sides of the interface. The isotope separation factor a,127 which is essentially the ratio of the vapour composition to the liquid composition, thus differs from unity entirely due to quantum effects (assuming rigid molecules). For rigid molecules simple considerations of zero-point energy lead one to expect the lighter isotope to be the more volatile.129 This is often found in practice; when an 'inverse isotope effect' is found, it is attributed to perturbations of the vibrational motions of the molecules due to intermolecular forces130 in the liquid. As a numerical example,127 a for 14N2/15N2 has been calculated at T = 70K using an atom-atom potential, with the result a = 1 + 0.008 + 0.004-0.002, where the second
1.2 MOLECULAR FLUIDS 13 and third te^ms arise from the force and torque terms in A.4) respec- respectively, and the last term, which has the opposite sign, arises from vibrational perturbations. For molecular gases, quantum corrections to the second virial coeffi- coefficient B and to the second dielectric virial coefficient have been calculated.131 For H2O and NH3 for example,132135 calculations using Stockmayer potentials give quantum corrections to B of about 5-10 per cent at room temperature. For these molecules the corrections are predominantly rotational, due to the small moments of inertia. We conclude that the quantum effects in the thermodynamic properties are usually small, and can be calculated readily to first approximation (see Chapter 6). For the structural properties (e.g. pair correlation function, structure factors), no detailed estimates are presently available for molecular liquids. For atomic liquids, and atomic and molecular gases, the relevant theoretical expressions for the quantum corrections are available in the literature.135"9 In later chapters we also discuss the quantum corrections to the dielectric and spectroscopic properties. 1.2.3 The pairwise additivity approximation The third basic approximation that we shall often introduce is that the total intermolecular potential energy °11(tnojn) is simply the sum of the intermolecular potentials for isolated pairs of molecules, i.e. where M(ri)a)ia)J) is the potential140 for a pair of molecules i and j (isolated from other molecules) with the centre of / at riy = r, — rf from i, and orientations &>; and &>,. The sum over i runs from 1 to N~ 1, while that of / is from 2 to N, i always being kept less than j in order to avoid counting any pair interaction twice. Fiquation A.13) is exact in the low density gas limit, since configurations involving three or more molecules can be ignored. It is not exact for dense fluids or solids, however, because the presence of additional molecules nearby distorts the electron charge distributions in molecules 1 and 2, and thus changes the intermolecular interaction between this pair from the isolated pair value (Fig. 1.3). More precisely, we should write <%i(tnojn) as a sum of isolated pair potentials, a sum of three-body correction terms due to the addition of an interacting t It is sometimes convenient to write this equation in the form In this form each pair potential is included twice in the sum; the factor one-half corrects for this.
14 INTRODUCTION 1.2 Fig. 1.3. The presence of molecule 3 nearby causes the charge clouds of molecules 1 and 2 to be distorted slightly (dashed lines) from their distributions for the isolated 12 pair (solid lines), and thus changes the intermolecular potential between 1 and 2. Similarly the 13 and 23 potentials are influenced by the presence of the third molecule. third molecule, a sum of four-body correction terms, etc. Thus A.14) where w(i/) = u(!•,•,&>;&>,¦), u(ijk) = u(itjiikijk<i)i<i)j<i)k), etc. The term u(ijk) is the additional part of the potential not included in the sum of the isolated pair terms u(ij) + u(ik)+u(jk), and so on. For inert gas fluids it seems that the series in A.14) can be truncated at the triplet term,20 but the situation is largely unknown for molecular fluids. For polar molecules with large polarizability the higher D-body, etc.) terms in A.14) may be important (see §§2.10 and 4.10). The influence of the three-body terms on physical properties has been studied in detail for atomic fluids;141'142 for these the long-range Axilrod- Teller143 triple-dipole dispersion term seems to be the most important part of the three-body interaction. Presumably this is because configura- configurations where three molecules overlap are rare, due to the strongly repul- repulsive nature of overlap potentials. The effect on the internal energy of the liquid is small even at the triple point, being only a few per cent of the total configurational energy; for less dense fluids the effect is smaller. However, some properties are more sensitive to the three-body forces. Figure 1.4 shows that the Axilrod-Teller three-body interaction has a large effect on the third virial coefficient, B3, and at the lower tempera- temperatures can have an influence equal to that of the two-body interaction. The three-body interaction also has a marked effect on some other properties for atomic liquids, for example the pressure and the surface tension. Table 1.4 shows the influence of three-body forces on the surface tension and surface energy for liquid argon, as calculated by Miyazaki et al.14S The
MOLECULAR FLUIDS 15 000 rr :ooo h 000 - ; 1 - - 1 1 1 1 t i 1 - 1 1 100 200 300 Temperature/K 400 500 7IG. 1.4. Third virial coefficients of argon. The circles are experimental data (Michels et al). The dashed curve is calculated with the Barker-Fisher-Watts144 potential alone; the dotted curve includes also the Axilrod-Teller143 interactions; the dash-dotted curve adds third-order dipole-quadrupole interactions, and the solid curve adds fourth-order dipole interactions. Here B3 is denned by the virial series equation of state for gases—see Chapter 3 (from ref. 20). leglect of three-body forces is seen to'lead to a 25 per cent error in the ;urface tension and a 10 per cent error in surface energy. Much less is known146 about the influence of three-body potentials in nolecular fluids, because the two-body potentials are not yet accurately cnown. (We note, however, that direct electrostatic potentials are exactly additive.) Table 1.4 Calculated and experimental surface properties for liquid argon at the triple loint. The pair potential used is that of Barker, Fisher and Watts144 (BFW) md the three-body potential is the Axilrod-Teller143 (AT) term (from ref. 145) Method Potential Surface tension (erg cm) Surface energy (erg cm ~2) Monte Carlo simulation Monte Carlo simulation Experiment BFW BFW + AT 17.7 14.1 13.35 38.3 34.7 34.8
16 INTRODUCTION 1.3 1,3 Scope of this book Figure 1.5 shows the p-V projection of the phase diagram for a simple pure substance, in which there is a single solid, liquid, and gas phase. There is theoretical147 and experimental148 evidence that a clear distinc- distinction exists between the solid and fluid states; such is not the case for gases and liquids, since the two states become identical at the critical point C. Thus, different theoretical approaches are applied to fluids and solids, and we shall not treat the theory of solids1491 in this book. The theories to be considered later are suitable for gases or liquids, with the exception of the region immediately around the critical point.152 In the vicinity of the critical point large-scale fluctuations occur, leading to correlations be- between molecules which are much longer in range than the range of the intermolecular forces. Because these correlations occur over regions containing many molecules, the detailed nature of the intermolecular forces has little effect on the property behaviour. Thus, critical phenomena in widely varying types of fluids (inert gases, polar fluids, superfluids, etc.) show a remarkable similarity (a situation known as universality). In fluids away from the critical point, on the other hand, the intermolecular correlations have a range similar to that of the intermolecular forces. The fluid properties are then very markedly affected by the nature of these forces.157 As a result, different theoretical methods are used for fluids near and away from the critical point. We do not consider the theory of p Pr \\\ \\\ ll 1 \» 11 ll ll ll III Solid (\ \| I I L \ \ \ \ \ \ \ \ I \ —iLiquid/ N/ _ j/— v^ C f \ L-G S-G V Gas ~~ —rL ~— 7; — V Fig. 1.5. p-V projection of the p-V-T diagram for a pure fluid. Solid lines are phase boundaries, dashed lines are isotherms, horizontal dashed lines are tie-lines connecting coexisting phases, pt and Tt are triple-point pressure and temperature, and C is the gas-liquid critical point.
1.3 SCOPE OF THIS BOOK 17 the critical point region in this book, but refer the interested reader to the substantial literature that exists.3'158'64 The preceding remarks and those in § 1.2 define the scope of the present work, i.e. to non-critical fluid phases of small, nearly rigid molecules, whose quantum effects are usually negligible or small. Exam- Examples are given in Table 1.1. Since the theory of atomic fluids has been reviewed in many other piaces20-31-32-78111165166 we do not discuss these systems in detail except for particular problems such as PY theory for hard spheres, mixtures such as HCl-Ar, etc. Also, we discuss mainly molecular liquids, the discussion of molecular gases being brief because of the extensive treatments given elsewhere.7'8 In addition to excluding from discussion solids, freezing,20-31149167-168 nucleation,14 and critical phenomena, we also exclude long-chain molecules,851692 glasses,173174 liquid crystals,175"82 electrolytes,78165-166183 plasmas,31163-165 molten salts,165 liquid metals,166187188 reacting and dissociating liquids88-89171'190 and quantum liquids.111163196"9 Moreover, we shall not discuss dynam- dynamical (non-equilibrium) properties.31-78-2001 In Chapter 2 the intermolecular potential models used for molecular fluids are discussed. Relatively few ab initio quantum mechanical calcula- calculations are available for polyatomic molecules, and these are not usually of sufficient accuracy for property calculations. Two approximate potential models have been especially widely used. These are the generalized Stockmayer potential and the site-site potential. The generalized Stock- mayer potential is a (truncated) sum of spherical harmonic terms repres- representing the electrostatic, induction, dispersion, and overlap parts of the potential. It is correct at long range, and is particularly convenient for statistical mechanical calculations because angular integrations over the harmonics are simple. However, the truncated spherical harmonic expan- expansion may break down at short distances. The site-site potentials may give a more realistic account of the molecular shape, are a convenient rep- representation of highly anisotropic charge distributions, and are simple to use in computer simulation studies, at least for small molecules. However, they are incorrect at large intermolecular separations, and are difficult to use in most analytic calculations. We introduce in Chapter 3 the various molecular distribution functions, and discuss their properties. Theories of these functions are taken up in Chapters 4 and 5. The various forms of perturbation theories for molecu- molecular fluids are described in detail in Chapter 4 while integral equation methods are considered in Chapter 5. These chapters complete the fundamental theory. In Chapters 6-11 (Volume 2) we discuss applications of the theory to several specific problems, including thermodynamics of pure and mixed
18 INTRODUCTION 1.3 fluids, interfacial properties, the structure factor (as determined by neut- neutron and X-ray scattering), the dielectric constant, and the infrared, Raman, and neutron spectral moments. References and notes Historical references A. Partington, J. R. An advanced treatise on physical chemistry, Vol. 1 (Gases) and Vol. 2 (Liquids), Longmans, London A949 and 1951). B. Brush, S. G. The kind of motion we call heat, Books 1 and 2, in Studies in statistical mechanics, Vol. 6 (ed. E. W. Montroll and J. L. Lebowitz) North- Holland, Amsterdam A976). General references 1. Frankiss, S. G. and Green, J. H. S. Chemical thermodynamics, Vol. 1, Chapter 8 (ed. M. L. McGlashan) Specialist Periodical Report, Chemical Society, London A973). 2. JANAF, Thermochemical tables, Nat. Standard Ref. Data Series, Nat. Bur. Stand. (U.S.) 37, U.S. Department of Commerce A971). 3. Rossini, F. D. et al. Selected values of chemical thermodynamic properties, Nat. Bur. Stand. Circular 500 A952). 4. Ursell, H. D. Proc. Camb. Phil. Soc. 23, 685 A927). 5. Mayer, J. E. J. chem. phys. 5, 67 A937). 6. The quantum virial series was also worked out later.7*10 7. Mason, E. A. and Spurting, T. H. The virial equation of state, Pergamosi, Oxford A969); Maitland, G. C, Rigby, M., Smith, E. B., and Wakeham, W. A. Intermolecular forces: their origin and determination, Clarendon Press, Oxford A981). 8. Hirschfelder, J. O., Curtiss, C. F., and Bird, R. B. Molecular theory of gases and liquids. Wiley, New York A954). 9. Kahn, B. and Uhlenbeck, G. Physica 5, 399 A938). 10. Lee, T. D. and Yang, C. N., Phys. Rev. 113, 1165 A959). 11. Practical difficulties12 still remain in evaluating the quantum virial coeffi- coefficients for polyatomic molecules. 12. Larsen, S. Y. and Poll, J. D. Can. J. Phys. 52, 1914 A974). 13. However, the series has been utilized indirectly, through resummation methods (see the discussion of integral equation methods below, and in Chapter 5) and through models suggested by the series (e.g. the physical cluster and droplet models146 for condensation and nucleation). The virial coefficients have also been used to locate the critical point. 17~19 14. Abraham, F. F. Homogeneous nucleation theory, Academic Press, New York A974). 15. Fisher, M. E. Physics 3, 255 A967). 16. Dorfman, J. R. J. Stat. Phys. 18, 415 A978); Lockett, A. M. X chem. Phys. 72, 4822 A980); Gibbs, J. H., Bagchi, B., and Mohanty, U. Phys. Rev. B24, 2893 A981). 17. Majumdar, C. K. and Rama Rao, I. Phys. Rev. A14, 1542 A976); Estevez, G. A., Gould, H., and Cole, M. W. Phys. Rev. A18, 1222 A978). 18. Yang, C. N. and Lee, T. D. Phys. Rev. 87, 404, 410 A952).
REFERENCES AND NOTES 19 19. Kihara, V, Intermolecular forces, p. 44, Wiley, New York A978). 20. Barker, }'. A. and Henderson, D. Rev. mod. Phys. 48, 587 A976). 21. Barker, J. A. and1 Henderson, D. Can. J. Phys. 45, 3959 A967). 22. De Llano, M. and Nava-Jaimes, A. Mol. Phys. 32, 1163 A976). 23. Ruelle, D. Statistical mechanics: rigorous results, Benjamin, New York A969). 24. Barker, J. A. Lattice theories of the liquid state, Pergamon, Oxford A963). 25. Levelt, J. M. H. and Cohen, E. G. D. Studies in statistical mechanics, Vol. 2 (ed. J. de Boer and G. E. Uhlenbeck) p. Ill, North-Holland, Amsterdam A964). 26. Reed, T. M. and Gubbins, K. E. Applied statistical mechanics, McGraw-Hill, New York A973). 27. Lennard-Jones, J. E. and Devonshire, A. F. Proc. Roy. Soc. A163, 53 A937). 28. Stillinger, F. H. Adv. chem. Phys. 31, 1 A975); J. stat. Phys. 23, 219 A980). 29. Wheeler, J. C. Ann. Rev. phys. Chem. 28, 434 A977). 30. Watts, R. O. Statistical mechanics, Vol. 1, p. 1 (ed. K. Singer) Specialist Periodical Report, Chemical Society, London A973). 31. Hansen, J. P. and McDonald, I. R. Theory of simple liquids, Academic Press, London A976). 32. Smith, W. R. Statistical mechanics, Vol. 1, p. 71 (ed. K. Singer) Specialist Periodical Report, Chemical Society, London A973). 33. Boublik, T. Fluid Phase Equilibria 1, 37 A977). 34. As will be discussed in Chapter 4, the perturbation theory can in some cases be based on hard spheres as reference, so that the zeroth approximation for such liquids is a packing model for hard spheres (rather than a gas-like or solid-like picture). The theory is then a quantitative version of the van der Waals picture. 35. Longuet-Higgins, H. C. Proc. Roy. Soc. London A205, 247 A951). 36. Zwanzig, R. W. J. chem. Phys. 22, 1420 A954). 37. Pople, J. A. Proc. Roy. Soc. London A221, 498 A954). 38. Barker, J. A. and Henderson, D. J. chem. Phys. 47, 4714 A967). 39. Leland, T. W., Rowlinson, J. S., and Sather, G. A. Trans. Faraday Soc. 64, 1447 A968). 40. Reiss, H., Frisch, H. L., and Lebowitz, J. L. J. chem. Phys. 31, 369 A959). 41. Reiss, H. Adv. chem. Phys. 9, 1 A965). 42. Gibbons, R. M. Mol. Phys. 17, 81 A969); 18, 809 A970). 43. Boublik, T. Mol. Phys. 27, 1415 A974). 44. Reiss, H., in Statistical mechanics and statistical methods in theory and application (ed. U. Landman) Plenum, New York A977). 45. Wood, W. W. and Erpenbeck, J. J. Ann. Rev. phys. Chem. 27, 319 A976). 46. Berne, B. (ed.) Modern theoretical chemistry, Vols. 5 and 6, Plenum, New York A977); see reviews by Valleau and Whittington, Valleau and Torrie, Erpenbeck and Wood, and Kushick and Berne. 47. Streett, W. B. and Gubbins, K. E. Ann. Rev. phys. Chem. 28, 373 A977). 48. Binder, K. (ed.) Monte Carlo methods, Springer-Verlag, New York A979). 49. Vesely, F. Computerexperimente an Flussigkeitsmodellen, Physik Verlag, Weinheim A978). 50. Computer simulations are usually carried out classically. However a quan- quantum version of the Monte Carlo method has been developed. See ref. 48 and also Schmidt, K. E., Lee, M. A., Kalos, M. H., and Chester, G. V. Phys.
20 INTRODUCTION Rev. Lett. 47, 807 A981). 'Semiclassical' Monte Carlo can be done easily by adding the quantum corrections given by A.4) in terms of the classical mean-squared force and torque. Approximate versions of quantum molecu- molecular dynamics have been studied by Corbin, H. and Singer, K. Mol. Phys. 46, 671 A982). Modified molecular dynamics-type simulations for low density (binary collision) gases have also been carried out for studies of collision- induced absorption and collision-induced light scattering (Chapter 1 1); see Birnbaum, G., Guillot, B., and Bratos, S. Adv. chem. Phys. 51, 49 A982) and refs. therein. 51. Various 'intermediate' methods have been devised, e.g. 'Brownian dynamics'; see J. M. Haile and G. A. Mansoori (ed.), Molecular-based study of fluids, Advances in Chemistry Series, 204, American Chem. Soc, Washington A983)-particularly the papers by Mansoori and Haile, Evans, and Weber et al. See also H. C. Andersen [J. chem. Phys. 72, 2384 A980)] for proposals for molecular dynamics at constant temperature or pressure. For results see Haile, J. M. and Graben, H. W. J. chem. Phys. 73, 2412 A980); Abraham, F. F. Phys. Repts. 80, 340 A981). 52. Chapela, G. A., Saville, G., Thompson, S. M., and Rowlinson, J. S. J. Chem. Soc, Faraday Trans. 11 73, 1133 A977) . There are also problems in simulating dielectric properties for systems with long-range forces - see Chapter 10. 53. This assumes quantum effects are negligible; otherwise we test the potential model and the presence of quantum effects. As discussed in ref. 50, quantum effects can sometimes be allowed for. 54. Other examples are the long-time tails on correlation functions,'5''6 new transport coefficients (e.g. transverse current or shear wave correlation functions,57 rotational viscosity coefficients58), Lorentz model fluid proper- properties,ss properties of one- and two-dimensional fluids,55 hard sphere fluid properties at varying density,55 finite frequency components of viscosity and thermal conductivity,58 metastable and glass states of atomic liquids,59 higher-order and non-linear transport coefficients,60'61 cluster and nucleation studies,62^ the triplet correlation function,66 fluid properties at extreme state conditions,61'67'68 bulk viscosity,69 and non-local dielectric constants.7" We also note that, combined with computer graphics techniques, computer simulation gives us a vivid way of visualizing the complicated structures and dynamical processes in condensed phases and at interfaces. 55. Wood, W. W. in Fundamental problems in statistical mechanics III (ed. E. D. Cohen) North-Holland, Amsterdam A975). 56. Berne, B. J., Faraday Symposia of the Chemical Society, no. 11, 'Newer Aspects of Molecular Relaxation Processes', p. 48 A977). 57. Jacucci, G. and McDonald, I. R. Mol. Phys. 39, 515 A980), and references therein. 58. Evans, D. J. and Streett, W. B. Mol. Phys. 36, 161 A978). ' 59. Rahman, A., Mandell, M. J., and McTague, J. P. J. chem. Phys. 64, 1564 A976). 60. Wood, W. W. and Erpenbeck, J. J. ref. 46. 61. Hoover, W. G. and Ashurst, W. T., in Theoretical chemistry, advances and perspectives. Vol. 1, Academic Press, New York A975); see also Evans, D. J., Phys. Rev. A23, 1988 A981). 62. Harrison, H. W. and Schieve, W. C. J. chem. Phys. 58, 3634 A972). 63. Lee, J. K., Barker, J. A., and Abraham, F. F. J. chem. Phys. 58, 3166 A973).
REFERENCES AND NOTES 21 64. Rao, M.,; Berne, B. J., and Kalos, M. H. J. chem. Phys. 68, 1325 A978). 65. Mandell, M. J., McTague, J. P., and Rahman, A. J. chem. Phys. 66, 3070 A977). ' 66. Raveche, H. J. and Mountain, R. D., in Progress in liquid physics (ed. C. Croxton) Chapter 12, Wiley, New York A978). 67. Pollock, E. C. and Alder, B. J. Phys. Rev. A15, 1263 A977). 68. Dewitt, H. E. and Hubbard, W. B. Ast. J. 205, 295 A976). 69. Hoover, W. G., Evans, D. J., Hickman, R. B., Ladd, A. J. C, Ashurst, W. T., and Moran, B. Phys. Rev. All, 1690 A980). 70. Pollack, E. L. and Alder, B. J. Physica 102A, 1 A980). 71. Thompson, S. M. and Gubbins, K. E. J. chem. Phys. 74, 6467 A981). 72. A more complete list of references for both liquid-vapour and liquid-solid interfaces is given in Chapter 8. 73. Sullivan, D. E., Barker, R., Gray, C. G., Streett, W. B., and Gubbins, K. E. Mol. Phys. 44, 597 A981). 74. A special case is rigid molecules, having only rotational and translational degrees of freedom; hence the somewhat inaccurate phrase 'rigid molecule approximation'. 75. Internal rotations can occur in many molecules, for example about C-C bonds in hydrocarbons. Thus in ethane, H3C-CH3, the second CH3 group can rotate relative to the first about the C-C bond. Such rotations are hindered in general, the degree of hindrance depending on the molecule concerned. See, for examples: Orville-Thomas, W. J. (ed) Internal rotation in molecules, Wiley, New York A974). 76. Herzberg, G. Spectra of diatomic molecules, 2nd edn, Van Nostrand, Prince- Princeton A950). 77. Herzberg, G. Infrared and Raman spectra of polyatomic molecules, Van Nostrand, Princeton A945). 78. McQuarrie, D. A. Statistical mechanics, Harper and Row, New York A976). 79. For a review see e.g. Robin, M. B. Simple dense fluids (ed. H. L. Frisch and Z. W. Salsburg) p. 215, Academic Press, New York A968), and ref. 80. 80. Gray, C. G. and Welsh, H. L. 'Intermolecular Force Effects in the Raman Spectra of Gases', in Essays in structural chemistry (ed. A. J. Downs, D. A. Long, and L. A. K. Staveley) MacMillan, London A971). 81. Kohler, F., Wilhelm, E., and Posch, H. Adv. mol. relaxation Proc. 8, 195 A976). 82. Lascombe, J. (ed.), Molecular motions in liquids, Reidel, Dordrecht A974). 83. Oxtoby, D. Adv. chem. Phys. 40, 1 A979); 47 (Pt. II), 487 A981); Ann. Rev. phys. Chem. 32, 77 A981). 84. See for example: Stanton, G. W., Clarke, J. H., and Dore, J. C. Mol. Phys. 34, 823 A977); Walford, G., Clarke, J. H., and Dore, J. C. Mol. Phys. 33, 25 A977); Powles, J. G., Mol. Phys. 42, 757 A981); Sullivan, J. D. and Egelstaff, P. A. Mol. Phys. 44, 287 A981); Sullivan, J. D. and Egelstaff, P. A. J. chem. Phys., 76, 4631 A982). 85. Sinanoglu, O., in The world of quantum chemistry (ed. R. Daudel and B. Pullman) p. 265, Reidel, Dordrecht A974). 86. Scott, R. A. and Scheraga, H. A., J. chem. Phys. 44, 3054 A966). 87. Ryckaert, J. P. and Bellemans, A. Chem. Phys. Lett. 30, 123 A975). 88. Jorgensen, W. L., J. phys. Chem. 87, 5304 A983). 89. Chandler, D. Farad. Disc. Chem. Soc. 66, 184 A978), and references cited therein.
22 INTRODUCTION 90. Abraham, R. J. and Bretschneider, E. Chapter 13 of ref. 75. 91. Bulanin, M. O. J. molec. Structure 19, 59 A973). 92. Benson, A. M. and Drickamer, H. G. J. chem. Phys. 27, 1164 A957). 93. Buckingham, A. D. Proc. Roy. Soc. A248, 169 A958); A255, 32 A960). 94. Berne, B. J. and Harp, G. D. Adv. chem. Phys. 17, 63 A970). 95. Lemberg, H. L. and Stillinger, F. H. Mol. Phys. 32, 253 A976). 96. Rahman, A., Stillinger, F. H., and Lemberg, J. J. chem. Phys. 63, 5223 A975). 97. Ryckaert, J. P. and Bellemans, A. Farad. Disc. Chem. Soc. 66, 95 A978). 98. Stillinger, F. H. Israel J. Chem. 14, 130 A975). 99. McDonald, I. R. and Klein, M. Farad. Disc. Chem. Soc. 66, 48 A978), and references cited therein. 100. Freasier, B. C, Jolly, D. C, Hamer, N. D., and Nordholm, S. Chem. Phys. 38, 293 A979); Herman, M. F. and Berne, B. J. Chem. Phys. Lett. 77, 163 A981). 101. Due to rotational invariance, u(rco!co2) in fact depends on fewer variables than is indicated by its argument. This fact is exploited in Chapter 2. 102. Mason and Spurling7 argue that diffraction effects are really excluded volume effects, at least for molecules with infinitely hard cores. 103. The quantity A, arises formally in C.78) in the evaluation of the partition function. 104. Prigogine, I. Molecular theory of solutions, North-Holland, Amsterdam A957). 105. Poll, J. D. and Miller, M. J. chem. Phys. 54, 2673 A971). 106. Kirkwood, J. G. J. chem. Phys. 1, 597 A933). 107. Larsen, S. Y., Kilpatrick, J. E., Lieb, E. H., and Jordan, H. F. Phys. Rev. 140, A129 A965). 108. Hynes, J. T., Deutch, J. M, Wang, C. H., and Oppenheim, I. J. chem. Phys. 48, 3085 A968). 109. For non-equilibrium properties an additional conditionlnR-""-111 must be satisfied. For the time correlation function (A@)B(t)), the time t must satisfy t»h/kT; stated alternatively in terms of the Fourier transform of <-A@)B(f)>, the frequency a> must satisfy w«kT/h (see Chapter 11). 110. Schofield, P. Phys. Rev. Lett. 4, 239 A960). 111. Egelstaff, P. A. Introduction to the liquid state, Academic Press, London A967). 112. The derivation and detailed references for this expansion are given in Appendix 3D and Chapter 6. 113. For infinitely hard core potentials, odd powers of ft also occur. See Sinha, S. K. and Singh, Y. J. math. Phys. 18, 367 A977) and references therein. 114. Van Kranendonk, J. Can. J. Phys. 39, 1563 A961). 115. Gray, C. G. and Taylor, D. W. Phys. Rev. 182, 235 A969). 116. Uhlenbeck, G. and Beth, E. Physica 3, 729 A936). 117. The reader will recall that for a one-dimensional harmonic oscillator the potential energy is |kx2, and therefore the force constant k is the second derivative of the potential. 118. This is derived in Appendix E. 119. We can ignore the factor of N (which is not small!) in A.4) since, as explained in Appendix 3D, it is ultimately the free energy change per molecule, (A - Acl)/N = -(feT/JV)ln(Q/QoI), which is of interest. 120. The operator Vm is related to the angular momentum operator I of (A.12) and (A.70) by l = -i Vm.
REFERENCES AND NOTES 23 121. For most cases where A.10) is satisfied the classical rotational kinetic condition, A.11^, is also satisfied. Liquid hydrogen is one of the few exceptions. 122. Weares, O. and Rice, S. A. J. Am. Chan. Soc. 94, 8983 A972). 123. Shipman, L. L. and Scheraga, H. A. J. phys. Chem. 78, 909 A974). 124. eT is related to Ar of C.81) by Ar = 0r/T. 125. Hansen, J. P. and Weis, J. J. Phys. Rev. 188, 314 A969). 126. Jancso, G. and van Hook, W. A. Chem. Rev. 74, 689 A974). 127. Thompson, S. M, Tildesley, D. J., and Streett, W. B. Mol. Phys. 32, 711 A976). 128. Casanova, G. and Levi, A., in Physics of simple liquids (ed. H. N. V. Temperley, J. S. Rowlinson, and G. S. Rushbrooke) p. 299, North-Holland, Amsterdam A968). 129. The zero-point energy is inversely proportional to the square root of the mass. The lighter component will have a greater zero-point energy, and therefore a greater 'escaping tendency', in general. 130. The perturbation of the vibrational motions of a molecule due to interac- interactions with its neighbours in the liquid often results in a mean frequency shift to a lower value (cf. Tables 1.2 and 1.3). Further, the frequency of the lighter isotope shifts more than that of the heavier one.93 Thus the lighter isotope molecule can be vibrationally excited more easily by collisions, and is therefore expected to take up less translational energy on collision; it can therefore be 'kicked' into the vapour less easily. An alternative view is to regard the larger negative intramolecular zero- point energy shift (arising from the larger intramolecular mode negative frequency shift) for the lighter isotope as a larger stabilization energy for the lighter isotope liquid. For discussion of this alternative mechanism, see F. H. Stillinger, in Theoretical chemistry: advances and perspectives (ed. H. Eyring and D. Henderson) Vol. 3, p. 178, Academic Press, New York A978). 131. Wang Chang, C. S. Thesis, University of Michigan A944); reported in ref. 8, p. 434. 132. McCarty, M. and Babu, S. V. K. J. phys. Chem. 74, 1113 A970). 133. Pompe, A. and Spurting, T. H. Aust. J. Chem. 26, 855 A973). 134. Singh, Y. and Datta, K. K. J. chem. Phys. 53, 1184 A970). 135. MacRury, T. B. and Steele, W. A. J. chem. Phys. 61, 3352, 3366 A974). 136. Nienhuis, G. J. math. Phys. 11, 239 A970). 137. Singh, Y. and Ram, J. Mol. Phys. 25, 145 A973); 28, 197 A974); Singh, Y. and Sinha, S. K. Phys. Reports 79, 213 A981). 138. Gibson, W. G. Mol. Phys. 28, 793 A974); 30, 1 A975). 139. Jancovici, B. Mol. Phys. 32, 1177 A976). 140. We shall further make the usual assumption that ("(rijco.co,))^^, where (¦ ' ¦)<„,<«, denotes an unweighted average over the orientations, decreases faster than r~3. This is necessary for the thermodynamic properties to exist; see, e.g. Miinster, A., Statistical thermodynamics, Vol. 1, p. 218, Springer- Verlag, Berlin A969). We note that although the dipole-dipole potential u^ falls off only as r~3, the effective central potential (see Chapter 4) -(uj»)»,»iftf falls off as r~S- I" Coulombic systems, where the electrostatic potential decays as r~\ the effective potential is a screened one that falls off faster than r~3. 141. Hirschfelder J. O. (ed.), 'Intermolecular forces', Adv. chem. Phys. 12 A967).
24 INTRODUCTION 142. Margenau, H. and Kestner, N. R., Theory of intermolecular forces, 2nd edn, Chapter 5, Pergamon, Oxford A971). 143. Axilrod, B. M. and Teller, E. J. chem. Phys. 11, 299 A943). 144. Barker, J. A., Fisher, R. A., and Watts, R. O. Mol. Phys. 21, 657 A971). 145. Miyazaki, J., Barker, J. A. and Pound, G. M. J. chem. Phys. 64, 3364 A975). 146. See introduction to Chapter 2 and §§ 2.10 and 4.10. 147. Landau, L. and Lifshitz, E. M. Statistical physics, 2nd edn, p. 260, Addison- Wesley, Reading, MA A969). 148. Rowlinson, J. S. Rep. Prog. Phys. 28, 169 A965). 149. Parsonage, N. G. and Staveley, L. A. K. Disorder in crystals, Clarendon Press, Oxford A979). 150. Califano, S. (ed.) Lattice dynamics and intermolecular forces, Academic Press, New York A975). 151. Kitaigorodsky, I. A. Molecular crystals and molecules, Academic Press, New York A973). 152. Although critical behaviour can be studied1 " '' using the integral equation methods of Chapter 5, one most often introduces totally different methods. We do not consider in detail the theory of the gas-liquid critical point for pure fluids in this book; however, we do discuss the location of gas-liquid and liquid-liquid critical lines in binary mixtures in Chapter 7. 153. Stell, G. Phys. Rev. B 1, 2265 A970). 154. Luks, K. D. and Kozak, J. J. Adv. chem. Phys. 40, 139 A978); Green, K. A., Luks, K. D., Jones, G. L., Lee, E., and Kozak, J. J. Phys. Rev. A 25, 1060 A982); Kumar, N., March, N. H., and Wasserman, A. Phys. Chem. Liq. 11, 271 A982). 155. Fisher, M. E. and Fishman, S. Physica 108A, 1 A981); Phys. Rev. Lett. 47, 421 A981). 156. Foiles, S. M. and Ashcroft, N. W. Phys. Rev. A 24, 424 A981). 157. Although the property behaviour in the neighbourhood of the critical point is similar for widely different types of fluids, we note that the location of the critical point and extent of the critical region in the p-V-T diagram are strongly affected by the nature of the intermolecular forces. The location of the critical points of mixtures is discussed in Chapter 7. 158. Stanley, H. E. Introduction to phase transitions and critical phenomena, Oxford, New York A971). 159. Domb, C. and Green, M. S., (ed.) Phase transitions and critical phenomena, Vols 1-6, Academic Press, New York A972-1977). 160. Sengers, J. M. H., Hocken, R., and Sengers, J. V. Phys. Today 30, A2), 42 A977). 161. Wheeler, J. C. Ann. Rev. phys. Chem. 28, 411 A977). 162. Sengers, J. V. and Levelt Sengers, J. M. H. Chapter 4 of ref. 66. 163. March, N. H. and Tosi, M. P. Atomic liquid dynamics, MacMillan, London A976). 164. Greer, S. C. and Moldover, M. R. Ann. Rev. phys. Chem. 32, 233 A981). 165. Watts, R. O. and McGee, I. J. Liquid state chemical physics, Wiley, New York A976). 166. Miinster, A. Statistical thermodynamics, Vols. 1 and 2, Springer-Verlag, Berlin A974). 167. Hoover, W. G. and Ross, M. Contemp. Phys. 12, 339 A971).
REFERENCES AND NOTES 25 168. Sherwood, J. N. (ed.) The plastically crystalline state, Wiley, New York A979). 169. Flory, P. J. Statistical mechanics of chain molecules, Wiley, New York A969). 170. Bixon, M. Ann. Rev. phys. Chem. 27, 65 A976). 171. Synek, M, Schieve, W. C. and Harrison, H. W. J. chem. Phys. 67, 2916 A977). 172. Chandler, D., in Studies in statistical mechanics, Vol. 8, p. 275 (ed. E. W. Montroll and J. L. Lebowitz), North-Holland, Amsterdam A982). 173. Discussion of the Faraday Society 50, 'The Vitreous State', A970). 174. Goldstein, M. and Sinha, R. (ed.), Ann. N.Y. Acad. Sci. 279, 'The Glass Transition and the Nature of the Glassy State', A976). 175. de Gennes, P. G. The physics of liquid crystals, Oxford University Press, Oxford A974). 176. Stephen, M. J. and Straley, J. P. Rev. mod. Phys. 46, 617 A974). 177. Forster, D. Adv. chem. Phys. 31, 231 A975). 178. Chandrasekhar, S. Liquid crystals, Cambridge University Press, Cambridge A977). 179. Brown, G. H. (ed.) Advances in liquid crystals, Vol. 1 A975-present) Academic Press, New York. See in particular the reviews by Leslie (Vol. 4, 1979) and Ericksen (Vol. 2, 1976). 180. Chandrasekhar, S. and Madhusandana, N. V., in Progress in liquid physics, (ed. C. Croxton) Chapter 4, Wiley, New York A978). 181. Luckhurst, G. R. and Gray, G. W. (ed.) The molecular physics of liquid crystals, Academic Press, New York A979). 182. Gelbart, W. M. and Barboy, B. Ace. chem. Res. 13, 290 A980). 183. Enderby, J. E. and Neilson, G. W. Adv. Phys. 29, 323 A980). 184. Friedman, H. L. Ann. Rev. phys. Chem. 32, 179 A981). 185. Hafskjold, B. and Stell, G. Studies in statistical mechanics, Vol. 8, p. 175 (ed. J. L. Lebowitz and E. W. Montroll) North-Holland, Amsterdam A982). 186. Baus, M. and Hansen, J. P. Phys. Reports 59, 1 A980). 187. Ashcroft, N. W. and Stroud, D. Solid State Physics 33, 1 A978). 188. Shimoji, M. Liquid metals, Academic Press, London A977). 189. Moelwyn-Hughes, E. A. The chemical statics and kinetics of solutions, Academic Press, New York A971). 190. Jolly, D. L., Freasier, B. C, and Nordholm, S. Chem. Phys. 25, 361 A977). 191. Eisenthal, K. B. Ace. chem. Res. 8, 118 A975). 192. Stillinger, F. H. Theoretical chemistry: advances and perspectives, (ed. H. Eyring and D. Henderson) Vol. 3, p. 177, Wiley, New York A978). 193. Hamann, S. D., in High pressure physics and chemistry (ed. R. S. Bradley) Vol. 2, p. 131, Academic Press, New York A963). 194. Entelic, S. G. and Toger, R. P. Reaction kinetics in the liquid phase, Wiley, New York A976). 195. Coker, D. F. and Watts, R. O. Mol. Phys. 44, 1303 A981). 196. Huang, K. Statistical mechanics, Wiley, New York A963). 197. Pines, D. and Nozieres, P. The theory of quantum liquids, Benjamin, New York A966). 198. Feenberg, E. Theory of quantum fluids, Academic Press, New York A969). 199. Campbell, C. E. Chapter 6 of ref. 6,6. 200. Gubbins, K. E., in Specialist Periodical Reports - Statistical Mechanics Vol.
26 INTRODUCTION 1, p. 194 (ed. K. Singer) Chem. Soc. London, A973). 201. Berne, B. J., in Physical chemistry, Vol. 8b (ed. H. Eyring, D. Henderson, and W. Jost), Academic Press, New York A971). 202. Steele, W. A. Adv. chem. Phys. 34, 1 A976). 203. Marshall, W. and Lovesey, S. W., Theory of thermal neutron scattering, Clarendon Press, Oxford A971). 204. Egelstaff, P. A., Gray, C. G., Gubbins, K. E., and Mo, K. C. J. stat. Phys. 13, 315 A975). 205. Berne, B. J. and Pecora, R. Dynamic light scattering, Wiley, New York A976). 206. Hynes, J. T. Ann. Rev. phys. Chem. 28, 301 A977). 207. Bauer, D. R., Brauman, J. I., and Pecora, R. Ann. Rev. phys. Chem. 27, 443 A976). 208. Mountain, R. D. Adv. mol. relaxation Processes 9, 255 A976). 209. Hynes, J. T. and Deutch, J. M., in Physical chemistry: an advanced treatise, Vol. 11B (ed. H. Eyring, D. Henderson, and W. Jost) Academic Press, New York A975). 210. Dahler, J. S. and Theodosopulu, M. Adv. chem. Phys. 31, 155 A975). 211. Boon, J. P. and Yip, S. Molecular hydrodynamics, McGraw-Hill, New York A980).
2 INTERMOLECULAR FORCES! . . . for the whole burden of philosophy seems to consist in this —from the phenomena of motions to investigate the forces of nature, and then from these forces to demonstrate the other phenomena;. . . Issac Newton A686), Preface to Principia Knowledge of the intermolecular potential for simple (i.e. monatomic) molecules30 has increased greatly in recent years. For polyatomic molecules,117 on the other hand, such knowledge is still rather meagre, and much more is needed. One needs to know (i) what is the pair potential? (ii) how important are the triplet and other multibody poten- potentials in liquids? These multibody potentials have been studied very little for polyatomic liquids (see refs. 28-35 and §§ 1.2.3, 2.10, and 4.10), and are usually taken into account, if at all, by an effective pair potential.369 There have been, especially at short range, relatively few theoretical evaluations of the pair potential for diatomic or polyatomic molecules (see, e.g. refs. 21-7 and 40-55a). The most reliable existing knowledge has been obtained from binary collision experiments, or, for the long- range part of the potential, from measurements of properties of single molecules. Examples include molecular beam scattering,26a-561 induced birefringence,17'61" pressure and dielectric virial coefficients,6'62 and collision-induced absorption62a (including gas dimer spectra,63 which can also be studied by beam resonance spectroscopy66'67) which yield values for the parameters (e.g. Lennard-Jones constants, polarizabilities, dipole moments, quadrupole moments, octopole moments, etc. - see also Appendix D) occurring in the expressions for the intermolecular poten- potentials. The shape of the repulsive core of the potential can be inferred approximately from the molecular structure and charge density as deter- determined experimentally, for example68121 by electron and X-ray diffraction or by quantum calculations.72"8 As an example of the last point we show in Fig. 2.1a contour map for the theoretically calculated charge density of N2, the prototype molecule for simple nonpolar molecular fluid studies. Over 95 per cent of the total electronic charge is contained within the outermost @.002 au) contour, and the dimensions of this contour are sometimes used to define a theoretical size of the N2 molecule. The dimensions shown on Fig. 2.1 t The history of this subject is discussed in refs. 1 and 2.
28 INTERMOLECULAR FORCE: -0.002 Fig. 2.1. Electron charge density contours in a plane containing the nuclei for the N molecule, in atomic units A au = e/al, see Appendix D). (From refs. 14, 75.) agree roughly with dimensions obtained experimentally from Lennard- Jones diameters in gases (virial coefficients and viscosity) and so-callec van der Waals radii79 from X-ray diffraction studies of solids. Of course the charge density, as in Fig. 2.1, is simply related to the intermolecula- potential79b only for a test charge (see § 2.4.1 or ref. 79a). For two rigk interacting molecules, the electrostatic energy is related in a fairly compli- complicated way to the charge densities of the two molecules, particularly a short range (see §§ 2.4.5 and 2.8); formally the energy is just a Coulomr integral over the two charge densities (see B.283)), but the explici dependence on the separation and orientations of the charge densities i: complicated. Nevertheless, one feels intuitively that in some sense tht shape of the charge density contours should be qualitatively similar tc intermolecular equipotential contours (cf. Figs 2.1, 2.2). In addition to the methods mentioned above, several other experimen- experimental methods on gases have some potential for determining anisotropk intermolecular forces, e.g. transport coefficients,80 nuclear spin relaxa- relaxation,81 pressure-broadening of infrared,82 Raman83-84 and microwave8 spectral lines, double resonance,61-86 infrared and Raman band spectra moments,83-87 collision-induced light scattering,88 ion-molecule scatter- scattering,89 neutron and X-ray diffraction,89" and others.80-803 Values of tht intermolecular potential parameters obtained from some of thest
INTERMOLECULAR FORCES 29 methods may be subject to inaccuracies introduced by approximations made in the theory, The same holds true for pair potential parameters determined from fits to liquid and solid datas9b (see Fig. 1.1 and discus- discussion), where there is the additional uncertainty introduced by multibody potentials. A long-range objective is to obtain a single potential for a given system which will fit all the gas, liquid, and solid data; such a potential has been found for some monatomic systems.4>4a'10 A shorter term objective may be to construct simple model potentials which explain at least qualitatively, and perhaps semi-quantitatively, the various experi- experimental data. We consider molecules with no internal rotation, and which are in their ground electronic and ground vibrational states.(see § 1.2.1). The pair potential i^ra^o^) can thus be assumed to depend only on the inter- molecular separation r and on the molecular orientations a)l and co2; throughout we adopt the convention that r points from molecule 1 to molecule 2. Here the vector r is r=rd4> and the orientations are <u, = <f>,OiXi (Euler angles) and refer to some space-fixed coordinate frame. (The Euler angles are defined precisely in § A.2.) If the orientations are referred to r as polar axis, the notation u(r(o[oj2) is used. The potential energy is expressed in terms of the intermolecular separation between molecular centres, which can be chosen arbitrarily for each molecule. In §3.1.2 in considering the kinetic energies, we introduce the centre of mass in order to separate the translational and rotational kinetic energies. One then often chooses the molecular centre for potential energy calcula- calculations also as the centre of mass, so that all molecular parameters (mo- (moments of inertia, multipole moments, etc.) relate to one origin. If one chooses a different centre, it should be kept in mind that individual intermolecular force parameters (e.g. multipole moments, polarizabilities) can depend on the choice of centre (see § 2.4.3 and Appendix C), although the total potential does not of course depend on this choice, and some of the force parameters are in fact origin-independent. In this chapter we first discuss some models and general aspects of the pair potential (§§2.1-2.3). At long range the interaction energy is classified^8 as multipolar (electrostatic), dispersion, and induction energy, and at short range as overlap (electrostatic and exchange) energy. These contributions to the pair potential are discussed in §§ 2.4-2.7. Expansions of the atom-atom model potential are considered in § 2.8 and con- convergence of the spherical harmonic expansion (particularly the multipole series) is discussed in § 2.9. Finally we briefly discuss multibody interac- interactions (§2.10). It perhaps should be pointed out that magnetic129'32*149 multipolar, dispersion, and induction intermolecular forces are usually negligible. The molecule NO, for example, has a permanent magnetic dipole m due to
30 INTERMOLECULAR FORCES 2.1 electron orbital and spin motion, as well as a permanent electric dipole ijl. The ratio of the magnetic to electric dipole-dipole interaction energy is (see §2.4.5) umju^~(m2lr3)j{ix2lr3) = {mlixJ. The magnitudes of the dipole moments are ix~eaa (i.e. 1 au, where ao = h2/mce2 is the Bohr radius), and m~eh/2mcc (i.e. lau=lBohr magneton). Hence ml\i~ e2/hc = a (the dimensionless fine structure constant = 1/137). Thus the ratio is umm/u^ ~ 1CT4. Other7 relativistic [i.e. O(a2), O(a4), etc.] terms are also negligible; for exceptional cases see, e.g. discussions of dispersion interactions in optically active molecules,89c and spin-orbit effects in NO.89f (Retarded dispersion interactions, which are O(a~J) at large r, are discussed in § 2.6.) The molecule O2 also has a magnetic dipole moment by virtue of a non-zero electron spin (S = 1). For O2-O2 interactions8911 a spin-dependent overlap (Heisenberg-type89e exchange) potential has been invoked to explain the low temperature crystal structure and antifer- romagnetism. This interaction is not a magnetic one, however, but a much larger one of basically electrostatic origin (see also § 2.7) cast into the form of an effective spin-spin interaction. 2.1 Model potentials Because accurate theoretical potentials are few for polyatomic molecules, statistical mechanical calculations are usually done with model potentials. A particular model may be purely empirical [e.g. atom-atom], or semi- empirical [e.g. generalized Stockmayer], where some of the terms have a theoretical basis. In this section we briefly discuss in turn a number of such models; further discussion of some of these and other models is given by Mason and Spurling.6 2.1.1 Generalized Stockmayer model6-7¦1214-90~2a This model consists of central and non-central terms. For the central part one assumes a two-parameter central form u(r) u(r) = sf(rla) B.1) where e and u set the energy and distance scales respectively. The classic example is the Lennard-Jones (LJ) A2, 6) form where s and u are the well depth and diameter respectively. Occasionally an (n, 6) potential is used in place of a A2,6), thereby introducing another adjustable parameter. The long-range non-central part contains in general a truncated sum of multipolar, induction and dispersion terms
2.1 MODEL POTENTIALS 31 Fig. 2.2. Equipotential contours for (ufrojjojj)),^ (i.e. the pair potential averaged over all orientations of molecule 2), in a plane zx containing the nuclei of molecule 1 (which lie along z) and the centre of molecule 2, for four different models for N2-N2 interactions. Crosses indicate the locations of the nuclei of molecule 1 (bond length / = 1.102 A). The outer dimensions of each of the boxes shown are 13.0x13.0 A. Starting from the innermost contour, the contours in each case correspond to (u)<1Jk = 100 000, 50 000, 10 000, 5000, 1000, 500, 0, -20, -40, -60, -80, -95, -80, -60, -40, and -20 K; in the case of (c) the —95 K contour is not present, and for (a) neither the -80 nor the -95 K contours are present- For each negative value of (u^/fc there are two contours, which in some cases form closed loops, as in (a), (b), and (c). Thus, in (a) the closed loop corresponds to —60 K. The figures are for the following models: (a) generalized Stock- mayer, U + S-model of eqn B.4) with928 e/k = 87.5K, a = 3.702 A, 8 = 0.15; (b) atom- atom U model of eqn B.5) and B.6), with" 6NN/k =37.32 K, o-NN = 3.310A, 1 = 1.090 A; (c) Corner model of eqns B,8)-B.10), with117 en/k = 119.4 K, ero = 3.37A, I* = 0.146; (d) Gaussian overlap model of eqns B.8), B.12), and B.13) with16 E0/k = 94 K, <r0 = 3.37 A.,x = 0-257. (From ref. 92b.)'
32 INTERMOLECULAR FORCES (see §§ 2.4-2.6) so that the total potential is u(ra),&>2) = B.3 The short-range angle-dependent overlap part uov(ra>i&>2), representing the shape or core of the potential, is modelled, for diatomics for examplt by the leading (I = 1, 2) terms of the type (see § 2.7) Sr~12P,(cos 0[). Her^ P, is the Legendre polynomial, 6[ is the polar angle of molecule i relative to the intermolecular axis r (see Fig. 2.9), and 8 is a strength coefficient Figure 2.2(a) shows some equipotential contours for a LJ central term plus / = 2 components of this overlap model Ccos20I + 3 cos202-2) B.4 for N2-N2, averaged over the orientations a>2 of molecule 2. For fluic calculations there are at least three (e, cr, S) adjustable parameters, dt- pending on whether the multipole moments and polarizabilities entering «muii> "ind- and "dis are known from other sources. Further spherica harmonic terms can easily be incorporated into uov, but this may lead U an unwieldy number of adjustable parameters of the S-type. Of courst for a given system, some of the terms in B.3) may be negligible. The model is correct at sufficiently long range and is convenient for use in analytic (see Volume 2) and computer simulation calculations. The anisotropic part of the potential is oversimplified in neglecting othe" terms at long and short range, and in assuming that simple additivit- holds at intermediate range. (It is sometimes useful to introduce ar Short | 6 7 r/au Intermediate Long Fie;. 2.3. Approximate division of the He-He potential into snort, intermediate, am" lonu-range parts A au of energy = e2la0). (From ref. 93.)
2.1 MODEL POTENTIALS 33 -40 -120 5 6 rlk Fig. 2.4. Argon-argon pair potential. The solid curve is the potential of Barker, Fisher, and Watts4 and the broken curve is the LJ A2,6) potential with elk = 119.8 K and rr = 3.405 A. (From ref. 94.) intermediate range, between the short and long ranges, which is indicated roughly on Fig. 2.3 for He-He; theories for the intermediate range for atoms are reviewed in refs. 1, 10, 27, 93; for calculations on molecules, see, e.g. refs. 79d, 93a.) The central part B.2) is also undoubtedly somewhat in error. Modern rare gas potentials are known to have a greater well depth and to be shallower in the tail than the LJ potential, as shown in Fig. 2.4. 2.1.2 Atom-atom and site-site models14-20-20*-20*3-26-95-103 In this model one assumes that the intermolecular potential is a sum of interatomic potentials uafi between the atoms which constitute the two molecules: B.5) where rafi is the separation between atoms a and j3 of molecules 1 and 2 respectively. Thus there are 4 atom-atom terms for N2-N2, 25 terms for CH4-CH4, 144 terms for C6H6-C6H6, etc. The atomic potentials uafi are usually taken of LJA2, 6) form, B.6) but other forms, e.g. r ", (exp, 6), .fused hard sphere, and Kihara15 (§2.1.4) have also been used. (For larger molecules, e.g. hydrocarbons, one hopes15-20'102' to find parameters eafi, arafi that are transferable, so
34 INTERMOLECULAR FORCES 2.1 that predictions can be made for cases not used in the fitting.) When the interaction sites a and /3 are taken as not coincident with the nuclei (and often when they do coincide), the model is referred to as a site-site potential. This lack of coincidence, of course, introduces more adjustable parameters. For small molecules the atom-atom model is convenient for use in computer simulation studies.104 For larger molecules (e.g. C6H6) there are too many terms uafi for it to be practical if the number of interaction sites is taken equal to the number of atoms. As it stands, the dependence on molecular orientations in B.5) is implicit, so that it is difficult to use in analytic perturbation calculations of the type discussed in Chapter 4. The latter two difficulties may be obviated in some cases by expanding B.4) in spherical harmonics105 (see § 2.8). The model can be extended to non- rigid and very large molecules (§ 2.11). The atom-atom model may give a qualitatively correct short-range shape or overlap potential. Figure 2.2(b) shows some equipotential contours forNz-Nz.Themodelhas.however.anumberofdefects.14-15-20"-25-27-441051 The principal deficiency is the form of the potential at long range. Due to the model's neglect of bonding charge density73 (i.e. the elec- electronic charge which has shifted from the atoms to the bonds upon molecule formation) the atom-atom model neglects entirely all the long- range multipole and induction interactions, and treats incorrectly the long-range anisotropic dispersion interactions. For example, the leading long-range anisotropic term in the LJ atom-atom potential for N2-Ar varies with r as r~8 (see § 2.8), whereas the correct London anisotropic dispersion term varies as r~6 (see § 2.6). In some cases it also gives the wrong angular dependence of the dispersion forces. This can lead107 to predictions of incorrect dimer structures. There are also other deficien- deficiencies, related to the assumed pairwise additivity109>109a of the model, to the exact shape of the equipotential contours,44'110 and to the use of atom- atom or bond-bond models when non-localized electrons26'111 (e.g. ir electrons) are present in the molecule (as in C6H6). Some of these deficiencies can be rectified (see below). 2.1.3 Point charge models5-40-51-79*-111-3-206 For H20-H20 interactions, many multipoles appear to be important, so that the electrostatic part of the anisotropic potential has been rep- represented by point charge models, both planar and non-planar. The electrostatic potential is thus chosen to bet «(ro,1a,2) = S^e B.7) t Throughout this chapter and the rest of the book we use the esu system of units. See Note on units, p. xiv, and Appendix D.
2.1 MODEL POTENTIALS 35 where a and J3 label charge sites in molecules 1 and 2 respectively. Such models have also been used for other so-called hydrogen-bonded fluids,112 e.g. HF and NH3, and for other cases as well.51 These models can be thought of as site-site potentials of the charge-charge type, and are suitable for computer simulation.104'112 Examples are given in § 2.9, including the ST2 potential for H2O-H2O interactions. All the above models can be refined and combined14-15-241023-114115 in various ways. Thus one can add the leading multipole interaction to the atom-atom potential.114115 For QH6-C6H6 bond-bond models have been constructed,2013-116 involving multipolar, induction, dispersion, and overlap interactions between the six CH bonds of each molecule. For H2O-H2O5-47-116a and other cases,5012 theoretical potentials have been fitted by a combination of atom-atom and charge-charge terms. 2.1.4 Pseudo-atomic models In these models a pseudo-atomic expression of the type B.1) is used for u(iaIw2) with either the potential parameters (Corner, Gaussian over- overlap, or purely empirical models) or the distance variable (Kihara model) replaced by a function of the orientations. Like the atom-atom and site-site models their principal virtue is in providing a reasonable descrip- description of the molecular shape. They usually have fewer adjustable parame- parameters than the atom-atom models, especially for more complex molecules; in addition, they are more convenient in computer simulations of complex molecules such as benzene, for which 144 atom-atom terms might be needed as opposed to one for the pseudo-atomic case. Apart from this, the pseudo-atomic models suffer from the same defects as the atom-atom models,.in that they fail to yield the correct long-range dispersion and multipolar behaviour (although the dispersion part may be qualitatively correct in some cases). (i) Corner's four-site model6-8-117 Corner proposed modelling an intermolecular potential u(roIoJ) by a pseudo-atomic form B.1), where e and a are assumed to be angle dependent. Thus the LJ form B.2) would become i \ a t ,|7o-(a>ifaJ2a))Y2 /cr(qIftJ(o)>\61 u(ra>1a>2) = 4e(a>1a>2a>)l I I -I I I B.8) where <ou a>2 and a> denote the orientations of molecule 1, molecule 2, and the intermolecular axis r respectively. For symmetric linear molecules Corner assumed a LJ site-site potential with four sites per molecule, each with site-site parameters ess and crss. By comparing the exact site-site potential with B.8) he was able to find approximate empirical expressions
36 INTERMOLECULAR FORCES 2.1 for e(aIoJa)) and cr{aI(o2o))- These relations are +ll*s[c[s'2c'2c'-ill*s'l2s?cnY, B.9) s^2c;2c22c'2]' B.10) where s'j = sin0';, c5 = cos0';, c' = cos($( —<?2) [see Fig. 2.9], eo=16ess, cr0 = crss and I* = 2*(l/cr0), with I the intramolecular distance between the first and fourth sites. The above expressions are valid for l*?? 1.5, and are not necessarily unique or the simplest that could be found from the numerical fitting. It is a three-parameter model (e0, cr0, I*), or perhaps a two-parameter one if bond-length information is used to fix I*. It is suitable for virial coefficient calculations6'8'117, and for computer simula- simulation. In Fig. 2.2(c) we show some contours of ("(rwjO)^),^ for N2-N2; the parameters117 were obtained by fitting virial data. (ii) Gaussian overlap model16AS-118 We consider an axially symmetric molecule whose charge density p(r) is assumed to be Gaussian, p(r)~exp[—(jc2 + y2)/<r2 — z2/cr2] where x, y and z refer to the principal axes, z being the symmetry axis. The contour surfaces of p are ellipsoids of revolution, with cry and cr± the 'diameters' of the distribution parallel and perpendicular respectively to the sym- symmetry axis. If two such distributions, separated by r and having orienta- orientations o>i and o>2, interact, the Coulomb energy is rigorously given by M(ra>ia>2)= J dr! dr2p1(rI)p2(i2) ^ — rj. (Here a), = 6^ is the orientation of the symmetry axis of distribution i.) Electron-electron exchange ener- energies have a similar but more complicated two-centre integral form.8-27 One now assumes, as seems a reasonable first guess (see, however, refs. 93 and 121), that at short-range where the distributions overlap slightly, u is approximately proportional to the overlap volume integral of the two distributions, u(r<t>i<t>2) ~ J driPi(*i)P2(ii ~~')• This integral can be evaluated exactly,118 and has a Gaussian form uCrw^J = e(o»1w26)) e-r2M<">°vJ B.11) where e(o)iW2w) and a-(aIaJw) are angle-dependent strength and range parameters given by e(a>io»2a») = eo(l - *2c?2)-*, B.12) or(a»1a»2a») = cro(l - x2c?2)Kl - x(c? + c\) + *2BcjC2c12 - c?2))^ B.13)
2.1 MODEL POTENTIALS 37 where <70 = \/2crx and ^ = [1 - (cx/cr^^/tl + (cr±/cr||J] is an anisotropy parameter. If ni, n2 and n denote unit vectors along cou <o2 and &> respectively, then the three cosines in B.12) and B.13) are given by c1 = n1.n, c2 = n2.n, c12 = n1.n2. Note that in this model e(oIaJa)) is independent of w; as one expects, e(w1oJo)), which measures the overlap at zero separation, is a maximum when the molecules are parallel, and a minimum when they are perpendicular. The model is constructed to give a reasonable angle dependence to u(tct)iCt)-2), but the Gaussian r-dependence in B.11) is not realistic. One then replaces the latter by a more realistic one, e.g. the LJ form B.8). If the r-dependence is thus fixed, the model is a three-parameter one (e0, a0, x), unless one reduces this number by using bond lengths and van der Waals radii to estimate cr0 and/or x- It is valid for oblate as well as prolate shapes of arbitrary anisotropy (whereas the Corner model discussed above (which could be extended) is valid only for prolate molecules of restricted anisotropy), and is convenient for use in computer simula- simulation.119 It has been applied16-45 to H2, N2, CO2, and C6H6 interactions. Figure 2.2(d) shows some equipotential contours for N2-N2, averaged over molecule 2's orientations. The parameters were obtained by fitting virial and solid state data. The shape of the molecular core is seen to be modelled correctly, at least qualitatively. It is interesting that the attrac- attractive dispersion potential for N2 is also given correctly, again at least qualitatively, since the Gaussian model dispersion potential —4e(aIa>2a>)[cr(a>1«2a>)/r]6 is most attractive, for a given separation r, essentially where cr is largest (since e depends only on the angle between Wj and cd2), i.e. in the end-to-end orientation. The true dispersion potential is most attractive in this orientation since linear molecules are more polarizable along their bonds than across it. Similarly, for two oblate C6H6 molecules the model gives the largest attraction when the two planar molecules are side-to-side, which is correct since benzene is most polarizable for a direction in its plane. It is a flexible model which can be extended in various directions, e.g., to (a) non-identical molecules,118>118a (b) asymmetric linear molecules120 (by incorporating odd-order spherical harmonics cu c\, etc. in B.12) and B.13)), (c) general shape molecules,118 (d) better approximations1203-121 than the basic one of the model, that u is proportional to the overlap integral of p1 and p2; this leads to a dependence of e on a>, as well as on a), and w2. (iii) Kihara core mode/6-8-15-16-24-121* In this model the molecules are envisaged as convex hard cores (e.g. spherocylinders, ellipsoids, etc.) which may or may not be surrounded by longer range force fields. The pair potential M(roIoJ) is assumed to be a
2.2 function of p, where p = p(rco1w2) is the shortest distance between the cores for the configuration (rw1co2)- F°r the case that force fields are assumed to surround the cores, the LJ form is often employed, where e0, cr0 are angle-independent parameters. Because of the difficulty in calculating pOro)!^) f°r a given configuration of a pair of molecules, the general model has not been used until recently122 for theories for dense fluids; its use has been restricted to virial coefficient calcula- tions.6'8'24-123 The pure hard core model has been used for computer simulations123-124 and in scaled particle theory125-126 (Chapter 6). The analytic work (virial coefficients, scaled particle theory) makes use of the 'mean curvature' of the core, so that Kihara core potentials are usually restricted to convex cases (see ref. 127 and Chapter 6). An example of a useful non-convex core is the fused hard sphere model discussed in Chapter 5 (which we have classified as an atom-atom model). 2.2 Isotropic and anisotropic parts of the potential For some applications it is convenient to separate the pair potential into isotropic and anisotropic parts M(roIoJ)=M0(r)+Ma(rftIco2) B.15) where the isotropic potential m0 depends only on r. This decomposition is obviously not unique, but the most convenient choice for u0 is the unweighted orientational average of the full potential: uo(r) = <u(r<o1<o2)>mi(a2 B.16) If =73 B.17) where da) = sin 0 dO d(f> or sin 0 AO d<f> dx for linear or non-linear molecules respectively; hence fl is 4ir for linear or 8tt2 for non-linear molecules (see (A.28) and (A.81)). By a linear molecule we mean one where the equilibrium positions of all the nuclei lie on a straight line, so that the molecule has complete axial symmetry. In our definition, therefore, molecules like ethylene and ammonia are non-linear. The choice B.16) may not be optimal if u(r<t>i<t>2) contains a very hard anisotropic core (see
2.3 SPHERICAL HARMONIC EXPANSION FOR THE POTENTIAL 39 §§4.6, 4.9 and 5.4.9a). The anisotropic part ua in B.15) is, defined as u — u0, and, for the choice B.16), has the important property l<J2 = 0. B.18) All anisotropic potentials defined by B.15) and B.16) satisfy B.18). As we shall see for neutral molecules, multipolar and some terms of the other potentials also satisfy the stronger condition = 0 B.19) and will be referred to as 'multipole-like' potentials. 2.3 Spherical harmonic expansion for the potential1273 As will be seen in later chapters it is extremely useful in theoretical calculations to have a spherical harmonic expansion128 for the potential M(ra)!aJ). For detailed discussion of the use of spherical harmonic methods, together with the properties of the harmonics, the reader is referred to Appendix A and § 3.2. Here we merely note that analytic statistical mechanical calculations are easily carried out since the proper- properties (integration, differentiation, rotation, etc.) of the spherical harmonics are so well established. Also a given potential model will often contain only a few harmonics (e.g. each multipole interaction contains only one harmonic of the type lxlzl; see § 2.4.5). In the following sections we expand in turn the multipolar, induction, dispersion and overlap interac- interactions. Here we discuss the general properties of such expansions. For molecules of general (non-linear) shape we write the expansion as Ylm(cor B.20) where C^iM; m1m2m) is a Clebsch-Gordan (CG) coefficient, Dlmn(co)* is a generalized spherical harmonic, and Y|m(a>) is a spherical harmonic, all in the conventions of Rose (see Appendix A). As explained! in Appendix t The main point of the argument127" is that, for fixed IJ^ln^^ the unique (within a factor) invariant combination of products Dl&.D^n, is I C(l1l2l;m1m2rn)Dh*ntD'fin2Yfm. m(s) Similarly the unique invariant formed from products of three ordinary spherical harmonics [used for linear molecules in B.23)] is Zm(si C(/,/2/; m,m2m) YlimiYhmiY^. These are generalizations of the addition theorem tor products of two spherical harmonics, Xm Ylm(aj)Yjl(o)') = invariant. For further discussion see notes 128a, 128b of this chapter, and examples near (A.195), (A.196) of Appendix A. The simplest example of these symmetry arguments occurs following eqn B.75).
40 INTERMOLECULAR FORCES 2.3 A (see argument following (A. 193)) the CG coefficient, which contains all the m-dependence of the expansion coefficient, occurs in B.20) to ensure that u is invariant under simultaneous rotation of Wj, o>2 and w> where Wj = faOtXi (Euler angles), and to = 0<? (polar angles) refer to an arbitrary space-fixed coordinate frame (Fig. 1.2b). The quantity u(lj2l; nxn2; r) is referred to as the reduced expansion coefficient (as opposed to the complete expansion coefficient uilJJ; nxn2; r)C{\x\2\; m1m2m)). As we shall see, the sum ?m(s) CD*D*Y* is in general complex, so that u(lil2l', W1W2; r) W'H in general be complex to ensure u(raIw2) is real. [In the linear case B.23), however, it will turn out that ?m(s) CYYY* is real, so that u(/1i2/; r) is real.] For linear molecules, which are axially symmetric, the interaction energy u must also be invariant under rotations of each molecule about its symmetry axis. With the choice of the body-fixed frame JC^Zf with z, along the symmetry axis of molecule i, this means that u must b^ independent of the third Euler angles Xi and Xi (see Fig- 2-5). Thus, since D'mj,(j)8x)* depends on x as exp(m^) (see (A.64)), we must have u(l1l2l;n1n2;r) = Using (A.105), B.21) B.22) \2l + V we see that B.20) reduces to a sum over three ordinary spherical z Fig. 2.5. Geometric interpretation of the Euler angles: 6<f> are the polar angles of the body-fixed z axis, and x is the 'twist' angle about z. (Cf. § A.2 and Fig. A.6 for a precise definition of the Euler angles.)
2.3 SPHERICAL HARMONIC EXPANSION FOR THE POTENTIAL 41 Rg. 2.6. Geometry of interaction for two linear molecules. <o, = ().<?. denotes the orienta- orientation of the symmetry axis of molecule i, and <o denotes the orientation of the intermolecu- lar axis. All orientations refer to the space-fixed axes XYZ. harmonics u(l1l2l;r)C(l1l2l;m1m2m) i>l>«) B-23) where a>t = $,-0; now denotes the polar angles of molecule i (Fig. 2.6) and ?;00;r). B.24) The invariant ?m(s) CYYY* in B.23) must occur for linear molecules1283, and is discussed in Appendix A (see (A.195a)). The expressions B.20) and B.23) are sometimes referred to as 'invariant expansions' since they hold in an arbitrary coordinate system (see also (A.195,195a)). The expansion coefficients u(lj2l; n^; r) satisfy other general con- constraints. Since u is real, we can put u - u* in B.20) and withthe help of (A.66) and (A. 133) we find (A.^O ^ t^'4^ = 1) where n = — n. If the molecules are identical, u is invariant under inter- interchange of the molecules, i.e. under aI-^w2, o>2-* <ax, <o—*—<o (where —a> = (tt — 6, it + 4>) is the inverted direction of «). This is illustrated in Fig. 2.7 for the case of linear molecules. Using (A.47) and (A.134) we
42 INTERMOLECULAR FORCES 2.3 (a) (b) Fig. 2.7. The configurations of two identical linear molecules, (a) before, and (b) after interchange (i.e. ojj->o>2, <*>2~*°>i> g>—*~g>)- find u(hl2l; nxn2; r) = (-) ^; r). B.26) The energy u is invariant under inversion of the axes. (Inversion (x—> — x, y —* — y, z—> — z) is here done passively. It can also be done actively (see Figs 2.8, and A.5, and also Figs A.2, A.3 for an analogous discussion of active and passive rotations)). For molecules of general shape this does not lead to a condition on u^-J^l; n^n-i, r), since under inversion a right-handed (R) molecule changes into a left-handed (L) one, and vice versa. One can then only derive a relation between the expan- expansion coefficients for right- and left-handed molecules. (See ref. 128c for an analogous discussion of wave functions, and also eqn (A.103)). In cases where the R and L forms of a molecule are identical (i.e. congruent or superposable), one can find an equivalent rotation for the inversion operation, and thereby find a condition on the us. The simplest example is the linear molecule where inversion F<j)) -* (it - 6, tt + <j>) can be carried out by a rotation of ir about an axis perpendicular to the symmetry axis. Using (A.47), we see that inversion [i.e. «!—»¦— u>u&J~*~w2, a> —>— w] introduces a Fig. 2.8. Right- and left-handed forms for the molecule Cabcd. The two forms are obtained from each other by inversion in the origin. (An equivalent method is to reflect one molecule in a vertical mirror through its origin and replace the second molecule with this image; the two forms are then mirror images of each other, in a vertical mirror midway between them.)
2.3 SPHERICAL HARMONIC EXPANSION FOR THE POTENTIAL 43 factor (-)'.+!*+l into B.23), so that w(?1/2/;00; r) = 0 unless/h + k+l is even. From B.25) we also see that u(/i/2';00; r) 'S rea' f°r linear molecules. Planar molecules also have identical R and L forms. In fact, the simple molecules for which we give explicit calculations in this book all have sufficient symmetry for the R and L forms to be identical. The classic case of non-identical R and L forms is the molecule Cabcd, where C = carbon. Lactic acid, for example, corresponds to the case a = COOH, b = OH, c = H, d = CH3, (see Fig. 2.8). Parity conditions for other (sym- (symmetrical) shapes can be derived from (A. 103). Because the intermolecular forces129 are the same for pure fluids of R and L molecules, no differences in thermodynamic properties can occur. Binary AR/AL mixtures,130 how- however, will in general have properties differing from the pure AR and AL fluids. (Here AR denotes a right-handed A-molecule.) Also, mixtures of the type AR/BR will have different properties from the type AR/BL. The pure R and L fluids do differ in their optical properties (optical activity etc; see, e.g. Barron, ref. 11 of Appendix C). For molecules of other symmetrical shapes (e.g. tetrahedral CH4, octahedral SF6), rotational constraint conditions on the uil^l; nxn2; r) can also be written down.12>13>10'5128d Examples are given in §§ 2.4.5, 2.7, and 2.8. In some applications (e.g. structure factor calculations, see Chapter 9) space-fixed axes are important, whereas in others (e.g. thermodynamic calculations) one can choose the polar axis along the intermolecular axis. For example, for linear molecules, we use (A.55), (or i ( \rY8mo B-28) to reduce B.23) to the formt M(K»i»2= I u(.hhm\r)Yhm(a>[)Yhm{a>'2) B.29) where w'j denotes the orientation of molecule i relative to the inter- intermolecular axis (see Fig. 2.9), and f mmOMhy-r). B.30) Since Y[m@<?) depends on 4> as exp(im$) (see (A.2)), B.29) is seen to t Note that some authors18-2791 use (it-62, -<!>i) rather than polar angles F'24>'2). Also note that since CiJihh' mlm2m-3) = 0 unless each ImJsij, for fixed /]/2, m in B.29) runs over the B^+1) values m = -l<,...,+l^, where i< is the smaller of lx and l2. Simple counting shows that, as must be the case, this is the same (i.e. 21,- +1) as the number of / values in B.23), for ( in the range 1 = |/, -12\, ...,lx + l2.
44 INTERMOLECULAR FORCE 2.3 Fig. 2.9. The intermolecular axis coordinate system (the 'r-frame'), with polar (Z') axis along r. depend only on D>[~- (f>2); this is to be expected since u is invariant under rotation of the two molecule system about the intermolecular axis. Since CG coefficients are real and uil^l; r) is real, we see from B.30) that uilj^m; r) is real; it also satisfies u(M2m ;/•) = «(*! !2m;r), B.31) since C(ltl2l; mmO) = (-)i^'^'C(l1l2l; mmO) (see (A.133)), and lt + l2+l is even. For identical molecules B.26) and B.30) (together with the definition B.24) and the use of (A.133) and (A.134)) give u(/1Z2rn;r) = (-I>+I*H(Mim;r). B.32) Conversely, if the M(?1J2m;r) are known, we can solve B.30) for "OiM; r)- Multiplying both sides of B.30) by C(lj2l'; mmO), summing over m, and using the orthogonality relation (A.141) for the CG coeffi- coefficients, gives u(^f2m;r). B.33) If molecule 2 is spherical (e.g. HCl/Ar) B.29) simplifies further since u(r(i)[aJ) is independent of a>2 = 02$, and of (f>[. Hence we have 12= m = 0 so that using (A.2) and (A.56) we get uirO'} — / u(l' t)P (cos 0') B 29a) i where u(l; r) = Dir)~1Bl + 1)'m(/00; r) and P, is the Legendre polyno- polynomial. In terms of the space-fixed axes coefficients u(I1{2{; r) and w(/1/2/;n1n2;r) we also have u(l; r) = DTr)~iBl + l)u(l0l; r) = Dir)-l{2l+l)iu(lM;00;r). In a similar way we can refer the general expansion B.20) to the intermolecular axis frame. The expansion coefficients uil^l; nxn2; r) and* u{lx\2m; n^n2; r) are again related by B.30) and B.33).
2,4 MULTIPOLE EXPANSIONS 45 The dependence of the coefficients w(Z1?2Z; tiin2', r) on the' choice of molecular centre is discussed below separately for the multipolar, induc- induction, dispersion and overlap interactions. Finally we note that since the unweighted orientational average of a spherical harmonic of order lj=0 vanishes (see (A.38), (A.92)) <DLFi)*>M=<Yht(«)>.=0, B.34) the decomposition B.15) of the potential into isotropic and anisotropic parts is accomplished automatically using a spherical harmonic expansion. Thus the lj2l = 000 term gives the isotropic part u0, uo(r) = Dtt)-*h@00; 00; r) B.35) and the terms with ^^^000 constitute the anisotropic part ua. Those terms with both Z1; /2^0 satisfy B.19) and are the multipole-like parts of 2.4 Multipole expansions When two molecules are sufficiently far apart that the overlap of their charge clouds is negligible, a part of the interaction energy can be interpreted as being electrostatic and due to Coulomb forces between the two charge clouds. This contribution to the interaction energy arises even for rigid (non-polarizable) molecules, and can be evaluated classi- classically by calculating the field acting on one molecule due to the other [see § 2.6 and Appendix C for a quantum discussion]. This field is usually approximated by a multipole series, which is valid if the distance r between the two molecular centres is greater than the dimensions of the molecular charge clouds (see § 2.9). The multipole interactions can be attractive or repulsive, depending on the relative orientations of the two molecules, and angle-average to zero with the exception of the charge-charge term (see B.19)). We first briefly remind the reader of some basic electrostatics (§ 2.4.1). In § 2.4.2 we then discuss the one-centre expansion problem, where the field outside a charge cloud is decomposed into multipolar contributions. In § 2.4.3 we deal with properties of the multipole moments and in § 2.4.4 consider the multipolar decomposition of the interaction energy of a charge cloud with a given external field. The results of these preliminary • sections serve to introduce the multipole moments, and also are used in later sections of the book. In § 2.4.5 the two-centre expansion problem is considered, where the electrostatic interaction energy between two charge clouds is decomposed into multipole-multipole interactions. In refs 130a the corresponding results in two-, four-, and d-dimensions are derived.
46 INTERMOLECULAR FORCES 2.4 2.4.1 Review of basic electrostatics The force F2 exerted on a charge q2 due to charge qx is given by Coulomb's law (in esu-see Appendix D and Note on units, p. xiv) as B.36) B.37) where f = r/r is a unit vector along r, r = r2 — it is the vector from qx to q2 and E(r) is the field at r due to qt; the force Fj on q1 due to q2 is given by Fi = —F2 (Fig. 2.10). When q1 and q2 have the same sign the forces are repulsive; for charges of opposite sign the forces are attractive. The force F2 can be expressed in terms of the electric field E(r), or the potential field <j>(i). The electric field E at r (in this case due to a single charge qx taken at the origin for simplicity) is defined as the force acting on a unit positive test charge placed at the position r, and is given from B.36) as E(r) = ^f. B.38) We can now calculate V. E and VxE from B.38); we get (proof below) V.E = 4ttP, B.39) VxE = 0 B.40) where p(r) = q1S(T) is the charge density due to qu and 5(r) is the Dirac delta function (see Appendix B). Equations B.39), B.40) are the funda^ mental Maxwell equations of electrostatics, and, together with the bound- boundary condition E(r-»<»)-*0, are equivalent to Coulomb's law B.38). Because the fields due to a number of source charges qlt q2, ¦ ¦ ¦ at ri, r2,... are superposable, the relations B.39), B.40) are true in general. To derive B.39) and B.40) we first rewrite B.38) as B.41) Fig. 2.10. Equal and opposite electrostatic forces Ft and F2 on two point charges separated by r, for the case that qt and q2 have the same sign.
2.4 MULTIPOLE EXPANSIONS 47 Equation B.40) then follows immediately because of the identity Vx V<E> = 0, where 4> is arbitrary (see (B.42)). [A direct calculation from B.38) also quickly gives B.40)]. Equation B.39) follows from B.41) when we use the relation (B.81), i.e. V2 - =-47i-S(r), B.42) (Refs. 131, 133 contain more detailed discussions.) Because E is curlless (eqn. B.40)), a general theorem132 states that E(r) can be represented as the gradient of a scalar potential 4>(i), E = -V<fc B.43) where the minus sign is included for convenience (see below). In general one finds <j> from the general equations B.45) or B.49) given below. In this case (f> is obvious by inspection of B.41) and is given by *(r) = ^. B.44) 4>(t) quite generally has the physical significance133 of the potential energy (relative to infinity) of a unit positive test charge at r. To see this we calculate the work (w) done against the electric field in bringing a unit positive test charge from infinity to point r, w=-| E.dl B.45) where dl is a line element of the path taken between infinity and r. Since E is curlless (i.e. conservative), w is independent of which path is chosen,132 a fact which we verify shortly. Substituting B.43) in B.45) gives w=\ V^.dl. Using the fact that V4>.dl dxAyA v dx dy y dz where d(f> is the total differential, we get -['¦ as stated above. We have chosen <?(<») = 0. [For the point-source case
48 INTERMOLECULAR FORCES 2.4 B.38) one easily verifies explicitly that B.45) gives w = qjr. It is simplest in this case to choose the straight path along the radius from infinity to r, so that dl = f dr, where dr is the (negative) radius increment.] Of course (f> + C, where C is a constant, is an equally good potential, since (f> and (f> + C give rise to the same field E = — V(f>. The choice B.44) means we assume for simplicity ^O30) = 0, as mentioned above. From B.39) and B.43) we derive the Poisson equation for the potential V2<? = -4irp B.46) which reduces to Laplace's equation f V2<? = 0 B.47) ^ in the source-free region of space. (The latter is r > 0 for a point source at the origin.) Using the relation (cf. B.42)) -rO, B.48) one easily verifies that the solution of B.46) for a general source p(r) is B.49) If the source p consists of discrete charges qt at positions i;, p(r) = ZqiS(r-ri), * B.50) i then B.49) reduces to Equations B.49) and B.51) agree with what one would write down intuitively using B.44) and the superposition principle. One often works with the potential (f> rather than the field E, both because of the physical significance of <\>, and because a scalar field is in general simpler than a vector field. [The analogous magnetic scalar potential due to a localized current distribution, and its multipole expansion, can also be derived.132a] 2.4.2 Held outside a charge distribution13-18-19133'135 For simplicity we first consider a discrete charge distribution with charges labelled! i = l, 2,... (see Fig. 2.11); the final results are easily t In this section it is convenient to use r, as the position of charge i in some molecule, whereas in other parts of the book we shall often use rf as the position of the centre of molecule i. Similarly in § 2.4.5 rtj is the separation between qt and q,, and not the intermolecular separation. This will not lead to confusion since in the final results the charge positions r( do not appear explicitly.
2.4 MULTIPOLE EXPANSIONS 49 Rg. 2.11. The field point r and the source point Tt; other charges (not shown) q2, %... are atr2>r3... . generalized to continuous distributions by replacing the sum over discrete charges by an integral over the charge density (cf. B.49) and B.51)). The electrostatic potential 4>(i) at a point r=(r, 0<t>) = (r, w) due to source charges qu q2 ... at r^ r2 ... is Y.i qj\t-ti\, (see B.51)), where the rt are referred to some arbitrary space-fixed frame, with origin chosen some- somewhere inside the distribution (Fig. 2.11). The multipole series involves an expansion of |r—rj in powers of rjr. This expansion can be derived using either Cartesian or spherical tensors.! Both approaches are useful. The Cartesian method is elementary for the lower order (dipole and quadrupole) terms and gives a simple physical picture of the lower order multipole moments; however, the spherical tensor method is easier to use both for the moments themselves and in statistical mechanical applica- applications (see Chapters 4, 5 and Vol. 2) when higher moments and general theorems must be considered. We first discuss the multipole expansion in terms of Cartesian tensors, and follow this by the spherical tensor derivation. The relations between the Cartesian and spherical multipole tensors are given at the end of this section. (i) Cartesian tensor derivation18-19-134*-1*5 The multipole expansion in terms of Cartesian tensors is obtained by expanding |r—T;!" in a Taylor series in r,-, so that the electrostatic potential is given by134b t Cartesian and spherical tensors are introduced in Appendices B and A, respectively, and the relations between them are also discussed in Appendix A.
50 INTERMOLECULAR FORCES 2.<t or 4>(r) = qT<0)(r) — |x. T^'Or) +10 : TB)(r) -§O • T<3)(r) +... B.52) where 7*" = 1/r, and TA) = V(l/r), TB) = W(l/r), T(I) = V'(l/r) B.53) with V' = VV ... V (/ factors). The contraction notation used in B.52) is defined in § A.4.2 of Appendix A and in §B.l of Appendix B. The multipole moments q, |a, 0, O, etc. are given by B.54) and so on. Here q is the charge, |a the dipole moment, 0 the quadrupole moment, and O the octopole moment. These Cartesian tensors are symmetric but not traceless (e.g. 0a3 = 6^, but ?a ^^O). The gradient tensors T0', I*2' etc. can be written out explicitly, TA)(r) = V(l/r) = f f A/r) = -ir2, B.55). or = V(-r/r3) = r3r-"r-lr-3 = Cfr- l)r3, B.56) and so on, where f = r/r is a unit vector along r, and we have used (B.47), Vr = l. The relation B.56) is valid for r^O, cf. B,42). The tensors T® for l>2 are symmetric and traceless,1353 and therefore irreducible - see § A.4.3 - and vary with r as r~<l+1). The Cartesian quadrupole moment 6 has six independent components, but one of these is redundant. This redundancy arises because we can add an arbitrary term of the type Al to 6, since Al:W(l/r) = AV2(l/r) = 0, (r>0). Thus, the term Al produces no r~3 field external to the distribution. Conventionally we take i to make the quadrupole moment traceless. Thus, we can write the
2.4 MULTIPOLE EXPANSIONS 5 quadrupole term <j>2 in B.52) as135b "¦ 4>2(r) = §e:T<2)(r) = 11*^-^1) :T<2>(r) i = K>:T<2)(r) B.5"/ where Q is the traceless quadrupole moment, defined by i-r?lI. B.58 i The components of Q are Q,e=5lqiCwip-'-2M> B-59 and in particular = \ I*Cx2- r?), Qxy = 11 qtxiyi. B.60 i For a continuous charge distribution the summation over qt in B.58) i replaced by an integration over the charge density p(r), Q = ||drp(r)Crr-r2l). B.61 If B.57) is used in B.52) the expression for the potential becomes <t>(r) = qT<0)(r) -|a . TA)(r) +|Q: T<2)(r) +..., B.62 Successive terms in this expansion are approximately in the ratio a/i where a is the size of the molecule, so that convergence is optimal fo r»a (see §2.9). The corresponding expression for the electric field E(r) = -qT^Or) + ji. T<2)(r) - |Q: T<3)(i) +.... B.63 The field line patterns1350""" produced by the charge, dipole, quadrupoli and octopole are sketched in Fig. 2.12. The field line patterns are sucl that the direction of the field at any point is given by the tangent to th< field line through that point, and the magnitude of the field is equal to thi flux density or number of lines passing through unit normal area. (Sei refs. 131, 131a, 133, 135c, for detailed general discussions of field lines. Equipotential surfaces, corresponding to the terms in B.62) can also bf drawn; they are everywhere perpendicular to the field lines. At the smal intermolecular separations occurring in liquids, the electric field terms it
52 INTERMOLECULAR FORCES 2.4 (a) (b) (c) (d) Fig. 2.12. Sketches illustrating field lines from (a) positive ion, (b) dipole, (c) axial quadrupole, and (d) tetrahedral octopole. (From ref. 135e.) B.63) can be large by normal laboratory standards, e.g. of order 10 J au, where 1 au = e/ao = 51.3x 108 V/cm, (see Appendix D for discussion of atomic units), compared to 104-105 V/cm attainable in static macroscopic experiments. (Dynamic fields due to focused giant-pulse laser beams135f\ can easily reach much higher values, e.g. 107V/cm.) The traceless oc- octopole (ft) and hexadecapole (<&) moments are given explicitly by19-27 (we assume a continuous charge density for simplicity of notation) \ J drp(r)[5 V g j drp(r)[35r + ry8ap)], B.64) *,-5r2(rarp8yS ' rpfy8aS + rprs8ay + ryrs8ap) B.65) The general traceless symmetric (or irreducible) Cartesian multipole
2.4 MULTIPOLE EXPANSIONS moment of order I is19'27 Q%... = ^f f drp(r)r2I+1 —^— (!) B.6( I! J dra dre.-. \rJ where there are I indices for Q<0. Equation B.66) is written moi compactly in vector form as drp(r)''2!+1T(I)(r). B.6' The general term in the expansion B.62) is1927.135 ^(r) = ^^Q<0.T<«(r) B.6* where B1-1)!! = B/- l)BZ-3)... E)C)A), and the large dot indicates full contraction (yielding a scalar) of the two rank-J tensor: Q«)tiy...T%y...- The order of the contracted indices a, /3, etc. is immateri; here since Q(!) and T*0 are symmetric tensors; see (B.14). (ii) Spherical tensor derivation**'13'133 To obtain the desired result quickly we first give a short derivatio based on the generating function for Legendre polynomials and th addition theorem for spherical harmonics. We then give a derivation frot first principles based on electrostatic arguments. The latter also intrc duces symmetry arguments of a type which is useful generally (e.g. i §§ 2.3, 2.8, 2.4.3, 2.4.5, A.I, and B.5.) Short derivation. We start with the trigonometric cosine law for |r—r(| = [(r-r,J]* (see Fig. 2.11), (r-riJ = (r-ri). (r-r;) = r2+r?-2rr; cos yt where y{ is the angle between r and rf. Thus we have With the help of the generating function for Legendre polynomials (A.9a), (l-2cosyf+f2)^=? P,(cosTy, (M<1), B.70 1 = 0 where P, is the Legendre polynomial of order /, B.69) can be written as B.71
54 INTERMOLECULAR FORCES 2.4 Using the spherical harmonic addition theorem, (A.33), F,(cos Ti) = (^-) I Y,m(Wi) Ylm(a>)* B.72) where m runs over the values —I,... ,+l, and o>j and a> are the directions of rf and r, respectively, we see that B.71) becomes QlmYfm(o>)/rl+1 B.73) where Qim is the mth component of the spherical multipole moment tensor of order /t of the charge distribution (in the space-fixed frame, see Fig. 2.11), and is given by Qm=Z*riyi«(«i)- B-74) i Properties of the Qlm are discussed below. They are complex quantities in general (see definition (A.2) of the Y,m). As we see from B.68) and B.73), in both the Cartesian and spherical^ tensor expressions, <f>(i) is a sum of multipolar constituents <fr(r), where 4>i(t) (being a scalar) is a contraction of two rank-/ tensors, the one a property of the charge distribution (the multipole moments Q(l) or Q,), the other characterizing the geometric dependence on the field point r (i.e. the T<0(r) or Y,(a>)lrl+1). The corresponding tensors, Q(l) and Q, and T^r) and Y,(&))/rl+1, can be simply related, either in general (using the methods of Appendix A), or in particular cases (see end of this section). First principles derivation. The electrostatic potential at a point r = (r, 6<f>) = (r,o)) due to charge q1 at rt is qj\r—rjj. Choosing an arbitrary coordinate frame with origin somewhere inside the distribution (Fig. 2.11), we expand |r—rjl in spherical harmonics V1 V A"i,m/_ .\y /, . \ V I, *\* A 7C\ I — L-. L-m -<*1 I \M • / ¦* I m V^I/ ¦* ImV.^7 \^"'^J 1,1 where A™/?(rxr) is an expansion coefficient to be determined and the complex conjugation is included for convenience (recall Yfm = (-)mY,E). Since Ir-r^ is invariant under simultaneous rotation of r and r1; and since the unique invariant combination of products of two spherical harmonics is (see (A. 181)) ? Y,Bl(o>1)Y,m(o>)*, tWe note that some authors13-25-102"-133-136* include a factor of Dir/B/ + l))i (or other factors7), and/or a complex conjugation in the definition B.74). For example refs. 12 and 134 use definition B.74) whereas ref. 13 defines QIm in terms of the Racah spherical harmonics C,m (see (A.ll)).
2.4 MULTIPOLE EXPANSIONS 55 we see that A™1™ must vanish for /x / I and m1 / m, and is independent of m. Hence we get r^- = lA,(r1r)Y,m(ftJl)Yi*(o,) B.76) where Afar) is independent of m. We also note that Afor) must be real and symmetric in rx and r since jr—fij^1 is real and symmetric in i^ and r. [The quantity Ir-rJ" is also invariant under translations (i.e. origin shifts). The resulting conditions on the Afar) are complicated, but are not needed here.] For |r—Til/0, |r —r^ is a solution of Laplace's equation for the variables r and rx: V2|r-r,| = V? Ir-r^^O, (see B.42)). We recall (see (A.21), (A.23) or ref. 133) that the fundamental solutions of Laplace's equation Vr<?(R) = 0 are of the two forms RlYlm(O), Ylm(O)/R'+\ B.77) Because of B.77) and the dependence on the product of Ys in B.76), the de- dependence of Ai (rxr) on r1 and r must therefore be a linear combination of one or more of the four types r[r~iUl\ r\r\ r7(l+1>r(l+1), and rx{l+1hl. How- However, because of the boundary conditions that |r—I'll must remain finite as the source point tx -> 0, and must also approach zero as the field point r —> oo5 we can rule out all but the first type. Hence, the dependence of the expansion coefficient Afar) on the radial variables must be of the form A1(r1r) = (ri1/r'+1)Ai B.78) where A, is a factor independent of rt and r. The remaining numerical constants A, can be found by considering a special case, e.g. when Tt and r are both parallel to the polar axis (&)! = <o = 00). Using (A.55) we see that for this case B.76) with B.78) becomes On the other hand a direct binomial expansion yields B-80) Comparison of B.79) and B.80) gives 4tt
56 INTERMOLECULAR FORCES 2.4 Hence we get |_ . I = L, L, J T^T Ylm(<°l) Y*m(<»)- B.82) lr~rll Im V/l + 1/ T This is valid for r>r1; for r< ra (not needed for the multipole expansion) we interchange rt and r in B.82), because of the symmetry in r and rx mentioned earlier. The total electrostatic potential <?(r) at r due to all the charges qt of the distribution is obtained by summing over i the terms like B.82), <f>(t) = Ei <Ii/l*-*il- This again gives B.73) with the multipole moments defined by B.74). For a continuous distribution of charge the definition B.74) is replaced Qim = jdrp(r)r'Y,m(a,) B.83) where p(r) is the charge density at the source point r. [We note that Q,m can also be expressed in terms of the spherical harmonic expansion coefficients ftm(r) for the charge density, defined by p(r) = lim Pim(r) Y?m(a>), according to Qlm = J r2 drr!plm(r).] For a molecule in the ground electronic state, for example, the charge density is a sum of nuclear pn and electronic pe contributions. If the nuclei are held at fixed positions rn these charge densities are given by (we ignore electron spin133a for notational simplicity) B.84) where 5(r) is the Dirac delta function, En denotes a sum over the nuclei, Zne is the charge of nucleus n, t^e is the ground electronic state wave function for the fixed nuclear configuration (rn), and dr^ indicates an integration over (N— 1) of the N electron positions re. [These expressions are intuitively obvious, and can also be derived from the expression (cf. (B.54)) Qm =<LqirJYIm(o>i)>> where <.. .) = Jdrf^* ... ^ denotes the ex- pectation value over the electronic state i^e.] If the molecule is in the ground vibrational state, B.83) must be averaged over the nuclear positions (rn) in the ground vibrational state, keeping the molecular orientation (as defined, e.g., by the 'average' principal axes) fixed. This gives the so-called permanent multipole moment.17 In this book we treat the rotation of the molecule classically (cf. Appendix 3D, and § 1.2.2.). In
2.4 MULTIPOLE EXPANSIONS 57 a quantum treatment, if a linear molecule for example is in an angular momentum eigenstate |JM) (see Appendix 3D for notation) for example, one may wish to calculate the expectation value (JM\ Qlm |JM) (the quantum permanent moment for state \JM)), or one of the off-diagonal elements {JM\ Qlm \J'M'). The latter are the so-called transition multipole moments, which describe transitions induced by collisions, or by radia- radiation. There is a vast literature concerning molecules or processes where such matrix elements are relevant. Typical examples occur in refs. 17, 56, 57, 61, 63, 65, 80-6, 140, 148. The moments with I = 0,1,2,3,4 correspond to the monopole (charge), dipole, quadrupole, octopole, and hexadecapole. There are extensive tables of molecular dipole moments and quite a few quadrupole moments are also known. A few experimental values of octopole and hexadecapole moments have been obtained in recent years from, for example, collision-induced absorption spectra. A short selection of mul- multipole moments is given in Appendix D. The spherical multipole moment tensor of order I has B1 + 1) indepen- independent components Q!m (corresponding to m =—i to + 1). Thus the dipole moment (I = 1) has three, the quadrupole moment A = 2) has five, the octupole moment (I = 3) has seven, and so on. The correct number of independent components thus emerges automatically for the Qims, in contrast to the situation for the Cartesian moments. Other general theorems will be seen to be proved easily with the Qms and they will be found convenient for calculations for abritrary I; results are derived as easily for the hexadecapole, say, as for the dipole. The reason is that there are simple rules (Appendix A) for integrating, differentiating and rotating arbitrary spherical harmonics. Of course the Cartesian compo- components, particularly for I = 1 and I = 2, are easy to visualize, to list in tables (being real), and to work with. The components Qlm are complex in general (exceptions include Qlo and B.89)) and have no direct physical significance in static problems in general (see, however, B.89)). In dynamic problems,133'136 however, these components are responsible for circularly polarized multipole radiation, just as the Cartesian components are responsible for linearly polarized radiation. Also, in quantum radia- radiation (and collision) problems, the matrix elements (JMj Qlm \J'M') of the spherical multipole moment operators are very easily calculated using the Wigner-Eckart theorem;136b for examples, see the references cited two paragraphs above. Relations between spherical and cartesian components. In order to find the relations between the spherical multipole components Q,m and their Cartesian counterparts, we write out the r|Y,m(&)j) in B.74) in terms of Xj, yf and zt (see (A.62a) or (A.63)). This gives the following explicit
58 INTERMOLECULAR FORCES 2.4 relations 136a for the lower Qims: ; B.85) B.86) B.87) and so on, where i = -s/-l. (For Qn, Q21, etc. where m<0, we can use Oim=(-)mOL; see B.91).) The general"relations B.87) simplify in the principal axes frame of the molecule as discussed in § 2.4.3. 2.4.3 Properties of the multipole moments General properties A number of properties are immediately evident from the definition B.74). As we have discussed in §2.4.2 there are B1 + 1) independent components in general (i.e. for a general shape molecule in a general frame of reference). This number can be reduced by transforming to the principal axes! frame xyz of the particular multipole. Thus for the dipole, choosing z along ji leaves one independent component, /x2 = /x. The three independent components then correspond to the three parameters /x and 0<t>, the two angles giving the direction of the body-fixed z axis relative to some space-fixed frame. For the quadrupole, choosing the principal axes diagonalizes Q, 0 B.88I and since Q is traceless only two independent body-fixed components t Principal axes frames are discussed, e.g. in ref. 137 for the moment of inertia, in ref. 135b for the quadrupole moment, and in Appendix C for the polarizability. If the molecule has symmetry, the principal axes frames for the various multipoles can be taken as coincident with a common set of symmetry axes.
2.4 MULTIPOLE EXPANSIONS 59 exist, Qxx and Qyv say; Qzz is then fixed as -(Qxx + Qyy)- In a general frame XYZ, the five independent components correspond to the follow- following five free parameters: Qxl, QTO, and the three Euler angles 4>Q\ specifying the orientation of xyz relative to XYZ. Similarly three of the independent components for the octopole correspond to the orientation of xyz with respect to XYZ, and so on. The number of independent body-fixed components can be reduced13'17'19 if the molecule is of sym- symmetrical shape as we shall see below. To illustrate the simplifications that occur, we consider first in detail a molecule having a quadrupole moment, and use the principal body-fixed axes frame for the quadrupole moment, hereafter called simply 'the principal axes frame'. Thus B.88) applies, with Q^, Qyy and Qzz all different for a general (non-axial) molecule. For some polar non-axial molecules, e.g. H2O, SO2 and H2S, the dipole moment |a lies along one of the (quadrupole) principal axes. In these cases we call this the z-axis, so that /x = /xz and /xx = /Xy = 0 (and also Qu = Qu = 0). For polar non-axial molecules of more general shape (e.g. HCOOH), however, such a simp- simplification will not usually occur, so that /xx, /Xy and /xz are all non-zero. For the principal axes frame B.87) simplifies to B.89) We note in particular that in the principal axes frame Q2i and Q21 vanish, while Q22 and Q22 are real. Thus, in the spherical tensor formalism Q20 and Q22 are the axial and non-axial parts of the quadrupole moment respectively. [In using these terms we have in mind a molecule with some axial symmetry (e.g. ethylene Q>H4 with z along the CC double bond).] The corresponding axial (or prolateness) and non-axial Cartesian parts of Q are Qzz =JdrP(r)[z2-i(x2 + y2)] and Q^-Qyy =§JdrP(r)U2-y2), respectively. For non-polar, non-axial molecules142 (e.g. planar molecule ethylene CJH^-see Appendix D or Fig. 3.1 for structure figure), where the quadrupole-quadrupole term is the longest-range interaction, the non-axial nature of the interaction may be of particular importance (see Chapter 6). For axial molecules, the x and y directions are equivalent so that Qxx ~ Oyy, Q22 and O22 aH vanish, and we are left with the single spherical component Q20, or the single independent Cartesian component Qzz (with Qxx = Qyy — — \QZZ since Q is traceless). Further examples are given Q-=fcJ Q*
60 INTERMOLECULAR FORCES 2.4 below of the reduction of the number of independent multipole compo- components due to symmetry. Using (A.47) in B.74) we see that under inversion of the coordinate axes (r -> -r) we have On - (-)!Qm- B-90) Thus the dipole is odd under inversion, the quadrupole even, etc. From (A.3) and B.74) we see Q?m = (-)mQi2! B.91) where m = -m. From the definition B.74) it is clear that if the charge distribution is rotated rigidly, the Qlm will transform as the Ylm((o) (see (A.41)). Thus, using a passive (change of axes) rotation (see Figs. A.2, A.3 for active versus passive) we have B.92) m and conversely (see (A.43)) Qm=X?»mm,(O)*Qim, B.93) m' where Cl = <f>Ox denotes the Euler angles of the rotation carrying the old frame XYZ into coincidence with the new frame X'Y'Z'. In particular, if we introduce a frame X'Y'Z' = xyz fixed in the molecule (a so-called body-fixed frame) we have Qm=lDL(fi)*Qn B-94) n relating the body-fixed components Qtn to the space-fixed components Q,m; fi in B.94) denotes the orientation of the molecule in the space- fixed frame. Spherical molecules We note that for ground-state atoms, spherical ions, or other spherical charge distributions, B.93) implies that the Qim for f >0 are all zero. This is because for Z/0 B.93) gives a non-trivial transformation of Q,m under rotations, whereas for a spherical distribution, p(r) and hence Q,m must transform into itself. [An alternative proof is to use the definition B.83) (p(r) = p(r) here), Qim = Jr2drr'p(r)Jd<oY,m(f),
2.4 MULTIPOLE EXPANSIONS 61 and the property (A.38) of the Yim.] Thus the potential B.73) reduces to the monopole term, <?(r) = qlr, i-e. the potential for a spherically symmet- symmetric charge distribution is the same as that for a point charge at the origin. Newton originally proved the same result for gravitational potentials. For neutral atoms we thus have 4>(t) = 0. (We have assumed that p lies entirely inside r; for real atoms, p(r) and <j>(i) decay exponentially at large r (see § 2.9), but we ignore this contribution to <f> here.) Incidentally, we also see that knowledge of all the multipole momwits Qm for a general molecule is not equivalent to knowing the charge density p(r), since p(r) and p(r) + po(r) [with p0 spherically symmetfic] have the same set of multipole moments (for f>0). If, on the other hand, we expand the exponential in the Fourier transform p(k) = j exp(ik • r)p(r) dr, i.e. P(k) = J P(i) dr+ ik. jrp(r) dr-Jkk: Jrrp(r) dr+..., we obtain the Taylor series for p(k), and therefore see that knowledge of all the Cartesian moments B.54) (which have non-zero trace) is equival- equivalent224 to knowledge of p(r). The Q,m in general are thus a measure of the non-sphericity of the charge distribution. We now examine molecular symmetries less than spherical. Linear molecules One usually chooses the body-fixed axes to coincide with the principal axes in order to obtain the simplest set of Qin. For example, for a linear molecule we have Qln = 0 for n/ 0 if the principal axes are chosen with z the molecular symmetry axis. This is because under a rotation of angle a about the body-fixed z axis, Qln transforms as (see (A.64), or recall that Yjm(^) depends on 4> as exp(<m^>)) Qln -> e-™QiB. B.95) Thus only Qlo is completely invariant under rotations about z; the other Qln must vanish since t properties of axially symmetric molecules must be invariant under rotations about z. From B.74) and (A.2) the component Qo is given explicitly by t This is sometimes referred to as Neumann's principle1371"; the symmetry of a molecular property (e.g. p, Qlm, a, etc.) must be at least as high as that of the molecule itself. (A property symmetry can be higher than that of the molecule; e.g. for CH4, with tetrahedral molecular symmetry,« has complete spherical symmetry (see next section and Appendix Q.)
62 INTERMOLECULAR FORCES 2.4 where Q = I<J,r!P,(cosflt) B.97) i and P[ is the Legendre polynomial. \ Thus for axially symmetric molecules, there is only one independent multipole moment for every order I, namely Qi0. The quantities Q in B.97) are usually referred to as 'the' multipole moment of order I. For 1 = 0, 1, 2, 3, 4, one usually uses the notations Qo = q, Qx = n, Q2 = Q, Thus for the quadrupole moment (discussed explicitly above) we have Q = Q2 = T. <*r?P2(cos 6,) B.98) i or, in terms of Cartesian components, Q = Qzz=lZqiCzf-rf). B.99) i Some authors8131-133135 omit the factor \ in B.99) or in B.58), but it arises naturally and is to be preferred. We also note that B.99) can be written Q =Y,i<li(zf~xf) when the equivalence of the x and y directions is used. In a point-charge model with all the charges on the z axis this further reduces to Q =Y,i<lizf- From B.94), B.22) and B.96) we see that for linear molecules the space-fixed components take the simple form134 Qb» = Qnn(a>) B-100) where oi = 0<j> denotes the orientation of the symmetry axis of the molecule. There is a simple alternative derivation134 of B.100). In general QimWx) depends on the three Euler angles specifying the orientation of the molecule (Fig. 2.5), but it is clearly independent of the third Euler angle x f°r a linear molecule. Thus Qim(8<f>) is, for a linear molecule, a function of the same variables as YlmF4>), and must therefore be propor- proportional to Ylm(Q<$>) in order to transform as Yim@<?) under rotations. (Recall that Qlm, being an irreducible spherical tensor, always transforms as Y,m, i.e. according to the Dl law.) Putting Qm = Q,Y!m(&>), we find the proportionality constant Q by comparing the m = 0 components of this relation and B.74), in the body-fixed frame, giving again B.97). The simplest proof of all is obtained on using the general theorem for tensors given at the end of the second paragraph following. For a symmetric linear molecule (e.g. H2, CO2) we have Q = 0 for odd values of I, due to inversion symmetry (see B.90) or B.97)).
2.4 MULTIPOLE EXPANSIONS 63 For the dipole and quadrupole, the Cartesian analogues of B.100) are ji = /xn, B.101) Q=QCnn-l)/2 B.102) where n is a unit vector in the positive direction of the symmetry axis z of the molecule. Equation B.101) is obvious. To prove B.102) we can use the transformation relation [(A.123), (A.216)] between the space-fixed and body-fixed principal axes components of Q, and the expressions for the latter components, Qzz = Q, Q^ = Qyy = -|Q. Alternatively, analog- analogous to the simplified proof for spherical tensors given above, we can argue that Q = Q(n) must be proportional to Cnn-1), since the latter is the only (up to a factor) symmetric traceless second-rank tensor function of n that we can construct. [Q clearly depends only on n for axial molecules.] To find the proportionality constant A, we consider the body-fixed zz compo- component of Q = A.Cnn—1), which gives A=§QZ2. A still simpler proof is obtained by invoking the general tensor theorem that if a tensor relation A.P = B«p holds in one coordinate system (XYZ say), it holds in all (e.g. X'Y'Z'), so that Aa'P' = Ba,e: Equation B.102) is a tensor relation (i.e. both sides are tensors), and is obviously valid in the principal axes, and is therefore valid in all axes. [The general theorem used137de is valid for tensors of any rank, and also for spherical components.] Analogous relations hold for other traceless second-rank molecular properties; see Appendix C for the polarizability anisotropy. Relations analogous to B.101), B.102) for the higher order Cartesian multipole moments can be derived27 (cf. preceding remark and (C.28)); in general we find1303 Q<!)= QP<0(n), where P<0(n) is a tensor Legendre polynomial35 (denoted Y(l) on p. 493). For the octopole and hexadecapole moments we choose fi = fi2ZZ = Q3 and <6 = <1>ZZZZ = Q4 as the independent body-fixed compo- components. Other molecular symmetries For B.100) to hold, it is not necessary that the charge distribution have complete axial symmetry; a p-fold axis will suffice, for example, if p > I. For a p-fold axis of symmetry (the z axis say) the charge density, and hence molecular properties like Qln, are invariant under a rotation of angle Btt/p) about z. But (see B.95)) Qn transforms into exp[—mBT7/p)]Qin under this rotation. The phase factor will be unity only if n = 0, ±p, ±2p, etc. Hence, since \n\ =? I, if p > I only the n = 0 case will have a unit phase factor, and hence only Qi0 can be non-zero. As an example, there is only one independent quadrupole component for NH3, which has a threefold axis. H2O, on the other hand, has only a twofold axis, and therefore has two independent quadrupole components, con- contrary to what is sometimes assumed. The proofs of the above results using
64 INTERMOLECULAR FORCES 2.4 Fig. 2.13. The allene molecule C3H4. The two CH2 groups lie in the two perpendicular reflection symmetry planes crd (zx and zy). The three twofold symmetry axes are marked Cj. The point group is D2d. (From ref. 137a.) spherical components134 are much simpler than those using Cartesian components.138 The molecules HC1 and NH3, which have one independent body-fixed Q component, belong to the point symmetry groups Co^ and C3v respec- respectively.1373 An example of a molecule where (wofold symmetry is suffi- sufficient1383 to yield one independent Q component is allene (Fig. 2.13), C3H4. Here there are three perpendicular twofold axes, and two perpen- perpendicular reflection planes; the molecule belongs to the point group1373 D2d- From the symmetry of Fig. 2.13 we see that the x and y directions are equivalent, so that Qxx = Qyv, thus yielding one independent body-fixed component of Q. This, therefore, is a case where p-fold symmetry, with p = I, is sufficient to yield one independent multipole component of order. I. If the charge distribution has additional elements of symmetry besides a p-fold axis, it may also be possible in some cases13'17'19138 to characterize the multipole components of order / > p by a single scalar quantity. Thus for a charge distribution with tetrahedral symmetry Td (e.g. CH4), which has three twofold and four threefold rotation axes (Fig. 2.14), one finds
2.4 MULTIPOLE EXPANSIONS 65 Fig. 2.14. Body-fixed axes (xyz) for a molecule with tetrahedral (Td) symmetry. the following non-vanishing Qln for I = 3 and 4: B.103) ^-V<& B.104) where i = V— 1 and 'the' octopole and hexadecapole moments are defined by fi = - I drp(r)xyz, B.105) B.106) The quantity il is the same as that defined by Stogryn and Stogryn,138 and is equal to the Cartesian flxvz (see B.64)); 4> is the same as the hexadecapole defined in ref. 138, and is the Cartesian ^xyjix (see B.65)). The body-fixed frame to be used in B.103)-B.106) is shown in Fig. 2.14. To prove B.103) and B.104) one uses the group theoretical139 results that the functions in the subspacest I which are invariant under the symmetry operations of the tetrahedral group Td are: B.107) B.108) B.109) B.110) tBasis functions in the subspace / are the Ylm, for m =—!... + !. Note that, therefore, x2 + y2 + z2 (which is invariant under all rotations and therefore under Td) does not belong to the 1 = 2 subspace, since x2 + y2+z2~r2Y00, and therefore belongs to 1 = 0. (i) 1 = 1 subspace : no invariant function; (ii) / = 2 subspace : no invariant function; (iii) I = 3 subspace : one invariant function (>p3), ^3 = xyz ~ r3[ Y32(<o) - Y32M]; (iv) / = 4 subspace : one invariant function (</»4),
66 INTERMOLECULAR FORCES 2.4 There are various elementary methods89"-137" of deriving B.107)-B.110), including simply verifying that t^3 and ifi4 are indeed invariant under the 24 symmetry operations of Td. The formal group theory methods (e.g. those using projection operators896'139) are more systematic, and show as well that B.107)-B.110) are unique and complete. [We note that ifj3 and particularly i/»4 (because of its relevance to the octahedral or cubic group Oh - see below), which are ubiquitous in solid state and molecular physics, are often referred to as (particular) 'cubic (or Kubic) harmonics'.89e Just as (see Appendix A) the spherical harmonics Y,m provide a basis for the irreducible representations of the full rotation group RC), so the com- complete set of cubic harmonics provide a basis for the irreducible represen- representations of the cubic group Oh. Thus \p4 plays the role in Td in the I = 4 subspace (i.e. it is invariant) that Yoo plays in RC).] The results B.103) and B.104) follows from B.109) and B.110) respectively when the definition B.83) of Qln is used. Choosing polar coordinates (r, a>) to evaluate the integral we have Qln = Jdrp(r)r'Yln(a>) B.111a) = fdrrV fd<oYb,(w)p(r,<o). B.111b) The angular integral in B.111b) is the expansion coefficient in an expan- expansion of the charge density p(r,(o) in spherical harmonics (see also remark on this expansion on p. 56). Since the charge density is invariant under Td, it can contain 1-3 harmonics only of the type tfe [eqn B.109)], and 1 = 4 harmonics only of the type ijj4 [eqn B.110)]. The relative value Q32:032 thus follows from B.109), and the relative values Q44: Q44: Q40 from B.110). The explicit values for Q32 and Q44 (i.e. B.105) and B.106)) follow from B.111) when the rlYln(a>) in B.111) are written out in terms of x, y, z (see (A.63)). Thus, for tetrahedral molecules, from B.94) and B.103), B.104) one can write down134 an analogue of the linear molecule relation B.100), for the cases I = 3 and I = 4, e.g. Q3m = i(^)i(|)in[D3n2(co)*-D3n3(w)*]. B.112) An analogous relation can be written down168 for the third-rank polariza- bility A of Appendix C; the Cartesian version (analogous to the linear molecule relations B.101), B.102)) is given in refs. 18 and 19. The octopole moment is the first non-vanishing multipole moment for CH4, CF4 etc., and has been seen experimentally, for example in collision-induced absorption (see ref. 140 and refs. in Chapter 11 and
2.4 MULTIPOLE EXPANSIONS ff) Fig. 2.15. Body-fixed axes (xyz) for a molecule with octahedral (Oh) symmetry. Appendix D). The group theory results B.95) and B.96) show that the dipole and quadrupole moments vanish for tetrahedral molecules. These results can also be seen intuitively from Fig. 2.14. Thus |a vanishes since there is no single preferred direction for ft in which to point. The quadrupole moment Qafj, or Q2m, is a measure of the non-sphericity of the charge density, and vanishes for a spherical distribution (see earliei argument). The ellipsoid Q:rr = L*p QopVp associated with137 the second-rank tensor Q clearly degenerates to a sphere here, since the x, y z axes in Fig. 2.14 are equivalent. Thus the charge density is spherical as far as Q is concerned, so that Q = 0. [Alternatively, we have Qxx — Oyy = Qzz from the equivalence of the x, y, and z directions in Fig 2.14. But Q is traceless, Qxx + Qyy + Qzz = 0, so that 3QXX = 0, or Qxx = 0 and therefore also Qyy = Q2Z = 0.] For molecules with octahedral symmetry Oh (e.g. SF6), the first non- vanishing moment is the hexadecapole A = 4). The dipole and octopole vanish by inversion symmetry and the quadrupole moment vanishes foi the same reason as it did for Td symmetry. The relations B.104) and B.106" are valid for the hexadecapole moment, if the body-fixed axes are choser as in Fig. 2.15. The hexadecapole moment of SF6 has been seen ir collision induced absorption (ref. 140, Chapter 11 and Appendix D) anc dielectric constant62 experiments on gases (see also Chapter 10). There are tables17'19 showing the number of independent multipol* components for other point groups. Magnitude of the multipole moment In considering induction interactions (§ 2.5) and in statistical mechani cal calculations of thermodynamic properties142 (see Chapter 6), viria coefficients142 (Chapter 6), and infrared absorption coefficients140'14 (Chapter 11), for example, there arises naturally the invariant combina- combination Xm QfmQm = Sm |Oim|2- We therefore introduce the magnitude Q, o
68 INTERMOLECULAR FORCES 2.4 the multipole moment of order I, defined by ^l B.H3) where Q[m are an arbitrary set of space-fixed components. Since B.113) is an invariant (see (A.179)), we also have B.H4) where Qin are the body-fixed components. For the dipole Qi^fi B.114) becomes, in the 'dipole principal axes frame' (i.e. the frame where /x^ 0, /xx = /Lty = 0), ?o = /x2 B.115) where /x2 = Ea/x2(=|Xz (principal axes); we have used B.86) and Qn = For the quadrupole Q2^Q in its principal axes frame we have Q2 (Q y B.116) since Q20 and Q2±2 are real and Q2i = 0 (see B.89)). O can also be written in rotationally invariant Cartesian form Q2 = §Q:Q, B.117) or, in the principal axes frame, Q2 = i(Qxx-QyyJ+Q2z B.H8) = KQL+C&+QD. B.119) To derive B.117) and B.118) one can use the general transformation relation (A.230), or alternatively the specific relation B.89) to transform B.116) into B.118). Equation B.119) is derived from B.118) using Qxx + Qyy + Qzz = 0, or directly from B.117). For axially symmetric (linear) molecules we have Oi = |Oi|, where Q is 'the' multipole moment denned by B.97). For general shape molecules Q2 can sometimes be used142 as an 'effective axial quadrupole moment' (see Chapter 6). For tetrahedral molecules we have Q3 = (^)i|n| and Q4 = 0r)^ |4>|, where fl and 4> are the octopole and hexadecapole mo- moments B.105), B.106). The relation Q4 = D^)i |<J>j is also valid for oc- octahedral molecules.
2.4 MULTIPOLE EXPANSIONS 69 Origin dependence We now discuss the transformation properties of the multipole mo- moments under translation. As we did for rotations, we use the passive (change of axes) point of view. (Active and passive rotations and inver- inversions are discussed in Appendix A.) Consider (Fig. 2.16) a second, parallel, space-fixed frame X'Y'Z' whose origin O' is shifted by a from origin O of XYZ. Relative to X'Y'Z' the coordinates of a charge qt of the distribution are rj, where r; = ^-a. B.120) The transformation B.120) can now be applied to the Cartesian or spherical multipole moments. The former is simple17 for I = 1 and ( = 2 so we discuss it first. If |a' denotes the dipole moment relative to X'Y'Z' we have from B.54) » V > V i \ V V f* = Li *Jiri = 2-, QiW ~a) = Z. fli1' ~S Zi Qb or Similarly, from the definition B.58) of Q' and B.120) we get Q' =Q-B(|ia + «|i)-(|i .a)l]+iqCaa-a2l) B.121) B.122) Note that Q' is still symmetric and traceless, as it should be. We see that the change in |a involves q, and the change in Q involves |a and q. If q = 0 and n / 0, then n' = n and Q' / Q, whereas if q = |a = 0 and Q / 0, then |t' = 0 and Q' = Q. In general (see below) only the first non-vanishing moment is independent of origin. To discuss the general case we use the spherical tensor moments. We denote the multipole moments relative to O and O' by Qim and Q'lm, Fig. 2.16. Translation of the coordinate system by a.
0 INTERMOLECULAR FORCES 2.4 respectively. Then, using the notation Ylm(a>r)= Yim(r), we have Q!m-X<F;lYlm(n) i = Z*|ri-a|'Ylm(ri-a). B.123) i Introducing the notation /lm(r,a) = |r-a|!Ylm(r-a) B.124) for the quantity in B.123), we expand /(m(r, a) in spherical harmonics, /im(r,a) = X X fhijx, a)C(l1l2l;m1m2m) xYlim»y,!B2(a) B.125) where the CG coefficient in B.125) ensures that the right-hand side transforms under rotations in the same way as the left-hand side, i.e. according to Dl (see § A.4.1). The quantity r''YIm(rO is clearly a potential function in r7 (see (A.21)), and therefore in r since V'2 = V2. From the symmetry of r and a in r7 = r—a, r'!Y(m (rO is also a potential function in a. Hence the radial dependence of fi^Jj, a) must be of the form rlla'2 (and not, e.g. of the form r-(I.+ua-(!2+1) since flm is finite as r —* 0 or as a —* 0.) We also must have lx +l2 = /, so that the dimensions are (length)' on both sides of B.125). Thus for fixed I, the sum in B.125) runs over the finite number of lx values lx = 0,1,..., I, where l2 = I — li'. (*,«)= I I B.126) To find the remaining constants fhl in B.126) we consider a special case: we put r and a parallel to the polar axis, and choose the m =0 component. Equation B.126) reduces to ;000) B.127) 4tt I \ 4tt Using the binomial expansion B.127a)
2.4 MULTIPOLE where I2=l-lu and the value (A. 162) for C(lj2l; 000), we get by comparing B.127) and B.127a) The derivation just given for B.126) is that of ref. 13, wherein references to other derivations can also be found. Substituting B.126) into B.123) we have xQ,imiY,2m2(a) B.129) where J2—'~'i- (The factors fi^CH^l; mlm2m) can also be expressed explicitly143 in terms of binomial coefficients.) From B.129) we see that the change in the multipole moment of order / in shifting origin from O to O' involves the lower order multipoles. In particular, the first non-vanishing multipole moment is independent of origin. For the case of a linear molecule an origin translation through distance a along the symmetry axis transforms the scalar moment B.97) into Qi= Q + X(~)'~'1(n(I-l1)!)a'~'lQ>' B-l29a) For example, the dipole Qt = fx. and quadrupole moment Q2^Q trans- transform to ** ** aq' , B.130) Q'=Q-2atx + a2q where q = Qo is the net charge. The relations B.130) can also be derived from the Cartesian definitions, ix = n,z, Q = QZZ, using B.121) and B.122). For a neutral polar molecule like HC1, there is no ambiguity when one speaks of 'the' dipole moment, since it is independent of origin. The quadrupole moment of HC1, on the other hand, depends on origin and the origin corresponding to an experimental value will depend on the particular experiment used to measure it. The latter origin is not neces- necessarily the centre of mass, so that the conversion relation B.130) may have to be used if one wants a centre of mass value. Thus in the induced birefringence experiment144 (Chapter 10), the measured values of Q(HC1) and Q(DC1) are the same, so that Q is definitely not referred to the centre of mass. The actual origin in this experiment is not simply related tc either the centre of mass or the centre of dipole (see Chapter 10 anc
72 INTERMOLECULAR FORCES 2.4 Appendix D). The origin where Q' = 0 is referred to17 as the 'centre of dipole' by analogy with the centre of charge (the point in an ion where lx' = 0). (For a general shape molecule it is not possible27 to find a point where Q = 0.) Another consequence of B.129) is that a single multipole at one origin generates an infinite number when referred to a new origin. For example, a charge q at O generates with respect to O', q' = q, ix' = -aq, Q' = a2q, etc. and a dipole fi atO generates with respect to O' and so on. The relations B.131) are useful in calculating higher molecular moments from the 'bond-dipole' or 'atom-dipole' models (see Appendix D). Values of the multipole moments If not zero for symmetry reasons, the values of the molecular multipole moments for the small molecules listed in Appendix D are usually of order one atomic unit, i.e. ea0 for dipoles, ea2, for quadrupoles, etc., where e is the proton charge and a0 the Bohr radius. In exceptional cases the values may be somewhat smaller, either due to a near symmetry, or to a near cancellation of nuclear and electronic contributions (e.g. jx(CO) and Q(O2)). 2.4.4 Interaction energy of a charge distribution with a given external field In this section we derive the multipole expansion for the interaction energy u of a non-polarizable charge cloud in a given external field, characterized by a potential field <?(r), « = I<Ji<K'.) B-132) t (see § 2.4.1). Such an expansion will be used in several applications later, for example in deriving expressions for the multipole-multipole and induction parts of the intermolecular pair potential (§§2.4.5, 2.5, 2.10), and in the theories of the dielectric constant (Chapter 10), infrared absorption (Chapter 11), and molecular polarizability (Appendix C). (i) Cartesian tensor method18'19 We expand the potential <?(r) due to the external field in a Taylor series about an origin within the charge cloud: ^... B.133) B.134)
2.4 MULTIPOLE EXPANSIONS where E = — Vcj> is the electric field and the subscript 0 indicates at the origin. Substituting B.134) into B.132) we find the i decomposition of the interaction energy in Cartesian form, u = q<j>0-p. Eo-?8 : (VEH +.... We see from B.135) that the charge interacts with the pote dipole with the electric field, the quadrupole with the ele< gradient, and so on. As in § 2.4.2 we can replace 58 in B.135) I B.58)) since A1:(VEH = A(V.EH = 0, there being no source charges in the region r~0. Thus we hav< u = #0-(A. E0-^Q : (VEH+.... The general term in the expansion is19.27'135 where B/-l)!! = Bl-l)Bf-3).. .E)C)A), Q(" is defined b E(D= -(V'0)o, and the large dot in B.137) indicates a full cc (yielding a scalar), Q%y..E%y._. The relation B.136) has been derived here for static fields. A zation to dynamic fields can be given.145 Force and torque. In computer simulation studies of moleculj with the molecular dynamics method, one needs the force F and on a molecule due to the fields of the neighbouring molecules. r and torque with respect to the molecular origin O are derh F= — Vu and t = — Vmw, where V is the gradient operator for the of O and Va is the angular gradient operator about O. The n decompositions of F and t result from using B.136) for u. Altern we can use the definitions together with the Taylor expansion for E(r), E(r) = E0 + r . (VEH + ^rr: (WEH+. to give
74 INTERMOLECULAR FORCES 2.4 where, as before, we have introduced Q rather than 8. The mixed double product T.xU in B.142) is defined by (B.25). The reader can construct simple point-charge models to visualize intuitively B.141) and B.142). For example, for a simple dipole (charges +q and — q separated by 0 one can see why the torque |x.xE0 vanishes for (I parallel to Eo, and why the force (a . (VEH vanishes if E is a uniform field. Similarly a point-charge model of a quadrupolar molecule shows why a quadrupole is oriented in a non-uniform field (see discussion following A0.168)). If the molecules are polarizable, additional terms must be added to B.136), B.141), B.142), (see Appendix C). (ii) Spherical tensor method Choosing an origin inside the charge distribution we expand the field 4>(t) =<f>{r, to) in spherical harmonics, *« = I<km(r)Y?,(a»). B.143) lm We have included the complex conjugate in B.143) to make manifest the fact that <f> is a scalar (see (A.179)). Since <j>(i) satisfies Laplace's equation with the boundary condition <?@) finite, the r-dependence in B.143) is r' (see (A.21, 23)) so that we have 4>« = I 4v'YfJ") B.144) lm where the <?,m are constants. The constants <?,m in B.144) are irreducible tensors (see § A.4.1) characteristic of the field and must satisfy <fom = (-)"*<?&, since <?(r) is real. If the field <?(r) has any symmetry propreties this must be reflected in the 4>tm. For example if <?(r0<?) is independent of angle <?, we have <f>lm = 0 for The <plm are determined in the usual way by an angular integral over 4>(i) (see (A.30)). Alternatively,13'145" one can relate the <foms to the field and its gradients at the origin by comparing B.144) with the Taylor expansion of <f>(i) about the origin, B.133). We find (see below for proof) <foo = D7r)tyo, B.145) B.146) ^o B.147) and so on, where Vm are the spherical components (A.48), (A. 167) of the
2.4 MULTIPOLE EXPANSIONS 75 gradient: VO = VZ, V±1= =F2~s(Vx±iVy). The relation B.145) is obvious from B.144) and B.133). To derive B.146) we transform the Cartesian scalar product r.(V<?H in B.133) to spherical component form using (A.225): a.,r.V4> = ?r*Vm4>. B.148) m With the help of (A. 165) this becomes and comparison of B.149) and B.144) gives B.146). To derive B.147) we transform rr: W<f> to reducible spherical components using (A.228), and then to irreducible spherical components using (A.231), rr:W4> = ?(rr)*m(VV4>Jm, B.150) m where (rrJm =1 CA12; nvm)Vv, B.151) v-v 1 rf. B.152) In B.150) we have used the facts that (W<?H0 = (Wq)lm =0. Using (A. 165) and (A.37) we write (rrJm as (rrJm = (y) B.153) Substituting B.153) and B.152) in B.150), multiplying by one-half, and comparing with B.144) gives B.147).
76 INTERMOLECULAR FORCES 2.4 The explicit values for the <?2m are B.154) where, for example, Vq<? = d2$/dz2 = -dEJdz. In terms of the electric field Em = — Vm<?, we see that 4>i is essentially the spherical tensor E, B.155) and 4>2 is essentially the / = 2 irreducible part of the spherical tensor V?, J0. B.156) Substituting B.144) into B.132) we find the multipolar decomposition of the interaction energy u in spherical tensor form, ,mQ?m, B-157) where the Q,m are the multipole moments B.74). Again we see that the charge interacts with the potential, the dipole with the field, the quad- rupole with the field gradient, etc. If needed, spherical tensor expressions120 for the force F and torque t on the molecule in the external field, analogous to the Cartesian expres- expressions B.141), B.142), can be derived in one of three ways: (i) by transforming B.141), B.142) to spherical tensor form using the methods of § A.4.3, (ii) from F^ = -V^m and t^ = -(Vm)Am, with u given by B.157), (iii) substituting the spherical harmonic expansion for E(r) [obtained from E = — V<?, B.144), and the gradient formula (A.49)] into B.138), B.139). 2.4.5 Multipole interaction energy between two charge distributions The total electrostatic interaction energy of two rigid charge clouds (Fig. 2.17) is given by a sum of terms of the type q-^q^r^ (cf. B.44)), i.e. u=l^ B.158)
2.4 MULTIPOLE EXPANSIONS 77 Fig. 2.17. Interaction geometry for charges qt and q2- XYZ is a set of arbitrary space-fixed axes. where i and / run over the charges of distributions 1 and 2 respectively, and ri; is the separation between q{ and fy. If the two charge distributions do not overlap, we can resolve B.158) into multipole-multipole con- constituents m(i,2, in which the multipole of order lx of distribution 1 interacts with the multipole of order l2 of distribution 2. We can derive these multipole-multipole terms by two methods. In method (i), using the results of § 2.4.2, we calculate the fields of the various multipoles of molecule 1 at the position of molecule 2. Then, using the results of § 2.4.4, we calculate the interaction energy of the various multipoles of molecule 2 in these fields. This method can be carried out using either Cartesian or spherical tensors. In method (ii), we expand the quantity ry1 = |r+r,- —Til, either in a double Taylor expansion in rf and r,- (thereby generating a Cartesian tensor expression), or in a spherical harmonic expansion in r( and f;- (thereby generating a spherical tensor expression). We shall illustrate method (i) using Cartesian tensors, and method (ii) using spherical tensors. Of.course, once one has derived a Cartesian tensor expression for w(i,2, the spherical tensor expression can then be derived by transforming the Cartesian tensors to spherical form, and vice versa. We give one example of this below. (i) Cartesian tensor expressions We illustrate method (i) above by calculating uiu the dipole-dipole interaction energy. According to B.63), the dipole ^ at the origin (Fig. 2.18) produces a field E(r) at the position r of dipole p^ equal to E(r) = fix. T^Or). Hence wu = -^ • E (see B.136)) is given by u^-^l^-.T^r) B.159) where we have used the fact that T*2' is a symmetric tensor. Using B.56) we can also write un in a form commonly employed, »u =-(^i^2/'i3)[3(n1 .n)(n2.n)-n! .n2] B.159a)
78 INTERMOLECULAR FORCES 2.4 Fig. 2.18. Interaction geometry for the dipole-dipole interaction energy un. where nf = ttJiM and n = r/r are unit vectors along |ij and r, respectively. Other terms ulll2, ul2 (dipole-quadrapole), u22 (quadrupole-quadrupole), etc. are derived similarly. For the total interaction energy u =Xi,i2 m(i,2 we find : T<2)(r) + .... B.160) From B.68) and B.137) we derive the general term19-27-135 in the series B.160), B.161) where Q(" is defined by B.67) and the large dots in B.161) indicate a full contraction (yielding a scalar), Q^...'^:feI...e,2e™...Q?k^... • (We can be somewhat cavalier about the order of the indices here, since the tensors are all symmetric, cf. (B.14).) (ii) Spherical tensor expressions We select typical charges ql and q2 of the two charge clouds (Fig. 2.17), having Coulomb energy qiq2/r12, and consider expanding r^ in spherical harmonics. This is the simplest of the so-called two-centre expansion problems, and was first solved by Carlson and Rushbrooke,146 who used r in Fig. 2.17 as polar axis, and subsequently transformed to general axes. There have been a number of rederivations.147 The derivation here13-134 uses general axes from the outset, and fully exploits the invariances and
2.4 MULTIPOLE EXPANSIONS 79 scaling properties of the interaction q\q^jr12. We use the invariance of qtq2/r12 under rotations, inversion and complex conjugation. The interac- interaction is also invariant under a number of other transformations not needed here (e.g. charge conjugation qf —» — q{, charge interchange, and transla- translations). Similar methods are useful in other two-centre expansion prob- problems, e.g./overlap, induction, and dispersion interactions (ref. 148 and § 2.3), atom-atom interactions (§ 2.8) and magnetostatic interactions.149 We expand the function rj~2 = \t+t2—r1\~1 m terms of products of spherical harmonics of the orientations (ou &>2 and to of iu r2, and r respectively, referred to an arbitrary space-fixed coordinate frame (Fig. 2.17), — =1 I AT/S'TOw) '12 Ijl^l mim^m xYllBlW^WY?M B.162) where the * is included for convenience (see below, or (A.195a)). Because r^l is invariant under simultaneous rotation of ri, r2 and r, the expansion coefficient A in B.162) must contain a CG coefficient C(Zxy; mim2m), A'E'fttw) = C(hl2l; mlm2rn)Aw(r1r2r) B.163) where the factor A^^r^r) is independent of the ms. The reason for the presence of the CG coefficient in B.163) is explained in detail in § 2.3, and in Appendix A (see (A. 195a)), the essential point being that the rotationally invariant combination of products of three spherical har- harmonics is Xn« CYYY*. We also remember that the CG coefficient in B.163) has built into it two constraints (A.130), (A.131) on the Is and ms. The quantity r12 is also invariant under simultaneous inversion of r1; r2 and r. Using B.27) we see that a factor (-)'>+^+I is introduced into the expansion so that AlxlJj]T2r) must vanish unless (h + l2+l) is even. Because ri\ and ?ms CYYY* (for l^ + ^+l even) are real, the coefficient Aisijjir2r) must be real. Further, r72 is invariant under the interchange of the points 1 and 2, i.e. under the transformation! tx —»r2, r2 —> rls r —» —r. This gives AuJjw) = (-)'Al2lll(r2r1r). B.164) We now use the non-overlapping condition (r>rl + r2) and the fact that t Four more results similar to B.164) can be derived from the invariance of rtl under the transformations (r^r)—»(—rr2—O, (rjrrj). (~*i*t^), and (—n^rj. We do not give these relations here as they correspond to configurations where the two charge clouds overlap. (The overlapping region r^r1 + r2 is discussed in ref. 8).
80 INTERMOLECULAR FORCES 2.4 r\~l is a potential function to show that the non-vanishing A(ll2i corres- correspond to l = lt + l2. For r12^0, the quantity ri2 =\t+i2 — ii\~1 is a solution of the three Laplace equations in iu r2 and r. Since the fundamental solutions of Laplace's equation Vr<? = 0 are of the forms B.77), and because of the product of Ys in B.162), the dependence of Aw(rir2r) on rlt r2 and r must be one or more of the eight types r\lrl?rl, r['r^r~u+1), ri(ll+1)r22r\ etc. However, because of the boundary conditions for the region r>ri + r2 that r^l must remain finite as tx or r2 tend to zero, and must also approach zero as r—>°°, we can rule out all but the second type. Hence the dependence of the expansion coefficient Ax^Jj-j^r) on the radial variables must be of the form (cf. similar argument in § 2.4.2) A,, yfory) = {rWflrM)Ahhx B.165) where the remaining constants Alll2, are independent of all the rs. We also see from B.165) that we must have l = lx+l2, in order that the right-hand side of B.162) have the dimension of inverse length. [A scaling argument gives the same result.] Putting A[lWl+,2 = AljB we there- therefore get I ll[2(}?) ? ihh l<o2) Y* (a) B.166) where I = lt +12. The remaining constants A,ji2 can be found by considering the special case where rx, r2 and r are all parallel to the polar axis. Using (A.55) we see that for this case B.166) reduces to = Z Alll2(r[^/rl+1)C(lJ2l; 000) Comparison of B.167) with the direct binomial expansion T2+r-rx = I (- )'2 mi , (ri1r?/r''+1*+1) B.168) gives A _(-)'= r D7rKBi + l)! hh~ 21+11B/,
2.4 MULTIPOLE EXPANSIONS 81 when (A.162) is used. (The unsymmetric phase ( —)'2 arises from the choice of r as pointing from distribution 1 to distribution 2.) The total interaction energy B.158) between the charge distributions is thus given by w=L,i2 mm2 where xQlim,Q,jm!Yf» B.170) r '~ where the space-fixed multipole moments Qlm are defined by B.74). ullh is the electrostatic interaction energy between the multipoles of orders lx and l2 of charge distributions 1 and 2, respectively, un being the dipole-dipole interaction, ul2 the dipole-quadrupole interaction, and so on. The fact that there occurs only the maximum value (lx + Z2) of I allowed by the triangle relation among lly l2 and I is characteristic of the multipole expansion of the Coulomb interaction.134 For other cases, such as the induction, dispersion and overlap interactions, terms corresponding to smaller values of I also occur148 (see §§ 2.5-2.8). The dependence of ulih on the molecular orientations can be made explicit by substituting in B.170) the relation B.94) between the space- fixed axes components Qim and the body-fixed axes components Qlm which gives for general shape molecules «ia = A^ I (QliniQhJrl+1) I C(l1l2l;m1m2m) o))* B.171) where &>j = (p^Xi is the orientation of molecule 1. For linear molecules a simpler expression results from using B.100), «Ma = Alli2(Q,1Qi2/r1+1) ? C(l1l2l;m1m2m) mi(Ti2m X YlimiM YhmzM Y*lm (ay) B.172) where now &>j = Ojfa denotes the orientation of the symmetry axis of molecule i. The expressions B.171) and B.172) have the standard rotationally invariant forms B.20) and B.23) respectively. As discussed in § 2.3 these expressions reduce still further in the intermolecular frame (i.e. the r- frame (Fig. 2.9), with polar axis along r). For example B.172) takes the standard form B-29), m iW B.173) where w'j= 0/<?; denotes the orientation of molecule i in the r-frame.
82 INTERMOLECULAR FORCES 2.4 Standard expansion coefficents. As pointed out above the expression B.171) has the standard spherical harmonic expansion form B.20), with coefficient umMJ; nin2; r) = Ahh(QhniQhJr'+1), B.174) where Q,n is the body-fixed axes multipole component and Alih is the constant B.169). Using B.96), B.103), and B.104) we write out explicitly the lowest order terms in B.174) for pairs of identical molecules of linear, tetrahedral, and octahedral shape: linear: MmultA12; 00; r)= -2F7r/5)^2/r3), B.175) wmultA23; 00; r) = 2A5^/7)^01^), B.176) MmultB24; 00; r) = (f)G07r)*( Q2/r5), B.177) tetrahedral: «multC36; 22; r) = -MmultC36; 22; r) = A68/5)C3Tr/91)K^2/r7), B.178) octahedral: «mu,,D48; 00; r) = A4/5)*uraultD48; 40; r) = 6A4307r/17)K4>2/r9). B.179) The other non-vanishing coefficients for these cases, uB13; 00; r), uC36; 22; r), and uD48; 04; r) can be obtained from those written down together with the symmetry property B.26). Explicit angle dependence. In analytic calculations (see Chapters 4, 5 and Vol. 2) we use the spherical harmonic forms B.171) or B.172) for uhh, since we can employ the integral relations for spherical harmonics and the sum rules for CG coefficients (see Appendix A) to simplify the calcula- calculations. Alternatively we can use the Cartesian form B.161) together with the properties of Cartesian tensors given in Appendix A. For some numerical calculations (e.g. computer simulations, exact virial coefficient evaluations) we need the explicit angle dependence. This is obtained by writing out either B.171) (or B.172)) or B.161) in terms of Euler angles (or polar angles for linear molecules). In space-fixed axes the dipole-dipole term uu is Mh = -(nW^O^Ca- c12) B.180) where c, is the cosine of the angle between (a,- and r, ci=ni.n = cos 0j cos 0 + sin 0( sin 6 cos($j — <?), and c12 is the cosine of the angle between m and ftj, c12 = nt. n2 = cos Ox cos 02 + sin 0t sin 02
2.4 MULTIPOLE EXPANSIONS 83 Here n1 = (A1/M.1, n2 = iju,/^, and n=r/r are unit vectors in the directions of cox^O^i, <o2=024>2 and co = 6<p respectively. The direction of r is chosen from molecule 1 to 2 in the intermolecular frame (Fig. 2.9) with polar axis along r, so that B.180) reduces to "u = -0*Wr3)Bc^-S;S2c') B.181) where c'; = cos0j, Sj=5in0j, and c' = cos(<p[ — 02). For axially symmetric molecules, (or molecules with at least threefold symmetry axes) the dipole-quadrupole term u12 in space-fixed axes can be obtained either from [B.160), B.102)] or B.172), with the result «i2 = l()x1Q2/r4)(c1Ec|-l)-2c12c2). B.182) In the r-frame (Fig. 2.9) this reduces to u12^(fi1Q2/r4)(c'lCc22-l)-2s[s2c2c'). B.183) The quadrupole-dipole term u21 is obtained from B.182) or B.183) by interchanging the subscripts 1 and 2 and multiplying the complete expres- expression by (—1). For axial molecules the quadrupole-quadrupole interaction energy can be derived similarly in space-fixed axes as B.184) which reduces in the r-frame to B.185) The minimum energy orientations are shown in Fig. 2.19 for the three interactions un, u12, u22. Thus, for example, two linear quadrupoles prefer a 'T' configuration, which can be understood intuitively using a point charge model such as + = +. These minimum energy configurations are of importance for orientational structures of gas dimers,107 liquids (see Chapter 9) and solids.14>20b24>150 For specific molecules there are often additional 'competing' interactions, such as shape and dispersion, (a) (b) (c) Fig. 2.19. The minimum energy orientations for (a) dipole-dipole interaction (assuming H1H2>0), (b) dipole-quadrupole interaction (assuming m.1Q2>0), (c) quadrupole- quadrupole interaction (assuming QiQ2>0).
84 INTERMOLECULAR FORCES 2.4 Fig. 2.20. 'Parallel-staggered' configuration for (CO2J dimer. (Simplified pictures of the molecular shape and quadrupole (point-charge model) are shown.) Note the electrostati- electrostatically favourable configuration. which may lead to preferred orientations different from those favoured purely electrostatically. For example,148 for N2-N2 the quadrupole- quadrupole interaction is larger than the dispersion interaction, whereas the opposite is true for O2-O2. For dispersion interactions the preferred orientation at fixed r is end-to-end, as opposed to the T orientation for quadrupole-quadrupole interactions. The molecular shape influences the preferred orientations by control- controlling the distance of closest approach.27-1493 For example, carbon dioxide dimers (CO2J have been observed to be non-polar (|a = 0) in electric field deflection experiments. This rules out a T-structure, which, being non- symmetric, possesses a dipole moment induced by intermolecular forces (see Chapter 11). Barton et af.151a postulate a 'parallel-staggered' struc- structure, Fig. 2.20; the smaller centre-to-centre distance r compensates for the fact that the QQ energy in this configuration is 25 per cent higher than that for the T configuration for the same value of r. For further examples, see refs. 51, 102b, and Figs. 2.27-2.29. For condensed phases (liquids and solids), packing considerations also play a role in deciding the preferred nearest neighbour orientations - see comments in § 4.1 and Chapter 9. For two non-linear molecules the explicit expressions become unwieldy. For example, in the r-frame, u22 depends on five angles (<?i20i/Vi02,V2) where <M2=<?j-<?2 (ref. 151b gives the explicit expression). It may be advantageous in this case to program directly B.171) (in the r-frame), with subprograms for CB24; m1m2m) and d^n@) based on (A.163) and (A.115). However, as we have noted, for the calculation of quantities like (u;,;,)^^, {ul2)aia,z, (Hl2)mi(Oj, etc. the explicit expressions are not needed. Transformation between Cartesian and spherical tensor expressions. The pedestrian way to show that the Cartesian B.161) and spherical B.170) tensor expressions for the multipole-multipole interaction energy are equal is to write both out explicitly in terms of angles. This was done in the preceding section for un, u12, u21 and u22. A more elegant way is to transform one expression into the other using the transformation relations of § A.4.3. We illustrate there the method (and an alternative simplified version) with the dipole-dipole interaction mu (see (A.245), (A.247)).
2.5 INDUCTION INTERACTIONS 85 The general interaction uhh is discussed, using a more complicated approach, in ref. 151. 2.5 Induction interactions In § 2.4 the molecular charge distribution was assumed for simplicity to be rigid (non-polarizable). In reality, the charge distribution will adjust in response to an applied field, i.e. real molecules are polarizable. When an arbitrary (non-uniform) external electrostatic field E(r) is applied to a molecule, a dipole moment mnd is induced (see (C.I)) (%nd = «.E B.186) where E = E@) is the field at the molecular centre and a is the molecular polarizability tensor. (In Appendix C we discuss the properties of a, and in Appendix D we list values of the principal components where these are known.) There is an associated interaction energy uind (the induction energy, see (C.2)), ^ = _^. E? B18?) This interaction energy with the induced dipole moment is in addition to the interaction energy of the field with the permanent moments discussed in § 2.4.4. [In reality there are additional (non-linear) terms in B.186), B.187) and additional induced moments involving the higher-order polarizabilities (e.g. A) of the molecule. These are discussed briefly in Appendix C, but are usually unimportant for induction intermolecular forces (cf. §2.6 on A-type dispersion forces).] The energy uind is negative, since" B.187) simplifies to uind = —%[axxE* + ayyE* + azzEl) in the a principal axes where a is diagonal, and the diagonal principal components axx, etc. are positive (see Appen- Appendix C). If E is due to a neighbouring molecule, the relations uind<0 and uind ~ r~" (see below) imply that the induction forces between molecules are attractive, for all orientations. Using B.186) or B.187) we can estimate the magnitude of inter- intermolecular induction interactions. Consider for example two dipolar molecules with dipole moments m and {Jl2 separated by r. The potential field <t>(i) at molecule 2 due to fij is (see B.62) and B.55)) <?(r)~ m/r2, so that the electric field E(i) = -V<f> is E(r)~ fijr3. The induced dipole moment in molecule 2 is thus ixind~a2E~a2ii1lr3. The interaction energy between jxj (the inducing dipole) and jxind (the induced dipole) is (see B.159)) uM~ tx^^Jr3, or «W ^i«2)x1r6 B.188) where uiLav_ is the ((x1a2)x1) contribution to uind. (The minus sign follows from the argument in the paragraph above.) We thus see that the ratio of induction to permanent dipole-dipole energy is u^Ju^ ~ a/r3. (The exact expression B.211) for u^^ differs from B.188) only by a geometric
86 INTERMOLECULAR FORCES 2.5 factor.) Similarly we can show that the OiaQv induction energy is of order air3 times the Q,Qy permanent multipole-multipole interaction energy. We show in Appendix C that a~(cr/2K, where a is the molecular diameter, so that for r~cr, we expect a/r3—1CT1. More precise estimates show that a/cr3 is in fact typically about 0.05. Thus the induction energy is often small for isolated molecular pairs, and therefore for low density gases. For liquids (see § 4.10) it turns out that induction interactions can make a significant contribution to the free energy and pressure, .even when the contribution to the internal energy is small. Multibody induction interactions (see §§ 2.10 and 4.10) are also important in liquids, and can enhance these effects. Also, for mixtures such as HCl-Ar (see Chapter 7), there are no permanent multipole-multipole interactions for the unlike pairs, so that the induction terms are relatively more important. Explicit expressions for the induction interactions can be derived classically from B.187) using either the Cartesian18-19-152 or spherical141 tensor methods. In Appendix C we find it convenient to use the spherical tensor method, as this yields expressions in standard spherical harmonic form B.20). (We also outline in Appendix C a quantum derivation, using Cartesian tensors.) We summarize below the results of the calculations of Appendix C. In § 2.10, on the other hand, we find it convenient to use Cartesian tensors in discussing the multibody induction interactions. We can decompose the total induction interaction energy into terms like B.188) v , V M1OW Mind = L UWaM'+ L "I,'*,!*" B.189) li'li" Wh" where M,i.a2,i» is the energy due to the multipole of order l[ of molecule 1 inducing a dipole in molecule 2 via a2, which in turn reacts back with the multipole of order l'[ of molecule 1. The terms uWail2» arise similarly from the polarization of molecule 1 by molecule 2. In §2.10 we derive a Cartesian tensor expression for the dipolar term ulQ2l. In Appendix C we derive the general term Mi,^,. in spherical tensor form. Spherical harmonic expansion coefficients In Appendix C we write the term u,ia2l!in standard spherical harmonic form B.20), with coefficient B.190)
2.5 INDUCTION INTERACTIONS 87 where {; j i} is a 9/ symbol (see Appendix A), Q[ini is the body-fixed (see B.94)) multipole moment component of molecule 1, and cthrt2 is the body-fixed irreducible polarizability component of molecule 2. As discus- discussed in Appendix C (see also (A.229)), the (body-fixed) irreducible spheri- spherical components aln (where I = 0 or 2) are derived from the (body-fixed) reducible spherical components a^v by oIb=2cA1I;/iwi)o|1V. B.191a) The latter in turn are derived from the (body-fixed) Cartesian compo- components aap by (see (A.227)) <V=I^otV*«0 B.191b) where U^ is the unitary transformation matrix (A.220). The component ain = aoo is proportional to the mean polarizability a, a=l,(axx + ayy + azz) B.192a) where axx, etc. are the principal components (see below, or (C.22), (C.23)). For linear molecules, only the n = 0 component of a2n is non- vanishing and a20 is proportional to the polarizability anisotropy y = cx||— ax, where a||=azz, and aJ_ = axx = ayy (see (C.26), and note also (C.27)). For linear molecules we thus have a=i(au + 2aj, y = aira±. B.192b) The energy uWaiV, arising from the polarization of molecule 1 by molecule 2, can be written down from u^^* by inspection [(C.68)]. When we remember that r points from molecule 1 to 2, and the symmetry property (A. 134) of CG coefficients, we see that the spherical harmonic expansion coefficients Ui^^l^l; n^; r) are obtained from B.190) by interchanging the subscripts 1 and 2, and multiplying by the phase (—)'1+l2 (or simply by (-)'2 assuming Ii = 0,2). Equation B.190) holds for two molecules of arbitrary shape and for "arbitrary multipolar induction. We now consider some special cases in detail. (a) The isotropic term For l'i=l'i, Uwa2h" contains an isotropic term. This is obtained from B.190) with (lj2l) = @00). The polarizability «00 is given by B.191a), B.193)
88 INTERMOLECULAR FORCES 2.5 The reducible spherical components a^v are given in terms of the Cartesian aap by B.191b) so that, with the help of (A.221) and (A.226), we find = I<W B-194) a Hence from B.193) and B.194) we have aoo=-C)*a B.195) where a — Tr(a)/3 is the mean polarizability B.192a). The 9/ symbol in B.190) has the value (see (A.297) and the 9/ symmetry properties) ru i ii+n \l[ 1 /I + l[ = [C)BII + l)BII + 3)]-*. B.196) U 0 0 J Substituting B.195) and B.196) in B.190) gives «i,w,'@00; 00; r) = -(^(^j^a^r^4' B.197) where Q, the magnitude of the multipole moment of order I, is defined by B.114). Equation B.197) shows that there is a non-vanishing isotropic part of the l[al[ induction potential (see B.16) and B.35)), which is attractive, t^r^'. B.198) The total isotropic induction interaction between a pair of molecules will be ^ ) = I uJU^rH I u?w.(r). B.199) For polar molecules the leading term will be the dipole-induced dipole contribution {l[a2l[ = ^a2l and /i/' ll) B.200) while the quadrupole-induced dipole terms are Hq-oM = -iQfaz^-iCW-8 B.201) where Q is the magnitude of the quadrupole moment B.119).
2.5 INDUCTION INTERACTIONS 89 (b) One molecule is spherical ¦ ¦ For molecules with spherical, tetrahedral, or octahedral symmetry, a is isotropic. Thus, if we consider the induction interaction for, say, H2O-Ar or N2-CH4, the polarizabihty a2 of molecule 2 is isotropic, and therefore contains only an 12= 0 part. The l2-0 term in B.190) can be reduced to simpler form. The polarizability a00 is given by B.195). For J2 = 0, the 9; symbol reduces to a 6; symbol (see (A.292)): V'1 1 l'1+l\ L 0 I J B.202) + 1 l'{ Substituting B.195) and B.202) in B.190) and using (A.155) gives x r-('i'+ir+4) x«2 I C(l[V[lx;n[n'[nl)Ql^Ql^- B.203) (c) A linear and a spherical molecule If molecule 1 is linear and molecule 2 is spherical (e.g. HCl-Ar), we must have nI = ni = nx = 0 in B.190). Using B.203) this gives ;+3I J xC(i;+111+11; 000) xQ,ja2Qirr-(i;+'i'+4) B.204) We now specialize B.204) further to the dipolar and quadrupolar cases.
90 INTERMOLECULAR FORCES 2.5 The isotropic terms ule,2l@00;00; r) and u2o22@00;00; r) are given by B.197). The coefficients of the anisotropic terms are: 6 > B-205) ulaJ101; 00; r)'= -C7r)Hf)(M1a2Qi)'-7, B.206) ula22C03; 00; r) = -(u/7)l(?)(mi«2Qi)^7, B-207) u2a22B02; 00; r) = -E7r)Hi)(Q1a2Q1)r-8, B.208) <,22D04; 00; r) = -(^(IXQ^QOr-8. B-209) The coefficients for the interaction u2a2l are equal to the corresponding terms in B.206) and B.207). For computer simulation work one needs the complete potential, which can be reassembled from the harmonics. Using the intermolecular axis coordinate system (Fig. 2.9) we get "ind=«ia2i + "ia22+«2i+M2a22 + (a2-* a1) + ... B.210) where (a2 -* at) indicates four similar terms with a2 replaced by a,. The four terms Uia2i, etc. are given by ulail = -hix\a2{3c[2+l)r'6, B.211) "ia22=-3Mi«2QiC;3r7, B.212) «22=-lQ?«2Ec;4-2c;2+l)r8 B.213) where c{ = cos0i, Q is defined by B.99), and w2<»2i = Mi<*22- The four terms ulaii etc. are obtained from the corresponding terms B.211), etc. by interchanging the subscripts 1 and 2, and multiplying the cos30 terms by (-1). For analytical calculations the spherical harmonic form of the potential is the simplest to use. (d) Two linear molecules For two linear molecules all n^O terms in B.190) vanish, so that we have r(?;+i)a'1;+i)B;;+i)B^+i)B;;+3)B/';+3)B/1+i)B;2+i)l L B1+1) J B.214)
2.6 DISPERSION INTERACTIONS 91 with Qi defined by B.97), aOo ~ a and a2o ~ y (see (C.41) and Appendix C glossary). The explicit angular dependence can be obtained, if needed, from B.23), B.24), and B.214) (see refs. 18 and 19 for the dipole and quadrupole cases). Origin dependence The dependence of the interaction or the «(Zxy; n1n2; r) on the choice of molecular centre is obtained simply from the dependence of the multipole moments on origin (§ 2.4.3) and from the fact (Appendix C) that the polarizability a is origin independent. 2.6 Dispersion interactions181819 In contrast to the multipole and induction interactions, the dispersion and overlap interactions cannot be calculated completely classically. One must solve the Schrodinger equation for the electronic ground-state energy E(rn) for the molecular pair, for a fixed nuclear configuration (rn). The intermolecular potential «(rn) is E(rn) minus the energies of the isolated molecules. The concept of a continuous intermolecular potential «(rn) is based on the adiabatic or Born-Oppenheimer approximation,1'8 in which it is assumed that as the nuclei move, slowly compared to the electrons, the electron cloud adjusts adiabatically, always remaining in the ground state corresponding to the instantaneous nuclear configuration. Thus, the nuclei move in the average field of the electrons, and the electrons follow the motion of the nuclei adiabatically. As discussed in Appendix C, at long range one neglects exchange effects, and expands the intermolecular Coulomb interactions between the electrons and nuclei in powers of r (the multipole expansion), where r is the intermolecular separation, and then treats the multipole terms as a perturbation. The unperturbed system is two isolated molecules. The final result is a series for the energy in r~", with powers arising both from the multipole expansion and the perturbation expansion. The first-order perturbation terms are the interactions between the permanent multipole moments derived classi- classically in §2.4.5. Part of the second-order terms involve the permanent moments of one of the molecules and the polarizability of the other, and correspond to the induction interactions derived classically in § 2.5. The other second-order terms are called the 'dispersion energy', since they involve the two molecular polarizabilities (polarizability also occurs in the dispersion theory (refractive index) of light). The dispersion forces are attractive1523 for all orientations, and are present whether or not the molecules possess multipole moments. At short range, electron exchange is important, and the multipole and perturbation expansions are invalid,
92 INTERMOLECULAR FORCES 2.6 so that one must solve the Schrodinger equation by a different method (e.g. Hartree-Fock, variational, configuration interaction, electron gas model, etc.) for the overlap interaction energy (see § 2.7). Although the dispersion and overlap terms cannot be derived classi- classically, they can be interpreted semi-classically in various ways. Thus, from the Hellmann-Feynman theorem (see ref. 79a) one shows that the force acting on a given nucleus when molecules interact can be obtained from the classical electrostatic attraction of the total electron charge density of the system and the electrostatic repulsion of the other nuclei; quantum mechanics (including electron spin and exchange effects in general) is needed only to furnish the electron charge density. Thus at long range, attractive dispersion forces come about for Ar-Ar, for example, because electronic charge shifts slightly into the internuclear, or 'bonding' region, thereby pulling each nucleus electrostatically toward this region. Another8-153 well known semi-classical picture of the dispersion in- interactions is based on the ground-state fluctuations of the charge density. Consider two s-state atoms for simplicity. From a classical, or from a semi-classical Bohr-orbit point of view, the charge density in atom 1 fluctuates due to the orbital motion of its electrons. At any instant there is a dipole moment fij say. This produces a field E ~ m/r3 at the position r of atom 2. This field polarizes atom 2, giving an interaction energy u12 ^a2E2. Hence u12 ^a2ii\r~6. We now average this instantane- instantaneous interaction over a long time. The net energy udis = (u12) *s "du—«2<M?>^6- B.215) The expression B.215) has the same sign and r-dependence as the properly derived dispersion interaction (see below and Appendix C). The above argument also demonstrates that it is the correlated motion of m and (*2 ~ a2(»Vr3) which is responsible for the dispersion interaction. Hence a Hartree-Fock calculation of the energy, which neglects electron correlation, will miss the dispersion energy. Further rough arguments connect {(xi) to al4 First we replace (...) in B.215) by a quantum mechanical expectation value over the ground electronic state. The polarizability a for a spherical atom is given by (see (C.15)) B216) Replacing the excitation energy En-E0 in B.216) by an average value E gives iZ .<n||i|0>. B.217) The sum in B.217) can be extended over all states n now, since
2.6 DISPERSION INTERACTIONS 93 (O|ft|O) = O for a spherical atom. Using the completeness relation (B.53), tn\n)(n\ = l, we get «~|<M2> B-218) where (.. .) = <0|... |0). [The relation a ~ (i*2)/E can also be obtained with a more classical argument. Consider the simple classical model of the atom of Fig. C.I, i.e. a positive nuclear charge +e surrounded by a uniformly charged sphere of radius a and total charge -e. A simple classical calculation (Appendix C) shows that the polarizability of this model is a = a3. Since the nucleus-electron Coulomb energy E of this electrostatic model of the isolated atom (Fig. C.I) is E~e2/a, we can also write a ~ a2e2/E. If we now think of the electron as fluctuating over the atomic volume to produce the long-time uniform charge density, then (ae) is ofthe order of the instantaneous dipole moment /x, and hence a~{jx2)/E, where (p2) is the mean fluctuation in the dipole moment.] Note that both relations a~a3 and a~l/E (where E~ ionization energy) are intuitively reasonable. Combining B.218) and B.215) gives B.219) which resembles closely the exact London model expression (see below and Appendix C) B.220) where El2 = E1E2/(E1 +E2), with Et the average excitation energy of atom i. (Et is usually taken to be equal to the ionization energy for rough estimates. For the rare gases24 this is accurate to 10-40 per cent. Note that El2 = EJ2 for identical atoms.) A number of other approximate expressions for udis have been proposed.6-27-102" We now discuss the exact expression for udis. We have taken a2 to be the static polarizability in the above qualitative physical argument. In a more careful treatment154-24-161 we can decom- decompose the fluctuating field E(t) into Fourier components E(o>), and then use the corresponding dynamic polarizability a2(a>). The dynamic polarizability can also be introduced via the exact second-order perturba- perturbation theory method of calculation of dispersion forces, as we describe in Appendix C. We find that for two atoms, the London static polarizability expression B.220) gets replaced by the exact expression ¦f dco"ai(Uo")a2(uo") B.221)
94 INTERMOLECULAR FORCES 2.6 where a(i<o") is the dynamic polarizability a(w) at the imaginary fre- frequency a) = ico", defined by (C.47). a(f«u") is a real positive well-behaved (monotonically decreasing) function of w". Precise theoretical and semi- empirical estimates and bounds for the strength of the dispersion forces for atoms have been made10-27 using B.221). For molecules, the complete dynamical polarizability tensor a(iw") is involved (see (C.48)) which is not known for many molecules, so that only in a few cases have precise calculations been made.55?102b>155'155b It is thus necessary in most cases to fall back on the static polarizability approximation. For CO2~rare gases, for example,155 with ionization energies used to compute El2, the latter gives a value for the harmonic coefficient udisB02; 00; r) [see below] which is about 25 per cent too large as compared with the dynamic polarizability result. Retardation effects In the above discussion we neglected retardation effects in the propaga- propagation of the field; for large distances (i.e. r>A, where \=2ttcI<o is the wavelength corresponding to the electronic frequency to), the induced dipole gets out of phase with the inducing dipole, thereby weakening the interaction. The detailed calculations156'157 show that for r» Ao, where Ao is the wavelength corresponding to the characteristic electronic frequency <oo of the molecule, the dispersion interaction energy u is of order B.215) multiplied by (A0/r), so that the dispersion energy changes to a u ~ r form at very large r. The factor (A0/r) can be understood in a rough way by remembering133 that the static r~3 field of a dipole changes to the form r~3 eikr in the dynamic case, where fc = B-jt/A) is the mag- magnitude of the wave vector corresponding to the wavelength A of the field. If we average the phase factor ei2kr (the factor of 2 arises since the interaction energy is proportional to E2; alternatively, the field must propagate from molecule 1 to 2 (a distance r), and then back again) over all wave numbers up to a characteristic maximum fc0, using : I (AVIV \ a'2kr Jo =| (dfc/fco)ei2 we get <ei2fcr)~(fcorr1~(Ao/r) if fcor»l. This argument is but slightly better than a dimensional one but the actual calculation156157 is much more complicated, due in part to the fact that the retarded dipolar field133 contains terms in r~~2 eikr and in r~x eikr, in addition to the term we have considered. For small molecules, Ao is typically of order 102-103 A, so that retarda- retardation effects are of no importance since u is essentially zero anyway at these large distances. Retardation effects are of importance for large (e.g.
2.6 DISPERSION INTERACTIONS 95 biological159'160) molecules, for macroscopic objects158-161-162 (e.g. capacitor plates), and perhaps for colloids.161-163 Retardation effects associated with permanent multipole interactions can also almost always be neglected, since the multipole moments vary in time with the vibrational and rotational frequencies of the molecules. The latter are in the infrared, so that retardation would not set in until r~A~106A, where u is negligible. Higher-order dispersion interactions The dispersion interactions considered so far involve the dipole-dipole polarizability0 a, and vary as r~6. Molecules also have higher-order (dipole-quadrupole, etc.; see Appendix C) polarizabilities, and allowance for these introduces intermolecular interactions varying as r~~'', r~8, etc. The rigorous method of calculating these terms parallels that for the ordinary ar~6 dispersion terms (see Appendix C). The qualitative physical argument given above for the ar~6 terms can also be extended to include these terms. Thus the Ar~7 dispersion term arises as follows (A is the dipole-quadrupole polarizability, describing the dipole induced by a field gradient-see Appendix C). The fluctuating dipole ix1 of molecule 1 produces a field (~/x1r) and therefore a field gradient (—/^r) at molecule 2. The field gradient induces a dipole /u.2~Az(Mir~4) m molecule 2, which reacts back with the original dipole with interaction energy u~/u,1/u.2r~3~^2/lxir~7- Averaging over the Hi fluctuations and using B.218) thus gives u~E1a1A2r'7 B.222) which has the correct form (see below). It turns out that this term can be attractive or repulsive, depending on the orientation of molecule 2, and, in fact, contains no isotropic part. We can similarly derive the form of the r~8, etc. terms, as well as other intermolecular dispersion properties, e.g. three-body potentials (§ 2.10), collision-induced dipole moments1363 (Chapter 11), and collision-induced polarizabilities (Appendix 10D). For non-spherical molecules, the exact expressions indicated schemati- schematically by B.219) and B.222) involve the polarizability tensors a and A. For molecules with inversion symmetry (e.g. Ar, H2, SF6), A vanishes so that the dispersion terms vary as r'6, r~8, etc. and the angle dependence involves spherical harmonics with / even. For molecules lacking a centre of inversion (e.g. HC1, CH4) A does not vanish, so that the dispersion terms vary as r, r~7, r~8, etc. with both even and odd spherical harmonics present. Since a is isotropic for CH4, the Ar term is the longest-range anisotropic term for CH4-CH4 and CH4-Ar, and contains spherical harmonics (see below) of order 1 = 3. This interaction has been invoked to explain the mean squared torque (Chapter 11) derived from
96 INTERMOLECULAR FORCES 2.6 the pressure dependence of vibration-rotation bands167 of CH4. The A tensor may also be responsible for at least some of the high-frequency collision-induced rotational absorption and light scattering (Chapter 11) seen in CH4 (see ref. 168 for a summary of these and other relevant experiments; further references are given in Appendix D). Pressure broadening of HC1 infrared169-169" and Raman169a lines in dilute HCl/Ar is also sensitive to the Ar interaction. For NH3-Ar, the Ar~7 interac- interaction is ^-dependent (where x 1S the third Euler angle for NH3) and is therefore at least partially responsible (along with octopolar-induction and overlap forces) for the An —±3 collision-induced rotational transi- transitions of the symmetric top NH3 molecule, seen in double resonance experiments.86 Here n is the quantum number corresponding to the body-fixed z-axis component of angular momentum J (see CD.48) and discussion there). In contrast to a, A is origin dependent (see Appendix C), so that the strength of the AT1 forces depends on the choice of molecular centre. Except for centro-symmetric molecules, the Ar7 forces cannot be trans- transformed away, but can be 'minimized' with the appropriate choice of centre. For CH4, the obvious optimum choice is the geometric centre. For HC1, choosing the centre such that A|| = |AX eliminates the 1 = 3 har- harmonics, whereas the choice of centre such that A^= -2A± eliminates the / = 1 harmonics (see below). The latter origin has been termed169 the 'centre of dispersion force'. For HC1 it is about 0.1 A from the centre of mass.toward the H atom. Just as for the ar interactions, more accurate Ar'7 interaction strengths are obtained by introducing the dynamic polarizability A(iw"). This has been considered explicitly1558 for CH4. For interactions involving the octahedral SF6 molecule, such as SF6-SF6 or SF6-Ar, the leading anisotropic dispersion interaction1690 varies as Cr~8 + Er~8 where C and E are the fourth-rank quadrupole-quadrupole and dipole-octopole higher polarizabilities respectively (denoted by a22 and a13 in (C.36); see also (C.6), (C.7), Appendix C glossary, and Appendix D). Explicit expressions for molecules The expressions B.220) and B.222) are valid for atoms. In §§ C.3.1 and A.4.3 we derive some of the relevant expressions for the ar'6 and Ar'7 dispersion interactions for general shape molecules, in terms of the Cartesian and spherical tensor components of a and A. The explicit angle dependence is sometimes needed, so that we give it here for linear and tetrahedral molecules. For linear molecules the static polarizability approximation for the ar~6 dispersion energy was derived independently by London,170 using the
2.6 DISPERSION INTERACTIONS 97 Drude (harmonic oscillator) model,8 and by de Boer171 without the Drude model. Buckingham18'27 (see also Kielich19), whose derivation we follow in Appendix C, has generalized these results to molecules of arbitrary shape, to include the Ar, etc. terms, and to include the proper dynamic polarizabilities. Neglecting anisotropy in the ionization energy Ef (or Drude model oscillator frequency - see Appendix C) we find in the static polarizability approximation for linear molecules the ar terms \r x.(s[s2c'-2c[c2J ] B.223) where c'j = cos0'j, s'^sinfl';, c' = cos(<p'1-^J), 0'4>' refer to the inter- molecular frame (Fig. 2.9) and a^ = azz, a± = axx^ayy are the principal axes static polarizability components (see Appendix C). Et2 is defined earlier. For the Ar terms we have the corresponding expression -2a1A2|1c23-fa1A2xCc2-2c23)] B.224) where a = f(a||+2ax) is the mean polarizability, and A^ = AZZZ, A±^AXXZ are the two independent principal components of the third rank A tensor. There are also171a terms of the type Aly2r'!> where y = a$ — a± is the polarizability anisotropy, in addition to those in B.224). In principle E12 in B.224) could differ from E12 in B.223), but can be taken equal for estimates of u?K. As noted above, and as will be seen again below when we list the spherical harmonic components, «JS contains both isotropic (attractive) and anisotropic parts, whereas u^s contains only an aniso- tropic part. For a linear molecule-spherical molecule case (e.g. HCl-Ar) one obtains the relevant expressions by setting a2§ = a2x and A2j = A2X = 0. For two tetrahedral molecules (e.g. CH4-CH4), or a tetrahedral- spherical pair (e.g. CH4-Ar), «2is is given by the expression B.220) for atoms, since a is isotropic. «^s, on the other hand, while vanishing for two atoms, is given by m?s = -|Ei2(8)r^7[A1a2 cos e'xl cos 0'yl cos 6'zl - <*!A2 cos 6'x2 cos d'y2 cos 6'z2] B.225) for two tetrahedral molecules. Here A = Axyz is the principal axes (Fig. 2.14) xyz component, and cos d'x is the direction cosine of the body-fixed x-axis (Fig. 2.14) with respect to the z'-axis in the intermolecular frame (Fig. 2.9). (A has one, e.g. Axyz, independent principal axes component for tetrahedral symmetry.) Thus Q'xl is the angle between xl and z', B'x2 is the angle between x2 and z', etc. The expression for CH4-Ar, say, is
98 INTERMOLECULAR FORCES 2.6 obtained from B.225) by setting A2 = 0. Again u?is is purely anisotropic, <«&>.,., = 0. As discussed earlier, more accurate strengths of wdls and «^s are obtained by allowing for E\*2j=Ef2, E\2±E\2, or for dynamic polarizabilities. Spherical harmonic expansion The above expressions B.223), etc. can easily be put148 in our standard spherical harmonic forms B.20), B.23). Alternatively we can specialize the general expression B.234) below to linear or tetrahedral molecules. (a) Linear molecules The isotropic term u(Ij!2{; nxn2; r) = «@00;00; r) is the same as for two atoms (see B.35) and B.220)), udis@00; 00; r) = -Dir)*| E12axa2r6. B.226) The anisotropic {lj2l^000) harmonic coefficients are given by (we must have n1 = n2 — 0 for linear molecules) ; 00; r) = udisB02; 00; r) = - (f)*(§)§ «disC03;00; r)= - udisB20; 00; r) = - (y)'(|)f HdjsB22;00; r)= -(|f 3 - _7 2 12 1 2 !12Yl«2''~6, E«r>,r', E y1y2^6, B.227) B.228) B.229) B.230) B.231) udisB24; 00; r) = - (ff)*(f)f finYxYzr6 B.232) where a = (a,, + 2a J/3, 7 = (a,, - ax), A = (A,, + 2Ax)/3, T = A,r |AX. The OH coefficients are obtained from the 101 values B.227)-B.229) by inter- interchanging the subscripts 1 and 2, and multiplying by (-)'. We see from B.232) and B.177) that the 224 dispersion term has the same angle dependence as the quadrupole-quadrupole interaction, but is of opposite sign in the most common case that yxy2 has the same sign as QiQ2; see also p. 84.
2.6 DISPERSION INTERACTIONS 99 (b) Tetrahedral molecules The ar~6 term is purely isotropic with coefficient given by B.226). The Ar~7 term is purely anisotropic with coefficients MdisC03; 20; r) = - i(^f ¦— | E^A^r7 B.233) and three other [C03; 20); @33; 02), @33; 02)] non-vanishing ones; the C03; 20) coefficient is minus the value B.233), and the @33; 0±2) coefficients are obtained from B.233) by interchanging the subscripts 1 and 2 and multiplying by (Tl). The straightforward way to put the first term, u?s(l), of B.225) in spherical harmonic form,168 and hence derive B.233), is to use the r-frame (z' long r) and the cosine law (A.34) to transform variables e'xe'yB'z -» 6'x', where 9' = 6'z. Here 6'x' are the last two of the three Euler angles <j)'6'x'- (The interaction is independent of <j)' here since m^sA) depends only on the orientation of molecule 1.) Alternatively one can transform the Cartesian tensor expression18 involving AaPy into the spherical tensor one involving Am (or a12Cm) in the notation of Appen- Appendix C) using the methods of Appendix A. A simpler method here is to note that for CH4-Ar, say, u^s(l) depends on orientation <j>'6'x' in the same way that the octopole-charge multipole interaction does, since A and 11 = Q3 are both third-rank tensors. Because «^s(l) is independent of <f>', we need only the Q3m dependence for m = 0, which is given by B.112). Thus we have = (y)V«[Y32@'x')- YAd'x')] B.233a) where f(r) is to be determined and we have used (A.106). We can determine f(r) by comparing B.225) and B.233a) for a simple orientation where m^sA) is non-zero, e.g. where molecule 2 (i.e. r) is in the A11) direction relative to the body-fixed xyz axes of the CH4 molecule (see Fig. 2.14). Then using (A.62), the fact that the polar angles of r in xyz are F', tt-x') [see p. 459, and Fig. A.6], and the relation uC03;20;r) = Dir/7)*/(r) gives B.233). (c) General shape molecules In §A.4.3 we derive the general coefficient for the ar6 dispersion interaction,
1(X) INTERMOLECULAR FORCES 2.7 where {...} is a 9/ symbol and a,n are the irreducible spherical compo- components of a (see definition following B.190), or Appendix C glossary). We note that the strength of the terms B.220), B.226), B.227), etc. is sometimes estimated from the empirical Lennard-Jones (LJ) potential g'Vmg §E12aia2 = 4eo-6. B.235) However, the value 4e<r6 gives a strength coefficient C6 which is often about a factor of 2 larger than the correct C6, since the long-range LJ term -4e(<r/rN is being made to represent the complete series -C6r~6- C8r~8+... (see also Fig. 2.4 and discussion). 2.7 Overlap interactions171b As explained in the introduction to § 2.6, when two molecules overlap, the intermolecular potential is found by solving the Schrodinger equation for the electronic ground-state energy, for all possible nuclear configura- configurations. In first approximation, the electronic energy, which we term the 'overlap energy', can be separated into electrostatic and exchange con- contributions. The electrostatic part arises from the Coulomb interaction of the two unperturbed charge clouds, including electron-electron, electron- nucleus, and nucleus-nucleus contributions (see § 2.8 for an example). The exchange part is of quantum origin, and arises because the electrons must obey the Pauli exclusion principle; when two charge clouds overlap, electrons with parallel spins must avoid each other, and do so by distorting their motions such that they involve higher energy orbitals (the lower ones are filled), at least for closed shell systems, thus giving a molecule-molecule repulsion. The exchange energy is often larger than the electrostatic energy, except for very short range (r —* 0) where the nucleus-nucleus repulsion will usually dominate. Alternatively, on the Hellmann-Feynman picture (see introduction to § 2.6) of the repulsive forces, (for Ar-Ar, for example, at short range) the electrons are forced out of the internuclear region (due to the electrostatic and exchange effects just discussed) into the 'antibonding' region at the far sides of the nuclei, thereby pulling each nucleus electrostatically in a direction away from the other nucleus. The total overlap energy is a rapidly (exponen- (exponentially) varying function of r, and can usefully be fitted by a simple exponential or power law, at least over the small range of r relevant at ordinary temperatures. The angle dependence reflects the molecular shape, as discussed below. Because of electron exchange and possibly charge transfer (see below), the molecules lose their identity upon in- interaction, so that one cannot hope to express the overlap interaction rigorously in terms of properties of the isolated molecules, in contrast to the situation for the long-range multipolar, induction and dispersion interactions (see, however, the atom-atom model of § 2.1.2). Thus there
2.7 OVERLAP INTERACTIONS 101 is no general theory at short range, and each pair must be treated separately. For a given configuration (rw,^) of two molecules, the overlap interac- interaction energy wov(rw1w2) just described is usually repulsive. In some systems (see below), charge transfer (chemical bonding) can occur, giving rise to an attractive overlap energy (for not too small r). These charge transfer effects are, of course, of great importance in some systems172 for reaction and thermodynamic equilibria, and are sometimes referred to172 as 'specific' or 'chemical' interactions, to distinguish them from the 'univer- 'universal' or 'physical' interactions. 'Hydrogen bonding'21>2227172al72b and other 'associations' or 'complexes', to the extent that these structures are due to electrostatic, exchange, and polarization forces, are included within the scope of the physical interactions. The relative importance of charge transfer forces is not known with certainty in many cases of interest,21-51 but is being actively studied in e.g. H2O-H2O,21'22'51C6H6- I2,173174HC1-HF51'107 and halogen-halogen20"'51 systems. In the interests of simplicity, we shall in this book neglect charge transfer interactions for the systems studied unless there is compelling evidence to do otherwise. Besides the standard perturbational and variational methods, the methods used to compute the short-range overlap potentials include the Hartree-Fock, configuration interaction, and electron gas methods (see refs. 21-7 and 41-55). In a few cases42'175'175 (e.g. N2-Ar, N2-N2, HF-HF) the angle dependence has been fitted by a spherical harmonic expansion, thereby rendering the potentials convenient for statistical mechanical perturbation theory calculations (see Chapter 6). In other cases47112 (e.g. H2O-H2O, HF-HF) the potential has been fitted by atom-atom and charge-charge terms, yielding potentials conve- convenient for computer simulation. Since the accuracy of theoretical angle- dependent overlap potentials is known in so few cases, one in general falls back on model potentials, which contain adjustable parameters to be obtained by fitting some data. We discussed a number of model overlap potentials in § 2.1, including the generalized Stockmayer, atom-atom, Corner and Gaussian models. The equipotential contours were compared in Fig. 2.2 for the case of N2-N2. In this section we discuss the spherical harmonic expansion of the generalized Stockmayer model, and in § 2.8 we do the same for the atom-atom model. 2.7.1 Generalized Stockmayer overlap model As explained in § 2.3, we can always expand the overlap potential "ov= «ovOrwiw2) in an infinite spherical harmonic series, = wov@00)+uovA01)+wov@11) + ... B.236)
where mOv('i'20= Z "ovCiM; n1n2;r)C(;i/2/;m1m2m) x DiA.Jw.rDkJ^)* Y,m(")*. B.237) The term /]/2( = 000 constitutes the isotropic part u°v of the overlap potential (see §2.2) and the terms /^/^OOO constitute the anisotropic part Mov The additional anisotropic terms in B.236) include lj2l = HO, 111, 112, 220, 221,... Because of the triangle condition (A.131) on the CG coefficients, terms such as 100, 113, etc. are absent. Pople91 suggested as a model the series B.236) truncated with a few leading harmonics, the particular harmonics chosen depending on molecular shape (see below). It is hoped that by so truncating the series and force fitting the strength parameters one obtains a reasonable ap- approximation to the shape, while retaining a model that is easy to work with. We discuss a few simple shapes in turn. Unsymmetrical linear molecules (HC1, CO, N2O, etc.) The simplest thing one can try here is retaining just the 1^^1 = 000, 101, and Oil harmonics. In most cases,92b however, one needs to include also at least the 202 and 022 harmonics, to obtain reasonable shapes. In addition, to model the 4>' dependence (Fig. 2.9), one will need harmonics /,/,/ with both /, and l2 non-zero. For linear molecules we must have nx = n2 = 0 in B.237) (see § 2.3). In the intermolecular coordinate frame (Fig. 2.9) the 101 term of B.237) reduces to wovA01) = uovA01;00; r)? CA01; mOm) which, with the help of (A.55), (A.62), (A.107), (A.I 14), and (A.162), reduces to C N1 — 2uovA01;00;r)cose;. B.238) 4tt/ If the isotropic overlap term is taken to be of LJ r~12 model form, i.e. , B.239) then it is convenient to preserve the r~12 power-law form in the strength coefficient uovA01; 00; r) of the anisotropic 101 term, and to write for the
first two factors in B.238) ; 00; r) = 4eS,(^), B.240) thereby defining the anisotropic overlap (shape) parameter Sj. Thus we have movA01 ; 00; r) = (y^eS,^). B.241) If molecule 1 is non-spherical and molecule 2 spherical (e.g. HCl/Xe), the anisotropic term wov @11) will be absent, so that wovA01) alone is the leading anisotropic term. If both molecules are non-spherical, both A01) and @11) terms will be present, where wov@11) = (-^Mmov@11; 00; r)cos 6'2. B.242) \4tj7 If the two molecules are identical we also have (see B.26)) uov@11;00;r) = -uovA01;00;r). B.243) For the simplified A01) + @11) anistropic overlap model we then have MSv(ra)ia)i)=«ovA01)+«O?@11) (ci-ci) B.244) where c-= cos 0\. If the two molecules are non-identical, B.244) can be generalized to (Slac;-Slbc^) B.245) which allows the w[ and u>2 dependences to be of different strengths. A particular case is Slh = 0, corresponding to a spherical molecule 2. The angular function (cos 0J— cos 62) occurring in B.244) has the extreme values 2 (for 01 = 0, 02 = rr) and -2 (for 6\ = it, 02 = 0). If we assume the total overlap potential wov@00)+ uovA01) + uov@11) is posi- positive (see remarks in the introduction to this section), we get two in- inequalities, (l + 281)>0 and A —281)>0, corresponding to the extreme values of the angular function. Hence 81 must be within the limits -4<8,<i B.246) For a diatomic AB molecule, for example, positive §j values correspond to choosing the positive direction of the molecular symmetry axis in the
104 INTERMOLECULAR FORCES 2.7 (a) (b) Fig. 2.21. Two identical AB molecules in their most repulsive orientation (the neighbour- neighbouring B end configuration is assumed to be the most repulsive). Thus the anisotropic overlap potential, uovA01) + uov@11), should have its maximum positive value in this orientation. In case (a) the body-fixed z axis points from A to B so that (cos 8[—cos Oj) = 2, and 8, should be positive so that this configuration is the most repulsive. In case (b) z points from B to A, so that a negative 8, is needed. direction A —> B, where the (B-end)-(B-end) configuration is taken to be the most repulsive (see Fig. 2.21). As mentioned previously, a more realistic overlap model is obtained by including the 000, 101, Oil, 202, and 022 harmonics. Expressions for the B02) and @22) potentials are included in B.4), and the spherical har- harmonic coefficients are given by B.248) below. When the 202 or 022 harmonics are present, the constraint B.246) need not apply. Similar expressions can be written down for the anisotropic model potential for other molecular symmetries. For pairs of identical molecules we find: Symmetrical linear molecules (N2, CO2, etc.) «ov = "ov@00) + wovB02) + uov@22) +..., B.247) wovB02; 00; r) = uov@22; 00; r) = 2(yj5S24e(-Y2. B.248) Tetrahedral molecules (CH4, CC14, etc.) "ov = uov@00)+ wovC03) + wov@33)+..., B.249) uovC03; 20; r) = i(ffis34efy, B.250) u,,vC03; 20; r) = -uovC03; 02; r) = -movC03; 20; r) = uov@33; 02; r), where i = V-l. B.251) Octahedral molecules (SF6, etc.) wovD04) + uov@44) +..., B.252) movD04; 40; r) = (yj'MjQ B.253) movD04: 40; r) = (^)iMovD04; 00; r) = (i)iMov@44; 00; r) = wovD04; 40; r) = uov@44; 04; r) = Mov@44;04;r). B.254)
2.7 OVERLAP INTERACTIONS 105 In these expressions the St are shape parameters, and n = —n. Assum- Assuming the total overlap potential is repulsive yields the restrictions -0.25<S2<0.5, B.255) -0.474 <S3< 0.474, B.256) -0.299<S4<0.448. B.257) For linear molecules the body-fixed axes are chosen with origin in the molecular centre, and z axis along the symmetry axis. In B.255) §2>0 arises for prolate (rod-like) and §2<0 for oblate (plate-like) molecules. The limits' B.255)-B.257) and B.246) must be modified for pairs of non-identical molecules, or if further harmonics are included in the models. For example, for HCl/Xe, if only the 101 anisotropic term is included we get -1<8,<1. B.258) For tetrahedral AB4 molecules we have chosen body-fixed axes as in Fig. 2.14. Positive values of §3 indicate that the most repulsive orientation occurs when AB bonds from the two molecules are colinear (Fig. 2.22). For octahedral AB6 molecules we have chosen body-fixed axes as in Fig. 2.15. Values §4>0 indicate that the most repulsive orientation occurs when AB bonds from the two molecules are colinear. As mentioned, the shape parameters 8 are usually treated as adjusta- adjustable. One can also try to correlate 8 with the ratio of molecular dimensions using, e.g. the atom-atom model spherical harmonic expansion (§ 2.8), or using bond lengths and van der Waals radii directly. 175a'c For linear molecules there is also a correlation17513 of molecular length/width ratio obtained in a particular way from the 0.002 au contour (Fig. 2.1) and the dimensionless polarizability anisotropy y/a. It is thus possible in some cases to make a rough correlation between the anisotropies of the dispersion r~6 and overlap r~12 potentials. Such a correlation is enforced in some of the other model potentials, e.g. Gaussian overlap and atom- atom, but the correlation may be incorrect in some cases. o } ) o- ^ Fig. 2.22. The most repulsive orientation for two tetrahedral AB4 molecules occurs when AB bonds from the two molecules are colinear and x'\ — X2, although the dependence on x[ and X2 is probably weak.
106 INTERMOLECULAR FORCES 2.8 Origin dependence The set of coefficients u{lvl2l; "i^; r) refers to particular origins O, and O2 in the two molecules. We may wish to shift origins, due to, for example, a shift in centre of mass position arising from an isotope substitution in the molecules, or due to a need to shift to the centre of mass from the geometric centre, etc. Shifting to new origins OJ and O2 will give rise to a new set of coefficients u'(J.[l'2l'; n[n.2, r'), where r' is the separation between O[ and O2. For the multipolar, induction and disper- dispersion interactions the relation between the two sets of coefficients simply involves transforming the multipole moments and polarizabilities to the new origins (see §§ 2.4.3, 2.5, 2.6). For the overlap models of this and the next section, more general transformations of u(l-il2l; nrn2; r) can be worked out, using spherical harmonic or Cartesian (Taylor series) methods. The simplest example is a linear AB molecule with origin shift along the symmetry axes. For AB/C interactions (or the lower order har- harmonics of AB/CD interactions) the transformation of u(l0l; 00; r) [which is equivalent to u(l;r) in B.29a)] has been worked out explicitly.1763 As a result of such a transformation the new parameters (e', cr', S[) can be obtained directly from the old ones (e, cr, St), without a refitting of data. The new model potential will, however, involve new powers of r (e.g. r~13) not present in the old one. 2.8 Spherical harmonic expansion oi the atom-atom potential The atom-atom model potential B.5) was discussed in §2.1.2. Here we wish to derive explicitly the spherical harmonic expansion B.20) for this potential. Such an expansion, where possible, is useful in making explicit the dependence of u(T(a1<tJ) on the molecular orientations, and for analytic and computer simulation calculations. The difficulty (see below) is that for many cases of interest the convergence is too slow to be of practical use. It is always possible to find the expansion coefficients u(/,/2/; n,n2; r) in B.20) by numerical integration.178 Sack179 and others180 have shown that analytic expressions for the expansion coefficients can be derived for many cases of interest (e.g. r~? and exp(-raP/aap)). We follow the treatment of Downs et al.10S which utilizes the results of Sack. We give explicit results for molecules of various shapes, where the' atom-atom or site-site terms in B.5) have the LJ form B.6). The results for the purely repulsive r~12 form are also of interest as a model for the shape, or overlap forces, and are easily obtained105 from the LJ result by omitting the r6 contributions.
2.8 ATOM-ATOM MODEL SPHERICAL HARMONIC EXPANSION 107 Fig. 2.23. Geometry of the atom-atom interaction, a and |3 are atoms (or interaction sites) of molecules 1 and 2. The molecular centres Ol and O2 are chosen arbitrarily, and XYZ is some arbitrary space-fixed coordinate frame with origin at O1. We assume the site-site term uae involves a sum of scalar functions f(rap)= rap (as in B.6) or B.259)) of the site-site separation raC = |r + rp-ra|, see Fig. 2.23. To expand f(rap) in terms of space-fixed-axes spherical harmonics Ylm(w) of the orientations <oa, we and w of ra, re and r respectively, we take advantage of the fact that rap [and hence f(rap)] is invariant under rotations. Using the same argument as in § 2.4.5 [or § 2.3, or § A.4.2], we write the expansion in the rotationally invariant form f(l1l2l;raV)C(lll2l;m1m2m) B.260) where the CG coefficient C(/1i2i; mlm2m) contains all the m- dependence, and fil^l; rarsr) l% tne reduced expansion coefficient. The latter satisfies the further symmetry restrictions [see § 2.4.5] (a) fihh^W^^Q unless (I1 + I2+O = even (parity rule), (b) f(lil2l\rar&r) is real, and (c) f(l\l2l',rarpr) = { — )lf{l2lll;rsirar). For power-law forms /(raP)= Q", dimensional or scaling considerations show that f{lj2l; rarpr) can be written in the form r~nf(lj2l; rjr, re/r), where / is dimensionless. The coefficient f(lj2l; r^r) is found explicitly from the differential properties of f(rap). For the case f(rap) = ra?, f(rap) satisfies the relations (for r y2 -n = ap vr _ 1 -v .-(n + 2) B.261) For the case n = 1, rap is thus a potential function, as is well known and was discussed in detail in § 2.4.5. Sack179 (without using invariance arguments) has discussed the general case. For the non-overlapping region r> ra + r& (note that non-overlapping here refers to the nuclear, or interaction, sites whereas in § 2.4.5 we were dealing mainly with electron
sites), j(lsl2l; rarpr) is given by 2'A+ 2 ''i + ^ + 2V'r2/' ), B.262) where A = (/, + 12 + 1I2, A = A-/ = (/,+ /2-0/2 and the shifted factorial (a)s is given by (a)s=a(a + l)...(a + s-l) B.263) with (aH = l; note that (l)s = s! The Appell function F (a kind of two variable hypergeometric function1828) is defined as We are interested in intermolecular separations r~cr (where <r is the molecular, as opposed to the atom-atom, diameter). Hence the validity condition r> ra + re for B.262) is approximately 2rmax< <r, or 2rmax/cr< 1, where rmax is the largest intramolecular centre-to-site distance. (Sack179 has also derived results for the overlapping region r<ra + r0.) For a homonuclear diatomic molecule, for example, with origin at the centre, this condition becomes /*<1, where l* = l/cr with / the bond length. For the two-term LJ form B.6), the expansion for M(rai]ai2) thus becomes u{r«ox«o2) = y; mxm2m) B.265) x Ylimi(<O Ylimj(aK)Ylm(aj)* where the expansion coefficient u^H^l; rarpr) is related to fil^l^l', V^r) of B.262) by "apCiM; rartir) = 4eali[allfl2(l1l2l; rarpr)-o-lpUihkl; rarpr)] B.266) where /12 and f6 are given by B.262) with n = 12 and n = 6, respectively. For identical molecules the wafi(/i/2/; rarpr) satisfy the symmetry relation l1l; rprar). B.267) The dependence of B.265) on the molecular orientations u, and a>2 can be made explicit by introducing body-fixed axes, which are fixed relative to the molecule. The Ylm transform under rotation according to
(see (A.43)) Yi»(<O = I DU^f Yin{<) B.268) where &>i = <l>\6iXi is the orientation of the body-fixed axes of molecule 1 relative to the space-fixed frame, a>'a is the orientation of ra in the body-fixed frame, and D'mn((o)* is the generalized spherical harmonic. Substituting B.268) and the analogous expression for Ylm(wp) in B.265) gives the standard rotationally invariant form B.20), with coeffi- coefficient w(/i/2'; ii^; r) given by u(lxl2l; nxn2; r) = ? uap(lj2l; rarpr)Yhni(co?Yl2n2(co'p). B.269) One easily checks that these coefficients obey the general reality and identical molecule conditions B.25) and B.26); they are related to the intermolecular frame (Fig. 2.9) expansion coefficients m(!1I2w; nxn2; r) by B.30), B.33). For linear molecules u(rco1co2) cannot depend on the third Euler angles xi and \2 so tnat "GiM; »i«2; r) vanishes unless nl = rc2 = 0; in this case the expansion reduces to a sum over products of three ordinary spherical harmonics, as in B.23). The atom-atom potential u(ru>1aJ) contains an isotropic part uo(r), corresponding to Z^^OOO in B.269), which is universally non- vanishing, and an anisotropic part ujjra)iaJ) comprising the lj2lj= 000 harmonic terms. Isotropic part The isotropic part uo(r) is given by (see B.35)) Mo(r) = D7r)-iM@00; 00; r) = Dir)"* I ua/3@00; r^r). B.270) We plot in Fig. 2.24 uo(r) for homonuclear diatomic AA molecules, for various values of reduced bond length I* = l/cr, wheref I = 2rmax and <r is the molecular diameter defined by wo(cr) = 0. For /*/0, wo(r) is steeper, shallower, and has a larger diameter than for the LJ potential with /* = 0. For the case Z* = 0.29 (corresponding to N2) u0 is close to a A5,6) potential, and for I* = 0.49 (corresponding to Br2 and Cl2) it is close to a B0,6) potential. t Elsewhere (see § 2.1.4, and Chapters 3-5) we have used other definitions for the reduced bond length /*.
110 INTERMOLECULAR FORCES 2.8 /*-0 0.2 0.34 0.44 Fig. 2.24. Isotropic part of the atom-atom LJ A2,6) potential for diatomic AA molecules, for various reduced bond lengths 1*. (From ref. 105.) Anisotropic Part We list below the expressions for the leading non-vanishing anisotropic coefficients u{lil2l; nxn2\ r) for pairs of identical molecules of simple shapes. (a) Homonuclear diatomic AA molecules. We choose the body-fixed axes with origin at the geometric centre and such that z is along the symmetry axis, so that we find IT B.271) We note that u(lj2l; 00; r) = 0 if /, or l2 is odd. (b) Heteronuclear diatomic AB molecules. We choose the body-fixed axes such that z is along the symmetry axis with positive z in the direction B to A. The origin can be in an arbitrary position between the
2.8 ATOM-ATOM MODEL SPHERICAL HARMONIC EXPANSION 111 sites. Then, using B.269) and B.267), we get + (-)'2uAB(lll2l; /ArBr) + (-)'=MAB(/2;,/; rArBr) + (-YuBB(hl2l\rBrBr)l B.272) (c) Tetrahedral AB4 molecules. We choose the body-fixed axes to coincide with the three twofold axes such that a B atom is in the A, 1, 1) direction (see Fig. 2.14). The lowest non-vanishing coefficients are B.273) with the other non-vanishing / = 3 terms uC03;20;r) and u@33;02;r) obtained using the general tetrahedral symmetry relations B.251), and uD04;40; r)= ~(ffi KbD04; rBrBr) + |uBAD04; rB0r)], B.274) w@44;04;r)=uD04;40;r) uD04;40;r) = uD04;40;r) B.275) M@44; 04; r) = u@44; 04; r). (d) Octahedral AB6 molecules. We choose the body-fixed axes to coincide with the fourfold axes (Fig. 2.15). The leading non-vanishing coefficients are uD04;00; r) =^-[mbbD04; rBrBr)+ aMbaD04; rB0r)] B.276) with the other non-vanishing 1 = 4 coefficients wD04;40;r), MD04;40;r), u@44;00;r), u@44;04;r) and w@44;04;r) obtainable from the general octahedral symmetry relations B.254). The odd / terms vanish. We now list the non-vanishing coefficients for two simple cases of a pair of non-identical molecules: (e) Atom A with homonuclear diatomic BB. Choosing the origin of the body-fixed A axes to coincide with A itself and the body-fixed BB axes as in (a), we find that the coefficients vanish unless /, =0, l2 is even and / = l2. Then we have u@l2l2; 00; r) = B'^+1> uAB@/2/2; 0rBr). B.277) 2tt
112 INTERMOLECULAR FORCES 2.8 (f) Atom A with hetemnuclear diatomic BC. Choosing the A origin as in (e) and the BC body-fixed axes as in (b), we find that the coefficients vanish unless /,=0 and I = l2. Then we have u@/2/2;00;r) = Bl2+\)\ 477 B.278) Applications Downs et a/.105 studied the convergence of the spherical harmonic expan- expansion for pairs of N2, F2, Br2, Cl2, HC1, CH4, SF6, and for simple unlike pair cases. The full atom-atom LJA2,6) potential, as well as the atom- atom r'2 repulsive potential, were studied. The atom-atom potential parameters used are given in Table 2.1 and the reduced bond lengths in Table 2.1 Atom-atom LJ parameters c H(HC H(CH Cl S F a Ref. bRef. 'Ref C 51.2h ) — *> 5, p. 22. 185. 186. eo H 23.81 20.0c 8.6h B/fc/K Cl 72.9C 258.5C S 158.0a F — 93.1" 54.9a C 3.35b 1 1 1 H 2.995h 2.735C 2.813b a 3.044c 3.353C S — 3.700" 3 2 F — 200" 700" Table 2.2. In the calculations it was found that about 20-50 terms of the type x"y" give 0.5 per cent accuracy for the Appell function series B.264), for r ~ a and I* ~ 0.4. The latter is the largest value for which the spherical harmonic series is rapidly convergent (we saw earlier that the series converges for I*sl). Table 2.2 Reduced bond lengths I* = 2rmaju, where rmax is the largest centre-to-site distance, and <r is the molecular diameter defined by uo{cr) = 0. In all cases, the molecular centre is taken as the geometric centre, (or bond midpoint for HCl) HCI F2 Br2 Cl2 CH4 CBr4 CC14 CF4 0.29 0.32 0.42 0.49 0.49 0.52 0.58 0.58 0.59 0.63
:.8 ATOM-ATOM MODEL SPHERICAL HARMONIC EXPANSION 113 4 2 0 -2 -4 i 0.5 I 1 I 1 1 1.5 i 'K j/ (b) Convergence of the expansion for AA molecules in the parallel orientation o, = o2 = ir/2; tf>' = 0, where the prime denotes the intermolecular frame), (a) N2 (i* = ).29); (b) (* = 0.42; full potential; second-order expansion (terms to (, = l2 = 1); fourth-order expansion; sixth-order Expansion. (From ref. 105.) 7or /* values up to 0.34 good convergence is obtained for homonuclear liatomics by including harmonics up to Ii = l2 = 4 (the / values in i(lj2l; nln2; r) are determined by the triangle, /= /, +l2,..., |/j — 12\, and larity, (, + I2+/ = even, selection rules contained in C(lJ2l;000) in 2.262)). For /*s0.42 inclusion of up to /1 = /2 = 6 gives reasonable ¦onvergence. Figure 2.25 shows typical results for AA diatomic molecules or I* = 0.29 (N2) and I* = 0.42. For /*>0.42 convergence is much ;iower.182b Of the molecules of Table 2.2, convergence is thus rapid only for N2, F2 ;nd HC1. Downs et al.10S point out that the convergence can be improved ¦>y shortening the bond length (i.e. going over to a site-site model), ind/or perhaps increasing the atom-atom diameters. In some cases this nay give a sensible model, since the neglect of bonding charge density73 n the atom-atom model tends to produce in some cases, too asymmetric i charge distribution. Finally we note some possible extensions and alternative expansions, "he spherical harmonic expansion for an exponential atom-atom model, ¦xp(-ro(j/aop), which is also widely employed, has been derived. 179-I80a In he present expansion we have assumed that the site-site interaction is a ;calar function of the site-site distances rafi only. The present results
114 INTERMOLECULAR FORCES 2.9 would need to be extended to deal with, for example, a bond-bond model, if the bonds contain dipoles, quadrupoles, anisotropic polarizabil- ity, etc., as has been employed for C6H6-C6H6. Two alternative expansions have been proposed. The first1711*01" involves a Cartesian tensor expansion, using a double Taylor series of r,',p, generalizing the method mentioned in § 2.4.5 for ra^. The Cartesian expansion can be transformed180 into the spherical harmonic expansion using the methods of Appendix A. The second is a Gegenbauer polyno- polynomial expansion.184 We have seen (see B.262)) that in general the expan- expansion of r^p really involves two expansions, i.e. in spherical harmonics and in powers of r~\ For a given order lj2l of spherical harmonics, various powers of r arise. (The case r~p is exceptional (§ 2.4.5) in that a single power of r~', i.e. r~(l+1), where l=l, + l2, accompanies each harmonic /,/,/.) Alternatively one can expand from the beginning in powers of r'\ and a given power of /•"' involves a Gegenbauer polynomial of each orientation, which correspond to various spherical harmonic orders l,l2l. The Gegenbauer expansion can also'"' be transformed into the spherical harmonic expansion. 2.9 Convergence of the inverse distance and spherical harmonic expansions25-26-793-102"-135-146-175-187^2013 In the preceding sections we have for computational convenience usually represented the r-dependence of the multipole, induction, dispersion and overlap interactions u(raj1aj2) as a series expansion in r~\ and the angular dependence as a spherical harmonic expansion. We comment here briefly on the validity of such representations. The multipole, induction and dispersion interactions are expansions in r'\ and are believed202 to be asymptotic expansions for molecular charge densities which do not cut off sufficiently rapidly. This means203 that for every finite value of r, as one takes more and more terms in the series, the result at first converges toward the correct energy, and then, after N{) terms, begins to diverge from the correct energy. For a given system and a definite r, one wants to know (a) what is the optimum number of terms N,t, and (b) what is the error in using these No terms as an approximation to the energy? Few discussions of these questions have been given for non-spherical molecules. A simpler, and in fact more useful question is: how many terms are needed to represent the energy to, say, 1 per cent accuracy? The answer will depend on r and on the values of the multipole moments and polarizabilities. Usually only one or two multipole moments and polarizabilities are known for the molecules of interest, so that one is
2.9 CONVERGENCE OF EXPANSIONS 115 Fig. 2.26. Discrete charge models for (a) symmetrical (quadrupolar), and (b) unsymmetri- cal (polar) molecules. forced to truncate the series with the known terms. The question then is: after truncation with two terms, say, what is an error bound or estimate? This question clearly deserves study because of its importance for liquids, where the neighbouring molecules are in close proximity. The simplest model for investigating the convergence of the multipole expansion of the electrostatic energy is the point-charge model. The multipole expansion for the known charge distribution can then be directly compared with the exact electrostatic potential (i.e. with the sum of Coulomb terms). Figure 2.26 shows such point charge models for (a) a quadrupolar molecule and (b) a dipolar molecule; the exact potential in both cases is M(roIaJ) = I«^(rap) + I3i9' B.279) where ulaJe is a Lennard-Jones (LJ) site-site potential between sites a and |3; the aC sum is over all LJ sites, and the ij sum is over all charge sites. The multipole moments of the linear distribution of point charges are given by /*= I <?,-*,, Q = !<?¦*?. fi = Iq.z?. B-280) i i i etc. where q, is the charge located at z{ from some centre. In Fig. 2.26 the LJ sites are chosen to be on the positive charges in case (a), and on both charges in case (b); it is, of course, easy to generalize to the case where these two types of sites are all different. Figure 2.27 compares the exact potential of B.279) with the multipole expansion terminated at the first contributing (quadrupole-quadrupole) term. In spite of the large quad- rupole moment the two results agree very well for the favourable T orientation, and are quite good for other orientations; as expected, the agreement gets better at larger r values. For the T orientation this good agreement holds even when the charge separation is increased to /* = 0.8. We conclude that for many property calculations for quadrupolar molecules termination of the multipole series at the quadrupole- quadrupole term can give good results. The next contributing multipole
116 INTERMOLECULAR FORCES 2.9 u/c 2 0 -2 -4 (, \ + \\ V v i — v\—— v?f i _-——¦— \\ \l ^ 1.0 1.4 rla Fric;. 2.27. Intermolecular potential curves for symmetrical linear molecules interacting with a diatomic Lennard-Jones, atom-atom potential (with elongation. I* = Ha - 0.5471) plus electrostatic interaction due to the charge distribution of Fig. 2.26(a). The charge separation is /* = 0.5471, Q* = Q/(e<ts)' = 2, where f. and a are site-site LJ parameters, and the molecular centre is taken to be the geometric centre. Full curves are for the LJ atom-atom plus full electrostatic potential; dashed curves are for the U atom-atom plus quadrupole-quadrupole potential. The curves are for + = crossed (B\ = 0'2 = <t>' = tt/2), T = tcc(fl', =u/2, e'2 = <t>' = 0), and— = end-to-end (fl', = ft'2 = 4>' = 0). (From ref. 104.) II/C 2 0 -2 -4 -6 L |\' \.- l\^ 1^--' 1 1.0 1 4 rla 1.8 Fig. 2.28. Intermolecular potential curves for unsymmetrical molecules interacting with a diatomic IJ. atom-atom potential (I* — 0.5471) plus electrostatic interaction due to (he charge distribution of Fig. 2,26(b). The charges are at the LJ sites (reduced charge separation - I* = 0.5471) and the molecular centre is chosen to be the bond midpoint; the reduced multipole moments are jz* = ix/(e<r*)' = 2, Q* = 0. Full curves include the full electrostatic potential, dashed curves include the dipole-dipole term only. Arrows repres- represent dipole orientations. For the ^> and |—* orientations the electrostatic energy is zero.
CONVERGENCE OF EXPANSIONS 117 ¦ki. z.zv. iniermolecular potential curves for unsymmetricai molecules (charge distribu- distribution of Fig. 2.26(h)). The potential is the same as in Fig. 2.28, but now the molecular centre is different from the bond midpoint, so that z/f = O.l, Q* = -0.864. Here full curves are the exact potential, short dashes indicate the dipole-dipole term only, and long dashes indicate all multipole terms up to quadrupole-quadrupole. erm, should it be needed, would involve the hexadecapole moment, since he octopole moment vanishes for this charge distribution. Figure 2.28 ,nows a similar plot for a dipolar pair using the charge distribution of Fig. I.26(b); the molecular centre is again taken to be the bond midpoint z = //2), so that Q*= Q/(ecr5)' = 0. The agreement between the exact -lectrostatic result and the multipole series truncated at the first con- ributing term is still quite good, although not as good as for the madrupolar case. The agreement becomes poorer, however, as (a) the nolecular centre is moved away from the bond midpoint of the molecule, ;nd (b) the separation between the charges is increased. Figure 2.29 ;nows the effect of choosing a molecular centre that is appreciably lifferent B//= 0.1) from the midpoint (z//=0.5), so that now Q'* = 0.864. The convergence is seen to be much poorer; for the most irobable orientation quite good results are obtained by including all erms up to quadrupole-quadrupole, but the convergence is poor for the ess likely orientations. It is known146 that for discrete charge distribu- 10ns, for convergence of the multipole series, the intermolecular separa- lon must be at least as large as the sum of the radii of the two -ncapsulating spheres for the distributions (see Fig. 2.30). From Fig. 2.30 1 is clear that the region of convergence is largest when the charge listribution origins are taken as the geometric centres. It also seems easonable, and one can show rigorously by straightforward calcula- ion,204 that the bond midpoint gives the most rapid convergence of the
118 INTERMOLECULAR FORCES 2.9 Fig. 2.30. For convergence of the multipole expansion the condition is r > ra + rh, where ra, rh are the radii of the encapsulating spheres of the charge distributions, for the particular origins chosen, O[ and O2. multipole series for the distribution of Fig. 2.26(b). For this simple distribution, the multipole series converges when the intermolecular separation r exceeds /. For other choices of molecular centre the sphere inside which the series diverges is larger; thus if the centre is taken to be one of the charges themselves the multipole series converges only for r>2l. The above considerations show that the choice of molecular centre is important, and, at least for linear molecules, the bond centre will often be the best choice. Many experimental measurements of the multipole moments are referred to the centre of mass. For a molecule such as HC1, in which the centre of mass is very close to one of the nuclei, this is probably not the best choice as far as,convergence is concerned, and it may be preferable to convert the value of Q to some other choice of centre using B.130) (see §2.4.3). (The origin with respect to which Q'= 0, obtained from B.130), is termed the 'centre of dipole'.) As a further example205 of the convergence of the multipole series, we consider the ST2 point charge model206 for water (see Fig. 2.31). The Fig. 2.31. ST2 model of water molecule for H,O-H2O interaction. Here q = 0.2357e, <t = 3.10 A, 1,. = 1.0 A, /_ = 0.8A, flI = 2cos-'C~!) = 109° 28' is the tetrahedral angle. The two +q charges model the two H's. and the two -q's correspond to the two lone pair electrons. »
2.9 CONVERGENCE OF EXPANSIONS 119 potential is a LJ central potential plus a point-charge anisotropic poten- potential: u(ru>,a>2) = u, ,(/¦) +S(r) Y. -^ B.281) .. r.. i) 'i; where S(r) is a switching function which shuts off the electrostatic potential for r<2.016 A, thereby preventing divergences, S(r) = 0, (r-r,JCru-r,-2r) = 1, r,<r<r. r>r,, B.282) The LJ parameters are e/k =38.1 K, cr = 3.10 A, and the cut-off distances in the switching function are r, = 2.0160 A, ru = 3.1287 A. Using the body-fixed axes shown in Fig. 2.32 it is a straightforward matter to calculate the moments Q,n from B.74) (see Table 2.3), and then to compare the truncated multipole series with the exact ST2 potential (see Figs. 2.33, 2.34). It is found that if multipole terms up to hexadecapole (/, = /2 = 4) are included, the series reproduces the exact ST2 potential within a few per cent for all orientations. There have been a number of studies of the convergence of the multipole expansion using a more realistic continuous molecular charge density p(r). Examples102'-175-193-197'2013 studied include H2, HF, LiH, N2, CO2, C2H4, pyridine, and H2O. Thus, Ng et al."s have considered the 'exact' (first-order) electrostatic interaction energy between pairs of molecules of H2, HF, LiH, N2, and CO2. They have neglected overlap exchange effects and polarization effects (induction and dispersion). The Coulomb energy is then simply Pi(ri)p2(r2) u(rco iw2)= I dr. dr, B.283) Fig. 2.32. Body-fixed axes (xyz) for the ST2 H,O molecule.
120 INTERMOLECULAR FORCES 2.9 Table 2.3 Reduced spherical multipole moments 0,n = OJ(ec72l + 1)' /or the ST2 model t ! n 1 i 2 2 2 3 3 3 3 4 4 4 4 4 5 S 5 S 5 S 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 n 0 0 1 0 1 2 0 1 2 3 0 1 2 3 4 0 1 2 3 4 5 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 o,n 0 2.903788 0 0 0 -0.77908.3 -0.258475 0 -0.114232 0 -0.037299 0 -0.080460 0 0.022291 -0.007402 0 0 0 0.030963 0 0.003331 0 0.007299 0 0.006232 0 -0.003609 0.002732 0 0.001772 0 0.001957 0 -0.001630 0 t From ref. 205. where p,(r) is the charge density of an isolated molecule i. For H2 the charge density p(r) for an isolated molecule is obtained by using B.84) for a variety of approximate molecular wave functions for H2; one can then apply stringent accuracy tests, by comparing with the known accurate wave function of Kotos and Wolniewicz.207 The self-consistent field (SCF or Hartree-Fock) wave function is found to be quite accurate. For HF,
-20 -40 -60 - •id. z.jj. txact mz potential (solid line) and the spherical harmonic expansion up to 'i - '2 = 4 (dashed line) for all configurations (except no. 5, similar to no. 3) of Table 2.4. Here u* = u/e and r* = rfa. For configuration 2 the two curves are indistinguishable at this scale. (From ref. 205.) -20 -30 \ -40 1' -50 0.9 1.0 -60- -70- -90 L tig. 2.34. Exact ST2 potential (solid line) and the spherical harmonic expansion up to h = l2 = 4( "•¦ 5 ( ) and 6 (••¦•) for configuration 4 of Table 2.4. (From ref. 205.)
122 INTERMOLECULAR FORCES 2.9 Table 2.4 Configurations studied for the ST2 model @, = 109° 28'). Here 4>'6'x refer to the intermolecular axis frame of reference Number Configuration D>\,d\,x\) (<f>2> ^2, x'2) @, 77, 0) @, 0, 0) @,0,0) (i'°'°) H LiF, N2, and CO2 the SCF wave functions are used; it is felt by these authors that the quadrupole and hexadecapole moments calculated from p(r) are accurate to within a few per cent. In any case it is of interest by itself to compare the exact energy B.283) with its multipole expansion. The energy B.283) has been expanded in spherical harmonics and the exact intermolecular (see B.29)) spherical harmonic expansion coeffi- coefficients u(/,/2m; r) calculated as a function of r. There are two, somewhat related, questions of interest: (a) How rapid is the 'convergence' of the spherical harmonic expansion series of B.283)? (b) To what extent can one approximate the exact expansion coeffi- coefficients u(lj2m; r) by the multipolar values urnuI,(/1/2m; r), which one gets from B.283) in the manner of § 2.4.5 by neglecting the overlapping regions of the two charge clouds? Ng et a/.175 address mainly question (b). They calculate the exact coeffi- coefficients u(/,/2m;r) for /,/2 = 00, 20, 02, 22. The '22' term, uB2m\ r), corresponds to the angular dependence of the quadrupole-quadrupole
2.10 THREE-BODY INTERACTIONS 123 interaction; the multipole approximation umu]tB2m; r) varies as r~5, so that a deviation of uB2m;r) from an r~5 dependence at short range indicates the presence of an important charge overlap contribution. The 00, 20, 02 coefficients are absent for pure multipole interactions, so that their presence also indicates the importance of overlap. The results show that for the tightly bound molecules H2 and HF the two coefficients u{lxl2m; r) and wmu|t(/1/2m; r) are in good agreement (for 'i t4 0, '2/0), for the range of r of interest for property calculations, i.e. r>a. For N2, LiH and CO2, on the other hand, the charge densities are more diffuse (and prolate), as one expects from the negative values of the quadrupole moment. This gives rise to appreciable overlap contributions to the interaction energy for the range er<r<rmin, as well as at shorter range, where rmin is the position of the LJ well-depth minimum. Since these authors have neglected the short-range exchange effects in the overlap energy, one cannot obtain a realistic model for the overlap energy from their work (see, however, ref. 201a for N2), and one is forced to fall back on the empirical models of § 2.7. Their work does emphasize, however, that for some molecules, properties which derive a large con- contribution from the region <y<r<rmin will be sensitive to the overlap forces. Finally we note that it has been shown128 that for prolate and oblate shaped charge distributions, the speed and region of convergence of the multipole series can be improved by employing 'ellipsoidal multipoles'. There have been just a few similar studies of the convergence of the non-spherical long-range induction and dispersion interaction series,1026-2013 and the short-range overlap series.40'43176-2013 Thus for CO2-Ar, for example,176 one needs overlap harmonics M(AB/;r) with l\ =0, 2, 4, 6 to obtain about 1 per cent accuracy in the region r~@.9- l.l)o\ The convergence of the overlap series in the atom-atom model was discussed in § 2.8. Further work on the damping of non-spherical disper- dispersion forces at short range is required. 2.10 Three-body interactions1 2835 The multipole interactions are pairwise additive, but the induction, dis- dispersion and overlap interactions contain three-body (and higher multi- body) terms. Induction terms We first consider the multi-body induction terms.28-35 For simplicity we restrict ourselves to the case of permanent and induced dipoles (neglect- (neglecting higher multipoles) and to only the a-type polarizability. The field at molecule i due to all other molecules is given by B.63) for the purely
124 INTERMOLECULAR FORCES 2.10 dipolar case as E^Z't^.A, B.284) where the prime on the summation indicates ji=i, and T,, -T^r,,-) = V,V,(-) = Or.r, -l)r,3 B.285) where i = r/r, and A, is the total dipole moment of molecule /. The latter consists of a permanent part |i,. plus an induced part a( . E,, Ai=i*,-+o(.E/ B.286) The total electrostatic energy is given by t/clec = - i I' A. • T,,- . A, +11 E, . o, . Ef. B.287) 'J i The first term on the right-hand side of B.287) is the energy of all of the interacting dipoles Ai> while the second term is the work done to create the induced dipoles (cf. B.159) for the first term and ref. 2 of Appendix C for the second term). We wish to express Uelec in terms of the polarizabilities a; and the permanent dipole moments p.,. We must first solve B.284) and B.286) for Ef and Ai and then eliminate them from B.287). Substituting B.286) in B.284) gives Ei = I' T,7 • m, +1'T,, . a, . E,, B.288) i i Iterating B.288) we get + Z't,, .a, .Tjfc .afc .Tkl - |x, + B.289) ikl Similarly, (i, is obtained from B.286) and B.284) as A, = m +1 «i • T(I- . ^ + Z'«, . T,, . a, . T,fc . im +.... B.290) The primes on the summations in the above equations mean that the indices on each T must be different. Substituting B.289) and B.290) in
2.10 THREE-BODY INTERACTIONS 125 B.287) and collecting terms of order a0, a, a2, etc. gives "I ZVi • T,, . a, . Tjk . a,, . Tkl . |i, -.... B.291) ijkl In deriving this expression we have made use of the fact that T and a are symmetric tensors, so that T.(i = (i.T, o.fi^fi.a, etc. An alternative derivation of B.291) which gives an explicit form for the general term is given in the following section. The first term on the right-hand side of B.291) is the permanent dipole-dipole interaction, while higher terms (of order a, a2, etc. in the polarizability) are the induction contributions to C/eiec. The individual induction terms in B.291) have an obvious physical interpretation. Thus for the term of order a" a permanent dipole moment jl, produces a field which induces a dipole moment of . E, in molecule /', the field of which in turn induces a moment a.k . Ek in molecule k, etc., and the final induced moment (in molecule r) in this chain (of n induced moments) interacts with the permanent moment fis in molecule s; molecule s may or may not be the original molecule i (Fig. 2.35). The terms in B.291) can be subdivided into two-body,208 three-body, etc. terms as shown in Fig. 2.36. Thus, the O(a) term will contain two-body (i = fc) and three-body (i/fc) parts (Figs 2.36(a), (d)). The two-body part of O(a) has been worked out in §2.5; it is of the form (since T~r~3; we neglect numerical factors and the dependence on the B.292) Fig. 2.35. The general induction term of order a" in B.291). Solid and dashed arrows represent permanent and induced dipoles, respectively; bonds represent the T tensor. The two permanent dipoles M-, and |is interact via the chain of n induced dipoles in molecules
126 INTERMOLECULAR FORCES 2.10 :-body (b) 3 -body 4-body (g) (h) Fk. 2.36. The first few induction terms in B.291) for dipolar induction interactions, showing the order in polarizability. Solid and dashed arrows represent permanent and induced dipoles, respectively; bonds represent the T tensor. with a corresponding term involving \i\axr\^. The three-body part con- tains terms of the form B.293) The O(a2) term in B.291) will contain two-body (i = k, j = I), three-body (i = k or i = l or j = l), and four-body (i^k,l;j^ I) terms; these are shown in Figs 2.36(b), (e), (f), and (g), and contain terms of the forms B.294) B.295) — 3—3—3 fO OQA\ Mjntj ~" fX i^2^3M'l'"l2 ^" 13 ^23 ' \^" ^*VO/ it — it m rv it r~3r~?>r~3 A 797^1 inci pl^2"'tr'4 12 23 14 \?*.?*y i ) together with similar terms obtained by permuting the indices.
2.10 THREE-BODY INTERACTIONS 127 It is possible to generalize the above results by (a) including higher multipoles (by including extra terms in B.284), B.287)), and (b) including higher polarizability terms in B.286), B.287), in the same way as was done for the two-body case in § 2.5. Terms of the types B.292)-B.297) have been included by McDonald31 in perturbation theory calculations of the free energy. Singh3233 has generalized McDonald's calculation by using the generalizations of B.292) and B.293) to arbitrary multipoles, and then calculating the free energy by perturbation theory (see also §4.10). Dispersion terms The three-body dispersion terms (due to Axilrod and Teller, and Muto209) can be interpreted with the semi-classical model of § 2.6. Consider three spherical molecules for simplicity. The fluctuating dipole moment |x, of molecule 1 produces a field ~tiif[3 at molecule 2, and therefore a dipole ~a2(n!rj2). which in turn produces a field ~(a2tiir723)''23 at molecule 3, and therefore a dipole ii3~ a3{a2ii-ir^r23). The instantaneous interaction uX23~i*-i . (i3r73 is now averaged over the fluctuations in |x, to give uA23)~(/i2)a2a3rr23r733r233. B.298) The relation B.218) is used to write B.298) as B.299) which apart from a geometric factor has the form of the correct Axilrod- Teller expression1'24 for spherical molecules, uA23) = %EX23<Xia2a3ri2ri3r23O cos d, cos d2cos d3 + 1) B.300) where Ev23 = EXE2E3(E^ + E2 + E3)I(EX + E2)(El + E3){E2 + E3), and d, is the angle at molecule 1 of the triangle 123, etc. The Axilrod-Teller form B.300) is based on the static polarizability approximation. The generalization to include dynamic polarizability (cf. analogous discussion of two-body dispersion in § 2.6) is7 mA23) = Cft/7r)r123r,33r23C cos d, cos d2 cos d3+ 1) x j dw"a1(iw")a2(ia)")a3(iw"). B.301) The expressions B.300) and B.301) can be derived by a rigorous applica- application210 of the fluctuation argument sketched above, or by standard third-order quantum perturbation theory (see Appendix C for second- order dispersion energy calculations; dispersion energy is additive to second order).
128 INTERMOLECULAR FORCES 2.10 For identical molecules, note that in the polarizability approximation the coefficient of the three-body dispersion term B.300) is simply related to the coefficient of the two-body dispersion term B.220) by E123/.Ei2 = j. We also see that for the equilateral configuration for example, the ratio of three-body to two-body energies is M3/u2~(a/r3), which is typically of order 0.05 for r~a. The potential m3 can be attractive or repulsive, depending on the geometry (e.g. u3/u2 Wr3 for an equilateral config- configuration, and u3/u2—L-fealr3 for a linear configuration). For non-spherical molecules a is anisotropic and B.300) has been generalized by Kielich19-3"-35 to give «A23) = - 2E123(«i . T,2 . o2): (T23 . a3 . Tl3) B.302) where T^ is given by B.285) and E123 is the same as for the atomic case. In the (very rough) Lennard-Jones approximation, where we replace the two-body r~6 dispersion term by the LJ expression (see B.235)), we have ?,23 = 3E12/4 = 2ecr6/a2 for identical molecules. For non-identical molecules El23 is given in the LJ approximation by 3a1a2a3 (s, + s2)(s\ + s3)(s: where 11 1 1 st 4eucrfjak 4elfco-?ka/ 4e)kofkai where i/fc = 123, 231 or 312. B.302b) Overlap terms Three-body overlap forces for atoms are reviewed in refs. 1, 6, 53, and 211. For molecules, there are, for example,46206-212a Hartree-Fock calculations and various models for (H2OK. The (CH4K trimer has also been studied.214" Three-body interactions are strongly suspected to be of importance for the third virial coefficient of gases,94 thermodynamic properties of atomic liquids,1'4 and structural and elasticity properties of atomic solids.1-2" Hence, their study for molecular liquids is warranted (§4.10). 2.10.1 Alternative derivation of the induction energy B.291J'5~7 An alternative (matrix) derivation of B.291) is given here, which has advantages in deriving the general form of the energy, and in extending the result to include higher multipoles and polarizability, and/or an external field (see § 10.1.5). We first write out the tensor contractions in B.284), B.286) and
2.10 THREE-BODY INTERACTIONS 129 B.287) explicitly: B.303) B.304) B.305) ij a/3 i aC where i,j=l,...,N are molecule indices, and a, |3 = x, y, z are the tensor indices. We define Tu = 0 so that the sums over i and / are now unrestricted. The relations B.303)-B.305) are clearly in matrix multiplication form: E = T/2, B.306) (L = li+aE, B.307) l^eiec= -l/iTTja+5ETaE. B.308) Here T and a are 3Nx3N symmetric matrices with elements T**(:i and S^af3 respectively, and E, fi, and /x are 3Nxl column matrices, with elements Ef, fi.?, and /if, respectively. The notation A1 denotes the transpose of A, so that fiT and ET are lx3N row matrices. We wish to express the energy t/e]cc in terms of the molecular proper- properties /i and a. Thus we must solve B.306) and B.307) for E and fi, and eliminate them from B.308). Substitution of B.307) in B.306) gives B.309) Iterating B.309) now gives E=Tijl + TaTix + TaTaTfi + ... = T(I+aT+aTaT+...)n = TSix B.310) where S = I + aT+aTaT+... B.311) and I is the unit matrix, with elements /^/3 = 8,;8Qp. We can also write B.311) as an inverse matrix S = (I-aT)'\ B.312) We note for future use the identity S = I+aT(I + aT+aTaT+...) = I+aTS. B.313)
130 INTERMOLECULAR FORCES 2.11 From B.307) and B.310) we now have ll = p. + B.314) where we have used B.313). Substituting B.314) and B.310) in B.308) gives B.315) where we have used AT + BT = (A + B)T, (AB)T=BTAT, a = aT (since ?,,«"'' is symmetric in /', / and in a, /3), and the identity B.313). The expression B.315) can be written out explicitly, Uc^-kltfTfSfcul B.316) Hk --ilni.Ty.S,*.^. B.317) ijk Substituting the explicit expression B.311) for S in B.316) or B.317) gives the desired relation B.291) for L7elec. A somewhat briefer derivation is also possible.28 Since the local field at a molecule is a linear function of the permanent dipoles (see B.310)), one can write down B.315) immediately from B.310) by analogy with the usual charging argument (Appendix C, ref. 2) for deriving u = -{aaEl = -2«o V?nd. (with nind = a(,E0, or Eo = ao Vind) where u is the polarization energy of an induced dipole p.ind = a,,E(, of a single (monatomic) molecule in an applied field E() (see Appendix C). 2.11 Generalization to non-rigid molecules, etc. This book is restricted in scope to a discussion of small rigid molecules in their ground electronic states. We have written the pair potential as u(ra),w2). One can generalize in a number of directions including: (a) to non-rigid molecules214'21K2lg (b) to molecules with internal rotation222'(c) to large (e.g. long chain) molecules, which may be flexible,222 2:;' (d) to electronic excited state molecules, (e) to include covalcncy, and charge transfer effects. The first two generalizations are straightforward in principle; one lets u depend on coordinates describing the additional degrees of freedom involved. For example, the multipole moments Q( and polarizability a of a diatomic molecule are functions of
REFERENCES AND NOTES 131 the internuclear separation x. Similarly, the site-site model is simply adapted to the non-rigid case. The last three generalizations are reviewed in ref. 1. As for (c), for very large molecules the site-site plus charge- charge model is at present the only viable one. Under (d) one encounters the so-called long-range 'resonance' interactions.8 References and Notes 1. Margenau, H. and Kestner, N. R. Theory of intermolecular forces, 2nd edn, Pergamon Press, Oxford, A971). 2. Ref. B, of Chapter 1. 3. Critical reviews of modern inert gas potentials are given in refs. 4, 4a and 10, and more briefly in refs. 5, 94. Useful general surveys are given in refs. 1 and 6-10. 4. Barker, J. A. in Rare gas solids, Vol. 1, pp. 212-64 (ed. M. L. Klein and J. A. Venables) Academic Press, New York A976). 4a. Scoles, G. Ann. Rev. phys. Chem. 31, 81 A980). Aziz, R. in Inert gases: potentials, dynamics and energy transfer (ed. M. L. Klein), Springer Series in Chemical Physics, Springer-Verlag, Berlin A984), Chapter 2. 5. Watts, R. A. and McGee, I. J. Liquid state chemical physics, Wiley, New York A976). 6. Mason, E. A. and Spurling, T. H. The virial equation of state, Pergamon Press, Oxford A969). 7. Hirschfelder, J. O. (ed.), Intermolecular forces [Adv. chem. Phys. 12] Wiley, New York A967). 8. Hirschfelder, J. O., Curtiss, C. F., and Bird, R. B. Molecular theory of gases and liquids, Wiley, New York A954). 9. Buckingham, A. D. and Utting, B. D. Ann. Rev. phys. Chem. 21, 287 A970). 10. Maitland, G. C, Rigby, M., Smith, E. B., and Wakeham, W. A. Inter- Intermolecular forces: their origin and determination, Clarendon Press, Oxford A981). 11. There appears to be no comprehensive up-to-date review. Specific topics are covered in refs. 12-27. Older general surveys are given in refs. 1 and 6-9. 12. Egelstaff, P. A., Gray, C. G., and Gubbins, K. E. International review of science, physical chemistry, Series Two, Vol. 2 (ed. A. D. Buckingham) pp. 299-347, Butterworth, London A975). 13. Gray, C. G. Can. J. Phys. 54, 505 A976). 14. Scott, T. A. Phys. Reports 27, No. 3, pp. 89-157 A976). For more recent work on N2 see Murthy, C. S., Singer, K., Klein, M. L., and McDonald, I. R. Mol. Phys. 41, 1387 A980). 15. Raich, J. C. and Gillis, N. S. J. chem. Phys. 66, 846 A977). 16. MacRury, T. B., Steele, W. A., and Berne, B. J. J. chem. Phys. 64, 1288 A976). 17. Buckingham, A. D. in Physical chemistry, Vol. 4 (ed. H. Eyring, D. Henderson, and W. Jost) pp. 349-86, Academic Press, New York A970). 18. Buckingham, A. D. ref. 7, pp. 107-42. 19. Kielich, S. in Specialist periodical reports, Dielectric and Related Molecular Processes, (ed. M. Davies), Vol. 1, Chemical Society, London, pp. 192-387 A972). 20. Kitaigorodsky, A. I. Molecular crystals and molecules, Academic Press,
132 INTERMOLECULAR FORCES New York A973). 2()a. Califano, S. (ed.) Lattice dynamics and intermolecular forces, Proc. Int. School of Physics. Course LV, Academic Press, New York A975). See also Neto. N.. Roghini, R., Califano, S., and Walmsley, S. H. Chem. Phys. 29, 167 A978). 20b. Mason, R. Perspectives struc. Chem. 3, 59 A970). 21. Morokuma, K. Ace. chem. Res. 10, 294 A977). 22. Kollman, P. in Modern theoretical chemistry, Vol. 4, Chapter 3, Plenum, New York A978). 23. Faraday Discussions of the Chemical Society, No. 62, Potential Energy Surfaces A977). 24. Kihara, T. Intermolecular forces, Wiley, New York A976). 25. Van der Waals systems, Topics in Current Chemistry 93 A980). See in particular, the review of ab initio methods by A. van der Avoird, P. E. S. Wormcr, F. Mulder and R. M. Berns, p. 1. 26. Amos, A. T. and Crispin, R. J. in Theoretical chemistry. Vol. 2 (ed. H. Eyring and D. Henderson) Academic Press, New York A976), p. 1. 26a. Lawley, K. (ed.), Potential energy surfaces. Adv. chem. Phys. 42, A980). See in particular the reviews of R. J. LeRoy and J. S. Carley on van der Waals molecules, and H. Loesch on beam scattering. 27. Pullman, B. (ed.) Intermolecular interactions from diatomics to biopolymers, Wile\ A978). A. D. Buckingham and P. Claverie present general theoreti- theoretical reviews, and R. Rein and P. Schuster review specialized topics such as hydrogen bonding and biomolecular interactions. 28. Barker. J. A. Proc. Roy. Soc. A219, 367 A953). 29. McDonald, I. R. and Woodcock, L. V. J. Phys. C3, 722 A970). 30. Stogryn, D. E. J. chem. Phys. 52, 3671 A970). 31. McDonald. I. R. J. Phys. C7, 1225 A974). 32. Singh, Y. Mol. Phys. 29, 155 A975). 33. Shukla, K. P., Ram, J., and Singh, Y. Mol. Phys. 31, 873 A976). 34. Singh, S. and Singh, Y. Physica 83A, 339 A976). .35. Kielich, S. Physica 31, 444 A965). 36. Casanova. G., Dulla, R. J., Jonah, D. A., Rowlinson, J. S., and Saville, G. Mol. Phys. 18, 589 A970). 37. Dulla, R. J., Rowlinson, J. S., and Smith, W. R. Mol. Phys. 21, 299 A971). 35. Sinanoglu, O. ref. 7, p. 283. 39. Sinha, S. K., Ram, J., and Singh, Y. J. chem. Phys. 66, 5013 A977). 40. Ref. 8, p. 1031. 4 1. Yarkony, D. R., O'Neill, S. V., Schaefer, H. F., Baskin, C. P., and Bender, C. F. J.' chem. Phys. 60, 855 A974). 42. Alexander, M. H. and DcPristo, A. E. J. chem. Phys. 65, 5009 A976). 43. Gallup. G. A. Mol. Phys. 33, 443 A977). \ 44. McMahon, A. K., Beck, H., and KrumhansI, J. A. Phys. Rev. A9, 1852 A974). 45. Raich. J. C, Anderson, A. B., and England, W. J. chem. Phys. 64, 5088 A976). 46. The extensive Hartree-Fock work for H2O-H2O by dementi and cowor- kers is reviewed in ref. 47. 47. Clementi. E. Determination of liquid water structure, Springer-Verlag, Berlin A976). 48. Meyer, W. Chem. Phys. 17, 27 A976).
REFERENCES AND NOTES 133 49. Thakkar, A. J. Chem. Phys. Lett. 46, 453 A977). 50. Wormer, P. E. S., Mulder, F., and van der Avoird, A. Int. J. quant. Chem. 11, 959 A977). and references therein. 50a. Jorgensen, W. L. J. chem. Phys. 70, 5888 A979) and references therein; Jorgensen, W. L. and Ibrahim, M. J. Am. Chem. Soc. 102, 3309 A980). 51. Kollman, P. J. Am. Chem. Soc. 99, 4875 A977). 52. Magnasco, V. Mol. Phys. 37, 73 A979). 53. Clugston, M. J. Adv. Phys. 27, 893 A978). In addition to this review on electron gas methods, see the papers of Kim and coworkers, J. chem. Phys. 68, 5001 A978); 70, 4856 A979). 54. Ng, K. C, Meath. W. J., and Allnatt, A. R. Mol. Phys. 37, 237 A979). 55. Mulder. F., van der Avoird, A. and Wormer, P. E. S. Mol. Phys. 37, 159 A979). 55a. Representative of more recent ab initio papers are: Ree F. H. and Bender, J. J. chem. Phys. 73, 322, 4172 A980) [H2]; Ree, F. H. and Winter, N. W. J. chem. Phys. 73, 322 A980) [N2]; Ko-Tos, W., Ranghino, G. Clementi, E., and Novaro, O. Int. J. quant. Chem. 17, 429 A980) [CFLJ; Meyer, W., Hariharan, P. C, and Kutzelnigg, W. J. chem. Phys. 73, 1880 A980) [H2/Ar, etc.]; Tang, K. T. and Toennies, J. P. J. chem. Phys. 74, 1148 A981) [H,/Ar, etc.J.Habitz, P.. Tang, K. T. and Toennies, J. P. Chem. Phys. Lett. 85, 461 A982) [N2/He]; Burton, P. G. and Senff, W. E. J. chem. Phys. 76, 6073 A982) [H2], See also ref. 25. 56. Buck. U. Adv. chem. Phys. 30, 303 A975). 57. Reuss, J. Adv. chem. Phys. 30, 389 A975); Stolte, S. and Reuss, J. in Atom molecule collision theory (ed. R. B. Bernstein), Plenum, New York A979) p. 201. Thuis, H., Stolte, S. and Reuss, J. Comments at. mol. Phys. 8, 123 A979). 58. Rulis, A. M. and Scoies, G. Chem. Phys. 25, 183 A977). 58a. Tsou, L., Auerbach, D. J., and Wharton, L. J. chem. Phys. 70, 5296 A979). 59. Zandee, L. and Reuss, J., Chem. Phys. 26, 345 A977). 60. Buck, U., Huisken, F., Schleusener, J., and Pauly, H. Phys. Rev. Lett. 38, 680 A977); Buck, U., Schleusener, J., Malik, D. J., and Secrest, D. J. chem. Phys. 74, 1707 A981). 61. Faubel, M. and Toennies, J. P. Adv. at. mol. Phys 13, 229 A977); Toennies, J. P. Ann. Rev. phys. Chem. 27, 225 A976). 61a. See also Chapter 10. 62. Sutter, H. Specialist periodical reports. Dielectric and Related Molecular Processes (ed. M. Davies), Chemical Society London, Vol. 1 A972) p. 65; see also Chapters 6 and 10 for pressure and dielectric virial coefficients. 62a. References lo collision-induced absorption are given in Chapter 11; see also App. D. 63. LeRoy, R. J., Carley, J. S. and Grabenstetter, J. E. see ref. 23, and also 26a; Beswick, J. A. and Jortner, J. Adv. chem. Phys. 47, 363 A981). 64. Ewing, G. E. Can. J. Phys. 54, 487 A976); Blaney, B. L. and Ewing, G. E. Ann. Rev. phys. Chem. 27, 553 A976); Levy, D. H., Adv. chem. Phys. 47, 323 A981). 65. Dunker, A. M. and Gordon, R. G. J. chem. Phys. 68, 700 A978). 66. Howard, B. J. International review of science, Physical Chemistry, Series 2, Vol. 2 (ed. A. D. Buckingham) Butterworth, London, p. 93 A975). For more recent work (e.g. Kr/HCI) see, e.g. Hutson, J. M., Barton, A. E.,
134 INTERMOLECULAR FORCES Langridge-Smith, P. R. R., and Howard, B. J. Chem. Phys. Lett. 73, 218 A98A). For other recent work on dimers, see Faraday Discussions of the Chemical Society 73, A982), 'Van der Waals Molecules'. 67. Klemperer, W. see refs. 23, 107. 68. Coppens, P. and Stevens, E. D. Adv. quant. Chem. 10, 1 A977). 69. Stevens, E. D. Mol. Phys. 37, 27 A979). 70. Specialist periodical reports, Molecular Structure by Diffraction Methods 1 A973) Chemical Society London. 71. Becker, P. Physica Scripta 15, 119 A977). 71a. Bonham, R. A., Lee, J. S., Kennerly, R., and St. John, W. Adv. quant. Chem. 11, 1 A978). 72. Pople, J. A. in Modern theoretical chemistry 4 (ed. H. F. Schaefer) Plenum, New York A978). 73. Coulson, C. A. The shape and structure of molecules, OUP, Oxford A973). 74. Bader, R. F. International review of science, Theoretical Chemistry, Vol. 1 (ed. C. A. Coulson and A. D. Buckingham) Butterworth, London A975) p. 43. 75. Bader, R. F. An introduction to the electronic structure of atoms and molecules, Clarke, Irwin, Toronto A970). 76. Smith, V. H. Physica Scripta 15, 147 A977). 77. Streitwieser, A. and Owens, P. H. Orbital and electron density diagrams, Macmillan, New York A973). 78. Van Wazer, J. R. and Absar, I. Electron densities in molecules and molecular orbitals. Academic Press, New York A975). 79. Pauling, L. The nature of the chemical bond, 3rd edn, Cornell U.P., Ithaca, N.Y. A960) p. 237. 79a. The forces acting on the nuclei of the interacting molecules can, however, be calculated relatively more easily from the total charge density (after allowing for charge distortion due to the interaction) using the usual classical electrostatic relations; this is the statement of the Hellmann- Feynman theorem7'8790 (see also Appendix E). The Hellmann-Feynman route to intermoleeular forces appears, however, to be less accurate7'™ than the usual energy routes. [That the electronic ground-state energy (and therefore the intermoleeular potential) are calculable in principle from the total charge density seems clear from the Hellmann-Feynman theorem, and is a corollary of the Hohenberg-Kohn7t>e theorem. Approximation methods are needed to exploit this fact; see, e.g. §§ 2.4.5, 2.9, the electron gas/' and other so-called density functional theories.79'] 79b. Scrocco, E. and Tomasi, J. Topics in current Chemistry 42, 95 A973); Adv. quant. Chem. 11, 116 A978). 79c. Deb, B. M. Rev. mod. Phys. 45, 22 A973); Bamzai, A. S. and Deb, B. M., Rev. mod. Phys. 53, 95 A981); Deb, B. M. (ed.) The force concept in chemistry; Van Nostrand Reinhold, New York A981). 79d. van der Avoird, A. and Wormer, P. E. S. Mol. Phys. 33, 1367 A977). 79c. Hohenberg. P. and Kohn, W. Phys. Rev. B136, 864 A964); Mikolas, V. and Tomasek. M. Phys. Lett. 64A, 109 A977); for reviews see refs. 74, 79b and E. R. Davidson, Reduced density matrices in quantum chemistry, Academic Press, New York A976) pp. 57, 92. 79f. Payne, P. W. J. chem. Phys. 68, 1242 A978) and references therein; Harris. R. A. and Heller, D. F. J. chem. Phys. 62, 3601 A975). 80. Kestin, J. (ed.) AlP Conf. Proc. No. 11, Transport Phenomena, A973). See
REFERENCES AND NOTES 135 in particular the reviews by R. G. Gordon, and J. J. M. Beenakker, H. F. P. Knaap, and B. C. Sanctuary. 80a. In a few cases a potential model has been constructed from a 'multiprop- erty analysis' (i.e. fitting many data), as has been done for the rare gases.410 See e.g., the O2-Ar potential model of Pirani, F. and Vecchiocat- tivi, F. Chem. Phys. 59, 387 A981), and the H2O-H2O potential of Reimers, J. R., Watts, R. O., and Klein, M. L. Chem. Phys. 64, 95 A982). 81. Bloom, M. International review of science, Physical Chemistry 4 (ed. C. A. McDowell and A. D. Buckingham) Butterworth, London A973). 82. Rabitz, H. Ann. Rev. phys. Chem. 25, 155 A974). 83. Gray, C. G. and Welsh, H. L. in Essays in structural chemistry (ed. D. A. Long, A. J. Downs and L. A. K. Staveley) Macmillan, London A971) Chapter 7; see also Chapter 11 (Vol. 2 of this monograph). 84. Srivastava, R. P. and Zaidi, H. R. Topics in current physics 11, Springer- Verlag, Berlin A979), Chapter 5. 85. Birnbaum, G., see ref. 7. 86. Oka, T. Adv. at. mol. Phys. 9, 127 A973). 87. Gray, C. G. J. chem. Phys. 50, 549 A969). 88. See Chapter 11, and also ref. 168 for a resume. 89. Budenholzer, F. E., Gilason, E. A., Jorgenson, A. D., and Sachs, J. G. Chem. Phys. Lett. 47, 429 A977). 89a. Soper, A. K. and Egelstaff, P. A. Mol. Phys. 39, 1201 A980); Sullivan, J. D., and Egelstaff, P. A. Mol. Phys. 44, 287 A981). 89b. The literature here is vast. Representative recent work is: Soper, A. K. and Egelstaff, P. A. Mol. Phys. 42, 399 A981); Haymet, A. D. J., Morse, M. D., and Rice, S. A. Mol. Phys. 43, 1451 A981); Morse, M. D. and Rice, S. A. J. chem. Phys. 74, 6514 A981); Hinchliffe, A., Bounds, D. G., Klein, M. L., McDonald, I. R., and Righini, R. J. chem. Phys. 74, 1211 A981); Jorgensen, W. L. J. chem. Phys. 75,2026 A981); see also ref. 14 and Chapters 6, 7 and 9. 89c. Clugston, M. J. and Pyper, N. C. Chem. Phys. Lett. 63, 549 A979). 89d. English, C. A., Venables, J. A. and Salahab, D. R. Proc. Roy. Soc. A340, 81 A974). This interaction has recently been calculated ab initio: van Hemert, M. C, Wormer, P. E. S. and van der Avoird, A. Phys. Rev. Lett. 51, 1167 A983). 89e. Callaway, J. Quantum theory of the solid state, Part A, Academic Press, New York A974). 89f. Nielson, G. C, Parker, G. A. and Pack, R. T. J. chem. Phys. 64, 2055 A976). 90. Stockmayer, W. H. J. chem. Phys. 9, 398 A941). 91. Pople, J. A. Proc. Roy. Soc. A221, 498 A954). 91a. Eisenberg, D. and Kauzman, W. The structure and properties of water, Oxford U.P., London A969). 92. Berne, B. J. and Harp, G. D. Adv. chem. Phys. 17, 63 A970). 92a. Spurling, T. H. and Mason, E. A. J. chem. Phys. 46, 322 A967). 92b. Thompson, S. M., Gray, C. G. and Gubbins, K. E., unpublished. 93. Murrell, J. N. in Rare gas solids, Vol. 1, (ed. M. L. Klein and J. A. Venables) Academic Press, New York A976) Chapter 3. 93a. Geurts, P. J. M., Wormer, P. E. S., van der Avoird, A. Chem. Phys. Lett. 35, 444 A975). 94. Barker, J. A. and Henderson, D. Rev. mod. Phys. 48, 587 A976). 95. Sweet, J. R. and Steele, W. A. J. chem. Phys. 47, 3022, 3029 A967).
136 INTERMOLECULAR FORCES 95a. Eyring, H. J. Am. Chem. Soc. 54, 3191 A932). 95b. Miiller, A. Proc. Roy. Soc. A154, 624 A936). 96. Sandier, S. I., Das Gupta, A. and Steele, W. A. J. chem. Phys. 61, 1326 A974). 97. Barojas, J., Levesque, D., and Quentrec, B. Phys. Rev. 7, 1092 A973). 98. Powles, J. G. and Gubbins, K. E. Chem. Phys. Lett. 38, 841 A975). 99. Cheung, P. S. Y. and Powles, J. G. Mol. Phys. 30, 841 A975). 100. Singer, K., Taylor, A., and Singer, J. V. L. Mol. Phys. 32, 1757 A977). 101. Evans, D. J. Mol. Phys. 33, 979 A977). 102. MacRury, T. B. and Steele, W. A. J. chem. Phys. 66, 2262 A977). 102a. Mulder, F. and Huiszoon, C. Mol. Phys. 34, 1215 A977); Mulder, F., van Hcmert, M., Wormer, P. E. S., and van der Avoird, A. Theoret. Chim. Ada 46, 39 A977). 103. Hsu, C. S., Chandler, D., and Lowden, L. J. chem. Phys. 14, 213 A976). 104. Streett, W. B. and Gubbins, K. E. Ann. Rev. phys. Chem. 28, 373 A977). 105. Downs, J., Gray, C. G., Gubbins, K. E. and Murad, S. Mol. Phys. 37, 129 A979). 106. London, F. J. phys. Chem. 46, 305 A942). 107. Janda, K. C, Steed, J. M., Novick, S. E., and Klemperer, W. J. chem. Phys. 67, 5162 A977); Janda, K. C, Hemminger, J. C, Winn, J. S., Novick, S. E.. Harris, S. J., and Klemperer, W. J. chem. Phys. 63, 1419 A975); Steed, J. M., Dixon, T. A., and Klemperer, W. J. chem. Phys. 70, 4095 A979); Holmgren, S. L., Waldman, M., and Klemperer, W. J. chem. Phys. 69, 1661 A978). See also Klemperer's review, ref. 23. 108. Morris. J. M. Mol. Phys. 28, 1167 A974); Aust. J. Chem. 26, 649 A973). 109. Zwanzig, R. W. J. chem. Phys. 39, 2251 A963). 109a. Longuet-Higgins, H. C. and Salem, L. Proc. Roy. Soc. A259, 433 A961). 110. Krauss, M. and Mies, F. H. J. chem. Phys. 42, 2703 A965); Roberts, C. S. Phys. Rev. 131, 203 A963). 111. Coulson, C. A. and Davies, P. L. Trans. Farad. Soc. 48, 777 A952); Eisenberg, D. and Kauzman, W. The structure and properties of water, Oxford U.P., London A969). 112. McDonald, I. R. and Klein, M. L. Farad. Disc. chem. Soc. 66, 48 A978). For more recent work on HF-HF, see Redington, R. L., J. Chem. Phys. 75, 4417 A981). 1 13. Julg, A. Topics in current Chemistry 58, 1, Springer-Verlag, Berlin (cf. 84) A975). 114. Cheung, P. S. Y. and Powles, J. G. Mol. Phys. 32, 1383 A976). 1 15. Sweet, J. R. and Steele, W. A. J. chem. Phys. 50, 668 A968). 1 16. Evans, D. J. and Watts, R. O. Mol. Phys. 32, 93 A976). 1 Km. Schcraga. H. A. Ace. chem. Res. 12, 7 A979). 1 16b. Pack, R. T. Chem. Phys. Lett. 55, 197 A978); Rotzoll, G. Chem. Phys. Lett. 88, 179 A982). These authors lit B.8) to beam scattering data, by using truncated (at 1 = 2) spherical harmonic expansions to fit e(ti>tw2<^) and <r(coico2(o). 117. Corner, J. Proc. Roy. Soc. A192, 275 A948). 118. Berne, B. J. and Pechukas, P. J. chem. Phys. 56, 4213 A972). 1 18a. Salacuse, J., Gray, C. G., and Gubbins, K. E., unpublished. 119. Kushick, J. and Berne, B. J. J. chem. Phys. 64, 1362 A976). See also Modern theoretical chemistry, Vol. 6 (ed. B. J. Berne) Plenum, New York A977), Chapter 2. 120. Stone, A. J. Mol. Phys. 35, 241 A978).
REFERENCES AND NOTES 137 120a Gay, J. G. and Berne, B. J. J. chem. Phys. 74, 3316 A981). 121. Walmsley, S. H. Chem. Phys. Lett. 49, 390 A977). 121a. Sakai, K.. Koide, A., and Kihara, T. Chem. Phys. Lett. 47, 416 A977). 122. Boublik, T. Mol. Phys. 32, 1737 A976); Nezbeda, I. and Leland, T. W. J. Chem. Soc. Farad. Trans. II 2, 193 A979). 123. Monson, P. A. and Rigby, M. Chem. Phys. Lett. 58, 122 A978); Mol. Phys. 35, 1337 A978). 124. Rebertus, D. W. and Sands, K. M. J. chem. Phys. 67, 2585 A977). 125. Rigby, M Mol. Phys. 32, 575 A976). 126. Nezbeda, I. Chem. Phys. Lett. 41, 55 A976). See also Chapter 6. 127. Boublik, T. and Nezbeda, I. Chem. Phys. Lett. 46, 315 A977). 127a. We follow the method of refs. 134 and 148. 128. Stiles has suggested that an ellipsoidal harmonic expansion might be superior for molecules with large non-sphericity. See Stiles, P. J. Mol. Phys. 38, 433 A979) and references therein; Chebyshev polynomial expansions have been discussed by Monson, P. A. and Steele, W. A. Mol. Phys. 49, 251 A983). Gegenbauer polynomial expansions are discussed at the end of §2.8. 128a. To see, e.g. that Jmim;m C(M2'; m,m2m) Yjimi Y|2m2 Yfn is a rotational in- invariant, note that Zm,m2C(AB/; m1m2m) YlimiYi2m2 = Flm is a tensor of type /, m (i.e. transforms under rotations like Ybn, by definition of the CG coefficient), and that ?m FlmY^ is an invariant (addition theorem, or binary invariant). 128b. Some readers may find it helpful to consider the Cartesian tensor versions of the rotational invariants i//j (which occurs in ref. 128a, and in § 2.4.2) and i/>i,i2i, formed from spherical tensors (Appendix A), where •K.^ I C(ii;2!;m,m2m)Y,,,,,1(a)l)Y,2m2(aJ)Y^(a)). Consider the questions: what are the rotational invariants formed from (a) two vectors A and B, (b) three vectors A, B, and C? (Assume A, B and C have directions &),, a>2 and u>, respectively). The answers are: (a) A.B = YcAaBc and (b) A .BxC = ?ost Ee,s>AaB3Q. Thus (a) is proportional to i/>i (i.e. tpi for / = 1), and (b) proportional to i/>ni- The Cartesian analysis is cumbersome for (>1, however. Thus, for example, the Cartesian version of i//,12 is AB:CCC-C21) [see (A.246) and discussion, and E.62) and discussion]. Further Cartesian examples are discussed on pp. 485, 490, and in references cited there. The advantage of the spherical tensor formulation (as always-see p. 49) is generality, i.e. </'i,i2i is given in the general case by Ims C( )YYY*. 128c. Wigner, E. P. Group theory and its application to the quantum mechanics of atomic spectra, Academic Press, New York, p. 218 A959). 128d. Steele, W. A. Mol. Phys. 39, 1411 A980). 129. Craig, D. P. and Mellor, D. P. Topics in current Chemistry 63, 1, Springer- Verlag, Berlin A976). 130. Scott, R. L. J. Chem. Soc. Farad. Trans. II, 73, 356 A977); Wheeler, J. C. J. chem. Phys. 73, 5771 A980). 130a. Joslin, C. G. and Gray, C. G. Mol. Phys. 50, 329 A983); J. Phys. A. 17, 1313 A984). 131. Good, R. H. and Nelson, T. J. Classical theory of electric and magnetic fields, Academic Press, New York A971), Chapter 1.
138 INTERMOLECULAR FORCES 131a. Coulson, C. A. and Boyd, T. J. M. Electricity, 2nd edn, Longman, London A979). 132. Ref. 131, p. 51. 132a. Gray, C. G. Am. J. Phys. 46, 582 A978); 47, 457 A979). 133. Jackson, J. D. Classical electrodynamics, 2nd edn Wiley, New York A975) p. 34. 133a. See, e.g., Schuster27, Coulson", and Bader74. 134. Gray, C. G. Can. J. Phys. 46, 135 A968). 134a. Frenkel, J. Z. Phys. 25, 1 A924); Lehrbuch der Electrodynamik, Springer- Verlag, Berlin A926) vol. 1, p. 97. 134b. An alternative, but more cumbersome, way to obtain the Taylor series is to write Ir-r.r^Kr-riMr-riXT* as |r-r(|^ = [r2-2r - r, + rf]"' = r \\-2t .rjr2 + r2jr2~]'', and then to use the binomial expansion for A+ x) '. (Expressed in angle variables cos ¦/< = r . r,, this yields the generating function expression B.70).) 135. Jansen, L. Phys. Rev. 110, 661 A958); Physica 23, 599 A957). 135a. T"'(r) is tracclcss for general I since the trace involves V2(l/r), which vanishes for r>0 (see B.42)). (T*2)(r) is manifestly traceless from the explicit expression B.56).) 135b. Alternatively we can use the language of irreducible tensors (A.218). We decompose the 6 tensor into irreducible 1 = 0 and 2 parts, 6 = 6"" + 6'2'. Only 0'2' will contribute to 0 : T12', since T*21 is symmetric and traceless, i.e. pure I = 2 (see ref. 98a of Appendix A). 135c. Benedek, G. B. and Villars, F. M. H. Physics, Vol. 3, Electricity and Magnetism, Addison-Wesley, Reading, MA A979). 135d. Buckingham, A. D. Quart. Rev. Chem. Soc. 13, 189 A959). 135e. Cole, R. H. Prog, in Dielectrics 3, 47 A961). 135f. Long, D. A. Raman spectroscopy, McGraw-Hill, New York A977) p. 19. 136. Gray, C. G. Am. J. Phys. 46, 169 A978); Gray, C. G. and Nickel, B. G., Am. J. Phys. 46, 735 A978). 136a. Gray, C. G. and Lo, B. W. N. Chem. Phys. 14, 73 A976); Chem. Phys. Let;. 25, 55 A974). 136b. See, e.g. refs. 5, 8, 10, 12, 15 of Appendix A. 137. Goldstein, H. Classical mechanics, Addison-Wesley, Reading, MA A950) p. 151 [2nd edn A980), p. 198]. 137a. Herzberg, G. Infrared and Raman Spectra, Van Nostrand, New York A945) p. 6. 137b. Nyc, J. F. Physical properties of crystals, Clarendon Press, Oxford A957) p. 20. 137c. Birss. R. R. Symmetry and magnetism, North-Holland, Amsterdam, 2nd edn A966) p. 44. 137d. The theorem is self-evident geometrically (think of vectors A and B for simplicity) and nearly so algebraically. An equivalent statement is that if Tnl> = A,,,t - BaB vanishes in one coordinate system, it vanishes in all, and conversely if Tll(, is non-vanishing in one coordinate system it is non- vanishing in all. The latter statements are obvious from the transformation % relation (A.216) for tensors. The method of proof is identical for spherical components. The theorem is widely applicable in deriving tensor relations for various quantities; simple examples are the higher Cartesian multipole moment and polari/ability (cf. (C.29)) relations27 for molecules of symmet- symmetrical shapes. It is this theorem which is at the root of the fundamental
REFERENCES AND NOTES 139 property (a) of tensors mentioned on p. 476, both for our restricted defi- definition of tensors, and also in the more general cases mentioned on p. 477. 137e. For a discussion in the context of general tensor analysis, see, e.g. ref. 64 of Appendix A, p. 65. 138. Stogryn, D. E. and Stogryn, A. D. Mol. Phys. 11, 371 A966). 138a. Allene has an S4 axis (i.e. a fourfold rotation-reflection axis along the CCC bond-see ref. 137a), so the symmetry axis is four-fold in this sense. 139. Lax, M. Symmetry principles in solid state and molecular physics, Wiley, New York A974) p. 101. 140. Gray, C. G. J. Phys. B4, 1661 A971). 141. Armstrong, R. L., Blumenfeld, S. M, and Gray, C. G. Can. J. Phys. 46, 1331 A968). 142. Gubbins, K. E., Gray, C. G., and Machado, J. R. S., Mol. Phys. 42, 817 A981). 143. Steinborn, E. O. and Ruedenberg, K. Adv. quant. Chem. 7, 1 A973). Caola, M. J. J. Phys. All, L23 A978). 144. Buckingham, A. D. and Longuet-Higgins, H. C. Mol. Phys. 14, 63 A968). 145. Barron, L. D. and Gray, C. G. J. Phys. A6, 69 A973). 145a. Rowe, E. G. P. J. math. Phys. 19, 1962 A978). 146 Carlson, B. C. and Rushbrooke, G. S. Proc. Camb. Phil. Soc. 45, 626 A950). 147. References to other derivations can be found in ref. 134. 148. Gray, C. G. and Van Kranendonk, J. Can. J. Phys. 44, 2411 A966). 149. Gray, C. G. and Stiles, P. J. Can. J. Phys. 54, 513 A976). 149a. LeFevre, R. J. W. Dipole moments, Methuen, London, 3rd edn A953) p. 27. 150. Parsonage, N. G. and Staveley, L. A. K. Disorder in crystals, Clarendon Press, Oxford A978). 151. Rose, M. E. J. Math, and Phys. (MIT) 37, 215 A958). See also Yamamoto, T. J. chem. Phys. 48, 3193 A968). 151a. Barton, A. E., Chablo, A., and Howard, B. J. Chem. Phys. Lett. 60, 414 A979). 151b. Hosticka, C, Bose, T. K. and Sochanski, J. S. J. chem. Phys. 61, 2575 A974). 152. Evans, D. J. and Watts, R. O. Mol. Phys. 29, 777 A975). 152a. Dispersion forces are sometimes called van der Waals forces, since van der Waals introduced a phenomonological attractive intermolecular force in deriving his famous equation of state for dense fluids. Qualitatively, the existence of universal attractive long-range forces was evident from, for example, the very existence of low pressure liquids and from the phenomena of capillarity. Similarly, the existence of short-range repulsive forces, also discussed by van der Waals, was evident from the finite densities of liquids and solids. The nomenclature is not standard since the name 'van der Waals interactions' is also applied to the total long-range part of the potential, and even to the total potential. 153. Graben, H. W. Am. J. Phys. 36, 267 A968). 154. McLachlan, A. D. Proc. Roy. Soc. A271, 387 A963). 155. Pack, R. T. J. chem. Phys. 64, 1659 A976). 155a. Lekkerkerker, H. N. W., Coulon, P., and Luyckz, R. J. Chem. Soc. Farad. Trans. 73, 1328 A977). 155b. Langhoff, P. W., Gordon, R. G., and Karplus, M. J. chem. Phys. 55, 2126
140 INTERMOLECULAR FORCES A971). 156. The original work of Casimir and Polder,'57 and others, is reviewed by E. A. Power (see ref. 7, pp. 167-224). 157. Casimir, H. B. G. and Polder, D. Phys. Rev. 73, 360 A948). 158. Israelachvili, J. N. Contemp. Phys. 15, 159 A974). 159. Parsegian, V. A. Ann. Rev. Biophys. Bioeng. 2, 221 A973). 160. Gabler, R. Electrical interactions in molecular biophysics, Academic Press, New York A978). 161. Mahanty. J. and Ninham, B. W. Dispersion forces, Academic Press, New York A976). 162. Langbein, D. Theory of van der Waals attraction, Springer-Verlag, New York A974). 163. Faraday discussion of the Chemical Society, No. 42 A966) Colloid Stability in Aqueous and Non-aqueous Media; No. 65 A978) Colloid Stability. 164. Visser, J. Adv. in colloid and interface Sci. 3, 331 A972). 165. Van Olphcn, H. and Mysels, K. J. (ed.) Physical chemistry: enriching topics from colloid and surface science, IUPAC Commission 1.6 (Theorex, La Jolla, 1975). 166. Richmond, P. Specialist Peroidical Reports: Colloid science (ed. D. H. Everett) 2, 130 A975) Chemical Society London. 167. Gray. C. G. J. chem. Phys. 50, 549 A969). 16S. Buckingham, A. D. and Tabisz, G. C. Mol. Phys. 36, 583 A978). 169. Herman, R. J. chem. Phys. 44, 1346 A966). 169a. Fabre, D., Widenlocher, G., Thiery, M. M, Vu, H., and Vodar, B. Advances in Raman spectroscopy, Vol. 1 A973) (ed. J. P. Mathieu) Heyden & Son. London, p. 497. Gray, C. G., Thesis, Toronto A967). 169b. Kircz, J. G., van der Peyl, G. U. Q., van der Elsken, J., and Frenkel, D. J. chem. Phys. 69, 4606 A978). \ 169c. Isnard, P., Robert, D., and Galatry, L. Mol. Phys. 39, 501 A980). 170. London, F., ref. 106. See also ref. 8, p. 969 and ref. 1, p. 39, (molecules). The original London paper on atomic dispersion forces is: Z. Phys. Chem. Bll, 222 A930); see also Trans. Farad. Soc. 33, 8 A937). 171. de Boer, J. Physica 9, 363 A942). 171a. Bonamy, J., Bonamy, L., and Robert, D. J. chem. Phys. 67, 4441 A977). 171b. In the literature overlap interactions are also referred to as shape, steric, repulsive and exchange interactions. 172. Prausnitz, J. M. Molecular thermodynamics of fluid phase equilibria, Pren- Prentice Hall, Englewood Cliffs, N.J. A969) pp. 75-85. 172a. Coulson, C. A. Research 10, 149 A957); see also his book Valence, OUP, Oxford, 2nd edn A961) Chapter 13. [3rd edn by R. McWeeny, 1979]. 172b. Schuster, P., Zundel, G., and Sandorfy, G. (ed.) The hydrogen bond, Vols. 1-3, North-Holland, Amsterdam A976). 173. Mulliken, R. S. and Person, W. B. Molecular complexes, Wiley, New York A969). Person, W. B. in Spectroscopy and structure of molecular complexes (ed. J. Yarwood) Plenum, New York A973) p. 1. 174. Hanna, M. W. and Lippert, J. L. in Molecular complexes, Vol. 1 (ed. R. Foster) Crane, Russak, N.Y., A973). 175. Ng. K. C, Meath, W. J. and Allnatt, A. R. Mol. Phys. 32, 177 A976); 33, 699 A977); 38, 449 A979). 175a. Copeland, T. G. and Cole, R. H. J. chem. Phys. 64, 1741 A976). 175b. Winicur, D. G. J. chem. Phys. 68, 3734 A978).
REFERENCES AND NOTES 141 175c. Kong, C. L. J. chem. Phys. 53, 1516 A970). 176. Parker, G. A. Snow, R. L. and Pack, R. T. J. chem. Phys. 64, 1668 A976). 176a. Liu, W. K., Grabenstetter, J. E., LeRoy, R. J., and McCourt, F. R. J. chem. Phys. 68, 5028 A978). 177. Thompson, S. M., Tildesley, D. J. and Streett, W. B. Mol Phys. 32, 711 A976). 178. Sweet, J. R. and Steele, W. A. J. chem. Phys. 47, 3022 A967). 179. Sack, R. A. J. Math. Phys. 5, 260 A964). 180. Yasuda, H. and Yamamoto, T. Prog, theor. Phys. 45, 1458 A971). 180a. Yasuda, H. J. chem. Phys. 73, 3722 A980); 74, 6531 A981); Briels, W. J. J. chem. Phys. 73, 1850 A980). 181. van Rij, W. I. J. Phys. A8, 1164 A975). 182. Schroder, H. J. chem. Phys. 67, 1953 A977). 182a. Miller, W. M. Symmetry and separation of variables, Vol. 4 of Encyclopedia of mathematics and its applications (ed. G. Rota) Addison-Wesley, Read- Reading MA A977) Chapter 5. 182b. For the homonuclear diatomic case, similar conclusions were reached by Watanabe, K., Allnatt, A. R., and Meath, W. J. Mol. Phys. 42, 165 A981). 183. Huiszoon, C. and Mulder, F. Mol. Phys. 38, 1497 A979); 40, 249 A980). Dunkersloot, M. C. A. and Walmsley, S. H. Chem. Phys. Lett. 11, 105 A971). 184. Balescu, R. Physica 22, 224 A956); Prigogine, I. The molecular theory of solutions, North-Holland, Amsterdam A957), p. 263; the errors in these treatments are corrected in ref. 181. 185. Murad, S. and Gubbins, K. E. in Computer modeling of matter, (ed. P. Lykos) ACS Symposium Series No. 86, 62 A978). 186. Powles, J. G., Evans, W. A. B., McGrath, E., Gubbins, K. E., and Murad, S. Mol. Phys. 38, 893 A979). 187. Coulson, C. A. Proc. Roy. Soc. Edinburgh A61, 20 A941). 188. Brooks, F. C. Phys. Rev. 86, 92 A952). 189. Roe, G. M. Phys. Rev. 88, 659 A952). 190. Cusachs, L. D. Phys. Rev. 125, 561 A962); J. chem. Phys. 38, 2038 A963). 191. Dalgarno, A. and Lewis, J. T. Proc. Phys. Soc. A69, 57 A956). 192. Cohan, N. V. Mol. Phys. 17, 307 A969). 193. Pack, G. R., Wang, H., and Rein, R. Chem. Phys. Lett. 17, 381 A972). 194. Whitton, W. N. and Byers Brown, W. Int. J. quant. Chem. 10, 71 A976). 195. Young, R. H. Int. J. quant. Chem. 9, 47 A975). 196. Koide, A. J. Phys. B9, 3173 A976). 197. Mezei, M. and Campbell, E. S. Theor. Chim. Acta 43, 227 A977). 198. Kutzelnigg, W. ref. 23. 199. Ahlrichs, R. Theor. Chim. Acta 41, 7 A976). 200. Cole, R. H. Chem. Phys. Lett. 57, 139 A978). 200a. Brobjer, J. T. and Murrell, J. N. Chem. Phys. Lett. 77, 601 A981). 201. Jeziorski, B., Szalewicz, K. and Jaszunski, M. Chem. Phys. Lett. 61, 391 A979). 201a. Mulder F., van Dijk, G. and van der Avoird, A. Mol. Phys. 39, 107 A980); Berns, R. M. and van der Avoird, A. J. chem. Phys. 72, 6107 A980). 202. The reason the expansions are thought to be asymptotic is that the exact energy may contain terms which vanish faster than any power of r'1 (e.g. exp(-Kr)).
142 INTERMOLECULAR FORCES 203. More formally, the series representation Y.n = i c^r"" of f(r) is convergent if it approaches f(r) as N^>c° for given r, whereas it is asymptotic if it approaches f(r) as r^°° for given N. See, e.g., Morse, P. M. and Feshbach, H. Methods of theoretical physics, Vol. 1, p. 434, McGraw-Hill, New York A953). 204. Evans, D. J. unpublished. 205. Nicolas, J. Thesis, Cornell Univ. A979). 206. Stillinger, F. H. Adv. chem. Phys. 31, 1 A975). 207. Ko/os, W. and Wolniewicz, L. J. chem. Phys. 43, 2429 A965). 208. The two-body series can be summed exactly for isotropic a; see Bucking- Buckingham, A. D. and Pople, J. A. Trans. Farad. Soc. 51, 1173 A955), and Appendix 10D. 209. Axilrod, B. M. and Teller, E. J. chem. Phys. 11, 299 A943); Axilrod, B. M. J. chem. Phys. 19, 724 A951); Muto, Y. Proc. Phys. Math. Soc. Japan 17, 629 A943). 210. McLachlan, A. D. Mol. Phys. 6, 423 A963); McLachlan, A. D., Gregory, R. D. and Ball, M. A. Mol. Phys. 7, 119 A964). 211. Jansen, L. Adv. quant, chem. 2, 119 A965). 212. Lie, G. C. and Clementi, E. J. chem. Phys. 60, 1275, 1288 A974). 213. Campbell, E. S. and Mezei, M. J. chem. Phys. 67, 2338 A977). 214. Stillinger, F. H. and David, C. W. J. chem. Phys. 69, 1473 A978); Stillinger, F. H. Int. J. quant. Chem. 14, 649 A978). 214a. Clementi, E., Kotos, W., Lie, G. C, and Ranghino, G. Int. J. quant. Chem. 17, 377 A980). 214b. Novaro, O., Castillo, S., KoJos, W., and Les, A. Int. J. quant. Chem. 19, 637 A981). 215. Mandel, M. and Mazur, P. Physica 24, 116 A958). 216. Bottcher, C. J. F. Theory of electric polarization, 2nd edn, Elsevier, Amster- Amsterdam A973). 217. Wertheim, M. S. Mol. Phys. 26, 1425 A973). 218. Rahman, A., Stillinger, F. H., and Lemberg, J. J. chem. Phys. 63, 5223 A976). 219. McDonald, I. R. and Klein, M. L. /. chem. Phys. 64, 4790 A976). 220. Ryckaert, J. P. and Bellemans, A. Farad. Disc. Chem. Soc. 66, 95 A978). 221. Rebertus, D. W., Berne, B. J., and Chandler, D. /. chem. Phys. 70, 3395 A979). 222. Weber, T. A. and Helfand, E. J. chem. Phys. 71, 4760 A979), and references therein. 223. For a review of empirical site-site plus charge-charge potentials for large flexible molecular systems, see Olie, T., Maggiora, G. M., Christoffersen, R. E., and Duchamp, D. J., Int. J. quant, chem: Quantum Biology Symp. 8, 1 A981). 224. This statement should be qualified. See the corresponding discussion of spectral moments in Chapter 11.
I HI BASIC STATISTICAL MECHANICS Such inquiries have been called by Maxwell statistical. They belong to a branch of mechanics which owes its origin to the desire to explain the laws of thermodynamics on mechanical principles, and of which Clausius, Maxwell, and Boltzmann are to be regarded as the principal founders .... But although, as a matter of history, statistical mechanics owes its origin to investigations in thermodynamics, it seems eminently worthy of an independent development, both on account of the elegance and simplicity of its principles, and because it yields new results and places old truths in a new light in departments quite outside of thermodynamics. J. Willard Gibbs A902), Preface to Elementary principles in statistical mechanics In this chapter we introduce distribution functions for molecular momenta and positions. All equilibrium properties of the system can be calculated if both the intermolecular potential energy and the distribution functions are known. Throughout, we shall make use of the 'rigid molecule' and classical approximations. In the rigid molecule approximation the system inter- intermolecular potential energy cu^co™) depends only on the positions of the centres of mass r^ = r1... rN for the N molecules and on their molecular orientations <x>N = co1... coN; any dependence on vibrational or internal rotational coordinates is neglected. In the classical approximation the translational and rotational motions of the molecules are assumed to be classical. These assumptions should be quite realistic for many fluids composed of simple molecules, e.g. N2, CO, CO2, SO2 CF4, etc. They are discussed in detail in §§ 1.2.1 and 1.2.2; quantum corrections to the partition function are discussed in §§ 1.2.2 and 6.9, and in Appendix 3D. In considering fluids in equilibrium we can distinguish three principal cases: (a) isotropic, homogeneous fluids (e.g. liquid or compressed gas states of N2, O2, etc. in the absence of an external field), (b) anisotropic, homogeneous fluids (e.g. a polyatomic fluid in the presence of a uniform electric field, nematic liquid crystals), and (c) inhomogeneous fluids (e.g. the interfacial region). These fluid states have been listed in order of increasing complexity; thus, more independent variables are involved in cases (b) and (c), and consequently the evaluation of the necessary distribution functions is more difficult (see, for example, Fig. 3.2).
144 BASIC STATISTICAL MECHANICS 3.1 For molecular fluids it is convenient to introduce several types of distribution functions, correlation functions, and related quantities: (a) The angular pair correlation function g(r1r2co1co2)- This gives com- complete information about the pair of molecules, and arises in expressions for the equilibrium properties for a general potential. It is proportional to the probability density of finding two molecules with positions rx and r2 and orientations a^ and co2. (b) The site-site correlation function ^3(^3) is proportional to the probability density that sites a and C on different molecules are separated by distance ra&, regardless of where the other sites are (i.e. regardless of molecular orientations). These functions arise naturally in expressions for certain properties, e.g. the structure factor S(k) as measured by neutron diffraction. There the neutrons are scattered by the nuclear sites in the molecule. In principle the g^p can be measured for nuclear sites by neutron diffraction. The set of g^p always gives less information than knowledge of g(r1r2oo1aJ) = gA2). One can calculate g^p from gA2) but not vice versa. (c) Spherical harmonic coefficients of g(r1r2co1co2)-the g(M2Z; ^1^2; r) coefficients. If we have all the coefficients ({ — 0 to °°, ni=—li to 4-Z-) it is equivalent to knowing gA2). There are three main uses of these: (i) the harmonics have very convenient orthogonality properties in theoretical calculations, (ii) the coefficients provide a useful way to test theories, and (iii) some properties can be exactly expressed in terms of a single harmonic. Examples are measurements of the dielectric constant, which involves (P^cos y)), and measurements of the Kerr constant or de- depolarized light scattering, which involve <P2(cos y)). For linear molecules these involve the coefficients gA10;00;r) and gB20;00;r), respec- respectively. Here y is the angle between the molecular axes of two linear molecules and P{ is the Legendre polynomial of order I. In §3.1 we consider the definitions and properties of distribution functions in the canonical ensemble (fixed N, V, T). The spherical har- harmonic expansion of pair correlation functions is carried out in § 3.2; such expansions are often convenient in both theoretical and computer simula- simulation studies. Distribution functions in the grand canonical ensemble (fixed ia, V, T) are taken up in § 3.3; while the notation is more complex, this ensemble is of more general usefulness than the canonical. In § 3.4 we derive equations for the thermodynamic derivatives (with respect to ja, V, and T) and position and orientation derivatives of the distribution func- functions, using the grand canonical ensemble. In § 3.5 the most important equations are extended to the case of mixtures. Finally, in § 3.6 we derive expresssions for the distribution functions and equation of state that are valid at low density.
3.1 DISTRIBUTION FUNCTIONS IN THE CANONICAL ENSEMBLE 145 3.1 Distribution functions in the canonical ensemble 3.1.1 Distribution law and the partition function In this section we briefly discuss the probability distribution law, and the relation between the partition function and the thermodynamic proper- properties. For a detailed derivation and discussion of the expressions given below the reader should consult standard texts on statistical mechanics (see, for example, refs. 1-5). We consider a system of N identical molecules in thermal equilibrium in a volume V at temperature T. The probability Pn of finding the system in quantum state n at any instant is given by1'2 (see also Appendix B.2) Pn=Q~^/Q C.1) where |8 = 1/fcT, k is Boltzmann's constant, T is the absolute tempera- temperature, En is the energy for state n, and Q = Q(N, V, |8) is the canonical partition function, Q = I e-3E« C.2) n where the sum is over all quantum states. The Pn are normalized to unity, i.e. n = l. C-3) The thermodynamic functions are directly related to the partition function. The internal energy, U, is simply the average of En over all possible states n, i.e. or, using C.2), a(^) C-4) The relation between the Helmholtz free energy, A, and Q can be obtained from the Gibbs-Helmholtz thermodynamic relation, NV Using C.4) for U in C.5) and integrating over T gives • In Q dT
146 BASIC STATISTICAL MECHANICS 3.1 where the constant C must be independent of temperature. We prove rigorously in Appendix E that C must also be independent of the volume in order that p = —EA/9V)NT yield the known virial equation of state. That this statement is plausible is also made clear below (see discussion below C.85)), where we show that it is only if C is independent of V that C.6) yields the known equation of state, p = NkT/V, for a classical ideal gas. The expression for the entropy, S, is obtained from the definition A = U-TS. Using C.4) and C.6) for A and U in this relation gives S=l^j^ = ^+k\nQ-C. C.7) The connection between entropy and the probability distribution law can be obtained as follows. From C.1) we have In Q C.8) kT where we have used C.3) and C.4). From C.7) and C.8) we have S=-kYJPn\nPn-Q C.9) n which provides the connection between entropy and probability. The probabilities Pn are independent of the energy zero used, as is obvious from the fact that the Pn are physical quantities. To prove this we introduce energies E'n = En~E' referred to a new energy zero E'; we have from C.1) ' _ -(En-E')/JcT / V -(En-E')/kT n c / Zj C = Pn. C.10) We define the so-called absolute entropy by choosing C = 0 in the above equations. This choice6-7 of C makes S positive and extensive, and yields a simple expression for T-^0 (see below). It also ensures (see C.9) and C.10)) that S is independent of the energy zero used. In contrast, Q, U, and A are not absolute quantities, since their values depend on the energy zero. Although the choice C = 0 is universally used for the entropy zero, there is no corresponding universal choice for the energy zero; instead one chooses a zero based on convenience. In theoretical work on dense fluids of rigid molecules, for example, the conventional choice for
3.1 DISTRIBUTION FUNCTIONS IN THE CANONICAL ENSEMBLE 147 the energy zero corresponds to the molecules being at an infinite distance apart (^ = 0), with zero rotational and translational kinetic energy; this is the choice generally adopted in this book. For the calculation of ideal gas properties (see refs. 1-3 of Chapter 1), however, the energy of the molecular ground state is usually used as the energy zero. With C = 0, eqns C.6), C.7), and C.9) become A = -kTIn Q, C.11) C.12) S = -fc][Pnln-pn C.13) n and the equation of state is obtained from p = —(dA/dV)NT as Finally, we consider the entropy as the temperature approaches zero. In general there will be g0 degenerate states having the lowest energy Eo. From C.1) we see that as, T-^0 only these ground states can be occupied; for these states C.1) gives Pn = (l/g0), while Pn=0 for other states. Thus we have limS = -fcgo(—ln-) = fclng0. C.15) T^O \g0 go/ For non-degenerate ground states (e.g. a perfect crystal) we have g0 = 1 and S —* 0 as T —* 0. The entropy of most substances is of order Me, where JV~ 1023, so that unless g0 is extremely large the entropy at absolute zero is essentially zero. The choice C = 0 in C.9) is thus consistent with the third law of thermodynamics. Detailed discussions of the statistical mechanical basis of the third law of thermodynamics are given, for example, by Fowler and Guggenheim,8 Denbigh,9 Hill,3 and Wilks.10 The naive argument presented above is given in most textbooks, but is unfortunately somewhat misleading. The argument suggests that for C.15) to apply we must have T=sAe/fc, where Ae is the separation between the ground and first excited levels. For macroscopic systems this leads to T~ 10~15 K as the low temperature regime, for which the third law is valid. However, this is far below the temperature (T~ 1 K) where the lattice entropy is found to be vanishingly small in practice. The error
148 BASIC STATISTICAL MECHANICS 3.1 lies in neglecting the prodigious rate of increase of the number of states with energy, which more than compensates for the decreasing Boltzmann factor. A rigorous discussion11 requires consideration of the density of states in the limit V—>°°} with V-^oo taken first and the limit T—>0 taken subsequently. 3.1.2 Factorization of the distribution function and partition function Separation into classical and quantal parts. Using the model outlined in the introduction, we now assume that the Hamiltonian operator is separable into two independent parts,217 5if=^cl + ^qu C.16) where $?cl corresponds to coordinates that can be treated classically (the centre of mass and external rotational degrees of freedom) and $?qu to those that must be treated quantally (vibrational and internal rotational degrees of freedom). Equation C.16) implies that there are two indepen- independent sets of quantum states, corresponding to $?cl and $?qu, respectively, and obtained from the corresponding Schrodinger equations (B.56); i.e., the eigenstates \n) of %€ can be taken as products |rccl)|nqu) with corres- corresponding energy E^ + E^u. (States of the system will be denoted by \A) in Dirac notation, or by ijtA(x) in Schrodinger notation; see Appendix B.2.) Thus Pn and Q each factorize: Pn=P?PT, C.17) Q = QciQqu C.18) where pel = e-3E PT = e~3EQqu, Qqu = Z e-^s-. C.20) - n Equations C.16)-C.20) imply the neglect of interaction terms in the Hamiltonian between the vibrations (and internal rotations) and transla- tional and rotational motions. As discussed in § 1.2, such interaction terms can usually be neglected in considering equilibrium structural and thermodynamic properties. Because the intermolecular interactions are assumed to have no effect on the qu states, E^u is simply a sum of the individual molecular energies efu, and Qqu = < C.21) where qqu = Xt exp(— Csfu) is the corresponding molecular partition,func- partition,function. The quantal partition function, and hence contributions to physical
3.1 DISTRIBUTION FUNCTIONS IN THE CANONICAL ENSEMBLE 149 properties from the qu coordinates, are independent of density, and are the same for a liquid as for an ideal gas.1 We shall not consider these contributions further, but shall turn our attention to distribution functions for the cl coordinates. In classical statistical mechanics the probability distribution for the discrete states, P», is replaced by a continuous probability density P(tNpN<oNp™) for the classical states in phase space (positions, orienta- orientations, and the corresponding momenta of the molecules). We shall not enter into the rigorous arguments by which these two probability distribu- distributions are related. The derivations of the classical canonical distribution function from the quantum Wigner distribution function, and of the classical configurational distribution function from the quantum Slater sum, are given, for example, in refs. 2, 18, and 19. The closely related Wigner-Kirkwood expansion for the partition function is discussed in Appendix 3D. The probability density function is defined so that P(r"pNG>Np?) dr" dpN d'a>N dp^ is the probability of finding molecule 1 in the element (dti dpx d'co! dp^) about the point (riPxtoxp^), molecule 2 in (dr2 dp2 d'co2 dp^) about (r2P2w2P<o2)> • • • > and molecule N in (drN dpN d'wN dp^J about O^Pnovr^); more briefly, P(rNpNa)Np™) is the probability density of finding the configuration (r^p^co^p"). Here pN = PxP2 . . . pN are the momenta conjugate to the positions r" = r^ .. . rN of the molecular centres, o>N = a>1oJ .. . coN are the Euler angles (defined precisely in Appendix A; see also Figs 2.5, 2.6) giving the molecular orientations (a^ = (p^Xi for non-linear, or co; = 6^ for linear molecules, respectively), and p™=pOilp0>2... pO)N are the momenta conju- conjugate to o)N. We use d'co = d0 d<f> dx, as opposed to the element sin 6 d6 d</> d^, for which the symbol do> is used. The probability density is given byf C.22) where Z is the phase integral, Z = fdr^ dp" d'co" dp^ e-e^^'TO C.23) and H is the Hamiltonian function (the same function of the dynamical variables as is the operator $?cl), H = Kt + KT+<sU(rN(oN). C.24) ~We could have introduced a classical rigid molecule model at the start, and therefore begun with C.22); we chose to start with C.1) in order to show how one handles the quantized internal and external motions of the molecules, and how one extends the theory to non-rigid molecules.
150 BASIC STATISTICAL MECHANICS 3.1 Here %(rNwN) is the intermolecular potential energy, and the transla- tional and rotational kinetic energies are given by f/2m, C.25) KT=t I JL/24 C.26) t ? = 1 a=x,y,z where Jict is the component of the angular momentum referred to the body-fixed principal axis a for molecule i, Ia is the aa component of the moment of inertia tensor at the vibrational equilibrium configuration of the molecule (to a good approximation20'21), and m is the molecular mass. The principal axes here are the ones that diagonalize the moment of inertia tensor (the method of finding these axes is given, for example, in refs. 21-4). For molecules with symmetry these axes coincide with the multipolar and polarizability principal axes (see § 2.4.3 and Appendix C). Equation C.26) expresses Kr in terms of the J; rather than the indepen- independent canonical variables (co, pw). To obtain KT, and hence H, in terms of the p^s one uses the J — p^ relations; for a single molecule these are23'24 (see Appendix 3A): Jx = ~P<t> cosec 0 cos x + Pe sin x + Px cot 0 cos x, Jy = p^ cosec 0 sin x + Pe cos x~Px cot 6 sin x> h = Px- C-27) The distribution function C.22) is normalized to unity, f I Qr dp Q CO Clp^ F(Y p CO p^j = 1. {j.Zo) It should be noted that in writing C.24)-C.26) we have chosen the molecular centre to be the centre of mass of the molecule. The molecular symmetry Is often characterized in terms of the principal moments of inertia. Thus linear molecules (N2, CO2, etc.) have Ix= Iy= I, where the z-axis coincides with the molecular axis (for this case Iz is irrelevant since J2J2IZ =0, as J is perpendicular to z). Spherical top molecules (CH4, CF4, SF6, etc.) have all three moments of inertia equal, Ix =Iy=Iz=I. Sym- Symmetric top molecules (CH3C1, CHC13, NH3, C6H6, etc.) have two moments of inertia equal, Ix = Iyi= Iz. If all three moments of inertia are different (H2O, C2H4, etc.) the molecule is called an asymmetric top. The classical expression for the partition function Qd can be obtained as the classical limit of C.19), and is17 (see Appendix 3D) lZ C.29)
3.1 DISTRIBUTION FUNCTIONS IN THE CANONICAL ENSEMBLE 151 where h is Planck's constant and / is the number of classical degrees of freedom per molecule; it is 5 for linear, and 6 for non-linear, molecules. The factor (iV!) arises because the molecules are indistinguishable, whereas the integrations in C.23) treat the molecules as distinguishable; thus, there will be N\ possible permutations of the N molecules among the N sets of phase coordinates, and each of these permutations will be counted towards Z. The factor (hNf)~l in C.29) corrects for the fact that the phase coordinates cannot in reality be precisely defined because of the uncertainty principle (see also Appendix 3D). We note that Qcl is dimensionless (as is the quantum expression CD.1)). Factorization of the probability density For many purposes only the configurational probability density P(rNa)N) is required. Unfortunately, the probability density C.22) cannot be im- immediately factored into kinetic and configurational parts, i.e. because the Jts in C.26) depend on both the cos and p^s, as shown in C.27). Two routes are available for obtaining P(rN(oN) from (a) Integrate C.22) over the ps and p^s. Two variations on this approach are possible: (i) write Kx in terms of the pw using C.26) and C.27), and then integrate C.22) over the ps and p^s, or (ii) transform the integration variables from pw to J, and then integrate over the Js. Methods (i) and (ii) have been used by Mayer and Mayer,23 and by Van Vleck,25 respectively, to evaluate the partition function for a polyatomic ideal gas. Method (i) offers the advantage of working with the canonical variables pM, although the integrations are a little more straightforward in (ii). For the ideal gas °U = 0 in C.24), and P(rNpNa)Np^) and Z factorize in a straightforward way; for example, P(i*pNc*NpZ) = P(rN)P(pN)P(coNp^). For real fluids °U i= 0 and such a simple factorization is not possible. (b) A new distribution function P'(rNpNa)N3N) can be introduced, involving the JN in place of the p%. Although P' cannot be written down straightaway as a canonical distribution function, since the JN are not independent canonical variables (for example, there are Poisson bracket relations among them26), it is simply derived from the canonical function P(***pN<oNpS). This approach has the advantage that P'(rNpNwNJN) read- readily factorizes into kinetic and configurational parts; it also provides a more direct route to some results, e.g. the derivation of the mean rotational kinetic energy (see C.73)). In this section we adopt method (b). Before relating the distribution functions P(rNpNa)Np^) and P'(rNpNtoNJN), we first consider the simpler case of a distribution function P(x) for a single variable x. If a new
152 BASIC STATISTICAL MECHANICS 3.1 variable y is introduced, which is a function of x, y = y(x), then the distribution functions P(x) and P(y) are related by P(y)dy=P(x)dx. C.30) C.31) Equation C.31) states the obvious physical fact that the probability of finding a value of x in the range x tox + dx equals the probability that y lies in the corresponding range y to y+dy; here dy is found from C.30). Thus C.31) can be written P(y) = P(x) dy C.31a) The magnitude sign in C.31a) is necessary because the probabilities must be positive. This single variable result can be generalized to the many variable case.27 Suppose that we wish to transform a set of variables (xxx2 ... Xn) to another set (yxy2 ... yn), where each of the yt is a continuous, single-valued function of all of the xs, and vice versa. We further suppose that all of the ys are independent of each other, and that there are no singularities in the range of values considered. The single- variable relation dx = (dx/dy) dy is replaced by the following relation between corresponding volume elements dx1... dx^ and dyx... dyn: dxx... dx* = |j?(n)| dyx... dyn where ^(n) is the Jacobian of the transformation, given by the determinant <r(n,_a(x1x2...xn)^d(xn) ...yj 9(yn) dx7 dy dx1 dx2 dy2 dy2 C.32) dx1 dx2 dyn dyn and is assumed to be non-vanishing. Using P(yi---yJdy1...dyn=P(x1... we see that the generalization of C.31a) to the multiple variable case is tn\ 'C.33)
3.1 DISTRIBUTION FUNCTIONS IN THE CANONICAL ENSEMBLE 153 The absolute value of the determinant is taken to ensure that the transformed volume is always positive. Applying these results to relate P(i?pN<oNpS) and P'(rNpNcoNJN) we find dpN d'<oN dJN = N d'coN or where C.34) C.35) C36) is the Jacobian of the transformation JN —> p^. From C.27) Jf involves ordy Po>(, and not the p^. for /# i. It follows that the determinant in C.32) consists only of the single diagonal product, so that ^(N) factorises into a product of single molecule terms C.37) where $i^$ is given by sin 0 sin cos 9 Solving C.27) for p^, pe and px gives CA.2); using this in C.38) gives -sin 0 cos x sin x 0 cos^ 0 C.39) so that |^|= sin 0. Thus, from C.35)-C.39) we have P'(rNpNcoNJN) = P(rV^NP^)sin 61 sin 62 .. . sin 0N. C.40) From C.28) and C.34) we see that P' is normalized according to I dJ N C.41) It is convenient to introduce a distribution function P(rNpNcoNJN), where P(rNpNcoNJN) = P'(rNpNcoNJN)(sin 0X sin 02 . .. sin O^1 C.42) which is normalized with respect to the dcos (doj =d<j> d6 sin 0 dx) rather than the d'tos (d'co = d</> dO dx), i.e. J dr" dco N = 1. C.43)
154 BASIC STATISTICAL MECHANICS 3.1 From C.22), C.40), and C.42) we have explicitly C.44) It follows from C.43) and C.44) that Z, given by C.23), can also be written in terms of the new variables, Z = I dr^ dpN dcoN dJN e-PH^^»NIN) C.45) where H(rNpNcoNJN) is given by C.24)-C.26). The physical interpreta- interpretation of P(rNpNcoNJN) is that P(tNpN<oNJN) dr^ dpN d<oN dJN is the proba- probability of finding molecule 1 in the element (dr: dpx do^ dJx) about r1p1co1J1, molecule 2 in (dr2 dp2 dco2 dj2) about r2p2oJJ2, etc. It is a geometrically appealing distribution function, since it involves the 'solid angle' element dco = sin 6 dO d<f> dx, rather than the phase element d'oi = d6d<]> dx- From the decomposition C.24) of the Hamiltonian into translational, rotational, and configurational parts, we can immediately factorize the distribution function C.44) and the phase integral C.45): P(rV^NJN) = P(pN)P(JN)P(rNcoN), C.46) C.47) where P(pN) = e ^/2Zt, C.48). C.49) C.50) and C.51) C.52) Zc = | dr" dcoN e-3%(rN"N). C.53) Each of the Ps in C.46) is a normalized probability distribution function of its arguments. We see that the momenta pN and angular momenta JN are uncorrelated with the positions and orientations rNcoN; the r^ and coN are themselves correlated, however. We can further factorize P(pN) and P(JN) into products of single- molecule normalized distributions functions, P(pN) = P(Pl)P(p2). .. P(pN), C.54) P(JN) = P(JX)P(J2)... P(JN) C.55)
3.1 DISTRIBUTION FUNCTIONS IN THE CANONICAL ENSEMBLE 155 where P(p) = ep2/2m/Ztl, C.56) x \o) — C a oi <xj?^^y \D*D f) and Ztl=[dpe-3p2/2m, C.58) rl — i uj c » a. yj.jy) Hence the ps and Js of different molecules are uncorrelated. A final factorization is possible into products corresponding to compo- components, P(p) = P(Px)P(Py)P(pz), C.60) P(l) = P(Jx)P(Jy)P(Jz) C.61) where P(Px) = ep*/2m/Ztlx, C.62) P(Jx) = e-^/2VZrlx C.63) and + Ztlx=f dpxe-^/^ C.64) Zrlx = f +~dJx eW. C.65) Thus the components of p and J are uncorrelated. P(pxl), for example, is the probability density for molecule 1 to have X-component of momen- momentum pxl, irrespective of other molecules, etc. It should be noted that the components of p in C.60) are space-fixed, whereas those of J in C.61) are body-fixed. We have assumed non-linear molecules in C.61). For linear molecules (N2, HC1, CO2, etc.) the rotational Hamiltonian is J2/2I, and J is perpen- perpendicular to the molecular symmetry axis z. Thus Jz is no longer a degree of freedom, and C.61) is replaced by P(J) = P(Jx)P(Jy) C.66) where P(JX) and P(Jy) are given by C.63), with Ix=Iy=I. Similarly Zrl becomes Zri = ZrlxZrly C.67) where Zrlx is given by C.65). The explicit values of Ztlx and Zrlx are needed in deriving the
156 BASIC STATISTICAL MECHANICS 3.1 expressions for the partition function (see below), and are easily found using the standard integral )¦+°° / \i dxe""^ -J. C.68) W We get C.69) C.70) \ p / Average molecular kinetic energy As a simple example of the use of C.62) and C.63), we calculate the single molecule average energies <px/2m> and <J2/2JX>. Thus, using C.62) and C.64), we have r +°° 2 > = dPx^(Px)—~ J-oo 2 m a Ztlx. C.71) From C.69) we have Znx~B~K so that C.71) gives <px/2m) = |fcT. C.72) Similarly, using C.63), C.65), and C.70) we get C.73) The results C.72) and C.73) agree with the equipartition theorem,2*5 and are equally valid for liquids or perfect gases. This is because of the absence of correlation between the momenta and the coordinates in classical statistical mechanics. In quantum statistical mechanics the molecular coordinates and momenta are correlated, due to the uncertainty principle, and the equipartition theorem does not hold. For example, C.72) and C.73) are replaced by (to order h2) ^> C.74) C.75) where <FX> and (t2.) are the classical mean squared x -components of force and torque on a molecule, respectively. The proof of these relations is
3.1 DISTRIBUTION FUNCTIONS IN THE CANONICAL ENSEMBLE 157 given in Appendix 3D. A qualitative interpretation of the ft2 terms (in terms of zero-point energy) is given in Chapter 1. Similarly, in quantum statistical mechanics the pN and Is coordinates are correlated, so that one no longer has, for example, (Kt(U) = (KI)(aU), but (Kt°U) = {KI)(°U) + O(h2). C.76) The term of order h2 can be derived using the arguments of Appendix 3D, by writing K^=\{KtGUJrGllK^\{Ktall-GUK^. The first term gener- generates the classical limit (i.e. the first term in C.76)), and the commutator term generates O(h2) corrections using the methods of Appendix 3D. Factorization of the partition function. From the definition C.29) of the classical partition function Qcl, and the factorization of Z described in C.46)-C.70), we immediately obtain a factorization of Qcl into translational, rotational, and configurational parts: Qcl=QtQrQc C.77) where Qt = 7^7 dpN e-p ^ ^l2m = AtN C.78) h J with At=(z f, C.79) C.80) with 2/ prl \j , (non-lmear) 8tt2I and 1 (linear), C.81) C- In the above equations we have Cl= idco = 87r2(non-linear) = 4tt (linear).
158 BASIC STATISTICAL MECHANICS 3.1 The quantity At appearing in C.78) is the thermal de Broglie wavelength of the molecules (see §1.2.2), and in C.80) A7x = qr is the rotational partition function for a single molecule (see also CD.57) and CD.63); the notation qf is used there). In C.82) Qc is the configurational partition function. For spherical molecules it reduces to the usual expression, C.84) Moreover, C.80) and C.81) for Qr are the usual ideal gas expressions, and are the classical limit of the usual quantum partition function (see CD.40), CD.57), CD.63)). The configurational partition function is the only part of Q that depends on the intermolecular forces (and hence the only part that depends on the volume). In the ideal gas case (°U = 0, or °ll negligible due to infinite dilution) we have Qc= VN/Nl, and from C.14) the equation of state is p=NkT/V=pkT C.85) where p = N/V is the fluid number density. This expression is known to be true experimentally, and can be derived independently from the virial theorem (see Appendix E) or kinetic theory. The above derivation of the ideal gas equation of state therefore proves (classically) that the constant C in C.6) must be independent of the volume in the ideal gas limit (a more general proof of the volume independence of C is given in Appen- Appendix E). In writing C.24)-C.26) we chose the molecular centre to be the centre of mass, and the body-fixed axes to be the principal axes; the variables (r^, ojn) that appear in C.82) for Qc therefore correspond to this choice. However, the integral in Qc is clearly unchanged if some other choice of molecular centre or of body-fixed axes is used, since one integrates over all possible values of the r^ and coN variables. (A rigorous proof is possible28 by evaluating the Jacobian for an arbitrary change of integra- integration variables from rV to t'n(o'n.) Throughout this book we shall be primarily concerned with the relation between intermolecular forces and macroscopic equilibrium properties. It is therefore the configurational probability density C.50) and configura- configurational partition function C.82) that will be of central importance. The symmetry number In the above classical treatment we have neglected any possible sym- symmetry of the molecular state with respect to interchange of identical nuclei in the molecule. A full understanding of this point requires a quantum mechanical treatment, including a discussion of nuclear spin
3.1 DISTRIBUTION FUNCTIONS IN THE CANONICAL ENSEMBLE 159 )c - c H H (a) <b) Fig. 3.1. The rotational symmetry number a: (a) for nitrogen a = 2, since an indistinguish- indistinguishable orientation to that shown is obtained by a rotation of tt about an axis through the centre c and perpendicular to the molecular axis; (b) for ethylene cr = 4, since two equivalent orientations are available through a rotation of 77 of the molecule about z, and a further two by rotation of it about y. weights.29 However, a simple classical interpretation is as follows. For symmetrical molecules there will, in general, be cr indistinguishable molecular configurations obtained simply by rotating the molecule about its centre; cr is called the symmetry number. For a homonuclear diatomic molecule such as N2 we have a = 2, which is the number of congruent configurations differing by a rotation (Fig. 3.1). Similarly cr = 2 for H2O, 3 for NH3, 4 for ethylene, and 12 for CH4 (where there is a threefold symmetry about each of the four C-H bonds). In carrying out the integrations over the orientations in C.23) ancWater equations such identical molecular configurations have been treated as distinguishable, and each has been counted separately in the integration. In the classical treatment this can be corrected by inserting a factor cr in front of the integration over the orientations for each molecule. When the partition function is factorized, as in C.77), it is conventional to include these cr factors in the rotational partition function, so that in general C.80) is replaced by The introduction of cr"N here is similar to the introduction of (N!) in C.29) to account for the indistinguishability of identical molecules. The presence of the cr term has no effect on equations C.46)-C.53) for the probability densities. 3.1.3 Specific distribution functions30'31 The configurational probability density P(rNcoN) given by C.50) gives the probability that a specific molecule A) is at (ii<oi), another specific molecule B) is at (r2co2), etc. and is therefore termed a specific probability density. Properties can usually be expressed in terms of lower order or reduced (-, 3, or 4 molecule) specific distribution functions. Thus P(rhcoh) is the
160 BASIC STATISTICAL MECHANICS 3.1 probability density for finding the configuration (rhcoh), irrespective of the positions and orientations of the other (N—h) molecules. It is given by P(r\oh) = f drN"h C.87) where drN~h dcoN~h = drh+1 drh+2 ... drN d<oh+1... dcoN. From C.53) and C.87) it follows that P(rhcoh) satisfies the normalization condition I drhda>hP(rha)h) = l C.88) where drh da>h = drx.. . drh dcox... dcoh. If the molecules are uncorrelated, as in an ideal gas, then P(rhcoh) is a product of one-particle distribution functions P(r1co1)P(r2co2) • • • P(tha>h). It is convenient to define a specific correlation function of order h, gs(rhcoh), by P(r\oh) = P(rlWl)P(r2co2). . . P(rha>h)gs(rhooh). C.89) For ideal gases gs(rhcoh) = 1; its departure from unity in other cases takes account of correlations between the molecules. For the special case of an isotropic and homogeneous fluid the probability density P(r1co1) is inde- independent of both r1 and wx. From C.88) we obtain C.90) 3.1.4 Generic distribution functions30'31 For most applications (particularly those involving identical molecules) it is convenient to define a distribution function /(r^o^) so that /(rNcoN) dr^ dcoN is the probability that some molecule (not necessarily molecule 1) is in the element (di^ do^) about (i^cox), while at the same time some other molecule is in (dr2dco2) about (r2w2), and so on. Since there are N\ possible permutations of the molecules among a given set of coordinates, we have f(TNa>N) = N\ P(rNa)N) = N\ e-^^VZc C.91) so that f = N\. C.92) The function /(r^co^) is sometimes referred to as the generic distribution function. The reduced generic distribution function f(tho)h) is proportional to the probability density for observing any set of h molecules in the configura- configuration (rhwh), regardless of the positions or orientations of the others, and is
3.1 DISTRIBUTION FUNCTIONS IN THE CANONICAL ENSEMBLE 161 given by /V /(hh) ' /V f ' drN"h -h)l J The factor N\/(N-h)\ = N(N- l)(N-2) . . . (N- h + 1) is the number of ways of choosing h molecules from the N molecules of the system. From C.93) and C.92) we see that /(r^w^) is not normalized to unity, but instead f C-94) If the molecules are uncorrelated (e.g. an ideal gas) then /(rh<oh) is a product of the one-particle distribution functions /(r1co1)/(r2(o2)... /(rhcoh). It is therefore convenient to define a (generic) correlation function of order h (sometimes called the angular correlation function), g(r\oh) by /(rV) = /(rlWl)/(r2co2). .. /(rhcoh)g(r\oh). C.95) For ideal gases g(rhcoh) = 1, and its departure from unity in other cases is a measure of the effect of intermolecular forces on the molecular distribu- distribution. It is a simple matter to relate the generic correlation function g(l... h) = g(rha)h) to the specific one, gs(l . .. h), defined by C.89). From C.87), C.89), C.93), and C.95) we have (N-h)\ PA)PB) • • • PW&A • • • fc) =7(l)/B) • • • /(H)g(l • • • h). C.96) From C.87) and C.93) we see that /(I) = NPA), and using this in C.96) gives 1 N\ '" JNh(N-h)\ = gs(l... k)[1 + o(^)]' (h small). C.97) For many purposes where h is small the O(l/N) term in C.97) can be neglected. It is sometimes useful to introduce a function y(rhcoh), which we shall
162 BASIC STATISTICAL MECHANICS 3.1 call the "y-function", denned by C.98) In particular, we have ^^ C.99) where u(ro>1co2) is the pair potential. We note that exp(— fiu) is the pair correlation function for an isolated pair of molecules (see § 3.6), i.e., it is the direct correlation between 1 and 2. Thus y(ra>1(o2) can be thought of as expressing the indirect correlation between 1 and 2, due to effects of molecules 3, 4, etc. This function is especially useful in the theory of hard core molecules, since for given orientations it is a continuous function of r (even though u and g themselves are discontinuous). The function y(rw1co2) can also be interpreted32 as the pair correlation function for two 'cavities' (i.e. two molecules which interact with all the other molecules of the fluid, but not with each other), and as such lends itself to computer simulation,33 even for r values corresponding to the core region (r:?cr) of the potential. The region r^cr is unphysical, in the sense that the molecules never 'see' this region. However, the y values for r<cr are of interest for two reasons. Firstly, some perturbation theories, based on hard core potentials, require y(r<cr) (see ref. 34, and Chapter 4), and secondly, y(r = 0) and (dy/dr)r=0 can be related to thermodynamic properties for hard core potentials.32 Distribution functions for molecular centres, irrespective of orienta- orientations, are obtained by integrating the corresponding angular distribution function over orientations. These functions are denned by C.100) C.101) where an unweighted average over the orientations coi<o2 .. . coh is rep- represented by <¦ • .>«•. = <. • -U...^ jdcoh(.. .) C.102) with fl given by C.83). The centres pair correlation function, g(r), where C.103) is of particular importance, and for isotropic, homogeneous fluids it is function
3.1 DISTRIBUTION FUNCTIONS IN THE CANONICAL ENSEMBLE 163 Isotropic, homogeneous fluid. We first consider the special case of an isotropic and homogeneous fluid (Fig. 3.2); the probability density that a molecule has coordinates (i^cox) is then independent of the values of both t1 and a>l. From C.94), therefore, we have e r drx dcoi/^xcox) = /(r^) drx d(ox = Vft/Cr^x) = N C.104) or (a) (b) z=o (c) Fig. 3.2. The singlet and pair distribution functions for fluids that are (a) homogeneous and isotropic, (b) homogeneous but anisotropic, due to the presence of a uniform electric field E, and (c) both inhomogeneous and anisotropic, due to the presence of a plane surface at Z = 0. For simplicity the molecules are assumed to be linear. In case (a) the singlet distribution function /(l^/O^aii) is just p, the bulk number density (independent of both rt and w,); in case (b) /(r^) depends on u>1 (through 6l only) but not r1; while in case (c) /(r^) depends on both rl (through zx only) and o>1 (through 0t only). Similarly, the pair distribution function, /A2) = /(r1r2W1&J)> involves fewer variables in case (a) than in (b) or (c). Angles 0;, etc. are referred to Z as polar axis, while d\ and <?• are referred to Z'; $'= \<t>[-<t>'^.
164 BASIC STATISTICAL MECHANICS 3.1 where Q is Air for linear and Stt2 for non-linear molecules, respectively. For such isotropic, homogeneous fluids C.95) becomes /(rV) = i?g(rV). C.105) For isotropic homogeneous fluids the distribution functions /(rhcoh) are unchanged under rotation and translation of the set of h molecules, provided that the relative coordinates for the set remain fixed. We term these properties of the f(rhojh) rotational and translational invariance. Thus the two-particle distribution function /(r1oIr2w2) depends only on the separation and orientation coordinates r and co relative to rx and cox, and can be denoted by fitco^co^ or f(no). The angular pair correlation function g(ToI(xJ) plays a central role in the pairwise additivity theory of homogeneous fluids, i.e. where is a good approximation (see Chapter 1). We shall frequently be in- interested in some observable property (B) which is (experimentally) a time average of a function of the phase variables, B(rN(oN); the latter is often a sum of pair terms, B(rV) = I fed^ci,-) C.106) so that (B) is given by C.107) i.e. in terms of the pair correlation function gA2) = g(rco1co2). Examples of such properties are the configurational contributions to energy, pres- pressure, mean squared torque, and the mean squared force (see Chapters 6 and II).37 In C.107) (.. .L^ means an unweighted average over orienta- orientations (see C.102)). The last form of this equation is obtained by using C.93) and C.105). Here g(rcoxw2) is given by C.93) and C.105) as f dr3 . . . drN dco3 . . N ^c C.108a) = ^I <S(r;-ri) 6(r;-r2) S^-coJ S(a>;- <o2)> C.108b) where ri2 = r2~*i> and is normalized to unity (neglecting terms of order
3.1 DISTRIBUTION FUNCTIONS IN THE CANONICAL ENSEMBLE 165 3 4 to — 3 6 - 4 - 2 - - - \ll V / v '? 5 T i =0.3 *=0.85 *=1.29 -4- Fig. 3.3. Angular pair correlation function and intermolecular potential for two models of the anisotropic overlap interaction for homonuclear diatomic molecules: (a) diatomic Lennard-Jones potential, B.5); (b) isotropic Lennard-Jones plus first contributing spheri- spherical harmonics B02 + 022 terms), eq. B.4). Here I* = 11 a, p* = pcr3, pj = pa3, T*==kT/e and 8 are dimensionless bond length, density, temperature and overlap parameter, respectively. In (a) cr and e are the site-site parameters, while ae is the diameter of a sphere having the same volume as the hard diatomic molecule with each site of diameter a; in (b) cr and e are the spherical Lennard-Jones parameters. The three orientations are: || = 'parallel', @'x = 6'2 = \-a, <f>' = 0), |— = 'tee' (Q\ = \ir, 6'2 = <f>' = 0), and — = 'end-to-end' (d[ = 6'2 = <p' = 0). (From refs. 39, 40). N x) at large r for all angles; the brackets (. ..) in C.108b) denote ensemble averaging over the r- and <y •. The 8 function form is usually used in the theory of radiation scattering;38 the function 8(a)' — co) is denned in (A.32) and (A.85). Molecular dynamics calculations of g(joIoJ) for homonuclear diatomic molecules for several model potentials are shown in Figs 3.3 and 3.4.39>40 Two models for the anisotropic overlap (shape) forces (see Chapter 2) are compared in Fig. 3.3. The first, shown in 3.3(a), is the atom-atom diatomic Lennard-Jones model39 (see B.5) and B.6)). The second, shown in 3.3(b), is a generalized Stockmayer or Pople model, i.e. Lennard-Jones potential together with the first two spherical harmonic terms40 (see B.4)). For both models at small separations the parallel orientation is the most probable one, while the end-to-end orientation is least likely. However, there are marked differences between the two models. In the atom-atom model there is a strong differentiation of orientations accord- according to the distances they are most likely to occur at. In the Stockmayer model the distance of most probable occurrence is not much different for
3.1 DISTRIBUTION FUNCTIONS IN THE CANONICAL ENSEMBLE 167 1.3 r 0.5 (a) 2= =1.666 Hard sphere 2.5 (b) Fig. 3.5. The centres correlation function for a fluid interacting with a potential uHS + uQQ consisting of hard spheres plus point quadrupole-quadrupole interaction as determined by MCsimulation: (a) short-range behaviour; (b) long-range behaviour. Here Q2 = Q2/kTcr5. Only a few representative points are shown on the graph. (From ref. 43.)
/(rh+1coh+!) when the molecular groups A, 2 . .. h) and (h + 1, h + 2 . .. I) are widely separated is given for a homogeneous fluid by44'45 j3N\ap/3V dfjtW) p j3N\ap/3Vp dp )\p dp + O(N~2) C.109) where p = N/V is the bulk number density, and p is the pressure. Equation C.109) is useful in evaluating integrals of the type /¦ where B is some function of the rs and cos. Such integrals arise in expressions for the thermodynamic derivatives of the distribution func- functions, and in perturbation theory (Chapter 4). When integrated over the rs and cos, the term O(N-1) in C.109) makes a non-zero contribution. As an illustration of the use of C.109) we take the case of h = I = 1, and consider the integral I dt1 dr2 dcox dco2[f(r1r2ojloJ) - fir^^fi^^)]. C.110) Here / is the canonical distribution function defined by C.93), and from C.94) it follows that the integral C.110) is equal to —N. We now replace this canonical / for the finite N-molecule system by a function / which is identical to / at short separation distances, but which satisfies f(ih+lcoh+l)-+f(rha)h)f(iW) C.111) for large separation of the groups h and I. We then write fdl d2tfA2)-/(l)/B)]= f dl d2[/A2)-/(l)/B)] f •>r1 f •>r12>R where 1 means (i^cox), etc. If R is chosen so that for r12> f R [fA2)-/(l)/B)] is zero, but [fA2)-/(l)/B)] is given by C.109), we have Jdl d2[/A2)-/(l)/B)] = Jdl d2[/A2)-/(l)/B)] (^) f(MWM?)\ C.n2) p dp JV dp
168 BASIC STATISTICAL MECHANICS 3.1 /(rh+lo)K+l) when the molecular groups A, 2 . .. h) and (h+ 1, h + 2 . .. I) are widely separated is given for a homogeneous fluid by4445 J_/3p\ / #V)V 3/(r'o>')\ PN\dp)fAP dp )\P dp 1 ) C.109) where p = N/V is the bulk number density, and p is the pressure. Equation C.109) is useful in evaluating integrals of the type Jdrh+'do)h+l[f(r"+'a>h+l)-/(rtla>h)/(ria>1)]B(rh+1a>h+l) where B is some function of the rs and a>s. Such integrals arise in expressions for the thermodynamic derivatives of the distribution func- functions, and in perturbation theory (Chapter 4). When integrated over the rs and ws, the term O(N~') in C.109) makes a non-zero contribution. As an illustration of the use of C.109) we take the case of h = 1 = 1, and consider the integral dr! dr2d<t)i daJ[/(r1r2(»IaJ)-/(r1aI)/(r2w2)]- C.110) Here / is the canonical distribution function defined by C.93), and from C.94) it follows that the integral C.110) is equal to — N. We now replace this canonical / for the finite N-molecule system by a function / which is identical to / at short separation distances, but which satisfies /(r"^^1)-*/^")/^1) C.111) for large separation of the groups h and /. We then write fdld2[fA2)-/(l)/B)]= f J Jr,2 ,2-= f where 1 means (r,a),), etc. If R is chosen so that for r,2> R [/A2)-/(l)/B)] is zero, but [fA2)-f(l)/B)] is given by C.109), we have |dld2[YA2W(l)fB)] = |dl
3.1 DISTRIBUTION FUNCTIONS IN THE CANONICAL ENSEMBLE 169 As explained above, the left-hand side of this equation is equal to -N, and the right-hand side can be simplified by using C.104) for /(r^) and /(!>>;), and C.105) together with C.101) for /(r^ai,^), to give -l] C.113) where g(r) is the centres pair correlation function corresponding to /A2) and x = p~1(dp/dp)f5 is the isothermal compressibility. Equation C.133) is known as the compressibility equation of state. It should be noted that the g(r) appearing on the right-hand side of this equation approaches unity exactly as r—>oo. The grand canonical distribution function / defined in § 3.3 satisfies C.111). Similarly, the grand canonical g(r) defined in § 3.3 is equal to the g(r) that appears in C.113). In later parts of the book we shall usually use the symbol / for distribution functions obeying either C.109) or C.111), and distinguish between them only when integrals such as C.110) arise. Anisotropic, homogeneous fluid For such fluids the probability density that a molecule has coordinates (rjoii) depends on wl, but not on r,. Distribution functions for molecular orientations, irrespective of the location of molecular centres, play a central role in the theory of anisotropic fluids, and have been discussed, for example, by Buckingham.46 Of particular importance is the singlet probability density P(oj), given by P(oI) = -/(r1o>1). C.114) P (For an isotropic, homogeneous fluid P(a>i) = fr' from C.104).) Knowing P(<x>i) one can obtain the quantities4748 F, = <P,(cos ©!)>= I do>iP,(cos OJPM C.115) which are particularly relevant for linear molecules. (See Faber and Luckhurst49 for the generalization of C.115) to the case of non-linear molecules.) Ft is called the 'polarization' and F2 the 'alignment'. The F, are also referred to as the order parameters. Here P( is the hh-order Legendre polynomial, e.g., P,(jc) = jc, P2(x) = jCx2-1) (see Appendix A). The quantity Ft is involved in the dielectric constant, and the Kerr constant involves a linear combination of F, and F2 (see Chapter 10).50"M Anisotropic, inhomogeneous fluid In this case the probability density that a molecule has coordinates depends on both rj and wl. Examples that will be considered in
170 BASIC STATISTICAL MECHANICS 3.1 Chapter 8 are the gas/liquid and fluid/solid interfacial region. If the interface is parallel to the xy plane, then /(z,a>,) will be proportional to the probability density of finding a molecule at Zj with orientation a>,; similarly, the Fx{z) of C.115) will depend on z, and the pair distribution function of interest will be /(z1r12aIaJ). 3.1.5 Total and direct correlation functions The total correlation function h(x(i)i<o2) is defined by h(r12oIw2) = g(r]2w1(o2)-l. C.116) The total correlation function, like g(r,2«,a>2), measures the total effect of a molecule 1 on a molecule 2, at separation r,2 and with orientations o>, and a>2 (in contrast to the direct correlation function, defined below). It differs from g(r12o>,«2) in that it approaches zero (to within a term O(N ') in the canonical ensemble) in the limit ri2^"x>. Its deviation from zero provides a measure of the total effect of molecule 1 on molecule 2. In general, both h and g are of longer range than u (see Figs 3.3 and 3.4). The so-called direct correlation function, c, was first introduced for spherical molecules by Ornstein and Zernike.52 The total correlation between molecules 1 and 2 can be separated into two parts: (a) a direct effect of 1 on 2; this is short-ranged (having roughly the range of u) and is characterized by c, and (b) an indirect effect,t in which 1 influences other molecules 3, 4, etc. which in turn affect 2. The indirect effect is the sum of all contributions from other molecules averaged over their configurations. For non-spherical molecules c is a function of the variables (r]2w1oJ), and is defined for an isotropic, homogeneous fluid by53i = c(r12a>1&>2)+ p dr3<c(r13aIaK)fi(r32aKaJ))aK C.117) which is the generalization of the Ornstein-Zernike (OZ) equation to non-spherical molecules. The first term on the right of this equation gives the direct part, and the second gives the indirect part of h. The physical significance of this expression is most easily seen by eliminating h under t Two measures of indirect correlations in g have now been introduced, (h — c) here and y in S 3.1.4. These are not exactly equal in general, but are approximately related to each other in various theories (e.g. Percus-Yevick and hypernetted chain theories-see introduction to Chapter 5). ± For an inhomogeneous or anisotropic fluid, p in C.117) will depend on r3 or ai3 or both, and cannot be taken outside the integral. Direct correlation functions corresponding to other (e.g. singlet, triplet) correlation functions can also be introduced.54
3.1 DISTRIBUTION FUNCTIONS IN THE CANONICAL ENSEMBLE 171 (a) (b) (c) Fig. 3.6. Direct (a) and indirect (b), (c) effects of molecule 1 on molecule 2. One part of the indirect effect arises through molecule 3 and is given by cA3) cC2) averaged over the position and orientation of 3. A second contribution arises from molecules 3 and 4, and is given by cA3) cC4) cD2) averaged over positions and orientations of 3 and 4. The complete series C.118) is the sum of ail possible such graphs, (a) + (b) + (c) + ... . (O = fixed coordinate, • — integrated coordinate.) the integral using C.117), dr3 + p2 dr3 dr4<c(r13w,0K) + .... C.118) Thus, the second term on the right is the indirect contribution made up of direct effects between 1 and 3, and 3 and 2, and so on (see Fig. 3.6). For a dilute gas (binary collision region) all the indirect effects reduce to zero, and h = c =exp(—Cu)— 1 (see §3.6). Various authors5859 have discussed simplified physical pictures for cA2) in terms of an 'effective potential' between molecules 1 and 2. The name 'direct correlation function' cannot be interpreted too literally, since this would lead to c =/ = exp(—j3a)— 1, which is only true at low density; c is the direct correlation between 1 and 2 in the (average) presence of other molecules (see refs. 60 and 61 for a detailed discus- discussion). In any case the formal definition of cA2) is unambiguous, either in terms of the OZ equation, C.117), or in terms of diagrams, where cA2) is defined62 as (see Chapter 5) the irreducible part of h(l2). For a wide range of fluids the asymptotic behaviour of cA2) has been shown to be63 cA2)->-|3uA2) as ri2^°o C.119) where c —> -j3u means c/(-j3u)—>• 1. In deriving C.119) it is assumed: (a) that at large r,2 the potential uA2) vanishes faster than r\%, but no faster than a power law; thus C.119) does not hold for hard spheres, for example; (b) the state condition is not such that the compressibility approaches infinity, i.e. the fluid is not at or near a critical point. Equation C.119) can be made plausible by considering (a) low densities,
172 BASIC STATISTICAL MECHANICS 3.1 where cA2) = hA2)=/A2) = exp[-j3uA2)]-l, and (b) the Percus- Yevick and hypernetted chain approximations of Chapter 5. Correspond- Corresponding to C.119) one finds63 the asymptotic behaviour of h(\2) to be h(\2)->-(pkTxJ(iu(.\2) as C.120) where x is the isothermal compressibility. Of course if *•—»°° (i.e. the fluid approaches a critical point), the value of rJ2 where C.120) sets in moves to larger and larger values, and approaches infinity. For (rigid) polar fluids, where uA2) varies as r^i at large r12 (see B.159a)), C.120) must be replaced by6465 y /3uA2)/e C.121) where e is the fluid dielectric constant and y = D-7r/9)/3p(x2, with (x the dipole moment of one of the molecules. In Chapter 10 and Appendix 10A we give plausibility arguments for C.119) and C.121) for rigid polar fluids. The asymptotic behaviour of the site-site correlation functions K,iArap) °f § 3.1.6 can also be derived;66 for polar fluids they are O(rap) at large rn(i (see § 10.1). The direct correlation function for molecular centres, c(r), is defined by C.122) 2 - 1 - 0 - h(r) c(r) Fig. 3.7. Comparison of h(r) and c(r) for a typical liquid. (From ref. 67.)
3.1 DISTRIBUTION FUNCTIONS IN THE CANONICAL ENSEMBLE 173 It is to be noted that h(r) = g(r) — 1 and c(r) do not satisfy a separate OZ equation. It will be seen in § 3.2.2 that the spherical harmonic compo- components of /i(ro>,a>2) and c(ra)lct>2), of which h(r) and c(r) represent the isotropic, or !ty = 000, components, satisfy coupled OZ equations. We also note that the compressibility equation, C.113), can be written in terms of the direct correlation function as [l-pjdrc(r)] pkTx=[l-pJdrc(r)J . C.123) This relation is derived in Appendix 3E. In Figure 3.7 c(r) and h(r) are compared for a typical liquid; c(r) has both a shorter range and simpler structure than h(r). 3.1.6 Site-site correlation functions Correlation functions in r-space It is sometimes useful to introduce correlation functions for sites within molecules. These sites may be the nuclei themselves (as in the theory ef neutron diffraction from molecular fluids -see Chapter 9), or may be sites at arbitrary locations within the molecules (see § 2.1.2). The sites within a particular molecule are labelled a, a', a" . . ., and are at positions rc<«? 'ca' • • • from the molecular centre. Since the molecules are rigid, rm, rcc(', etc. are constants, and rca, rcc/, etc. depend only on the molecular orientation a>. The vector between sites a and C for different molecules is given by (Fig. 3.8) ra0 = r12 + rc20 - rcla C.124) where r12 is the vector between molecular centres. The site-site pair correlation function gc,p(ra0) is proportional to the probability density of finding the C-site of some molecule at a distance raB from the a-site of some different molecule. (Some authors, see e.g. Rao,68 and § 5.5.4, define gcpC^) as proportional to the distribution of /3-sites about an a-site, irrespective of whether the C- and a-sites are in the same molecule or not; the distribution functions defined in this way are often Fig. 3.8. Coordinates used to locate molecular sites a,
174 BASIC STATISTICAL MECHANICS 3.1 dominated by the intramolecular structure.) For a single-component fluid Rap(r) is defined for molecules of general shape by3869 gaeOas) = — ( ? S(ria -rcj ~rciQ) S(r,0 -rcj -rc/0)> C.125) P i#/ where rci and ija are the positions of the centre of molecule i and of site a in molecule i, respectively, with respect to an arbitrary coordinate system, and ia(i =im -ria, where i and / can be any pair of molecules, e.g. 1 and 2. The averaging in C.125) is over the rcis and rcjcts, the riO[s being fixed parameters. By writing the sum over i=fij under the averaging in C.125), we ensure that C.125) is also valid in the grand canonical ensemble of § 3.3. Carrying out the ensemble average in C.93) in terms of the angular pair correlation function g(r12w1oJ) gives go,0(''a0)=<g(rc<0+«-c,c,-rc20, a)iaJ)Ll(O2. C.126) The physical interpretation of this equation is that gcptap) is obtained by averaging g(r,2a>,a>2) over the orientations w, and o>2, keeping ra0 fixed. For linear molecules the above simplifies, because rcia determines the orientation a>j; thus rcja = rM(i)j where «; is a unit vector along the symmetry axis. We note that the centres pair correlation function g(r), defined by C.103), is a particular site-site pair correlation function. For a molecule containing s sites there will in general be ^s(s + 1) of the &tp(r<t(i) functions, because ^0a = gap; however, some of these functions may be identical because of symmetry. For example N2 and CO each have two nuclear sites; however, for N2 there is only one distinct site-site function gNN, whereas for CO there are three (gcc, gco, and goo)- It should be noted that g(rwjw2) contains more information than contained in the set of g^tag) functions. Thus ^(r^) can be deter- determined from g(rw!OJ) using C.126) above, but the converse is not true. When the molecules are assumed to interact with a site-site model intermolecular potential, some properties can be expressed in terms of site-site correlation functions (see Chapters 6, 11). For more general potential models only a few properties can be expressed in terms of the g,,(J(r); these include the isothermal compressibility (see C.113)), the structure factor38 (Chapter 9), the dielectric constant70 (Chapter 10), and the Kerr constant71 (Chapter 10), and for mixtures (Chapter 7) the partial molecular volumes and composition derivatives of the chemical potential. Of course, once the compressibility is known from the gcpS, then other thermodynamic properties can be obtained by appropriate ther- modynamic integration and differentiation. However, this may not be convenient; a given theory is usually accurate only over a finite region of the phase diagram. Besides the practical advantages, g(rwjw2) has a
:.l DISTRIBUTION FUNCTIONS IN THE CANONICAL ENSEMBLE 175 ¦onceptual advantage in that the orientational structure of nearest neigh- neighbour molecular pairs, second-neighbour pairs, etc. can be easily vis- lalized. In favour of the g^gS is the fact that they are clearly simpler, leing functions of only one variable, and furthermore are more easily generalized to non-rigid molecular fluids. We have seen that g(rw1w2) = gA2) cannot be obtained rigorously from he ga0(ra0)s; one can, however, derive approximate expressions for gA2) n terms of the g^g, either as an additive superposition (see E.257)), or a nuldplicative superposition72 - the so-called SSA, or site superposition ipproximation. In SSA one writes C.127) vhere g(r) is the centres correlation function. The SSA has been tested ¦.gainst computer simulation results for (a) fused hard sphere site-site potentials,74 (b) LJ diatomic site-site potentials,73 and (c) LJ + juadrupole-quadrupole potentials.75 In (a) SSA appears to give reasona- )le qualitative results for the G2 integrand (see E.245a)), while in (b) easonable results are obtained for the harmonic coefficients of gA2) leeded in calculating the pressure and mean squared torque.76 In (c), on he other hand, SSA fails even qualitatively. It would also appear that JSA is internally inconsistent in that, apart from the centre-centre g(r), he site-site g^s obtained from g(rw!w2) using the definition C.126) do lot agree with those used as input to C.127). In Fig. 3.9 is shown the site-site correlation function for a homonuclear uiatomic AA Lennard-Jones fluid.39 The main first peak at r~a is rla i>-j. j.y. oue-sue pair correlation function for homonuclear diatomic molecules interacting with a diatomic Lennard-Jones potential, from molecular dynamics. Solid curve is for molecules with no quadrupole moment (p* = 0.87, T* = 2.36, /* = 0.547); dashed curve (pJ = 0.91, T* = 2.20, !* = 0.547) shows the effect of adding a quadrupole moment, O* = O/(?iT5)' = 2.0. Reducing parameters defined as in Fig. 3.3. (From ref. 39.)
176 BASIC STATISTICAL MECHANICS 3.1 T T O 1 Flo. 3.10. The T-shaped orientation for diatomic Lennard-Jones molecules, for which there are two site-site distances of r2~l + cr. followed by a shoulder at a distance of approximately l + cr, where I is the intermolecular AA separation. This structure can be explained as follows. When two molecules touch, there is at least one atom-atom distance ~a, and two others between a and l+a. Beyond l+a there is, therefore, a sudden drop in probability, producing a shoulder at that distance. Adding a point quadrupole to the diatomic Lennard-Jones model sharpens the 1.5 1.0 0.5 Fig. 3.11. Site-site pair correlation function for homonuclear diatomic moiecuies miciac- ing with isotropic Lennard-Jones plus quadrupole-quadrupole potential, from molecula dynamics;40 p* = 0.85, T*= 1.294, Q* = 1.0, l* = 0.329.
3.1 DISTRIBUTION FUNCTIONS IN THE CANONICAL ENSEMBLE 177 Fig. 3.12. Possible configuration of a molecule A with two (B and D) nearest neighbours in quadrupole-preferred (T shaped) orientations. Molecule C is a second nearest neigh- neighbour to A. There are two characteristic site-site distances, r, = 0.85cr and r2= 1.16ct. For the cluster shown molecule A will have twelve site-site distances, of which four involve r, ('A.Di = 'A2Di = 'A2i)i = rA2B2) and five are approximately r2 (rAlo2 = rA1B1 = rA1B2 = rA2i«~rA2ci)- 'ne separations r, and r2 correspond roughly to the locations of the shoulder and peak in Fig. 3.11, while the ratio of the number of r2 to r, dis- distances E/4 -- 1.2) roughly equals the ratio of peak to shoulder height in Fig. 3.11. (From ref. 40). first peak, and the shoulder is transformed into a second peak. This effect is expected because of the increased probability of T orientations, for which there are two site-site distances approximately equal to I + a (Fig. 3.10). In Fig. 3.11 is shown gAA(r) for molecules interacting with an isotropic Lennard-Jones plus quadrupole-quadrupole potential. The site- site correlation function is significantly different from that for the diatomic Lennard-Jones case, showing a pronounced shoulder to the left of the main peak. This behaviour may be due to molecular packing of the type shown in Fig. 3.12. One can also introduce a site-site OZ relation and a site-site direct correlation function caB(rafi) which corresponds to ^(r^) (see §5.5). Correlation functions in k-space. For later discussions of the structure factor S(k) (Chapter 9), the angular correlation parameters Gt (Chapter 10), and the site-site Ornstein-Zernike relation (§5.5.2), it is convenient to transform C.125) to k-space. We first rewrite C.125) as - 1MA"! J S(r20 -ra0)> where the ensemble averaging over the molecular positions and orienta- orientations effectively averages over the site positions rlo, and r20; Ta(i is a fixed
178 BASIC STATISTICAL MECHANICS 3.1 parameter. Because of translational invariance of the fluid, the averaging is also effectively over all relative site-site separations r,Q20 = r2p "'k, so that C.125) can be written &.3('«s) = \77 (N(N~ DS(ri«2P " r«B)>- C.125a) P v We now calculate the Fourier transform, gc0(fe), S which, because of C.125a), becomes ^-l) e-'—>. C.129) Since we have (see Fig. 3.8) r]a20 =ri2 + rc23 ~rcln, we obtain from C.129) &*(*) = "T^ (N(N- l)eik-'" e-ik-'- eik'-»>. C.130) We now write out the average in C.130) with the help of the pair correlation function g(r12w1w2) (see C.108)), &,0(fc) = Jdr,2 eik-'.Kg(r12a,1a,2) e-ikjr- e1"--)^ C.131) 0 -'k-'-elk-'-)(Ui^. C.132) If we subtract jdr,2 eik-r« = B-n-KS(k) from both sides of C.131) we can also write C.132) as Kp(k) = <h(ka>1a>2) e-""- e""™)^ C.133) where fia0(r) = g»0(/-)-l and h(rw1aj2)=g(rw1w2)-1. As will be seen in later chapters, the advantage of C.133) over C.126) is that the integral over h(ka>,a>2) is unconstrained, whereas that over g(ro),oJ) is constrained such that ra3 is fixed. 3.1.7 Uniqueness theorem for the pair correlation function77 We now establish a uniqueness theorem for the pair correlation function, i.e. we show that different pair potentials give rise to different pair correlation functions. There is, therefore, a one-to-one correspondence between the pair potential uA2) and the pair correlation function gA2). 'Different pair potential' here means non-trivially different, i.e. differing by more than a constant. Potentials differing only by a constant have the
3.1 DISTRIBUTION FUNCTIONS IN THE CANONICAL ENSEMBLE 179 same associated forces, and give rise to the same pair correlation func- function. We consider the intermolecular potential energy %A(rN«N), %A =%0 + A(%-%0) C.134) which is such that aUK=0 = aU0, aUx = l=aU. We define where ZcA = Jdr" dwN e-3*\ C.136) Differentiating C.135) we get so that clearly — <%-%0), <0. C.138) If we integrate the quantity (d/dk)(aU~aU0)K between A =0 and A = 1 we get \-~(aU,-aU0)K C.139) dA so that from C.138) and C.139) we have (%-%,,)<(%-%o)o C.140) where (...) = (.. .)x = ,. From C.137) we see that the equality in C.140) holds if and only if aU — aU0 = (aU — aU0)^ for all configurations, i.e., only if cU-aU0 is a constant. We consider now the case when °li and °U0 are pairwise additive potentials, with pair potentials u and u(l, respectively. Then C.140) becomes or, 0 C.141)
180 BASIC STATISTICAL MECHANICS 3.2 where /A(ra>,a>2) is the pair distribution function corresponding to uK, "a i = «,/a i=/, Au = u-u(), and A/ =/-/,,. We consider the case where u and u() differ by more than a constant, so that the inequality in C.141) holds. The inequality can only hold if A/^0, i.e. if / differs from /„; Q.E.D. Thus, if only pairwise potentials act, there is a one-to-one correspon- correspondence between the pair potential and the pair correlation function. This is obvious for low density gases, where gA2) = exp[-|3uA2)], but is not obvious for dense fluids. The importance of this result is that if one knows a priori that three-body forces are negligible, then a unique pair potential corresponds to a given experimental pair correlation function. Equation C.141) can be applied in particular to the case where u(roIoJ) is a site-site model potential. For this case the theorem states that a unique set of site-site interactions {uuli} corresponds to a given set of site-site pair correlation functions {&,s}. Left unanswered by the above are the questions: (a) how does one actually find u given /? Closely related to this, how sensitive is / to changes in m? For this one needs some specific theory relating u and / (see Chapters 4, 5). (b) Given some experimental /, and no knowledge about three-body forces, is there any pairwise u which will produce the given /? (We only know that if there is one u, it is unique.) If three-body forces are not negligible, the discussion following C.140) must be generalized: if two- and three-body forces both operate, then the set {/A2),/A23)} must differ in some way from the set {/,,A2), /OA23)}, if u,,A2) differs from uA2) and uoA23) differs from uA23). 3.2 Spherical harmonic expansion of the pair correlation functions It is sometimes convenient to expand the angular correlation functions in spherical harmonics, using a procedure identical to that of § 2.3 for the potential. A given physical property is often related simply to certain spherical harmonics of the angular pair correlation function (see Chapters 6 and 10). Some theories are more easily given in terms of harmonics (see Chapters 4 and 5). The harmonic expansions also offer a convenient method for calculating correlation functions by computer simulation.3''" 3.2.1 The angular pair correlation function For a homogeneous isotropic fluid the function g(rw,oJ) can be expanded (cf. B.20) for the intermolecular pair potential u(koi(o2)) as C.142)
3.2 SPHERICAL HARMONIC EXPANSION 181 where C(/jf2/; mim^m), D'mn, and Y!m are the Clebsch-Gordan coeffi- coefficient, generalized spherical harmonic, and spherical harmonic, respec- respectively (see Chapter 2 and Appendix A). liquation C.142) holds for molecules of general shape with orientations referred to an arbitrary space-fixed reference frame. For linear molecules the coefficients g(/iM; nxn2\ r) vanish unless nl = nl=zQ (see § 2.3), and using B.22) we reduce C.142) to C.143) where r). C.144) The centres correlation function g(r), defined by C.103), is related to g@00; r) by g(r) = Di7)"'g@00;r). C.145) If the orientations are referred to the intermolecular frame, in which the z-axis is parallel to r (i.e. a> = 04>), then it follows from C.143) and B.28) thatt g(ra)',w2)= ? g(/1/2m;r)Y1,m(a.;)Yl2J«2) C.146) where the prime on <x>[ = f)[<p[ signifies the intermolecular frame angles, m = — m, and n i j- 1 \ J. ,l;r). C.147) Equation C.147) relates the spherical harmonic coefficients in the inter- intermolecular and space-fixed frames for linear molecules. This relation can be inverted by multiplying each side of C.147) by C(l,l2l'; mmO) and summing over m (cf. B.33)). After using (A.141) this gives :(lll2l;mtn0)g(lll2m;r). C.148) It is possible to derive general constraints (symmetry properties) for the harmonic expansion coefficients that are identical to those of § 2.3 (see B.24), B.25), B.31), and B.32)). The g(/1Brn;r) coefficients can be •The intermolecular harmonic coefficients g!('m(r) used by Steele78 and by Streett and Tildesley19 are related to ours by 4Trg,rm(r) = g{U'm; r). Their coefficient has the property that g()(H(r) = g(r), where g(r) is the centres correlation function.
182 BASIC STATISTICAL MECHANICS 3.2 calculated from g(r«Ja>2)> e-g- in computer simulation studies,39-79RO by using g(lll2m;r) = 16^2(g(r(o'l(o'2)Y*im(<o'1)YfIJ<o'2)Uu>i, C.149) which can be obtained from C.146) by using the orthogonality properties of the YIms. In computer simulations the intermolecular harmonics g((i/2m; r) have usually been calculated (see, however, ref. 79). However, the expressions for physical properties are particularly simple when space-fixed har- harmonics, g(([/2(; r), are used, often involving only a single harmonic (see ¦2.36 0.87 0.547 (a) (c) r/a Fig. 3.13. Spherical harmonic expansion coefficients of g(ru)[u>'2), referred to the inter- intermolecular frame, for homonuclear diatomic molecules from molecular dynamics; (a) diatomic Lennard-Jones potential, (b) diatomic Lennard-Jones plus point quadrupole, and (c) spherical Lennard—Jones plus point quadrupole. Reducing parameters are defined in captions to Figs. 3.3 and 3.4. (From refs. 39, 40.)
3.2 SPHERICAL HARMONIC EXPANSION 183 Chapters 6 and 10). The same properties require several intermolecular harmonics. Some of the lower order harmonic coefficients for homonuclear diatomic molecules are shown in Fig. 3.13. From C.145) and C.148) it follows that g@00; r) for the intermolecular frame is equal to 4irg(r), where g(r) is the radial distribution function for molecular centres. Also, the harmonic coefficients g(lxl2rn;r) for homonuclear diatomics vanish unless lx and l2 are even. For the diatomic Lennard-Jones potential the range and degree of angular ordering increases as the bond length I* increases. The degree of angular ordering also increases with increasing density. The effect of an added quadrupole is to make the B20) and B21) harmonics more negative. Molecular dynamics calculations of g(r<o[aJ) based on the series of C.146) terminated after various numbers of terms are shown in Fig. 3.14. Convergence is excellent for r*>l + /* (about 1.6 in this case), but is slower for small separation distances. Harmonic coefficients with m j= 0 are found to contribute little, indicating that g(r(o[<o2) is only weakly depen- dependent on the azimuthal angle <p[2. It has been suggested81 that the spherical harmonic expansion for y(roIoJ) may converge more rapidly than that for gCra^o^) (see also ref. 82). In a similar way one can expand other isotropic fluid distribution functions - e.g., the Mayer function, the direct correlation function, or the total correlation function. The expansion coefficients for a general quan- quantity /(ra),aJ) for a general shape molecule in the space-fixed and inter- intermolecular frames, f{l\l2l', n^n^; r) and f(lxl2m; nin2\ r) respectively, will 2.5 Fig. 3.14 The angular pair correlation function for a homonuclear diatomic fluid using the LJ atom-atom plus quadrupole-quadrupole potential, calculated from C.146) using various numbers of terms in the series. Based on molecular dynamics calculations (ref. 39) for the T-shaped orientation, 7* = 2.20, p* = 0.911, (* = 0.5471, Q* = 2.0; these latter symbols are defined in the captions for Figs. 3.3(a) and 3.4(a). (From ref. 39.)
184 BASIC STATISTICAL MECHANICS 3.2 be related by (cf. C.147) and C.148)) /((,Bm; n,n2; r) = ? ( J2C(/1B'; mmO)f(lll2l; «,n2; r), C.150) 7(/!/2/; mmO)f(lJ2m; ntn2; r). C.151) Similarly, for linear molecules the generalization of C.147) and C.148) are (cf. B.30) and B.33)) f(hl2m; r) = X (^^fcd^l; mmOmhhl; r), C.152) C,153) It has been suggested8384 that expansions in other complete sets of orthogonal functions (e.g. ellipsoidal harmonics, Chebyshev poly- polynomials) may improve the convergence and/or accuracy in some cases. 3.2.2 The Ornstein-Zernike equation The Ornstein-Zernike (OZ) equation, C.117), is the starting point for many of the theories of the pair correlation function (Percus-Yevick, hypernetted chain, generalized mean field, etc.) considered in Chapter 5. However, numerical solutions starting from C.117) are complicated by the large number of variables involved (four even for linear molecules). By expanding the direct and total correlation functions that appear in C.117) in spherical harmonics, one obtains a set of algebraically coupled equations relating the harmonic coefficients of h and c. These equations involve only one variable (r or k) in place of the four or more in the original OZ equation. The theories that are described later truncate the infinite set of coupled equations into a finite set, thereby enabling a reasonably simple solution to be carried out. The harmonic expansion of the OZ equation was first derived by Blum and Torruella.8'' We restrict our attention in this section to linear molecules. Full details of the derivation, for both linear and non-linear molecules, are given in Appendix 3B. We first Fourier transform h(rwiOJ) and c{ru>xu>2) to k-space according to /(ra>,a>2)=(— j jdke~ik-7(koIaJ), C.154) /(kw,ftJ) = jdr eik-'f(ia),aJ). C.155)
3.2 SPHERICAL HARMONIC EXPANSION 185 The convolution in r-space on the right-hand side of C.117) becomes a product in k-space, exactly as in the monatomic case: hikiOi^j = c(kwi«2) + p(c(k<o^(o3)h(k(o3<o2))^. C.156) This simplification of the OZ equation is one of the advantages of working in k-space. A second advantage is that some of the physical properties can be more concisely expressed in terms of the k-space harmonics h(l^l2l\ k)\ examples are the isothermal compressibility (Chap- (Chapter 6), the structure factor (Chapter 9), and the angular correlation parameters (Chapter 10). We now expand the k-space quantities fi(ko>io>2) and c(k«,a>2) in space-fixed spherical harmonics in the usual way: =J]/ai/2!;fc) X C(lsl2l-mim2m) x YIim,(a>,)Y^MY&fa) C.157) where a>k is the direction of k. (The k-space harmonic coefficients f(l^l2l; k) are related through a Hankel transform to the r-space coeffi- coefficients; see Appendix 3B.) Substituting C.157) into C.156) and simplify- simplifying (see Appendix 3B) gives the following harmonic component form of the OZ equation, h(lxl2i, k) = c(!,y; k) + Dirpp ? c(/,y; k) < o X ) l\ where ( I and j \ are 3/ and 6/ symbols, respectively (see Appendix A). Equation C.158) has the same structure as the original OZ equation, and couples together the harmonics of h and c. An important exception occurs in the limit k = 0 for short-range potentials. Only the coefficients h^^OjO) and c(/,/,0;0) are then non-vanishing, and these satisfy a decoupled OZ relation (see Appendix 3E). This does not hold for dipolar potentials (see Chapter 10). The OZ equation can also be written in terms of various other harmonics (see refs 85, 122, Appendix 3B, and Chapter 5), e.g., fil-t^m; k), f(lxl2l;r), or f(l^l2m; r). The k-frame spherical harmonics f(lil2m; k) lead to a particularly simple form of the OZ equation. In this frame we take the polar axis along k. The k-frame harmonics are related to the space-fixed harmonics used above by (cf. C.152)) B1 +1\- ——fCiUU; mmO)/(Z,Z2J; k) C.159) 417 /
186 BASIC STATISTICAL MECHANICS 3.2 Using C.159) to transform C.158) from (lj2l; k) to (lvl2m; k) we get (see Appendix 3B for details) h(/, J2m; k) = c(M2m; fc) + Dw)~V(-)™ I c(M3m; fc)h(y2m; k). C.160) This harmonic form of the OZ equation is remarkably simple, and quite useful in practice (see Chapter 5). 3.2.3 Relation between g^r) and g(M2(; r)86 In discussing the fluid structure factor S(fc) (Chapter 9) and the angular correlation parameters Gi (Chapter 10) it is useful to relate the site-site correlation functions gogM of § 3.1.6 and the spherical harmonic expan- expansion coefficients g(lj2l; r) °f the full pair correlation function g(roIoJ)- It is easiest to work in k-space, and begin with C.133). We substitute the spherical harmonic expansion CB.32) for ^(kcojb^) and the Rayleigh expansion (B.89) for exp(—ik.rcla) and exp(i'k.rc20) into C.133), to get ha(i(k) = X h(hl2l; n{n2; k)C(lJ2l; mlm2m)(-y<(.4m'<)Diri'i) x Vllmik)Y,;m;K)Y,m(<ok)* C.161) where the sum is over all the Is, ms, and ns, and where, for brevity, ra = rcic» ar|d wlct =a>cla. To carry out the averages over ©j and a>2 in C.161) we use the spherical harmonic transformation law (A.43) YJi<ola) = I DL(a>i)*n.(a>iJ C.162) M where gj1c( and u>\a denote the orientation of rcla in the space-fixed (XYZ) and body-fixed (xyz) frames respectively; w, denotes the orienta- orientation of xyz with respect to XYZ (i.e. the orientation of molecule 1). Substituting C.162) and the corresponding expression for Ylm(oJ&) into C.161), we carry out the averages over o>, and o>2 using (A.80). We also set <ok =00 in C.161) for convenience and use Y,m@0) = (B( + l)/4ir)iSm0; this is allowed since hafi(k) is independent of the direction of k for an isotropic fluid. We get Equation C.163) is general. For linear molecules C.163) simplifies. First we note that h(lj2l; nyn2; k) = 0 for nl, n2/0, and h(lxl2l; 00; k) is given in terms of h(lxl2l; k) by a relation analogous to C.144). Also, for
3.3 GRAND CANONICAL DISTRIBUTION FUNCTIONS 187 linear molecules, we choose the body-fixed z-axis such that <o'a = F'a4>'a) is either @4>') or (tt4>'), according as the a site is on the positive or negative z axis. Since then Ym(<oa) = (±)l[Bl + l)/4ir)f (according as <o'a is parallel or antiparallel to z), and since /((—x) = (—)'h(x), we can simplify C.163) to ha0(k) = Dir)-i ? i3I.+HBl1 x C{lyl2l; OOO)/,I(±kra)jl2(±fcr0)h(i1I2i; k) C.164) where, in /;(±fcra), the positive or negative sign is chosen if the a-site is on the positive or negative body-fixed z-axis respectively. Because of the transformation law C.159), the relation C.164) can also be expressed in terms of the k-frame harmonic coefficients h(lil2ni; k) of x;I,(±krj7l2(±fcr0)h(;i/2m = 0; k). C.165) The expressions C.163)-C.165) are useful in practice (see Chapters 9 and 10). Equation C.165) also sheds further light on why g(i(Oja>2) contains more information than the set of g^pWs; the harmonics with m/0 do not appear in C.165). 3.3 Distribution functions in the grand canonical ensemble While the canonical ensemble (N, V, T, fixed) offers the advantage of notational simplicity, there are many cases where the grand canonical ensemble ((x, V, T fixed) is more convenient; this is true, for example, in calculating properties which are related to fluctuations in the number density (isothermal compressibility, partial molal volumes in mixtures, etc.). 3.3.1 Distribution law and the grand partition function Consider an open, single-component system of volume V, chemical potential (x, and temperature T. In this section we assume that V is very large, to avoid the necessity of writing the limit V^oo repeatedly. The probability of observing the system at any instant with N molecules and in quantum state k is given by1 PNk=e~B(B"^N)/B C.166) where C = 1/fcT as before, H = H(/3jxV) is the grand partition function, H = Z e^N X e~BEk = I e^NQN, C.167)
188 BASIC STATISTICAL MECHANICS 3.3 and ON is the canonical partition function for a N molecule system, given by C.2). The probability that the system contains N molecules, irrespec- irrespective of the quantum state, is PN = I*W= = e3*NQN. C.168) The PNk and PN are normalized to unity, i.e. Nfc = l, C-169) IPN = 1. C.170) N The thermodynamic functions are directly related to the grand partition function. The average number of molecules (N) in the system is given by X C.171) Nk «- Nk or, using C.167), 1(^=) C.172) Similarly, by differentiating C.167) with respect to j3 at fixed (x and V, and using G = (x(N) = U + pV— TS, we find C.173) The thermodynamic quantity pV is the characteristic function for the variables C, (x, and V. Small changes in these quantities are related by the thermodynamic expression d(PV) = p dV+S dT+<N)d/x which follows from writing d(pV) = p dV+ Vdp, and replacing Vdp using the Gibbs-Duhem equation, Vdp = S dT + (iV)d(x. Thus, we have the thermodynamic identities: C.175) S C.176) C.177) 0V
3.3 GRAND CANONICAL DISTRIBUTION FUNCTIONS 189 From C.172) and C.177) we have, upon integrating over (x, where the constant C" must be independent of chemical potential (x. Since p is intensive, pV must be extensive, so that C, if it exists at all, must be linear in V. Moreover, from C.175) and C.178), we have (this also follows from C.178) and the intensivity of p). By the method used in Appendix E to derive the analogous canonical ensemble result, one can show that the term dC'/dV must vanish in order that C.179) yield the known virial equation of state; i.e. C must be independent of V. It therefore follows that C" must vanish if C.178) and C.179) are to be consistent. It is easy to show that the above results with C — 0 yield the known equation of state for a classical ideal gas. We consider a gas in the limit that V is large, so that the ideal gas equation of state should apply. For such a gas the molecules do not interact, and the canonical partition function QN has the form N QN=^j C.180) where q is the molecular partition function (c?, C.21) and § 3.1.2). With A=exp(/3(x), C.167) gives a= y Uq? = e*.=eP<PV-c-) C1 nTo Nl where C.178) has been used to obtain the last form. From C.172) and C.181) we have „ (^) =Aq = /3(pV-C) 0V so that PV = (N)kT+C C.182) which is the known result provided that C' = 0. With C' = 0 C.178) and C.179) become pV = -lnB C.183) 1/8lnH^ C.184) As noted earlier the pressure p is intensive, i.e. independent of volume.
190 BASIC STATISTICAL MECHANICS 3.3 Thus, from C.183) it follows that (l/V)lnH must be independent of volume; i.e. In 5 is linear in V, so that Hence dlnH/9V in C.184) can be replaced, if one wishes, by lnH/V. An expression for the entropy is obtained from C.176) and C.183) as C.186) The connection between entropy and probability is obtained by noting that we have, from C.166) ? Pm, In P^ = B ? PNfe(/j,N - Ek) - In H ? PNk Nk Nk Nk _lnS. dB From this result and C.186) we have S = -fc I PNk In P^ C.187) NIc which is the analogue of C.13) in the canonical ensemble. 3.3.2 Configurational distribution functions In classical statistical mechanics the probability distribution over states, PNk, is replaced by a probability density for the coordinates in phase space (see § 3.1.2). We shall confine ourselves to distribution functions for the configurational coordinates (rNa>N) only. For an open system it is not useful to define a specific distribution function as in §3.1.3, since molecules enter and leave the system at will. For an open system the generic distribution function f(rhwh) is defined to be proportional to the probability density for finding a set of h molecules in the configuration (rhoih), regardless of the total number of molecules N. It is given by f(rha3h) = Y.PNfN(rh<oh). C.188) N Here fN(rh(oh) is the canonical distribution function for a system of N molecules. We note that fN(rh(oh) vanishes for N<h in C.188). Using C.168) for PN, together with C.18), C.21), C.77)-C.82) for QN and
3.3 GRAND CANONICAL DISTRIBUTION FUNCTIONS 191 C.93) for fN, we get /(rVl) = h I tt^t; fdrN~h da>N~h e-e«^N> C.189) and uf I drN dpN d <oN dp™ < = 1^7 fdrN d<oN e^%('N<uN) C.190) which is the semi-classical limit of C.167). Here z is defined by - C.191) From C.189) and C.190) it is seen that ^) 0.192) where the averaging in C.192) is over N. The angular correlation function g(rhcoh) is defined as in C.95), and for isotropic, homogeneous fluids we have f(r^co1) = p/D,, C.193) h /(rfl<oh) = ^-g(rfl<ofl) C.194) where p = (N)/V. Distribution functions for molecular centres, and the total, direct and site-site correlation functions are defined as for the canonical ensemble (§§3.1.4 to 3.1.6). In the grand canonical ensemble the functions g(rKwh) approach unity exactly in the limit as all of the r,, —> °° (in contrast to the canonical correlation functions, where g —» 1 + O(N~1)).44'45'87 Similarly, when the molecular groups A ... h) and (h + 1 . .. I) are widely separated (cf. C.109)). For the free rotation case, where %(rN) is independent of the orienta- orientations, it should be noted that f(rhwh) defined by C.189) does not reduce simply to the usual f(rh), but to f(rh)/flh, where 1 x - -- . . ,*_„ _„„„..*, C.196)
192 BASIC STATISTICAL MECHANICS 3.4 with f drNe^(W1. C.197) 3.4 Hierarchies of equations for the distribunction functions A number of useful equations can be obtained by considering derivatives V of the distribution functions with respect to the independent variables on which they depend. The resulting expressions involve fluctuations in various observables, so that the equations depend in general on the ensemble. We use the grand canonical ensemble, in which case the variables are /u., j3, V, r,, w,, etc. For each derivative there is a hierarchy of coupled equations, in which each member of the hierarchy relates some distribution function, f{rhu>h) say, to distribution functions of higher order. The equations given in this section are a natural generalization88 of the work of Gibbs,89 Fowler,90 Yvon,91 Buff et al., Landau and Lif- shitz,93 and Schofield.94 The extension to mixtures is given by Buff et al.92 3.4.1 Thermodynamic derivatives for a general property It is convenient to first consider some dynamical variable B which is a function of the coordinates (rNpN&>Np^). The average value of B (in the grand canonical ensemble) is = X f drN dpN d'wN dpl.P(rNpNwNp^)B(rNpN(oNp^) C.198) where d'u> = dd d<fi dx, and P(rNpNa>Np") = g *^f e-fiH<r~p~"Nprf> C.199) and H is given by C.190). Examples of (B) include the average number of molecules, pressure, internal energy, or one of the distribution functions. In most of the cases of interest later B will be a function of the variables (rNtoN) only. In that case C.198) simplifies to (cf. §3.1.2) (B) = ? fdrN d@NP(rNwN)B(rN<oN) C.200) where 'N P(rNwN) = -/s^e-C*(r%"N). C.201) The quantity (B) depends on the thermodynamic variables |3, /x, and V if (B) is extensive, or on j3 and (x if <B> is intensive. Considering the
3.4 HIERARCHIES OF EQUATIONS 193 tx-derivative, we have from C.198) and C.190) ) = » 2.—.— sp" M^e B(r) -TTTT-^r dl e where T = (rNpNojNp™) and dr = dr* dpN d' wN dp?. Using C.198), we get =p[(BN)-(B)(N)]. C.203) The j3-derivative of (B) is similarly derived from C.198) and C.190) as finr) =^[<BN>-<B><N>]-[<BH)-<B><H>]. C.204) When B is a function of the configuration coordinates (rN&>N) only, this simplifies to () C.205) Va/3 /^v The derivation of the expression for the V-derivative of an extensive property (B) from C.198) and C.190) is longer, and is given in Appendix 3C. The resulting equation is /am C.206) where 9 is the pressure dynamical variable,95 given by ^ = fiV~^V' C.207) Equations C.203), C.204), and C.206) are the general results for the thermodynamic derivatives of (B). Several modifications of these equa- equations are of practical interest, and we briefly mention these in the remainder of this section. Experimental measurements are most readily carried out at constant density, pressure, or temperature. The chemical potential derivative in C.203) can be replaced by a density or pressure derivative by using /d(B)\ _ /d(B)\ /dp\ = \d^l /0V \ dp /pvW/sv V dp 2 (d(B)\ (d(B)\ ) pIT) C.208) dp /0V \ dp /nV
194 BASIC STATISTICAL MECHANICS 3.4 where the last two steps of C.208) follow because dix=-sdT + -dp C.209) P with s being the entropy per molecule. Thus, from C.203) and C.208) we have P2x(^) = p(^) =Hl(BN)-(B)(N)] C.210) \ dp /3V \ dp /0V which provides a means of measuring the fluctuation in BN. The derivative (diBy/dfi)^ that appears in eqn C.204) can be con- converted to the experimentally accessible (a(B)/a/3)pV by using =(d(B)\ /a<B>\ (dp \ a/3 )pV \ a/3 )^ V aP Using C.204) for 0<B>/3j3)M.v and (dpldft^v, and C.210) for (a<JB)/apHV, and noting that [(N2)-<N>2] = Vp2xlfi from C.210), we find from C.211) (~) =-?T-[{BN)-{BKN)][{NH)~(N){H)]-[(BH)-(B){H)] V a/3 JpV Vp2x C.212) or \ ap /pV Lvd/3/M. pj\ ap /0V C.213) Differentiating both sides of C.172) with respect to /3 at fixed /x, and using p = (N)/V, we have and using this in C.213), together with C.208), gives (lr) ^(t) (t) \ a/3 /pV pv v ap /0Vv dp /pv which suggests a method for studying the fluctuation in BH experimen- experimentally; here (H) is the observed internal energy of the system. When (B) is an intensive quantity (e.g. one of the distribution func- functions) it can only depend on two intensive properties (e.g. /3 and /x), so that the volume derivative (diBy/dV)^ vanishes. It is also possible to
3.4 HIERARCHIES OF EQUATIONS 195 show that the condition holds for an intensive (B) even when V is very large (V—»°°). (When (B) is extensive this becomes V(d(B)/dV) = (B), since (B) is linear in V.) Applying C.216) to C.206) leads to an 'intensivity condition', O. C.217) This equation can be applied to give a useful relation among the distribu- distribution functions, as shown in the next section. The methods given here are easily extended to higher derivatives such as d2(B)/d,x2,33<B W,... ,a2(Q)/a/32,... ,d2(Q)/d V2,..., d2(Q)/dlx dp,...,. We first introduce a more compact notation by rewriting C.203) and C.204) in terms of correlations of fluctuations: (—) = /3<ABAN>, C.218) (^) =-<ABAH'> C.219) where AB = B—{B) is the fluctuation in B, and H' = H — jjlN. The second order /x and C derivatives can be written in a similar form, = /32<(AB)(ANJ), C.220) C.221) ^-) =<ABAiV>-/3<ABANAH'>. C.222) The relations C.220)-C.222) can be derived by straightforward differen- differentiation of C.203) and C.204), or by iteration of (it) =C-)+^an>- C-223) \ d)X /pv voju,/ (ABAH> C.224) W which generalizes C.218) and C.219) to the case that B can depend on fx and C, in addition to the dependence on phase variables. (Note that AB depends on ju, and j3, even if B does not. Also note that C.220)-C.222)
196 BASIC STATISTICAL MECHANICS 3.4 are given here only for the case of B independent of /x and j3; these are easily generalized if further iterations are required.) These equations can also be extended to mixtures, and are then generalizations of the equations written down by Kirkwood, Buff and others.92-96 Thus, C.203) and C.204) become, for mixtures =t3[(BNj-(B){Na)l C.225) =I*U<BNa>-<B><Na>]-[<BH>-<B><H>] C.226) \ dp while C.206) is unchanged. Here /x represents (ju,aju,b . . . ju,r), where r is the number of components, and ju,' represents all /u.s except [xa. The chemical potential derivative in C.225) can be converted to the experimentally accessible derivative with respect to mole fraction (xa = (Na)l(N)) by using standard thermodynamic transformations (see Chapter 7). If one uses the canonical in place of the grand canonical ensemble the above fluctuation theorems are, of course, modified. Thus, there is no equation corresponding to C.203) for d(B)/dn, while the expressions analogous to C.204) and C.206) are C.227) dj3 The expression for (B) analogous to C.198) does not contain a summa- summation over N, and the canonical probability distribution function given by C.22) is used in place of C.199).97 Some special cases of these fluctuation theorems have been written down by others.89'9498 These cases include, for example, B = H, &, N, AH, AN, etc. The three best-known relations involve fluctuations in number density, energy, and pressure. The first is obtained by putting B = N in C.203) and using C.208) to give P2kTx = ^ <(ANJ> = V<(APJ) C.229) where p=(N)/V and Ap = (N/V)-«N>/V). The second relation is ob- obtained by putting B = H in C.227) and noting that (d(H}/dj3)NV = — kT2Cv, where Cv is the heat capacity at constant volume: kT2Cv = <(AHJ>. C.230)
3.4 HIERARCHIES OF EQUATIONS 197 The third results on putting B = *3> in C.228): After some manipulation," this can be put in the form 0.232) where xs is the adiabatic compressibility, C-233) Equations C.229), C.230), and C.232) relate the fluctuations in density, energy, and pressure to the thermodynamic functions of isothermal com- compressibility, heat capacity at constant volume, and adiabatic compressibility, respectively. Although the above derivations have been classical, an exactly parallel argument can be given for the quantum case; in quantum mechanics B is an operator. The equations are valid irrespective of whether B is a static or time-dependent88 quantity. 3.4.2 Thermodynainic derivatives of the distribution functions In this section we apply the general equations C.210), C.212), and C.217) of the previous section to the special case where (B) is the generic distribution function of order h. We choose B to be given by (cf. C.108b)) Bh= X 8M-r1)8(w'i-a>l)8(ifi-T2)8(a)'i-aJ)...8(T'p-rh)8(w'p-a)h) C.234) where the r'h etc. are the dynamical variables here, i.e. Bh =Bh(r'Nw'N), and the rhu>h are fixed parameters. From C.200) and C.189) we have C.235) Density derivative The quantity (BN) is evaluated using C.200) to give where use has been made of the symmetry of the integrand. Replacing N
198 BASIC STATISTICAL MECHANICS by (N~h) + h we get 3.4 ,=Iv. »J dr"" = drh+1 d«)h+,/(rh+1a>h+1) + ft/(r\oh). C.226) Furthermore, we have (cf. C.94)) <N>= |drdft)/(ra>) C.237N and substituting these two results into C.210), together with C.105, gives C.238 dp 0.12 0.08 0.04 0 -0.04 -0.08 -0.12 0.5 3 - Pressure Temperature 2.0 k/A Fig 3.15. Pressure (dashed line) and temperature (full line) derivatives or me structurt factor for liquid rubidium near its triple point. The lower part of the figure show: S(k) for comparison. (From ref. 100.)
HIERARCHIES OF EQUATIONS 199 vhich holds for an isotropic, homogeneous fluid. This is a hierarchy of oupled equations, the general member of which relates the distribution unction of order h to its density derivative and to the next higher listribution function. If C.238) is integrated over a^a^ . . . a>K, we obtain he compressibility hierarchy in terms of the centres correlation functions C.239) Jor the special case of h = 1 this reduces to the compressibility equation, 3.113). A density hierarchy for the direct correlation functions can also je derived.56 Comparisons of experimental diffraction data (neutron or X-ray) with heoretical estimates of the integral term in C.238) or C.239) provides a :l iG. 3.16. The isothermal pressure derivative of the static structure factor for krypton at 218 K, 60 atm, on the gaseous side of the critical point. The solid curve is drawn through the experimental data,102 the dashed curve represents the approximation103104 g(rr') = l + h(r) + h(r') + h(|r-r'|) = A(rr'), and the dash-dot curve represents the approximation g(rr') = A(n') + h(r)h(r')+ h(r')h(\i-Tr\) + h(\i-r'\)h(r) where h = g- 1 is the total correla- correlation function. Here g(rr') = g@rr') is the triplet correlation function. (From ref. 102.)
200 BASIC STATISTICAL MECHANICS 3.4 sensitive test of liquid theories. Diffraction measurements (see Chapter 9) yield the structure factor, S(k); for monatomic fluids (S(fc)— l)/p is the Fourier transform of the total correlation function, h(r). In Fig. 3.15 is shown the effect of temperature and pressure on the structure factor of a dense monatomic liquid near its triple point. The pressure derivative provides valuable information about the triplet correlation function, i.e. about density-density-density correlations. Both the temperature and pressure derivatives are needed to study the triplet density-density- energy correlations, as seen from C.213). Egelstaff et a/.101102 have used measurements of dS(k)/dp to test theories for the triplet correlation function gtr^r,). In Fig. 3.16 we show a comparison of data for krypton with an expansion of g^,^^) in terms of the pair correlation function. Two terms in this expansion (dash-dot curve in Fig. 3.16) were needed for qualitative agreement, while more than three terms were needed for quantitative agreement. Temperature derivative The fluctuation terms appearing in C.212) are evaluated by methods similar to those above, and are given by di^ dr2d + - drT dr2 dr3 dco, do>2 doj3u and l(BhH) - (Bh) <H>] = [<Bh%> - (Bh ¦I 1 x[/(rh+2o.h+2)-/(rho»h)/(rh+1.h+2ojh + Iojh+2)] where pairwise additivity has been assumed in the last two equations. (We
3.4 HIERARCHIES OF EQUATIONS 201 note that the term involving a sum over u(riJa>ia>,)/(rha>h) appearing on the right-hand side of this equation vanishes if h = 1.) Substituting these expressions into C.212) gives I —g(rh«h)) = p2l dr1 \dp /p U + 2Pj dr12 dr13<u(r12aj1(o2) r\oh) x \ (i- phg(i*o»h)) - fi I u( W,)g( + I pfdrh+1<u(rl.h+Ia>1a>h+1)g(i'l+1a»H+1)>oll^ C.240) Vo/ume derivative {intensivity condition) The fluctuation terms in C.217) are evaluated in Appendix 3C for the case where B is given by C.234). For an isotropic, homogeneous fluid this yields the following intensivity condition for the correlation functions:94 3 12 \#~i Si,- + 5P2 drh+1 drh+2(rh+lh+2 iL+1-h+2 = pf drh+1 J C.241) (The first term on the left of this equation does not occur for h = 1.) This equation is often useful in showing the compatibility of two different expressions for the same quantity (e.g. in thermodynamic perturbation theory). Thus, it can be used to show the compatibility94 of the (pairwise- force) pressure equation, CC.17), and the compressibility equation, C.113). For an isotropic, homogeneous fluid the pressure equation is,
202 BASIC STATISTICAL MECHANICS 3.4 from CC.17) and C.105), P P2 \a C 6 J H 6 J "\ drl2 From this expression and C.238) we have 1 If / du(rl2oi1<o2) d C.242) fdr / J 12V fdr 6J 12V" ar12 flp dr12< r12 f J , 2f, , / du(r12cQ|hJ) ,33., , + p2 dr12 dr3( rl2 [g(r3co3) - g(r12w,w2) However, using the consistency condition C.241) for h = 1, the last term in brackets can be simplified, so that (?) - C-243> which is the compressibility equation. The extension of the above hierarchy of equations to the time- dependent pair corrleation function is given in ref. 88. 3.4.3 Derivatives with respect to position or orientation coordinates A useful hierarchy of coupled integro-differential equations is obtained \ by differentiating the function f(ihojh) with respect to one of the rs, say r^ Thus using C.189) and assuming pairwise additivity, we find dr, or 3jdrh+1 j=2 o'l -/(rfl+1(ofl+1) = 0. C.244)
3.4 HIERARCHIES OF EQUATIONS 203 Assuming a homogeneous, isotropic fluid, and using C.105) and the notation Vl=d/drl gives J=2 + pPJdr(l+1<V1u(r1,,+1«1coh+1)g(r('+laJfl+1))@^1 = 0. C.245) Equations C.244) and C.245) each represent a hierarchy of coupled equations, the BBGKY (Bogoliubov, Born, Green, Kirkwood, Yvon) hierarchy.105 Equation C.244) for h = 1 provides a starting point for the study of the singlet distribution function /(i^co,) in an inhomogeneous system (see Chapter 8), while C.245) for h = 2 relates the pair and triplet angular correlation functions. The simplest method of closing this equa- equation is to assume the superposition approximation, g^^r^o^co,) = g(r12(o1(o2)g(ri3(o,@3)g(r23(o2(o3). C.246) This equation is exact at low densities, but may lead to errors at high densities.106 A hierarchy similar to C.244) can be obtained by applying an angular gradient operator, V^ say, to f(iho)h). This gives 1aJ'l41) = 0. C.247) The operator Vo, used here is related to the angular momentum operator L by (see Appendix A.2) The component (V(u)x generates rotations about x, etc. For the homogeneous, isotropic case C.247) becomes 1=2 h+1(VOJ,u(r1,(l+1col(oh+1)g(rfl+1aJh+1))OJh i = 0. C.248) Similar hierarchies can be derived for the direct correlation functions.54 3.4.4 Derivatives with respect to a parameter A In some cases the intermolecular potential energy is a function of some additional 'potential parameter' A, i.e. 6U = 6Ux(rNojN). One example of
204 BASIC STATISTICAL MECHANICS 3.5 such a situation is in perturbation theory (Chapter 4), where the reference system has a potential energy %A,-0 while the real system has aliK^i; properties (B) of interest might be the thermodynamic functions or g(r<o,<o2). In another example A might represent the strength of an electric field applied to the system; properties (B) of interest would then include the order parameters F, of C.115) (see Chapter 10). A third example is the Kirkwood hierarchy equations for the correlation functions (see ref. 19g of Chapter 5). The A derivative of some general property (B)K is readily found from C.200), C.201) and C.190) as where (. . .)A denotes an ensemble average for the system with potential aUx, and it is assumed that B is independent of A. When B depends on A a term (c)B/dk)K must be added to the right-hand side of C.249). Derivatives with respect to other (i.e. non-potential) sorts of parame- parameters are also of interest. Thus, in non-equilibrium problems A = ( (time) is of importance, and in quantum statistical mechanics A = h, A = m (molecular mass), and A = / (moment of inertia) are of interest (see Appendix 3D for examples). The case A = V (when we include a wall potential in °U) arises in the virial theorem (Appendix E); the derivative required is (dQ/dV)^ (grand canonical) or (OQ/dV)f3N (canonical), where O is the partition function. 3.4.5 Quantum thermal averages For the quantum thermal average (B) = Tr(PB) (see (B.59)), it is straightforward to calculate the /x and C derivatives, and the results are the same as in the classical case. The V derivative is less straightforward, but the result is also the same as the classical one in many cases (see, e.g. the virial theorem references of Appendix E). The A-derivative, on the other hand, is more complicated107 due to the fact that H and dH/d\ do not in general commute. The A derivative of the partition function can, however, be calculated easily (see Appendix 3D). 3.5 Distribution functions for mixtures The above equations arc readily extended to mixtures, and we mention only the most important of these expressions here. We consider a mixture containing NA,NB, . . ., NR molecules of components A, B,. . ., R at volume V and temperature T; here ?« Na = N. The generalization of C.82) for
3.5 DISTRIBUTION FUNCTIONS FOR MIXTURES 205 the configurational partition function is O= = n N\nK. |drN dwN e-^(^N) C.250) where °U now depends not only on the set of coordinates (rNa>N), but also on the assignment of molecules by species to these coordinates; i.e. °U is no longer invariant under permutations of all the molecules among the coordinates, but only under permutations involving molecules of the same species. (Only in an ideal mixture is °U. independent of the assignment by species to the various locations {r^}; see Chapter 7.) For homogeneous, isotropic mixtures, C.104) becomes C.251) where pa = NJV, and ila is 4tt or 8tt2, depending on whether the type a molecule is linear or non-linear. The canonical pair distribution function for an aC pair is given by - 8all) JdrN 2 dcoN-2 e^/Z, C.252) where 8ae is the Kronecker delta and Zc is given by C.53). Equation C.252) is the generalization of C.93) and C.105) for the case h = 2. (The function g^ in C.252) is the correlation function for two molecules of species a and /3, and differs from the g^ used in § 3.1.6; there a and /3 referred to sites in different molecules.) In the grand canonical ensemble the system is at V and T, with chemical potentials /xA, /u.B, . .., /xR. The pair distribution function is given by C.253) where the grand partition function is given by NdcoNe^. C.254) In these equations the sum is over NA, NB, . . ., NR, while the product is over all components A, B,C,. . ., R. Expressions for the derivatives of the mixture distribution function fctay (*hu>h) with respect to /x and j3 have been worked out by Buff et aj 92.108 ancj ^e jej-jvative with respect to a potential parameter is given
206 BASIC STATISTICAL MECHANICS 3.6 by Buff and Schindler.92 It is also possible to write some of these expressions in terms of direct correlation functions.109 3.6 Density expansions of the correlation functions and thermodynamic properties In order to calculate observable properties from the expressions given in subsequent chapters, a means of calculating the correlation functions g(rha>h) must be available. For liquids one usually uses perturbation theory (Chapter 4) or an integral equation method (Chapter 5). For gases at low to moderate densities, however, it is more useful to expand the correlation function in a power series in the number density. Detailed derivations of such expansions are given in many places (see, for example, refs. 110-5); in most cases such derivations are for simple atomic gases, where the pair potential is a function only of r. These derivations are easily generalized to molecular gases by replacing integrals over molecu- molecular positions by integrals over position and orientation, i.e. by making the transformation r 1 r r, d<o,. C.255) We do not give a detailed derivation of such expansions here, but simply outline a procedure for obtaining the first (two-body) term in the series for the pair correlation function g(ra>,&>2)- Higher order terms are simply quoted; they can be obtained by applying C.255) to the well-known results for atomic gases. Expansion of the pair correlation function g(ri2&>i&>2)S5gA2) about p = 0 yields gA2) = goA2) + PglA2) + p2g2A2) +.... C.256) Several methods are available for deriving the terms g0, g1;.. . in this expansion. The traditional method is that of Ursell116 and Mayer,117 who write the Boltzmann factor exp[-j3%] as a sum of terms (see below). In an alternative procedure, due to van Kampen,115 the factor exp[—j3%] is written as a product of terms; this avoids the need to consider graphs, reducible terms, etc. To derive the expression for goA2) we first introduce the Mayer /-function /((/)=/,¦, defined by /«=expH3u(i/)]-l. C.257) Assuming pairwise additivity, we then have exp(-p%) =11 A +/«) = 1 + I h + I I fnfir+- ¦ ¦ ¦ C-258)
3.6 DENSITY EXPANSIONS 207 Substituting this result in C.108a), we get IdxN gA2) = N(N-\)n2 C.259) where dXj =dr; dco{. Noting that N(N-l)-N2 for large N and carrying out the integrations in the usual way gives (see, e.g. eqn G.20) of ref. 4) gA2)=- /12 + O(P) 1 + O(p) and on taking the low density limit we get goA2) = limgA2) C.260) or = exp[-puA2)]. C.261) By considering higher order terms in C.259) it is possible to derive the expressions for gl and g2 in the pairwise additivity approximation. In writing down these higher order terms it is convenient to introduce a graph notation (this notation will also be useful in discussing integral equation theories in Chapter 5). In this notation open circles (O) are fixed points (not integrated over), filled circles (•) are integrated points (i.e. denotes integration over ri; and averaging over a>f), solid lines ( ) denote a Mayer /-function, and dotted lines (••••) denote 1+/ = exp(-j3u). Thus =/l o---o = = exp[-j3uA2)], C.262) C.263) C.264) etc. With this notation the expressions for g[ and g2 are C.265) C.266)
208 BASIC STATISTICAL MECHANICS 3.6 Equation C.256) can be used to derive density expansions for various equilibrium properties. For example, substituting C.256) in C.242) yields the virial equation of state, ^l+ZBnmr1 C.267) where Bn is the nth pressure virial coefficient, Bn = -k^drr(u'(xaIw2)gn^2(r<o1<o2))a,^ C.268) where u'=du/dr. The additive contributions to the first few virial coeffi- coefficients (B2 is given exactly in the pairwise approximation) are obtained from C.261), C.265), and C.266) together with C.268). After an integ- integration by parts, this gives 1 f ~0"<r''":'>- 1><U|U>2 dr, C.269) C.270) Although the region of convergence of C.256) is not known for a general potential, qualitative discussions of the convergence of the virial equation have been given by several authors, including Mason and Spurling.114 It is generally agreed that below the critical temperature C.267) converges for the saturated vapour (but not for the saturated liquid), except near the critical point itself. Above the critical temperature C.267) probably converges for densities up to the critical value (again this statement excludes the critical region itself). These rough guidelines presumably also apply to the expansion of g(r<a,«2). 3 The expressions given above are based on the three approximations described in Chapter 1, namely rigid molecules, no quantum effects, and pairwise additivity of the intermolecular potentials. An exception is the equation C.261) for g()A2), the low density limit of gA2), which is valid even when multibody potentials are present; similarly, C.269) for B2 is not dependent on any assumption of pairwise additivity. Experimental studies of low density gases (i.e. gases where C.261) applies) are there- therefore an important source of information on intermolecular pair potentials. Such studies include measurements of the structural (Chapter 9) and dielectric (Chapter 10) second virial coefficients, as well as the pressure second virial coefficient (Chapter 6). It is sometimes convenient to write
3.6 DENSITY EXPANSIONS 209 C.269) in the form B2 = -^J[e-Su«w-l]dr C.272) where ua(r) is a temperature-dependent central pseudopotential defined by This potential uo(r) is also used in later chapters for liquid state calcula- calculations (see §§ 4.6 and 5.4.9a). The extension of the above results to include three-body potentials and quantum corrections is straightforward. Thus, when three-body potentials are present the expressions for giA2) and B3 become C.274) 3 f 3 C.275) where gidd and B3dd are the pairwise additive contributions given by C.265) and C.270), respectively, and the non-additive contributions are given by # 1 AB3 = -—- Jfc . C.277) Here the shaded triangle indicates the three-body Mayer-function, fi]k = exp[-(Bu(ijk)] — 1, where u{ijk) is the three-body potential. Thus C.278) etc. From C.276) and C.277) we see that the study of third virial coefficients is an important source of information on three-body forces. Expressions for the O(h2) quantum corrections to the pressure virial coefficients can be derived from the expressions given in Chapter 6 for the quantum correction to the Helmholtz free energy. The explicit expressions arc obtained by substituting in these latter equations the density expansion, C.256) for gA2), into the general expressions given in Chapter 11 for the mean squared force (F2) and mean squared torque (t2), and noting that the density expansion for Acl is Acl = Ald + NkT ? Bnc] J-y C.279) where A'd is the ideal gas value and Bnd is the classical value of Bn. For
210 BASIC STATISTICAL MECHANICS 3A linear and spherical top molecules one finds that B2 is given by to order h2, where B2ci 's the classical value of B2 given by C.269), and <F2>0 = kT Jdr12<g0A2)V2uA2)>w C.281) <t H = fcT jdr12(g0A2) V^d^L,^ C.282) are the low density limits of (F2)/p and (r2)/p, and goA2) is the low density limit of gA2) and is given by C.261). For asymmetric top molecules the first two terms on the right-hand side of C.280) are unchanged, but the last term on the right, (t2H/7, is replaced by ^ C.283) where Ta is the principal (body-fixed) a-component of the torque, and Ia is the principal cm -component of the moment of inertia tensor. Appendix 3A Relation between the canonical and angular momenta The straightforward method118 of relating the principal axes components (Jx, Jv, Jz) of angular momentum to the canonical momenta (p^, pe, px) is to relate both sets to the angular velocity components (flx, fly, flz). As a result of the calculation one finds that the physical significance118 of the canonical momenta is as follows: p^ is the component of angular momen- momentum of the molecule along the space-fixed Z axis; p9 is the angular momentum component along the line of nodes (i.e., ON of Fig. 3A.1); px is the angular momentum component along the body-fixed z-axis. Such a physical interpretation is plausible as seen from Fig. 3A.I. With these results in hand, it is straightforward geometry (Fig. 3A.I) to write the ps in terms of the is; e.g. p^ = Jx cos(x, Z) + Jy cos(y, Z) + Jz cos(z, Z) C A. 1) where (x, Z) is the angle between Ox and OZ, etc. From the geometry of Fig. 3A.I or from the direction cosine matrix (A. 122) and (A. 123) we have: cos(x, Z) = -sin 6 cos x, cos(y, Z) = sin 0 sin x, cos(z, Z) = cos 0. In using (A. 122) and (A. 123) we note the following equivalences between the particular axes and angles used here and the more general ones in
3A CANONICAL AND ANGULAR MOMENTA 211 \ x' \ X' Z A t s Fig 3A. 1 The canonical momenta pl(l, pfl, px are the projections of the angular momen- momentum vector J along the OZ, ON, and Oz axes respectively. Here XYZ are the space- fixed axes, xyz are the body-fixed axes, tf>0\ are the Euler angles (sec Fig. A.6), and ON is the line of nodes (i.e. the line where the XY and xy planes intersect). Appendix A (see note on notation below eqn (A.71b)): Appendix A: xyz x'y'z' afiy Appendix 3A: XYZ xyz 4>QX- In a similar way we calculate p8 and px. The result is P<j. = Jx (^sin 0 cos x) + Jy s'i 6 sin x + K cos 6, pa = Jx sin x + Jy cos x, CA.2) px=J2. The inverse relation is Jx = p^i-cosec 6 cos x) + Pe sin \ + Px cot 6 cos x, Jy = pj, cosec 6 sin \ + pg cos x + Px("Cot 0 sin x), CA.3) Jz=PX which is C.27). [The reader will notice the identity of the transformations in CA.2) and (A.70). This is due to the fact that the operator generators of rotations correspond to angular momenta: i.e., corresponding to the angular momenta Jx, Jy, Jz are the rotation generators Lx, Ly, Lz, and correspond- corresponding to the angular momenta p^, pe, px are the rotation generators —id/d4>, — id/d6, —id/dx- Readers familiar with quantum theory can phrase the explanation slightly differently.]
212 BASIC STATISTICAL MECHANICS 3B Three final points are worthy of mention. The relation between the space-fixed components (Jx, JY, Jz) and the canonical momenta (p<j» Po, Px) can also be worked out; the results take the form (A.69). Secondly, the physical significance"9 of the body-fixed components (Jx, Jv, Jz) should be kept in mind. These are the components of angular momentum 'referred to' the xyz axes (i.e. the projections of J along the instantaneous body-fixed axes). They are not the angular momentum components 'in' the body-fixed frame, since the body is always at rest with respect to this frame. Finally, the above discussion assumes that the molecules are non-linear. For linear molecules (N2, HC1, CO2, etc.) with the body-fixed z axis chosen to lie along the molecular axis, only two Euler angles are neces- necessary, and these are the polar angles @, 4>) for z relative to XYZ. The angle x 'n Fig- 3A.I is not a variable, so that px and Jz do not arise. Thus CA.2) and CA.3) are replaced by P<b = ~JX sin 6, ^ x CA.4) Pe=Jy and Jx = p<()(-cosec0), CA.5) Jv = Pa- Appendix 3B Spherical harmonic form of the OZ equation85120 2 We consider in detail the case for linear molecules; in § 3B.3 we state the corresponding results for general shape molecules. 3B.I Space-fixed axes form The r-space form of the OZ equation for a pure fluid C.117) is where the rs and o>s refer to some set of space-fixed axes. We first Fourier transform h(ro>|O>2) and c(ra>io>2) to k-space according to f(rwlw2) = (~-^ jdk e-ik-7(kaj,oj2), CB.2) /(kojjojz) = j dr eik-'f(Tw,oj2). CB.3) The integral over r in CB.1), which is an r-space convolution, becomes
3B SPHERICAL HARMONIC FORM OF THE OZ EQUATION 213 a product in k-space, exactly as in the monatomic case: h(kwloj2) = c(ka>1&>2) + p<c(kaj1aj3)h(kaj3«2))<O3. CB.4) This step is reviewed in § 3B.4. In § 3.2.1 (see also §§ 2.3 and A.4.2) it is explained why the spherical harmonic expansion of rotationally invariant quantities such as h(ra>ia>2) and c(t<oi<a^) takes the form /(ro>lW2) = Y. fihW; r)<k,iA«>i«>2<»r) CB.5) where wr denotes the orientation of r, and *,y(wi«2«)= Z C(lJ2l;mlm2m)YUmi((ol)Yl2m7{(o2)Ylm((of CB.6) where C( ) is a Clebsch-Gordan (CG) coefficient. Parity arguments show (§2.3) that for linear molecules f{lll2l;r) = Q unless (ly + l2+l) is even, and the following symmetry properties can be derived (cf. B.26), B.25)): /;r), CB.7) f(lj2l;r)=f(lll2l;r)* CB.8) (i.e. /(Z,l2Z;r) is real). We can similarly expand the Fourier transforms h(ko}i(o2) and ^y) CB.9) where f(lj2l', k) are the expansion coefficients, wk denotes the orientation of k, and ^jl^l(w\°}2(i}) is the same rotationally invariant function CB.6). From CB.7), CB.8), and the Hankel transform relations between f(ltl2l; r) and f(hl2l; k) given in § 3B.5 we find the symmetry properties CB.10) [Eqn CB.11) is derived alternatively from /(ka>1o>2)* = /(-kaIaJ), which is the reality condition for /(rw,(iJ).] Note the difference between CB.8) and CB.11). We wish now to transform CB.4) into a relation between the harmonic coefficients h(lj2l; k) and c(lxl2l; k). To this end consider the product term in CB.4) aj2)>^. CB.12) Substituting the expansions CB.9) for c(ka)]oK) and h(ka>3o>2) and the
214 BASIC STATISTICAL MECHANICS 3B explicit form CB.6) in CB.12) gives 'Z l'il'il"; k)C(l[l3V; m\m'3m')C(l'il'il"; m3m'im") CB.13) where the sum is over all Is and ms. Using (A.3) and (A.39) to evaluate (. . .)„, and (A.36) to simplify Yrm.(a>k) Yr.m..(<wk) gives F(kw1w2) = PDt7)~1 Zi-r-cWil'; k)h(l'3lW, k) xC(l[l3l'; m[m'3m')C(l3W"; m'3m'2'm")Y,;m;(<o,)Yr.m»(a>2) i";; m'm"m)Y* (a.k) CB.14) where m= — m. Consider now the sum over the ms of the three CG coefficients in CB.14), xC(ri"l;m'm"m) CB.15) where V' denotes a sum over all ms except m[m'2'm. Converting the Cs in CB.15) to 3/ symbols using (A.139), using the 3/ symmetry properties and then applying the sum rule (A. 147) gives ' = [{21' i m'2' m/U" I' l3l Substituting CB.16) in CB.14) and again using the 3/-C relation (A. 139) gives ~ ~~ c(l[l3l';k)h(r3l2r;k) : X 'I ,!} X ;; CB.17) We see from CB.17) that F(kaj,aj2) has the standard form of expansion CB.9), with coefficients ?{1^1; k) given by c{lx\3V;k)h(l3U";k) r 1} CB'18)
3B SPHERICAL HARMONIC FORM OF THE OZ EQUATION 215 Thus we see from CB.4), CB.12), and CB.18) that the OZ equation in component form is -ip ? c(hl3l';k)h(l3l2l";k) Although this may seem complicated at first appearance, the simple OZ structure in CB.19) is apparent, and the equation is not difficult to use in practice (see ref. 122). A simpler form is derived in the next section. 3B.2 It-frame form The original OZ equation CB.1) or CB.4) has been written in CB.19) in terms of the k-space harmonics f(l\l2l; k). It is also possible to write it in terms of various other harmonics, e.g. the 'k-frame' harmonics f(\x\2m\ k) (to be defined below), the r-space harmonics f(lj2l; r) referred to space- fixed axes, and the r-space harmonics f(lj2m;r) referred to the inter- molecular axis r (i.e. the 'r-frame' harmonics). A detailed discussion of all these cases is given in ref. 85. (See also Chapter 5.) The 'k-frame' leads to a particularly simple form123 of OZ, so that we discuss this case here. For functions /(k<u,a>2) the k-frame is defined by choosing the polar axis along k. Referred to these axes the function f(k(i)[o}2) can be expanded as (compare C.146)) f{kw[o)'2)= ? /(Z,l2m;fc)YlimK)YIl!5(a>2). CB.20) The space-fixed and k-frame harmonic coefficients are related by (cf. C.152), C.153)) ihU fc) = I (^yJC(/i'2'; m^0)^'i^m; k), CB.21) f(hl2m;k) = y rC(/1/2/;mm0)/(/1/2/;fc). CB.22) | \ 477 / The /((i/2m; k) satisfy the symmetry relations f(lil2m;k) = f(lil2rn;k), CB.22a) which are derived from CB.22) and CB.10, 11), or from /(k<o,a>2)* = /(-kft^ojj and CB.20). Applying the transformation CB.22) to CB.19) and converting the C
216 BASIC STATISTICAL MECHANICS 3B to a 3/ gives m;k) CB.23) where F(hl2m;k) = D^r2p ? c(lJ3V; k)h(l3l2l"; k) l!'!"l3 x(-)'K2l' + l)Bl"+l)Bl + l){^ j2 f} CB.24) VO 0 O/Vm m 0/ Consider the sum in CB.24) X If we use the symmetry properties (A.283, 284) of the 6/ and the relation (A.281) we get ( )( ). OB.26) T \0 m ml\m 0 ml Thus from CB.26) and CB.24) we have F(/1l2m;/c) = D7r)-2p I c(lxl3l'; k)h(l3l2l"; k) h h)(l2 l" lA CB.27) m m/\m 0 m/ The sums over /' and I" in CB.27) can now be done with the help of CB.22) and the symmetry properties of the 3/ symbols: M3r;fc)( VO m m CB.28) m 0 m i CB.29) Combining CB.27), CB.28), CB.29) gives F(lil2m; k) = D-n-ylP(~)mYlc(lll3m; k)h(l3l2m; k). CB.30) i3
3B SPHERICAL HARMONIC FORM OF THE OZ EQUATION 217 Thus, from CB.23) and CB.30) the OZ equation becomes h(lxl2m; k) = c(lxl2m; k) + {4ttV p{-)m ? c(hl3m; k)h(l3l2m; k) CB.31) which is simple both in structure and appearance, and easy to use in practice (see ref. 122 and Chapter 5). 3B.3 Non-linear molecules The OZ forms analogous to CB.19) and CB.31) for non-linear molecules are derived in the same way as for linear molecules. The space-fixed harmonic expansion coefficients f(lxl2l; nxn2; k) are defined by writing /(kwioj2) as (cf. C.142)) X I I f(lxl2l;nxn2;k) CB.32) The k-frame coefficients f(lxl2m; nxn2; k) are defined by writing f{k(i>[a)'2) as (cf. C.146)) CB.33) The coefficients f(lj2l; nxn2; k) and f(lvl2m; ntn2; k) are related as in C.150), C.151). The symmetry properties of the space-fixed coefficients arC f(lll2l;nln2;k) = (-)l'+'i(l2lll;n2n1;k), CB.34) fihhU nin2; k) = {-)l'+l^n'+n-f{lxl2l\ nxn2; kf CB.35) from which one can work out the symmetry properties of the k-frame coefficients (cf. B.30), B.31)). The space-fixed axes form of the OZ is h(lj2l; nsn2; k) = c(lj2l; n{n2; k) + D77r1p x I I c(lj3r; n.ny, k)h(l3l2l"; n3n2; k) I,-., ,.,.. /r i" i\(i, u n @ 0 oil/" r J CR36) x(-y' B^i)@ 0 oil/" r J and the k-frame form is h{lil2m; nxn2\ k) = c(lxl2m; nxn2; k) + p(-)m x Y. (-T^h+iy'ciUkm; nxn3;k)h(l3l2m; n3n2;k). CB.37)
218 BASIC STATISTICAL MECHANICS 3B 3B.4 Fourier transform of a convolution In the OZ equation, or any many-body problem where chains of interac- interactions are considered, there occur convolutions of the type F(r12) = Jdr3c(r13)h(r32) CB.38) where r^i^-iv Using CB.2) we substitute the Fourier transforms of c(r,3) and h(r32) into CB.38) to get Htl2) = b-) I dk dk' e'kr- e"lk'-'=c(k)/i(k') I dr3 e-'*"*'-''. CB.39) From the relations (see (B.79), (B.80)) S(q) = S(-q) = (^-j Jdreiqj CB.40) where 5(q) is the Dirac delta function, and jdk'8(k-k')G(k') = G(k) CB.41) we see that CB.39) becomes / 1 V f F(r12)= — J dke"'k-'-c(k)h(k) CB.42) which, comparing with CB.2), shows that the Fourier transform of the convolution J dr3c(r13)h(r32) is c(k)h(k). 3B.5 The Hankel transform The r-space and k-space harmonic expansion coefficients defined by CB.5) and CB.9) are mutual Hankel transforms: tfCMzZ;*-), CB.43) CfcO/U.y; fc) CB.44) 7\ {Ztt) Jo where jt is the spherical Bessel function. To prove CB.43) we substitute CB.5) into CB.3) to get JJ; r) fdwr e^^^aW. CB.45) Substituting the Rayleigh expansion, (B.89), eikr= I 477iLiL(kr)YLM(Wt)*YLM(a>r) CB.46)
3C VOLUME DERIVATIVES AND INTENSIVITY CONDITION 219 into CB.45) and using the explicit expression CB.6) for t//^, we get x Jd<orYfm(<or) Yutto). CB.47) Using the orthogonality relation (A.27) of the Y(ms we find that /(ka^a^) has the form CB.9) with coefficients given by /(Mai; fc) = 4ml^drr2jl(kr)f(lll2l; r), which agrees with CB.43). The inverse relation CB.44) is proved in the same way. Appendix 3C Volume derivatives and the intensivity condition In this appendix we derive an expression for the volume derivative of some property (B), where B is a dynamical variable.94 We then consider the special case where (B) is a distribution function of order h. 3C.1 Volume derivative of a general property (B) To obtain the V-derivative of (B) we note that the r, integrations are over the volume, so that from C.198) we have where 1 = -77 f V • ¦ f VA'N e-*3"^"^'-»B(r'Vw'VS) CC.2) d V Jo Jo where JJ'dr denotes the integral JJ"dx J^'dy j^Tdz over the system volume, assumed here to be cubic. This integral can be evaluated using the method of Green and Bogoliubov,124 in which the following change to dimensionless variables rj is made, r,. = V*r,' (i = l,2,..., N) CC.3) so that dr, = V drj.
220 BASIC STATISTICAL MECHANICS 3C Thus CC.2) becomes I = — where JJ dr/=J(,dx' Jody' Jjdz' and we note that H and B depend on V through the rN coordinates. Transforming back to the original rN vari- variables, we get CC.4) Combining CC.1) and CC.4), we find By an exactly analogous argument to that used above to evaluate /, we can show that dlnH\ /N a Moreover, in the grand canonical ensemble, this derivative is related to the observed pressure p = (SP) by C.184). Here 0* is the dynamical pressure given by125 SP = (N/l3V)-(dHIdV) CC.7) Thus, from CC.4)-CC7), we get O CC8) To evaluate the quantity dH/dV that appears in these equations, we note that H depends on the rf only through att(rNa>N). If we assume pairwise additivity (eqn A.13)), we then have av
3C VOLUME DERIVATIVES AND INTENSIVITY CONDITION 221 From CC.3) we have ^V~W^3V CC9) so that we get -lrr -\ A<i(* , * , * \ Ofl 1 \~< 0 Li \I:;OJ:OJ;) Here ry = r^ — rb and we have used Hence, since r . V = ra/3r, we have Similarly, we have 8B 1 ^ dB When (B) is intensive the volume derivative (d(B)/dV)l3llL vanishes. Furthermore, we also have (^).O CC.2, which holds even when V-^°°. The proof of CC.12) is as follows.94 Any property (B) (extensive or intensive) can be viewed as a function of the variables /3, V, and (N), so that we have ^ av/3<N> la<N)/3v\ av,; If <B> is intensive then we have (a(B)/apH<N> = (a<B)/apKV = (a(B)/apH, since (B) depends only on C and p. Moreover, the derivatives (d(B)/dVH(N} and (d(B)/d(N)HV that appear in CC.13) are then simply
222 BASIC STATISTICAL MECHANICS 3C related to the density derivative by \ dp J0 Va<N>/3V = --{^) ¦ CC.14) p \dV /3<N> Also, from C.172) and C.184), together with C.209), we have Hbr = tH =p- CC15) Substituting CC.14) and CC.15) into CC.13) gives the desired condition CC.12), provided that (d<B)/dpK is finite. 3C.2 Volume derivative of the distribution function of order h We now derive the 'intensivity condition', C.241), for the case (B} = f(rhojh). To do this we substitute C.234) for B into C.217). The quantity (BhS?) is given by ~H n (N-h)l J* dw C aV where z is given by C.191). Replacing N by (N-h) + h in the first term, using CC.10) in the second term, and assuming pairwise additivity, we get ^77 1 drh+1dojh + 1r,.h+1 =- J V j_i J ofi,h + i If, , , , du 6VJ 6V h + i,h+2 xf(!>+2<0h+2). CC.16) (The second term on the right hand side of this equation vanishes for
3C VOLUME DERIVATIVES AND INTENSIVITY CONDITION 223 h = 1.) The quantity {?P) is similarly evaluated, P If, H~7T? dr' /3 6VJ CC.17) which is the pressure equation for a homogeneous, isotropic fluid. Equa- Equation CC.17) can also be derived from the virial theorem (see Appendix E). From CC.11) and C.234) it follows that where I, = [dre-0*wr.-S(r-r,), CC.19) J dr with <%(r) = <%(r!r2 . .. r, = r ... rN). Writing r . d/Bt = xd/dx + yd/dy + za/3z, we evaluate the xd/dx contribution to /, as Ilx = [dx dy dz e-ecu(xyz)xS'(x - x,)8(y - y,)8(z - z,) CC.20) where 8'(x) = dS(x)/dx. Using the standard relations (B.67) and (B.68) we get Ilx=-~(xe-^z> Thus, /, is equal to -0^'. CC.21) Substituting this result in CC. 18) gives Equations CC.16), CC.17), C.235), and CC.22) can now be substituted into C.217) to give a consistency condition for f(rh<oh). For the isotropic, homogeneous case, where C.105) applies, this gives C.241). A similar
224 BASIC STATISTICAL MECHANICS 3D derivation of this consistency (or intensivity) condition has been given for the spherical molecule case by Schofield.94 Appendix 3D The classical limit and quantum corrections We derive here the classical limit and the leading (order ft2) quantum corrections for the partition function (cf. A.3)). Expansions in powers of h for the partition function and the distribution functions were first derived by Wigner128 and Kirkwood,129 and have been reviewed in a number of places. 130~4 Simplified derivations have also been given;135136 our deriva- derivation is similar to that of ref. 136. We give the derivation in detail for the case of spherical top molecules. The results for other cases (linear, symmetric and asymmetric top molecules) can be obtained similarly (see the remarks in § 3D.2) and are given in Chapter 6. A qualitative interpretation of the results is given in Chapter 1. In § 3D.3 we discuss quantum corrections to the equipartition theorem (see C.74), C.75)). 3D. I Spherical top molecules The quantum partition function Q is given by (B.60), Q=Tre 3H CD.1) where Tr denotes the trace (over an arbitrary complete set of states-see argument following (B.63)), and H = K + °U is the Hamiltonian for the N interacting identical spherical top molecules, with al/. = °ll(rNwN) the inter- molecular potential, and K = /ft + Kr the total (translational plus rota- rotational) kinetic energy K = Xp2/2m + Xj2/27 CD.2) where the symbols p>, 3h etc. are standard and are denned in Chapter 3. The Hamiltonian operator, denoted here by H for brevity, is denoted by 2ecl in C.16). The derivation of the classical limit and quantum corrections to Q = Trexp[-|3(K+%)] is based on the Baker-Hausdorff identity.137 We recall that the proof of the exponential identity exp(A + B) = exp(A)exp(B) for numbers A and B is based on the fact that A and B commute. For example, if we expand all the exponentials, the second- order term on the left-hand side is \{A + BJ = |(A2+ B2 + AB + BA), whereas on the right-hand side it is {A2 + -2B2 + AB. These are equal only if AB = BA. Hence, for non-commuting quantities (e.g. operators, mat- matrices, etc.) we have eA+B^eAeB. CD.3)
3D CLASSICAL LIMIT AND QUANTUM CORRECTIONS 225 We can, however, express exp(A + B) as exp(A)exp(B) plus an infinite series of corrections involving the commutators [A,B] = AB~BA, [A, [A, B]], etc. For the case at hand we get136137 4[K,°llT +...}. CD.4) As we shall see, when the trace is taken the first term in CD.4), exp(— /3K)exp(— p^ll), generates the classical limit plus the intramolecular h2 quantum corrections (cf. A.4)), and the four commutator terms gener- generate the intermolecular h2 corrections. The higher-order commutators and powers in CD.4) would generate h4, etc. corrections, which we do not consider here. When we decompose K into translational plus rotational parts accord- according to CD.2), we then get three additive comtributions to Q = Trexp(—/3H): (i) Qj.- the classical limit plus intramolecular rotational corrections (from Trexp(—CK)exp(—C%)), (ii) O2: intermolecular transla- translational corrections (from the Kt commutators), (iii) Q3: intermolecular rotational corrections (from the KT commutators). The cross terms involv- involving [Kt, [Kr, %]] and [Kt, <%][Kr, <%] can easily be shown not to contribute to the O(h2) corrections. We take up these three contributions separately. (i) Classical limit plus intramolecular correction To evaluate Q,, where Q,=Tr(e SKe H, CD.5) we write out the trace138 of the product in terms of two complete sets of states, i.e. the coordinate eigenstates |rNa)N)=|r1a>1) |r2&>2) • • • I'm^n) (where Ir,^)^^) \<oj), etc.) and the momentum-angular momentum eigenstates |pN(Jmn)N) = |p1(J1min1)).. . |pN(JNmNnN)> (where \Pi(Jim1n1)) = \p1)\Jlminl), etc.), Qi= I fdpNdrNdoJN<rNaJNje-fiK|pN(Jrwn)N) (Jmn)" J x(PN(Jmn)N|e-fi*!rNaJN>. CD.6) (The state \Jtnn) is a common eigenstate of the commuting observables J2, Jz (space-fixed Z-component of J), and Jz (body-fixed z-component); the corresponding quantum numbers are J, m, and n respectively (see
226 BASIC STATISTICAL MECHANICS 3D CD.48)).) Since |pN> and \(Jmn)N) are eigenstates of K, and Kr respec- respectively, we have e-«K|pN(Jmn)N) = exp(-^Xi x exp(-0 X MJi + l)h2l2lj \pN(Jmn)N). CD.7) Similarly |rNa>'v) is an eigenstate of exp(—/3%), since °U depends only on the coordinates: e |r oj / — e \t o) /. pu.oj Using CD.7) and CD.8) with CD.6) and the relation (A | B) = {B | A)* gives V f N N N -PI., x e-fi 5:.-',<-',-i«2«' |(rN | pN)p |^n | (jmn)N)p. CD.9) The normalized single-molecule momentum eigenfunctions t//p(r) = <r | p) (see (B.50)) are given by139 p(^fl"-"h CD. 10) so that we have I</V(r)|2 = p. CD. 11) The normalized single-molecule angular momentum eigenfunctions ifcmn(d)) = (<i) | Jmn) are given by140 where Di,n(<w)* is the generalized spherical harmonic (see p. 460). Because of the unitary property (A.86) of the Ds, the t//,,,m satisfy IkJmnM|2 = n^. CD.13) Using CD. 11) and CD. 13) in CD.9) gives O -[yBJ+l^e-^»HN(^^-^^ Q^[L(.2J+1) e j j h3N {^2)N xj-^pjnmj-sw.")^ CD. 14)
3D CLASSICAL LIMIT AND QUANTUM CORRECTIONS 227 In § 3D.2 below we show that the single-molecule rotational partition function (the quantity inside the square brackets in CD. 14)) is given to order fo2 by qr= I B/+ IJ e-fW+lw'21 = qf f 1 +4^1 CD.15) j L 81 J where q,1 is the classical value [see also C.80) and C.81)], ,;' -11 <Sa dp. e"«""' - J5 Jdo, dJ e-«' CD.16) where d'co = d<f> dd dx, doj = d<? sin 0 d0 d^, and dpm = dp^, dpe dpx. From CD.14H3D.16) and the relation (l+x)N = l + Nx (see also remarks below CD.39)) we thus have to order h2 (^) CD.17) where Qcl =^ jda,N dJN d^ dpN e-^^^^ rN dpN e-0 s^/2( e^ z^/2m e^""^. CD. 18) Comparing CD. 18) and C.77) we see that CD. 18) is the correct classical limit, apart from the N! factor. To obtain this factor (and possibly symmetry number factors, etc.-see C.86) and ref. 138) one must use properly symmetrized states'292138 in evaluating the trace in CD.5). (ii) Translational corrections The translational contribution Q2 to Q is given by (see CD.2) and CD.4)) Q2 = Tr e-0K e-0*{-il32[Kt, %]-^3[Kt, [Kt, %]] + kP3[% [% KJ]+hP*[Kt,'*lf} CD. 19) where K"t = X,P?/2m is the translational part of K. (For atomic fluids CD. 19) is the only h2 quantum correction term.) We now evaluate the commutators in CD. 19). Using the identity141 [AB, C] = [A, C]B + A[B, C] CD.20)
228 BASIC STATISTICAL MECHANICS 3D we get for the first commutator Now the fundamental commutation relation of quantum mechanics is142 [Px>*] = ^'ft (with similar relations for [pv, y], etc. and [px, y] = 0, etc.). From the fundamental commutation relation there follows143 for the commutator of p with an arbitrary operator A = A(p, r, pot, a>) CD.22) where V = d/dr. Applying CD.22) to CD.21) gives [*?„<%] = -— ?v,<%.p, + p, .Vj%. CD.23) The second term in CD.23) can be replaced using pi.Vi(% = V,<% .p, +[p( ., V,<%]. Using CD.22) again to evaluate [pf ., V^] thus gives [/Cl,<tt] = --5]Vi<tt.pi--J-5;v?«fc. CD.24) m j 2m i The first term in CD.24) is O(ft); we anticipate the result that this term will not contribute to CD. 19) since (to O(ft2)) the trace arising from it is imaginary, whereas d is real. (The formal proof involves noting that Vt°ll .p, is linear in pf, so that the subsequent thermal average arising from Tr will annul this term to O(h2).) It is the second term141" in CD.24) which gives a non-vanishing O(h2) contribution to CD.19). The double commutators [Kt, [Kt, <%]] and [<%, [% Kj] can now be calculated from CD.24) employing similar methods. Using [pif p,] = 0 we find t, [Kt, %]] = --h— Z V,V,%: PjPi + O(ft3), CD.25) m (j X(V%J — /,vW\ CD.26) . m ; ^ where the tensor notation of §B.l is used. From CD.24) we also have h2 ^ The leading term in CD.27) arises from the square of the first term in CD.24); there is an O(h3) correction to this term due to the fact that (V,<&.p,)(V1<%.pj)/(Vj<%)(Vi%):pipi (cf. argument below CD.23)), but we do not need any terms of higher order than h2 for what follows.
3D CLASSICAL LIMIT AND QUANTUM CORRECTIONS 229 Since the four commutator terms in CD. 19) are O(ft2), we can replace the thermal average Tr(e~3K e~B°"A)/QC| by a classical thermal average (A)c] (see (B.59)), since the corrections will generate terms of higher order than h2. Thus we have to order h2 q2/q , = ^— Y <V2%) + —— ? (ViV,%: p,-Pi) 4m ; 6m „ where in CD.28) (...) denote classical thermal averages for brevity. Classically the momenta are uncorrelated with the positions (§3.1.2) so that, for example we have (Vf V,-% : p,.p,.) = <VfVj<U): <p,-Pi). CD.29) Since the momenta of different molecules are uncorrelated, we have (PiPj) = 0 for ii= j. Also, from the equipartition theorem, we have (p2x) = (pfy)= (pfz)= tnkT, and since the momentum components of a single molecule are uncorrelated we have (pixply) = 0, etc. All this information is summarized by (PiP,-) = mfcTSyl. CD.30) Using the results CD.29) and CD.30) in CD.28) together with the hypervirial relation (see (E.31)) CD.31) gives Q2lQc\= Z<(V^J)[4 + 6~3^8J. CD.32) III I The force on molecule i due to the other molecules is Ff = — V^. Since all the N molecules are equivalent we can write CD.32) as CD.33) where (F2) is the mean squared force on molecule 1. (iii) Rotational corrections The rotational contribution Q3 to Q is given by (see CD.2) and CD.4)) Q3 = Tr e~3K e ^{-|/32[/<:r, °U]-^C3[Kr, [Kr, <%]] + 5/33['%, [°U, KT]] + l(i4[Kr, °U]2} CD.34) where Kr = Vf I2121 is the rotational part of K.
230 BASIC STATISTICAL MECHANICS 3D The evaluation of the four terms in CD.34) exactly parallels the evaluation of the corresponding translation terms in CD.19). Replacing the commutation relation CD.22) is CD.35) where Vu> is the angular gradient operator.144 The analogue of CD.30) is (J,J,) = //cT8i,l, CD.36) and the angular hypervirial relation which replaces CD.31) is (see (E.32)) 2 J) CD.37) where Vf, is the angular Laplacian (see (A.71) and ref. 144). We can thus write down the result for Q3 by inspection of CD.33), ,m38) where t, -~Vo1i°U is the torque on molecule 1. Adding together CD. 17), CD.33), and CD.38) gives for Q = O, + Q2 + Q3 the result . CD.3* One can worry about the validity of the derivation of the series » CD.39), and the corresponding series for the free energy A—Acl = -/cTln(Q/Qcl) (cf. §6.9), since N is usually of order 1023, so that the terms of the series are usually not small. However, the series is valid for small N, and because A~AC] turns out to be proportional to N (i.e. extensive as it should be), and because the series is unique, it must be valid for large N also. Similar arguments are used to justify derivations of the virial series145 and the perturbation series (cf. §4.2). It is cumbersome to work out the higher-order (h4, etc.) terms of the series. The series is presumably an asymptotic one. 3D.2 Single-molecule rotational partition functions for general shape molecules It remains to derive CD. 15) for the single-molecule rotational partition function of a spherical top molecule. We shall formulate the problem a little more generally in order to include a brief discussion of the other cases of interest (symmetric top, asymmetric top, and linear molecules). The evaluation of the intermolecular quantum correction terms for these cases, involving the mean squared force and torque components, proceeds exactly as for spherical tops; the results are given in § 6.9. The rotational partition function for a single rigid molecule of arbitrary
3D CLASSICAL LIMIT AND QUANTUM CORRECTIONS 231 shape is qt = Tr e~3K" CD.40) where Krl is the rotational kinetic energy of a single molecule Ktl = aJl+bf$+cJl CD.41) where Jx, Jy, Jz are body-fixed principal axes components of J, and a = l/2/xx, etc. For a non-linear molecule there are three cases, (i) spherical top (a = b = c), (ii) symmetric top (a = b^c), (iii) asymmetric top (a, b, c unequal). For a linear molecule the term cJ\ is absent (i.e. KTl = a32, with a = 1/21). (a) Linear molecules The simplest example is the linear molecule. Since this case is discussed in many places131132146 we can run through the derivation fairly quickly. We write out the trace in CD.40) in terms of the eigenstates of the Hamiltonian Krl = a32, which are clearly also eigenstates of J2. These states can therefore be taken as \Jm), which are common eigenstates of the complete set of commuting observables J2 and Jz (where Jz is the space-fixed Z component): Jz\Jm) = mh\Jm). (The Schrodinger wave function forms of these relations are given in (A. 14).) Since m can take the values m = —J,..., +J, the energy levels are BJ+l)-fold degenerate, so that CD.40) becomes m2, CD.43) j since exp(-/3aJ2) |Jm> = exp(-/3aJ(J+ l)h2) \Jm) (see (B.58)) and (Jm\Jm)=\. The sum CD.43) cannot be done exactly, but can be evaluated approx- approximately with the help of the Euler-Maclaurin summation formula,147 148 ? f(n)= \ f(n)dn+Ma)+f(b)]-T2[f'(a)-f'(b)-] + ... CD.44) where the sum is from n = a to n = b in integer steps. Thus we have in this case - r- I f(J) = f(J) dJ + i/@) - hf @) +. ¦. CD.45) J=0 JO
232 BASIC STATISTICAL MECHANICS 3D where /(J) = BJ+l)exp(-aJ(J+1)), with a = /3aft2. CD.46) The integral in CD.45) can be done exactly (since BJ+l)dJ is the difTerential of J(J+1)) and gives rise to the classical value, qf = a~x. The leading [O(a) = O(h2)] correction arises from the terms f@) and /'@). The result is qr = a-'[l + |a + ...] CD.47) (b) Non-linear molecules For arbitrary non-linear molecules with rotational Hamiltonian CD.41) a complete set of commuting observables is149 {J2, Jz, Jz}, where Jz is the space-fixed Z-component and Jz the body-fixed 2-component of J. Because they commute this set of observables possesses a common set of eigenstates149 \Jmn), J2 \Jmn) = J(J+l)h2 \Jmn), Jz \Jmn) = mh \Jmn), CD.48) Jz \Jmn) = nh \Jmn). The complete set of states \Jmn) (with J = 0, 1, 2,. . . ; m = -J,. . . , +J; n = —J, . . . , +/) furnishes us with a basis for writing out the trace in CD.40). We take up separately the case of the spherical, symmetric, and asymmetric tops. (i) Spherical top molecules For spherical tops (a = b = c) the rotational Hamiltonian CD.41) is simply Kr, = aS2, with a = 1/2/. Hence the states \Jmn) defined by CD.48) are automatically eigenstates of Krl. Writing out the trace CD.40) in terms of these states thus gives (cf. CD.15)) qT= ?<Jmn|e-pa*|Jmn> Jmn = Y.BJ+lJcliaJa+>)fl2 CD.49) where we use the fact that the energy levels are BJ+ lJ-fold degenerate (cf. CD.43) and following remarks). The partition function is thus qT = Y.'j -o BJ + lJexp(-oJ(J+ 1)), with a again given by CD.46). Noting the identity J(J+ 1) = (J + ^J~k, we rewrite qr as X 2e-"<J+JJ. CD.50)
3D CLASSICAL LIMIT AND QUANTUM CORRECTIONS 233 Changing summation variable to j = J + h we have qr = 4 exp(a/4) Xr~' I2 exP(~a/2)' or because the summand is even in j, qr = 2e°'4 ? j2e-'. CD.51) /—— °° The sum in CD.51) can be evaluated approximately using the Euler- Maclaurin summation formula CD.44). Since the summand and all its derivatives vanish at / = ±o° we have150 CD.52) If we now expand the exponential in CD.52) as exp(a/4) = 1 + a/4, we obtain to O(ft2) CD.53) which establishes CD. 15). (ii) Symmetric top molecules By adding and subtracting a term aJ2, the symmetric top (a=b=/=c) Hamiltonian CD.41) can be rewritten as Krl = a32+(c~a)J2z, CD.54) where J2 = Jl + J2 + J2z. Thus, the states \Jmn) defined by CD.48) are again automatically eigenstates of KTl. The corresponding eigenvalues are obviously EJn = aJ(J+ \)h2 + (c - a)n2h2. Writing out the trace in CD.40) in the \Jmn) representation thus gives Qr= ie^(It" t e-^-O CD.55) where aa = (ih2a, etc. The sums in CD.55) can again be evaluated approxi- approximately using the Euler-Maclaurin relation CD.44). A straightforward but tedious calculation151153 gives to order h2 CD.56) Note that CD.56) reduces to CD.53) for the case of the spherical top
234 BASIC STATISTICAL MECHANICS 3D (iii) Asymmetric top molecules The energy eigenstates and corresponding energy eigenvalues are un- unknown in genera!154 for the asymmetric top (a, b, c unequal) Hamiltonian CD.41). We are thus forced to try to evaluate the partition function trace CD.40) without knowing the energy levels. The ft-expansion has been derived155 as an application of Kirkwood's general method12y (i.e. solving the Bloch equation with an h series expansion.) The result is (to order h2) () CD.57) cyclic V " /J where aa = Ch2a, etc. Note that CD.57) is symmetric in the moments of inertia (a = 1/2/^, etc.), as it should be, and reduces to CD.56) in the symmetric top case. The derivation in ref. 155 is fairly involved, so that we restrict ourselves to a heuristic argument. Starting with the Hamiltonian CD.41), we write qr using the Baker- Hausdorff formula (see discussion following CD.2)) as qT = Tr C-paJ- e-0W? e'^l + <€} CD.58) w here <€ denotes a series of commutators involving Jx, Jv, and Jz, e.g. [_J\, Jy], [Jj, [Jy, Jf]], etc. These commutators can easily be evaluated using CD.20) and the fundamental commutation relation [Jx,Jy] = -!frJz CD.59) (and cyclic permutations thereof). Note the minus sign in the body-fixed commutation relation CD.59); the space-fixed components obey the relation [Jx, JY]= + ihJz (see also p. 461, ref. 154 and discussion of analogous classical Poisson bracket relations in ref. 26). Just as in § 3D.1 one expects the 1 in {1 +c€} to give rise to the classical limit if? = Tr'2(aaabac)~l and *# to generate the quantum corrections. The first part proves difficult to prove rigorously simply, and it is here we invoke a plausibility argument. We wish to establish157 q ? = Tr e'paJ2' e~pbJ' c'^. CD.60) We use the fact that, at high temperatures where the classical limit is valid, many high J values are thermally populated. For the large (classi- (classical) J-values, we assume the BJ+1J states \Jmn), for m and n = - J,. . . , +J, are uniformly distributed in J-space. Thus, in a spherical shell of thickness (J+ \)h-Jh = h, there are approximately 4J2 states, so that the density of states is approximately 4J2/4ir(JtiJh= I/tt^3. We write out the trace in qt = Tr[exp(—/3K"r,)] in the \Jmn) representation, assuming
3D CLASSICAL LIMIT AND QUANTUM CORRECTIONS 235 that at large J (classical limit) the non-commutative nature of Jx, Jy, and Jz can be ignored, so that \Jmn) is approximately an eigenstate of KrX, e~pK-> |Jmn> = e-3<aJ*+w'+CJ? \Jmn) CD.61) where J2., etc. in CD.61) are classical numerical values. Approximating the trace sum by an integral, with the density of states l/-n-ft3 as deter- determined above, thus gives f^<^+M5+CJ5) CD.62) J Tin where dJ = dJx dJy dJz. The result CD.62) is equal to the classical1 value (cf. transformation C.23) —> C.45)) K- CD.63) where d'co = d<p dO dx, do) = dc/> sin 0 dd dx, and dpw = dp^ dp0 dpx. It is straightforward but tedious to work out the h2 quantum corrections by evaluation of all the commutators ^ in CD.58). Fortunately a short cut is available, analogous to that used by Stripp and Kirkwood.155 By studying the dependence on the inertia parameters a, b, and c of the commutators and thermal averages (using (aJ2) = \kT, etc.) in CD.58) we find h2 corrections proportional to a and be/a, (with similar terms involving b and c). Thus, because of the symmetry in a, b, and c we must have qjq? = I + \(a + b + c) + n(bc/a + ca/b + able) CD.64) where A and /^ are constants to be determined. We determine these by requiring that CD.64) reduce to the symmetric top result CD.56) for a = b. This quickly gives the desired result CD.57). 3D. 3 Quantum corrections to equipartition theorem The O(ft2) quantum corrections to the classical equipartition theorem results are given in C.74) and C.75). Here we give a simple derivation156 based on the ft expansion of the partition function and free-energy given in §§3D.l and 6.9. We start with the general quantum relation158 for the A derivative of the free-energy A, where A is any parameter upon which the Hamiltonian
236 BASIC STATISTICAL MECHANICS 3D H depends: 8^n CD.65) dA W/ where {...) denotes a quantum thermal average (see (B.55) or (B.59)). Here H = K, + Kr + aU(rN(j}N) is the total Hamiltonian, where, as always, Kt = I, Ku =T.i pf/2mt and KT = L Kri =!,„ Jfj2Iia are the kinetic parts. Equation CD.65) is also valid classically; classical special cases were derived in §§4.4 and 6.4 (see also §§3.4.4 and 3.4.5). We first apply CD.65) with A = m,, the mass of molecule 1. Noting that dH/Cm, =rKp?/2m,)/r)m1 = — m'{lKa where KtI = p?/2m, is the transla- tional kinetic energy of molecule 1, we get </Ctl> = -m,—A. CD.66) Similarly, choosing A =/,x, we get <KrU> = -/u-^-A CD.67) where KTix = J\J2l{x is the rotational kinetic energy of molecule 1 about the body-fixed x axis. From CD.66) and CD.67) we can obtain the h series for (K,,) and (KTlK) from the Wigner-Kirkwood series for A. As discussed in §§ 3D.1 and 6.9, the latter series is A=Acl + ^f XI^^ + ^^I—II^-^T-I + O^) CD.68) where Fla and ria are the body-fixed principal a-axis components of the force F, and torque t( on molecule i, Iia is the aa principal component of the moment of inertia I;, (. . ,)c, denotes a classical thermal average, and /,,//d/v is a cyclic permutation of IJIyIz; i.e. IJIeIy = IJIyIz, IV/IJX, or /z//x/v for a = x, y, or z, respectively. Carrying out the differentiations in CD.66) and CD.67) gives to O(h2) CD.69) CD.70) Equation CD.70) agrees with C.75). Equation CD.69) yields C.74)
3E DIRECT CORRELATION FUNCTION EXPRESSIONS 237 when we note that, by symmetry, (Ktl) = 3(Ktlx) and (F^) = 3(Ffx)- For brevity, in C.74) and C.75) we have dropped the subscripts 'cl' on (.. .)cl, and '1' on m,, lu (Knx), (Kr]x), {t\), and {F\) since these are the same for all N (identical) molecules. The relation CD.70) is valid for an arbitrary asymmetric top. For symmetric and spherical tops, and for linear molecules, it simplifies (cf. § 6.9). In the latter case, for example, we get cI+ . Appendix 3E Direct correlation function expressions for some thermodynamic and structural properties159^61 Many of the properties considered in this book can be expressed in terms of either the total or the direct correlation functions. In this appendix we derive the direct correlation function expressions for the quantities h(; these are defined below (see CE.6)). The best known of these quantities is hQ, which is simply related to the compressibility, but hx and h2 are also measurable, and are angular correlation parameters arising in the theory of the dielectric and Kerr constants (see Chapter 10). Other property expressions in terms of the molecular direct correlation function c(ro),w2) will be considered in later chapters, for example in the Kirkwood-Buff theory of mixtures (Chapter 7) and in surface properties (Chapter 8). There are also expressions for some properties in terms of the site-site direct correlation functions cali(r), e.g. the expression for isothermal compressibility in Chapter 6. For atomic fluids the isothermal compressibility x can be written in terms of h(r) = g(r)-l (see C.113) and following remarks): = l + p\drh(r) = l + ph@) CE.2) where f(k) is the Fourier transform CB.3) of f(r), CE.3) Jdreik-7(r). [We use the notation /(k) here, rather than /(k), in order that /@) be unambiguous]. Because of the OZ relation CB.4) we have l + ph@) = [1 —pc(O)], so that we can also write x m terms of the direct correlation function c(r), CE.4) = [l-pc@)]-\ CE.5)
238 BASIC STATISTICAL MECHANICS 3E For molecular fluids composed of linear molecules the quantities ht all have physical significance where s712))o,ia>2 CE.6) = p(h(Oa>1a>2)P,(cosy12)L,a>2 CE.7) and 7,2 is the angle between the symmetry axes of the pair of linear molecules (see Fig. A.4). As noted above, hih hl, and h2 are related to the compressibility A + h0 = pkT\), dielectric, and Kerr constants, respec- respectively. Below we shall prove that hL can be written in terms of the direct correlation function c(ra>,o>2), l+hl=(l-c,)-1 CE.8) where c,=p\ dr<c(ra>iaJ)P|(cos = p(c@oJlaJ)P1(cos 7.2)>oW CE.9) Alternatively, CE.8) can be written as (l + h,)(l-c,)=l CE.10) or h^d + cA CE.11) or 1-c, CE.12) As discussed below (see ref. 162) the relations CE.8) and CE.10)- CE.12) are not valid for / = 1 in the presence of dipolar interactions. The various forms CE.8), CE.10)-CE. 12) correspond to the various possible structures of the OZ relation (see Chapter 5). We also note that h( can be expressed in terms of the space-fixed spherical harmonic coefficients (h(ljl2l;r) or /i(/,/2(;/c) for linear molecules) of h(ra>io>2) or h(k(x>l(x>2), respectively (see C.143), C.157)); for linear molecules we have h = (~I{4ttT\21 + l)-ip f drh(»0; r) CE.13) = (-)ID7r)-'B/+l)-5ph(H0;0) CE.14) with identical relations between c, and c(U0; r) and c(U0; 0). The relation
3E DIRECT CORRELATION FUNCTION EXPRESSIONS 239 CE.13) is derived by substituting the expansion corresponding to C.143) for h(ra)!a>2), and the spherical harmonic addition theorem (A.33) 4t7 P,(cos 7l2) =—— Y YIm(oI)*Ylm(o>2) CE.15) into CE.6) and using the orthogonality relation (A.27) together with (A.38) for the Y,m, the sum rule (A.145), and the relation (A.139) between the CG coefficient and the 3/ symbol. The relation CE.14) is then obtained using CB.43) and the fact that /0@) = 1. The simplest method of deriving CE.11) is to consider the limit k —»0 of the harmonic form C.158) of the OZ equation. In the limit k —» 0, the quantity f(k(x>1a>2) must be independent of cok (the direction of k), so that the expansion coefficient f(lil2l; k = 0) in C.157) must vanish162 unless / = 0. We then must also have („ = l2 in order that /@a>jOJ) be invariant under rotations (i.e. depend only on the angle between o)t and aJ).163 As a result, we find that the OZ relation C.158) decouples in this limit, h(U0; 0) = c(«0; 0) + D-n-)^p(-)!B/+ l)^c(//0; 0)h(U0; 0) CE.16) where (A.285) has been used to evaluate the 6/ symbol in C.158). In view of CE.14), we see that CE.11) follows immediately. The relation CE.8) can also be derived directly from the form C.156) of the OZ equation, rather than from the harmonic form C.158). We rewrite the result to be proved, CE.8) or CE.10), as )=l CE.17) where for brevity we put hA2) = h(k = 0, w1a>2), (•••);—(¦••)„,, and P,A2) = Pi(cos 712). Equation CE.17) can be multiplied out and rear- rearranged to give <hA2)P1A2)>12 = <cA2)PIA2)>12 + p<cA2)P1A2)>12<hA2)P1A2))I2. CE.18) We compare CE.18) to <fiA2)P,A2)>12 = <cA2)P,A2)>12 + p<cA3)P,A2)hC2)>12, CE.19) which we obtain directly from the OZ equation C.156). To show the identity of CE.18) and CE.19) we use CE.15) to rewrite the last term in CE.19) as ^ cA3) Yfm(l)>,<fiC2)YImB)>2>3 CE.20) where yIm(l) = y[m(w1), etc. After averaging over aI; for an isotropic
240 BASIC STATISTICAL MECHANICS fluid (c(\3)Y*m(\))x is independent of the direction aK. Similarly (hC2)Y|mB)J is independent of o>3. Thus, we can drop the redundant averaging over co3 in CE.20). Also, choosing the polar axis Z along co3 in (c(\3)Y*m{l))i, we see that this quantity vanishes for m/0, since cA3) depends only on the polar angle 6X between u>x and oK (and not on the azimuth angle c/>,), and (exP("r'c/I))rf)] =0 for m/0. Using (A.2), we see that CE.20) and CE.19) give (hA2)P1A2)>12 = <cA2)PIA2)>12 + p<cA3)P,A3)>,<hC2)P1C2)>2. CE.21) Comparing CE.21) and CE.18), we see they are identical when we remember that the fluid isotropy enables us to apply an orientational averaging over co3 to each member of the product term in CE.21). This completes the derivation, which is seen to be based directly on the isotropy of the fluid. References and notes 1. Hill, T. L. Statistical mechanics, Chapters 2, 3, McGraw-Hill, New York A956). 2. Hirschfelder, J. O., Curtiss, C. F., and Bird. R. B. Molecular theory of gases and liquids. Chapter 2, Wiley, New York A954). 3. Hill. T. L. Introduction to statistical thermodynamics, Chapters 1, 2, Addison-Wesley, Reading, Massachusetts (I960). 4. Reed. T. M. and Gubbins, K. E. Applied statistical mechanics, Chapter 2, Appendix B, McGraw-Hill, New York A973). 5. McQuarrie, D. A. Statistical mechanics, Harper and Row, New York A973). 6. Uhlenbeck, G. E. and Ford, G. W. Lectures in statistical mechanics, p. 21, 23. American Mathematical Society, Providence, Rhode Island A963). 7. Schroclingcr. E. Statistical thermodynamics, 2nd edn, p. 15, Cambridge University Press A952). S. fowler, R. and Guggenheim, E. A. Statistical thermodynamics, p. 191, Cambridge University Press, Cambridge A939). 9. Denbigh. K. G. Chemical equilibrium, 4th edn, Chapter 13, Cambridge University Press. Cambridge A981). 1A. Wilks, J. The third law of thermodynamics, Clarendon Press, Oxford A961); also in Physical chemistry, an advanced treatise, Vol. 1 (ed. W. Jost) p. 437, Academic Press, New York A971). 1 1. Stuart, E. B., Brainard, A. J., and Gal-Or, B. (ed.) A critical review of ther- thermodynamics. Mono Book Corp., Baltimore A970). See the articles by R. B. Griffiths and P. M. Quay. 12. Callen, H. B., in Modern developments in thermodynamics (ed. B. Gal-Or), p. 201, Wiley, New York A974).
REFERENCES AND NOTES 241 f3. Klein, M. J. Scuolo Internazionale di Fisica "Enrico Fermi", Corso X, Varenna, Italy, p. 1 A959). 14. Casimir, H. B. G. Z. Phys. 171, 246 A963). 15. ter Haar, D. Elements of thermostatics, 2nd edn, p. 291, Holt, Rinehard & Winston, New York A966). 16. Huang, K. Statistical mechanics, p. 191, Wiley, New York A963). 17. Hill, T. L., ref. 1, p. 91. 18. Mori, H., Oppenheim, I., and Ross, J. Studies in statistical mechanics, (ed. J. de Boer and G. E. Uhlenbeck) Vol. 1, pp. 235-4, North-Holland, Amsterdam A962). 19. Hill, T. L., ref. 1, p. 56. 20. Herzberg, G. Spectra of diatomic molecules, 2nd edn. Van Nostrand, Princeton, New Jersey A950). 21. Herzberg, G. Infrared and Raman spectra of polyatomic molecules, Van Nostrand, Princeton, New Jersey A945). 22. Goldstein, H. Classical mechanics, 2nd edn, p. 201 ff, Addison-Wesley, Reading, Massachusetts A980). 23. Mayer, J. E. and Mayer, M. G. Statistical mechanics, 2nd edn, pp. 198, 199, Wiley, New York A977). 24. Wilson, E. B., Decius, J. C, and Cross, P. C. Molecular vibrations, pp. 281, 282, McGraw-Hill, New York A955) [Dover edn 1980] 25. Van Vieck, J. H. The theory of electric and magnetic susceptibilities, p. 34, Oxford University Press, Oxford A932). 26. See ref. 18 of Appendix A. The relations are {Jx, Jy}= -Jz, and cyclic permutations, where { , } denotes the Poisson bracket. This relation is sometimes called 'anomalous' (see Van Vleck, J. H., Rev. mod. Phys. 23, 213 A951)), since the sign is changed from the relation satisfied by space-fixed components, {JX,JY} = J7. The physical interpretation is, in fact, straightforward (see (A. 18) and discussion there, where the corres- corresponding commutation relations for the angular momentum operators are discussed). The 'mixed' Poisson bracket relations are {Jx, Jx} = \JX, /y}etc. = 0. 27. See, for example: Parzen, E. Modern probability theory and its applications, p. 330, Wiley, New York A960); Hildebrand, F. B. Advanced calculus for applications, p. 340, Prentice-Hall, Englewood Cliffs A962); Jeffreys, H. and Jeffreys, B. S. Methods of mathematical physics, 3rd edn, p. 182, Cambridge University Press, Cambridge A956). The two variable case is discussed in detail in Hildebrand and Jeffreys and Jeffreys for a general function, and from the point of view of probability distribution functions in Rice. O. K. Statistical mechanics, thermodynamics and kinetics, p. 19, Freeman, San Francisco A966). 28. Bearman, R. J. Mol. Phys. 34, 1687 A977). 29. McQuarrie, D. A., ref. 5, pp. 100-8, 135. 30. Hill. T. L., ref. 1, Chapter 6. 31. de Boer, J. Rep. Prog. Phys. 12, 305 A949). 32. Meeron, E. and Siegert, A. J. F. J. chem. Phys. 48, 3139 A968). See also Dcvore. J. A. and Mayer, J. E. J. chem. Phys. 70, 1821 A979). 33. Torric. G. and Patey, G. N. Mol. Phys. 34, 1623 A977). 34. Grundke, E. W. and Henderson, D. Mol. Phys. 24, 269 A972). 35. Hoover, W. G. and Poirier, J. C. J. chem. Phys. 37, 1041 A962). 36. Barboy, B. and Tenne, R. Mol. Phys. 31, 1749 A976). 37. Properties which cannot be expressed simply in terms of gA2), even for pairwise forces, include: (i) the entropy and free energy (see Chapter 6 for
242 BASIC STATISTICAL MECHANICS a coupling constant expression, or Morita, T. and Hiroike, K. Prog, theor. Phys. 25, 537 A961) for an expression involving an infinite series of integrals over g(l2)), (ii) the specific heat and thermal expansion coefficient (Egelstaff, P. A. Ann. Rev. phys. Chem. 24, 159 A973)), (iii) infrared and Raman spectral moments in the presence of collision-induced dipole moments and polarizabilities (Chapter 11). 38. Gubbins, K. E., Gray, C. G., Egelstaff, P. A., and Ananth, M. S. Mol. Phys. 25, 1353 A973). 39. Streett, W. B. and Tildesley, D. J. Proc. Roy. Soc. Lond. A355, 239 A977). 40. Haile, J. M., Gubbins, K. E., Streett, W. B., and Gray, C. G., un- unpublished data. 41. Wang, S. S., Gray, C. G., Egelstaff, P. A., and Gubbins, K. E. Chem. Phys. Lett. 21, 123 A973). 42. Patey, G. N. and Valleau, J. P. J. chem. Phys. 61, 534 A974). 43. Patey. G. N. and Valleau, J. P. J. chem. Phys. 64, 170 A976). 44. Lebowitz. J. L. and Percus, J. K. Phys. Rev. 122, 1675 A961). 45. Hiroike. K. J. Phys. Soc. Japan 32, 904 A972). 46. Buckingham, A. D. Disc. Faraday Soc. 43, 205 A967). 47. Buckingham, A. D. and Graham, C. Mol. Phys. 22, 335 A971). 48. Ramshaw, J. D., Schaefer, D. W., Waugh, J. S., and Deutch, J. M. J. chem. Phys. 54, 1239 A971). 49. Faber, T. E. and Luckhurst, G. R. Ann. Rept. Chem. Soc. 72, 31 A975). 50. We do not make the relation explicit in Chapter 10. For this see ref. 48. 51. Values of F4 have been obtained for liquid crystals from Raman and fluorescence polarization spectroscopies; see Pershan, P. S. in The molecu- molecular physics of liquid crystals (ed. G. R. Luckhurst and G. W. Gray), Academic Press, New York A981), and Kooyman, R. P. H., Levine, Y. K. and van der Meer, B. W. Chem. Phys. 60, 317 A981). 52. Ornstcin. L. S. and Zcrnike, F. Proc. Sect. Sci. K. ned. Akad. Wet. 17, 793 A914); reprinted in ref. 62. 53. Workman, H. and Fixman, M. J. chem. Phys. 58, 5024 A973). 54. Wertheim, M. S. J. chem. Phys. 65, 2377 A976). 55. Wertheim. M. S. J. math. Phys. 8, 927 A967). 56. Baxter, R. J., in Physical chemistry, an advanced treatise, Vol. 8A (ed. H. Eyring, D. Henderson, and W. Jost), Academic Press, New York A971). 57. Sullivan. D. E. and Stell, G. J. chem. Phys. 69, 5450 A978). 58. March, N. H. Liquid metals, Pergamon, Oxford A968); see also Enderby, J.. Gaskell, T., and March, N. H. Proc. Phys. Soc. 85, 217 A965). 59. Nelkin, M. and Ranganathan, S. Phys. Rev. 164, 222 A967); see also J. K. Percus in ref. 62. 60. Henderson, D. and Davison, S. G., in Physical chemistry, an advanced treatise, Vol. 2 (ed. H. Eyring, D. Henderson and W. Jost), p. 372. Academic Press, New York A967). 61. Rice. O. K., ref. 27, p. 349. 62. Frisch. H. L. and Lebowitz, J. L. (ed.), The equilibrium theory of classical fluids, Benjamin, New York A964). 63. Stell. G.. in Modern theoretical chemistry, Vol. 5. Statistical mechanics, Part A. Equilibrium techniques (ed. B. Berne), Plenum, New York A977). 64. Kuni. F. M. Phys. Lett. 26A, 305 A968). 65. Nienhuis. G. and Deutch, J. M. J. chem. Phys. 55, 4213 A971). 66. Chandler, D. J. chem. Phys. 67, 1113 A977); H0ye, J. S. and Stell, G. J. chem. Phys. 65, 18 A976).
REFERENCES AND NOTES 243 67. Henderson, D. Latin American School of Physics Lectures, Puerto Rico A978). 68. Rao, K. R. J. chem. Phys. 48, 2395 A968). 69. Chandler, D. and Andersen, H. C. J. chem. Phys. 57, 1930 A972). 70. H0ye, J. S. and Stell, G. X chem. Phys. 65, 18 A976). 71. H0ye, J. S. and Stell, G. J. chem. Phys. 66, 795 A977). 72. Hsu, C. S., Chandler, D. and Lowden, L. J. Chem. Phys. 14, 213 A976). 73. Quirke, N. and Tildesley, D. J. Mol. Phys. 45, 811 A982). 74. Ladanyi, B. M., Keyes, T., Tildesley, D. J., and Streett, W. B. Mol. Phys. 39, 645 A980). 75. Haile, J. M. Farad. Disc. Chem. Soc. 66, 75 A978). 76. Quirke and Tildesley73 have considered various modifications of SSA. These SSA modifications have only been tested for a few harmonic coefficients of gA2)~ those needed in calculating pressure and mean- squared torque-and only for diatomic LJ site-site potentials. 77. Henderson, R. L. Phys. Lett. 49A, 197 A974). 78. Steele, W. A. J. chem. Phys. 39, 3197 A963). 79. Hailc, J. M. and Gray, C. G. Chem. Phys. Lett. 76, 583 A980). 80. Monson, P. A. and Rigby, M. Mol. Phys. 38, 1699 A979). 81. Steele, W. A. Faraday Disc. Chem. Soc. 66, 138 A979). 82. Wang. S. S., EgelstafT, P. A., Gray, C. G., and Gubbins, K. E. Chem. Phys. Lett. 24, 453 0974). 83. Stiles, P. J. Chem. Phys. Lett. 30, 126 A975); Perram, J. W. and Stiles, P. J. Proc. Roy. Soc. Lond. A349, 125 A976). 84. Monson, P. A. and Steele, W. A. Mol. Phys. 49, 251 A983). 85. Blum, L. and Toruella, A. J. J. chem. Phys. 56, 303 A972); Blum, L. J. chem. Phys. 57, 1862 A972); 58, 3295 A973). We use a different notation from theirs for the /s, /s, ms, and ns. Our expansion coefficient (f) is related to theirs (/BT) by f(l,l2l- n,n2; k) = (-)''">+'D7r)'B/ + 1)-TUM; ntn2; kf. 86. Equations C.163)-C.165) have been derived independently by J. S. H0ye and G. Stell [J. chem. Phys. 66, 795 A977)] and by C. G. Gray [unpub- [unpublished, 1978; see also ref. 11 of EgelstafT, P. A. Farad. Disc. Chem. Soc. 66, 7 A978)], and probably by others [see, e.g., the various expressions derived for the closely related structure factor S(k) by Gubbins. K. E., Gray, C. G., EgelstafT, P. A., and Ananth, M. S. Mol. Phys. 25, 1353 A973)]. 87. Hill, T. L., ref. 1, Appendix 7. 88. Gubbins, K. E., Gray, C. G. and Egelstaff, P. A. Mol. Phys. 35, 315 A978). 89. Gibbs, J. W. Elementary principles in statistical mechanics, p. 201, Yale University Press, New Haven A902) (reprinted by Dover, 1960). 90. Fowler, R. H. Statistical mechanics, Chapter 20, Cambridge University Press, Cambridge A929). 91. Yvon. J. Actualites scientifiques et industrielles. No. 543, Hermann, Paris A937); Correlations and entropy in classical statistical mechanics, Pergamon Press. Cambridge A929). 92. Buff. F. P. and Brout, R. J. chem. Phys. 23, 458 A955); 33, 1417 A960); Buff. F. P. J. chem. Phys. 23, 419 A955); Buff, F. P. and Schindler, F. M. J. chem. Phys. 29, 1075 A958). 93. Landau. L. D. and Lifshitz, E. M. Statistical physics, 2nd edn, Chapter 12. Pergamon, Oxford A969). The quasi-thermodynamic approach to fluctuation
244 BASIC STATISTICAL MECHANICS theory used by Landau and Lifshitz is due to Einstein (A. Einstein, Ann. Physik, 33, 1275 A910)). 94. Schofield, P. Proc. Phys. Soc. 88, 149 A966). 95. In Appendix E we discuss another dynamical variable, 9' = — BH'h'IV, where H' contains a wall potential. 96. Kirkwood, J. G. and Buff, F. P. J. chem. Phys. 19, 774 A951). 97. Fluctuation relations in other ensembles (e.g. microcanonical, constant pressure, etc.) are given by: Ray, J. R. and Graben, H. W. Mol. Phys. 43, 1293 A981): Ray. J. R., Graben, H. W., and Haile, J. M. J. chem. Phys. 75, 4077 A981); Lado, F. J. chem. Phys. 75, 5461 A981). 9S. Hill, T. L., ref. 1., Chapter 4. 99. Klein, M. J. Physica, 26, 1073 (I960). There has been some controversy over the derivation of C.232) from C.231), as discussed by Klein. In part this controversy revolves around the question of whether the pressure fluctuations are a system property, like <(ApJ) or the pressure itself (as is implied by C.232)), or whether they depend on the nature of the forces between the molecules and the wall. The fluctuations in &' (see Appendix E and ref. 95 of this chapter) are given by kT ) [' '] ) [x,x ], i.e. by the first term of C.232). The quantities <(AS?J> and <(AS?'J> can both be measured in computer simulation. These fluctuations are both O(l//V) and arc therefore impossible to measure directly in real experi- experiments. The question of whether ((Ag^'J) is proportional to the fluctuation of the force on a wall is a subtle one (see Klein and also remarks in ref. 17 of Appendix E). 100. HgelstafF, P. A. The properties of liquid metals, p. 13, Proc. 2nd Int. Conf., Tokyo. 1972 (cd. S. Takcuchi), Taylor and Francis, London A973). 101. HgelstafF, P. A., Page, D. I., and Heard, C. R. T. J. Phys. C4, 1453 A971); Tcitsma. A. and Egelstaff, P. A. Phys. Rev. A21, 367 A980); EgelstafT, P. A., Teitsma, A., and Wang, S. S. Phys. Rev. A22, 1702 A980). 102. Winficld. D. J. and Egelstaff, P. A. Can. J. Phys. 51, 1965 A973). 103. Abe, R. Prog. Theor. Phys. 21, 421 A959). 104. Stell. G. Physica 29, 517 A963). 105. Yvon, J. La theorie statistique des fluides et Vequation d'etat, Act. Sci. et Indust. No. 203. Hermann, Paris A935); Born, M. and Green, H. S. Proc. Roy. Soc. A188, 10 A946); Bogoliubov, N. N. J. Phys. USSR 10, 256, 265 A946); sec also Bogoliubov, N. N. in Studies in statistical mechanics (cd. .1. deBoer and G. H. Uhlenbeck), Vol. 1, Part A, North-Holland, Amsterdam A962); both the earlier 1946 papers of Bogoliubov and the 1962 review are English translations of an earlier Russian work. The Kirkwood hierar- hierarchy [Kirkwood, J. G. J. chem. Phys. 3, 300 A935)] is equivalent, but not identical to C.245): see ref. 19g of Chapter 5. 106. Raveche, H. J. and Mountain, R. D., in Progress in liquid physics (cd. C. A. Croxton), p. 469, Wiley, New York A978). 107. Wilcox, R. M. J. math. Phys. 8, 962 A967). 108. Buff, F. P. J. chem. Phys. 23, 419 A955). 109. O'Connell, J. P. Mol. Phys. 20, 27 A971). 1 1A. Ref. 23. Chapter 13. 111. Hill, T. L. ref. 1, Chapter 5. 1 12. Hirschfelder, J. O., Curtiss, C. F., and Bird, R. B. ref. 2, Chapter 3. 1 13. Reed. T. M. and Gubbins, K. E. ref. 4, Chapter 7.
REFERENCES AND NOTES 245 114. Mason, E. A. and Spurling, T. H. The virial equation of state, Pergamon, Oxford A969). 115. van Kampen. N. G. Physica 27, 783 A961). 116. Ursell, H. D. Proc. Camb. Phil. Soc. 23, 635 A927). 117. Mayer, J. E. J. chem. Phys. 5, 67 A937). 118. Born, M. The mechanics of the atom, F. Ungar, New York A960). 119. Van Vleck, J. H. Rev. mod. Phys. 23, 213 A951). 120. Jepsen, D. W. and Friedman, H. L. J. chem. Phys. 38, 846 A963). 121. Wertheim, M. S. J. chem. Phys. 55, 4291 A971). 122. Gray, C. G. and Henderson, R. L. Can. J. Phys. 56, 571 A978); 57, 1605 A979). 123. The k-frame form of the OZ equation is identical to what is termed the "irreducible representation" in ref. 85. 124. Green, H. S. Proc. Roy. Soc. A189, 103 A947); see also Green, H. S. The molecular theory of fluids, p. 51, North-Holland, Amsterdam A952). This method was also derived independently by Bogoliiibov; see ref. 105. 125. The pressure dynamical variable is often written126127 as Sf>'=-dH'/dV, rather than as in CC.7). However, H' then includes a wall potential, whereas here the wall is handled through the integration limits (see Appendix E). 126. Reed, T. M. and Gubbins, K. E., ref. 4, p. 43. 127. Uhlenbeck, G. and Ford, G. W., ref. 6, p. 21. 128. Wigner, E. P., Phys. Rev. 40, 749 A932). 129. Kirkwood, J. G., J. chem. Phys. 1, 597 A933). 130. Ref. 93, p. 96. 131. Ref. 23, p. 444. 132. Ref. 5, p. 185 and references therein. 133. Friedmann, H., Adv. chem. Phys. 4, 225 A962); Physica 30, 921 A964). 134. St. Pierre, A. G. and Steele, W. A., Ann. Phys. (N.Y.) 52, 251 A969). 135. Hill, R. N., J. math. Phys. 9, 1534 A968); see also ref. 31 of Chapter 1, and Nienhuis, G., /. math. Phys. 11, 239 A970). 136. Powles, J. G. and Rickayzen, G., Mol. Phys. 38, 1875 A979). 137. One form of the Baker-Hausdorff identity is For a review of elegant methods of derivation, see ref. 107. A straightfor- straightforward if tedious method is to introduce a parameter A by and determine the coefficients C,, C2, ... by equating successive A deriva- derivatives of both sides at A = 0. 138. We note that Tr(AB) = ?„,<«! A \s){s\ B \a} in terms of the complete sets {\a)} and {|s)}, which follows by inserting unity A =?, \s)(s\, see the completeness relation (B.53)) between A and B in Tr( AB) = X« (a| AB \a). Note also that the product states |rN<wN>, etc. are used for convenience (see below CD.1)); we do not assume non-interacting molecules. We do ignore for now (see remark below CD. 18)) symmetriza- tion of the states arising from the boson/fermion character of the identical molecules, and also any possible intramolecule exchange effects (nuclear spin weights, etc.-see Fig. 3.1 and discussion). 139. Messiah, A. M., Quantum mechanics, Vol. 1, p. 306, Wiley, New York A961). 140. See refs. 2, 6, 18 of Appendix A, and also eqn (A.68).
246 BASIC STATISTICAL MECHANICS 141. Rcf. 139, p. 207. The identity is established trivially by writing out explicitly both sides. 142. Ref. 139, p. 206. 143. Ref. 139, pp. 207, 208. 143a. Alternatively, if we replace the first term in CD.23) giving 2m i in place of CD.24), a more detailed argument (e.g. writing out the trace explicitly in the momentum representation) is then needed to obtain the contribution ~(/r/2m) ?, V2(9/ to the O(h2) correction. 144. See (A.70). The differential operator L of (A.69), (A.70) is related to V,,, by L= -iV^,. We thus also have L2 =-V,2,, where L2 is given explicitly by (A.71). 145. Ref. 93, A958 edn) p. 219. 146. Mulholland, H. P., Proc. Camb. Phil. Soc. 24, 280 A928). 147. See the first edn of ref. 23, p. 153 and p. 431. 148. Jeffreys. H. and Jeffreys, B. S., Methods of mathematical physics, Cam- Cambridge University Press, Cambridge A946), p. 255; Wilf, H. S., Mathema- Mathematics for the physical sciences, Wiley, New York, A962) (Dover reprint, 1978) p. 119; Knopp, K., Theory and application of infinite series, Blackie, London, 2nd edn A951), p. 518. 149. See Landau, L. D. and Lifshitz, B. M., Quantum Mechanics, p. 373, Pergamon, Oxford A958) or refs. 17, 18, 20 of Appendix A, or Kroto (ref. 154 below). (A brief discussion is given on p. 461.) 150. it should perhaps be noted that, despite the fact that correction terms to the integral in CD.44) vanish in this case, the sum CD.51) is not exactly equal to CD.52). This is because the Euler-Maclaurin series CD.44) is an asymptotic expansion,MR which in this case misses (non-analytic) terms which vanish faster than powers of a = O(h2), e.g. exp(—a"'). To obtain the latter type of terms'^' (which do not concern us here) one can use the Poisson summation formula152 t f(n)= ? fB7rm) where /(k) = f, <ixc'kxf(x) is the Fourier transform of f(x). Non-analytic intramolecular exchange terms, if present,'" can be handled by similar methods.1'" 151. Kayser. R. and Kilpatrick, J. E., J. chem. Phys. 68, 1511 A978); 63, 5216 A975). 152. Courant, R. and Hilbert, D., Methods of mathematical physics, Vol. 1, Wiley, New York A953), p. 76. Morse, P. M. and Feshbach, H., Methods of theoretical physics, Vol. 1, McGraw-Hill, New York A963) p. 466. 153. Viney, I. E., Proc. Camb. Phil. Soc, 29, 142, 407 A933); Kassel, L. S., J. chem. Phys. 1, 576 A933). 154. See ref. 21, or Kroto, H. W., Molecular rotation spectra, Wiley, New York A974). 155. Stripp, K. F. and Kirkwood, J. G., J. chem. Phys. 19, 1131 A951). See also ref. 134.
REFERENCES AND NOTES 247 156. Byrns, F. L. and Mazo, R. M., J. chem. Phys. 47, 2007 A967); Gordon, R. G.. J. chem. Phys. 41, 1819 A964). 157. An alternative heuristic derivation of Tr e'1""—* J(d'o) dpjh*) e~pK" is to simply note the general one-particle result Tre~BH-H> J(d3p d3q/h3) e""CH<p''), where (p, q) are the canonical variables (see, e.g., Balescu, R., Equilibrium and non-equilibrium statistical mechanics, Wiley, New York A975), p. 120). 158. Equation CD.65) is sometimes referred to as the quantum statistical Hellmann-Feynman theorem, as it can be derived from the quantum dynamical Hellmann-Feynman theorem [ref. 7 of Appendix C] d{n\ H\n)/dk ={n\ dH/dk \n) by averaging this relation with Boltzmann weight Pn sexp(-KE,,)/O (see (B.55)). Alternatively, CD.65) can be de- derived more simply and directly by differentiating the partition function O=Trexp(-|3H). This gives dQ/dk = -13 Tr[exp(-CH) dH/dk]. Hence we have Q dQ/dk =-C 7r[PdH/dk] = -p{dH/dk), where P = exp(-|3H)/Q (see (B.59)). Since A = ~kT\n Q, we get dA/dk =(dH/dk). [Although the operators dH/dk and H (and therefore exp(-j3H)) do not commute in general, this causes no problems here (i.e. we need not worry whether to write d exp(-|3H)/flA as -|3 exp(-pH)(dH/dk) or as -j3OH/3A)exp(-j3H)) since the product exp(-j3H) dH/dk occurs under the trace, and Tr(AB) = Tr(BA) in general (see (B.63)). Compare remarks in § 3.4.5.] 159. Ramshaw, J. D. J. chem. Phys. 57, 2684 A972). 160. H0ye, J. S. and Stell, G. J. chem. Phys. 61, 562 A974). 161. Ramshaw, J. D. J. chem. Phys. 66, 3134 A977). 162. This vanishing for k —¦ 0 can also be seen formally since (see CB.43)) f(lJ7U;k)« j d^hikrmitUir) Jo and j,(kr)-^- O(fcr)', which vanishes for k —»0, except for ( = 0. An excep- exception occurs if f(l,l2l; r) is long-ranged, since the integral may then diverge, and /(/,J2/; fc —»U) may remain finite. This is the case for dipole-dipole interaction, where h(\\2; r) and cA12;r) vary at large r as r~3, so that ?A12; 0) and cA12;0)^0. As a result, the relation CE.8) is not valid for / = 1 for polar fluids. The correct relation for this case is derived in Chapter 10. 163. Alternatively, we can note that the CG coefficient C{lll2Q;m-im2Q) that appears in C.157) is zero unless (, = l2.
4 PERTURBATION THEORY It will thus be seen that I have never been able to consider that the last word had been said about the equation of state, and I have continually returned to it during other studies. Johannes D. van der Waals A910), Nobel Prize Lecture In perturbation theory1 one relates the properties (e.g. the distribution functions or free energy) of the real system, for which the intermolecular potential energy is aU(rNwN), to those of a reference system where the potential is %0(rNo)N), usually by an expansion in powers of the perturba- perturbation potential aU.-l=aU-°U.{). The first-order, second-order, etc. perturba- perturbation terms then involve both %! and the distribution functions for the reference system. In the sections that follow we first briefly discuss the historical back- background of perturbation theory (§4.1). As a simple example we then derive the u-expansion for the free energy (§ 4.2). This is followed by general expansions for the angular pair correlation function (§ 4.3) and the free energy (§ 4.4). The expansions developed in these latter sections are for an arbitrary reference system and arbitrary perturbation parame- parameter. We next consider some particular choices of reference system and perturbation parameter. We first consider the u-expansion (§ 4.5) further, for a potential having both attractive and repulsive parts, and also the /-expansion (§ 4.6) which uses a different reference fluid. This is followed by a description of methods for expanding the system properties for an anisotropic repulsive potential about those for a hard sphere potential (§ 4.7), and then the expansion for a general potential (attractive and repulsive parts) about a non-spherical reference potential (§ 4.8). We also discuss two approximation methods based on perturbation theory, the effective central potential method (§ 4.9), and generalized van der Waals models (§4.11). Non-additive potential effects are discussed in §4.10. Perturbation theories have been the subject of reviews for both atomic2 and molecular82 liquids. 4.1 Brief historical outline Perturbation expansions are closely related to inverse temperature expan- expansions, which have been a standard technique since the earliest days of statistical mechanics. As examples, we mention some early work on gases
4.1 BRIEF HISTORICAL OUTLINE 249 (virial coefficients,13"9 dielectric20'21 and Kerr22 constants), and solids (lattice phonon specific heat,23 polar lattice thermodynamics, electric and magnetic susceptibility properties,2425 alloy order-disorder transitions,26 ferromagnetism,27"9 and diamagnetism30). In considering perturbation theory for liquids it is convenient first to discuss the historical development for atomic liquids. The parallel de- development for molecular liquids is discussed below. The first use of perturbation theory for liquids appears to have been by Longuet- Higgins31 in 1951, who related the free energy of a non-ideal solution to that for an ideal one; all molecular pairs were assumed to be conformal, i.e., to obey the same two-parameter intermolecular potential law. The free energy was expanded in powers of <&,, the potential difference between the real and the ideal solution. The resulting theory worked well only for weakly non-ideal solutions.32 A few years later, Zwanzig34 showed how the free energy of a dense atomic fluid interacting with some isotropic pair potential u(r) could be related to that of a hard sphere fluid (with potential h0) by writing the potential as u — u^+u^, and expanding in powers of {uJkT). This was used to calculate the equation of state for a Lennard-Jones A2, 6) fluid. Good results were obtained at high temper- temperatures, although the calculations were sensitive to the choice of the hard-sphere diameter. The Zwanzig method does not take proper account of the "softness" of the repulsive part of the intermolecular potential (thus the perturbation will become infinite in magnitude for sufficiently small r). An improved method of doing this was proposed by Rowlin- son,35 who used an expansion in powers of a softness parameter, n~\ about n'1 = 0 (corresponding to a hard-sphere fluid) for a potential of the form u(r) = /(/¦""). This method gives good results for the repulsive, but not for the attractive, part of the potential. Subsequently, several workers3*^8 attempted to combine the strong points of the Zwanzig and Rowlinson approaches. The first successful approach for liquids was that of Barker and Henderson (BHK7 who showed that a second-order theory gave quantitative results for the thermodynamic properties of a Lennard- Jones (LJ) liquid. Even more rapidly convergent results were later ob- obtained by Weeks, Chandler and Andersen (WCAK8 by using a somewhat different reference system; in this case first-order theory gives good results for the dense liquid.39 The BH and WCA choices for the reference potential are compared in Fig. 4.1. The WCA choice corresponds to the repulsive part of the full potential. In both the BH and WCA theories the reference fluid properties are related to those of a fluid of hard spheres of diameter d, the prescriptions for d being somewhat different in the two cases.40 The development of perturbation expansions for the distribution func- functions has followed along similar lines to those described above for the free
250 PERTURBATION THEORY (b) (c) He 4.1. (a) The full potential for a pair of spherical molecules; (b) the Barker-Henuersor reference potential, ut)~u for r<cr, u(( —0 for r>cr; (c) the Weeks—Chandler-Anderset, reference potential, uo=u + e, for r<rm, uo = 0 for r>rm. energy. Kirkwood et al.41 applied perturbation theory to solve the ap- approximate Born-Green integral equation for the pair correlation function g(r). An exact expression for the first-order perturbation term in the expansion of the distribution function of order h was first derived by Buff and Schindler.42 Subsequently, Weeks et al.3S presented a theory of g(r) for fluids with purely repulsive forces, and suggested that the effect of attractive forces was negligible at high densities. Perturbation expansions for g(r) for a fluid having both attractive and repulsive potentials have been studied by Smith et al.43 and by Gubbins et al.44 Perturbation theory has also been used to calculate quantum corrections (see references given at the end of § 1.2.2) and the effect of three-body forces on the distribu- distribution functions (see §4.10). The pair correlation functions for the LJ potential and for various reference potentials used for atomic fluids are compared in Figs 4.2 to 4.4. For many atomic liquids near the triple point (see Fig. 4.2) the liquid structure is largely determined by the repulsive part of the intermolecular forces.45 From Fig 4.2 it is seen that g(r) for the dense LJ fluid, in which both attractive and repulsive forces are present, is very similar to g(r) for the WCA potential, where only the repulsive part of the forces is included; it is because of this similarity that the first-order WCA pertur- perturbation theory is successful (for a more precise discussion see ref. 96). Also included in Fig. 4.2 are the g(r) curves for the BH and McQuarrie-Katz36 reference systems; the latter consists of just the r~12 term from the LJ potential. These are appreciably different from the LJ g(r), and conse- consequently perturbation expansions about the BH and McQuarrie-Katz reference systems are less rapidly convergent for dense liquids than for the WCA reference.60 Although the attractive forces have little effect on
4.1 BRIEF HISTORICAL OUTLINE 251 3.0 2.0 - 1.0 U 1.0 2.0 3.0 Fig. 4.2. Comparison of pair correlation functions for a liquid state condition near the triple point (p* = pa* = 0.80, T* = kT/e = 0.720), from molecular dynamics (MD) simula- simulation: Lennard-Jones (LJ) fluid; Barker-Henderson (BH)reference;—•—¦— Weeks-Chandler-Andersen (WCA) reference; McQuarrie-Katz (MK) reference. Here r* = rla. g(r) at high densities,61 they do make large contributions to the ther- modynamic functions and dynamical properties; for example, the attrac- attractive forces are entirely62 responsible for fluid-fluid (gas-liquid, liquid- liquid, or "gas-gas") phase transitions, which are absent when the forces are purely repulsive. For less dense liquids (e.g. near the critical point), and particularly for gases, the attractive forces have a larger effect on g(r); since the molecules are no longer tightly packed, the attractive forces can cause significant structural effects. This is shown for a state condition near the gas-liquid critical point in Fig. 4.3, and for the dilute gas in Fig. 4.4. For gases at low temperatures none of the reference potentials proposed for liquids yields dilute gas g(r)s bearing any resemblance to that for the full potential. The brief historical outline given above is concerned with the develop- development of perturbation theory for atomic fluids. The first rigorous applica- application of perturbation theory to molecular fluids seems to have been made in 1951 by Barker,6364 who expanded the partition function for a polar
252 PERTURBATION THEORY 4.1 2.0 ¦ i 1.0 - 3.0 I-Ki 4 ?>. Comparison of pair correlation functions for a state condition near the critical point (p* = 0.35. T* = 1 350) from molecular dynamics. Key as in Fig. 4.2. fluid about that for a fluid of isotropic molecules. The first-order pertur- perturbation term vanishes in this series (neglecting induction interactions); Barker derived the second-order free energy term for the dipole-dipole interaction, and showed that it can be expressed in terms of ther- modynamic properties of the reference fluid. This was followed by a treatment of polar liquids by Pople65 in which perturbation terms were 4 ¦ 0 0.8 1.0 1.2 1.4 1.6 1.8 2.0 r* ]:i<i. 4.4. The pair correlation function in the low density limit, where g — e tiu, for the and various reference potentials at T* = 0.719. Key as in Fig. 4.2.
4.1 BRIEF HISTORICAL OUTLINE 253 evaluated within the framework of the cell theory; expressions for the second-order perturbation terms were also worked out for multipole interactions up to quadrupole terms. Barker66 extended this work to include polarizable dipoles, again using lattice theory to evaluate pertur- perturbation terms to second order. The expression for A2, the second-order contribution to the Helmholtz free energy, was first written in terms of reference fluid distribution functions for multipole-like potentials by Pople67 in 1953. This paper was the direct forerunner of Pople's much- quoted 1954 paper,19'68 in which he gave the rigorous expression for A2 for an intermolecular potential of general form. This involves two- and three-body integrals over reference system distribution functions; the three-body term vanishes for multipole-like potentials. In parallel with these developments, Cook and Rowlinson69 in 1953 pointed out that for certain types of orientation-dependent potential (e.g. dipole-dipole and certain other dispersion and overlap potentials) the free energy can be equated, to lowest order in the perturbation, to that of a fluid of isotropic molecules in which the molecules interact with a temperature-dependent effective potential. Extensive comparisons with experiment were made.6970 The choice of the isotropic potential used in the Barker-Pople expan- expansion (which we shall refer to as the u-expansion later in this chapter) causes the first-order term to vanish, and also results in simplification of the higher order terms. Comparisons with experimental data were limited to second virial coefficient calculations, because of the lack of any satisfactory theory or simulation results for dense fluids of isotropic molecules at that time. Later, using an approach similar to Pople's, perturbation theory for the angular pair correlation function g(raI&J) was initiated by Gubbins and Gray.71 The Barker-Pople approach for the free energy has also been extended, and detailed comparisons have been made with both computer simulations and experiment911; a Pade approximant to the free energy series proposed by Stell et al.72 is particularly success- successful. Further details of this approach are given in §4.5. Several other approaches have been suggested for molecular fluids. Some of these retain the use of an isotropic reference potential aU{), but adopt either a different choice of <%o3'74 or °f the expansion proce- procedure.7576 An alternative approach is to expand about some non-spherical reference potential.7778783 Such an expansion is often more rapidly convergent than the Barker-Pople series, but the perturbation terms are more difficult to evaluate numerically. For simplified model intermolecu- intermolecular potentials of the atom-atom or site-site form (see § 2.1.2), perturba- perturbation theory for the site-site pair correlation functions ga0(r) can be developed79 along similar lines to those described above for atomic fluids. Details of these various approaches, and references to more recent work on them, are given in §§ 4.6-4.8.
254 PERTURBATION THEORY 4.2 Although the evidence that the repulsive forces largely determine the structure of dense atomic liquids of the inert-gas type seems conclusive39 (as shown in Fig. 4.2), the situation is less clear-cut for the case of molecular liquids. Molecular liquids have an orientational structure, as well as a positional one, and in general these two structures are corre- correlated. Computer simulation results1180"82 suggest that both the repulsive and attractive anisotropic forces affect the orientational structure, at least for simple model fluids; for real fluids the relative importance of the long- range and short-range forces may differ from case to case.83 (See Figs 3.3(a) and 3.4(a), where the effects of both molecular shape and quad- rupolar forces on g(ru>lu>2) are shown.) Attempts to understand experi- experimental data for the structure factor S(fc) of molecular liquids8084"90 in terms of long-range versus short-range forces have been hampered by the difficulty in obtaining sufficiently precise data, and the lack of a successful theory that accounts for both the long-range and short-range forces. This is discussed in more detail in Chapter 9. As is also the case for atomic fluids, the thermodynamic properties are, of course, extremely sensitive to the long-range forces. 4.2 A simple example: the u-expansion for free energy As an example of perturbation theory we first give a simplified91 deriva- derivation of the Barker-Pople1963 expansion of the free energy. This series is considered in more detail in § 4.5. The configurational Helmholtz free energy is given by Ac = -/cTln Qc, with Qc given by C.82). We consider a total intermolecular potential energy °UX, <%x(rNWN)=<%0(rN) + A<%a(rNWN) D.1) where %, is the unweighted average of aU = aU.K = 1 over the orientations (see C.102)), N))<uN D.2) so that <%a(rNwN))<u~ = 0. D.3) We consider the ratio QcJQc(, of canonical partition functions, D.4) Ocx/Oco =.-<e-)o J drN do>N e-0*.< where Qc0 is the value corresponding to %, and (.. .H denotes a reference
4.2 THE u-EXPANSION FOR FREE ENERGY 255 system ensemble average. Hence AcK —Ac0 = —kTln(QcJQc0) is given by 3 ] D.5) where we have used D.3) to eliminate the linear term. Since the series [1 + O(A2) + O(A3) + ...] in D.5) contains no linear term, we have D.6) to third order in A. Hence, on substituting D.6) in D.5) and putting A = 1 to regain the real system (potential <%), we have ... D.7) where A2 = 4/3(<%2H, D.8) A3=l(i2(tUlH. D.9) We note that in the approximation D.6) we assume the quantity [O(A2) + O(A3)] is less than unity, and D.6) is, therefore, only valid for small values of A. Since A is extensive, the quantities (<%2H and (<%3H wiU be proportional to N, so that [O(A2) + O(A3)] will be small compared to unity only for very small A. However, because the Taylor series in A for ln(exp(-/3A<%a)H is unique, the result D.7) we have derived will be valid for larger values of A. If we now express <%a as a sum of pair potentials, and identify (as pair terms, triplet terms, etc.) the various possible types of terms in <%2 and °Ul, we can write D.8) and D.9) in terms of reference fluid distribution functions (see § 4.5). We note that the ensemble averages that appear in D.8) and D.9) are canonical ones for a system of N molecules in volume V, but they are unchanged on taking the thermodynamic limit. For higher order terms (A4, etc.) taking the thermodynamic limit is less straightfor- straightforward,91 and it is preferable to use the grand canonical ensemble (see Appendix 4A).
256 PERTURBATION THEORY 4.3 4.3 General expansion for the angular pair correlation function We consider a uniform fluid in which the potential energy is °UK, where A is some perturbation parameter (to be precisely defined later) such that <Uk =,> = %>, D.10) aUx = i=aU. D.11) Here %, and °U are the potentials in the reference and real systems, respectively. We now expand the angular pair correlation function g(r<O|O>2) in a Taylor series92 about A = 0 at fixed temperature and density94 (cf. Appendix 4A), g (tw i u>2) = go(rw2to2) + g i (rw! <u2) + g2(rw, «2) +... D.12) where g0 is the function for the reference system, g, = (dg/dAHpX=o is the first-order perturbation term,95 etc. The expression for g, is derived in Appendix 4A.4, and for the pairwise additive case it is -!/3p2 f dr3 dr4(P~^) [goA234)- goA2)goC4)]) \\ 3A /o 'o + ipfdr14dr4,((^4i)[g()C45)-g0C4)]) ] D.13) J W dA. /o I u,^u<oj where g,A2) = g,(r12a)i<u2), g()A2) = go(ri2WiW2), etc., and (du(ij)/d\H is the value of du(ij)/d\ at A = 0. Here (.. .)„, denotes an unweighted average over the orientations shown (see C.102)), and g0 is a correlation function for the reference system. An alternative perturbation expansion is obtained by expanding the indirect correlation function y(r,2&Ito2) defined by C.99). In place of D.12) one obtains - • •] D.14) where /3(———1 goGrwiw^ + g^rw!^) L \ 3A /o J D.15) with g, given by D.13). Perturbation theory for the direct correlation function c{tw-iCL>2) is discussed in §5.3.1.
4.4 GENERAL EXPANSION FOR THE FREE ENERGY 257 The above expression for g! is in terms of grand canonical averages, so that the /i-body correlation functions appearing in D.13) tend to unity when the h molecules form two groups that are far apart (see § 3.1.4 for a discussion of this point). Thus, there is no problem in evaluating the integral over [goA234)-goA2)goC4)] that appears in D.13); when the molecular pairs 12 and 34 are far apart this integrand vanishes, and we can calculate the integral using approximate theories (e.g. the superposi- superposition approximation, Percus-Yevick theory, etc.). Had g! been given in terms of canonical averages the resulting expression would lack the final fluctuation term that appears in D.13); the canonical correlation functions that appear in this expression no longer tend to unity at large separation of two subgroups of the h molecules, but tend to unity plus a term O(N~') instead (see C.109)). This O(N-1) term, when correctly evaluated for the term involving the four-body canonical correlation function, yields the final fluctuation term in D.13); this is shown in C.112). Early studies of perturbation theory used the canonical ensemble,34 so that the expres- expression for the second-order free energy term (corresponding to the first- order g[ term-see next section), while correct, was not suitable for numerical evaluation in general. 4.4 General expansion for the free energy Because of its importance in later chapters we now briefly consider the general perturbation expansion for the configurational Helmholtz free energy, Ac. This is96 Ac = A0 + Al+A2 + A3 + ... D.16) where Ao is the reference value, Ax = CAc/3AHVpjA=o is the first-order perturbation term, and so on. To evaluate these perturbation terms it is most convenient to use the canonical ensemble expression for Ac (cf. C.11)), Ac = -fcTlnQc, D.17) where Qc is the configurational partition function, given by C.82). The first derivative of Ac is, assuming pairwise additivity, D.18) where Zc = N!flNQc is given by C.53).
258 PERTURBATION THEORY 4.5 Thus we have96 A1 = ip2fdr1dr2((^)g0A2)) . D.19) J W dA /o ' <ol0>2 By further differentiating D.18) and taking the A —»0 limit it is easy to show that D-20) )g2A2)) . D.21) Jo ' o,,o>2 Thus, the term of order fc in D.16) involves all orders of g(r<Hi<o2) up to (fc-1) in D.12). Explicit expressions for A2, A3, etc. are obtained by substituting the expressions for g,, g2, etc. in the above equations. This is done for particular choices of reference systems in some of the sections that follow. 4.5 Further development of the u-expansion Barker,6366 Pople,19-65-67-68 and Cook and Rowlinson69 were the first to propose a perturbation expansion for molecular fluids. They considered the free energy, and expanded in powers of the anisotropic potential energy, <%a. For the pairwise additive case the expansion corresponds to the parameterization D.22) and Pople defines the isotropic reference potential u0 to be97 u0(r) = (u(rwi0J))o,^2 = -x2 I doyldto2u(T(o1oJ) D.23) where Cl is Air for linear or 8-n-2 for non-linear molecules, respectively. This same parameterization and reference system were first used for the angular pair correlation function by Gubbins and Gray.71 We refer to this expansion scheme as the u-expansion. From D.22) and D.23) it follows
4.5 FURTHER DEVELOPMENT OF THE u-EXPANSION 259 (^^)) D.24) and it is this property, together with the fact that the second and higher order derivatives of u with respect to A vanish, that leads to great simplification of the general expressions obtained in §§ 4.3 and 4.4. Thus, the most complicated terms in the expression D.13) for gi vanish, and the first-order contribution to the angular pair correlation function becomes 3 D.25) while the first three perturbation terms for the Helmholtz free energy are obtained from D.19)-D.21) as AX=Q, D.26) = 3P J dr1dr2<u D.27) dr, dr2(Ma(«-i2WiM2)g2(iri2«i«2))u)i<O2. D.28) Pople19-67 evaluated the A2 term for a general potential, while the A3 term was worked out more recently.7198 Substituting D.25) into D.27) gives for the A2 term ^ dr, dr2g0A2)(uaA2J)lt)itt,2 dr, dr2 dr3g0A23)(MaA2)uaA3))lo,o>2(U3 D.29) where «a(y') = ^(r^w,) and go(y) = goir^), etc. The expression for g2 is worked out in Appendix 4A.5. From D.28) and DA.45) the third-order contribution to the free energy is A3 = A3A + A3B+A3C + A3D+A3E D.30) where . A3A = t^/3VJ drx dr2g0A2)<uaA2KLitO2, D.31) A3B = i/32p31 dn dr2 dr3goA23)<MaA2)uaA3)uaB3))a)iOJ2ltK, D.32) A3C = ^/32P3J drt dr2 dr3goA23)(uaA2JuaA3))OJ,<020K, D.33) A3D =s|32p4j drx dr2 dr3 dr4g0A234)<uaA2)uaA3)uaA4))a)]<U2(U3(ll4, D.34) A3E = ^2p4j df! dr2 dr3 dr4g0A234)<uaA2)MaB3)uaC4))(Ui<O2<U3<O4. D.35)
260 PERTURBATION THEORY 4.5 Equations D.25), D.29) and D.30) are valid for a general anisotropic potential ua. If wa consists of a sum of spherical harmonic terms u(f,!2/) in which /,, l2 and / are all non-zero, then these equations simplify considei- ably; we refer to such potentials as multipole-like (all multipolar poten- potentials are of this type, but dispersion, overlap and induction potentials usually are not). For such cases we have (see B.19)) (Ma(ri2Mi<D2))., = <"a0ri2«l«2>a>2 = 0, D.36) which follows from the property B.34) of spherical harmonics. Thus for multipole-like ua, D.25), D.28) and D.30) simplify to D.37) dr, dr2g0A2)<uaA2J>Ui<-2> D.38) ^/3 V J dr, dr2 dr3goA23)<MaA2)uaA3)uaB3)LlUia>1. D.39) Other simplifications can occur in special cases. Thus when Ma is the dipole-dipole interaction A3A vanishes since (waA2K)<j)|CO2 = 0 due to the odd parity of the dipole-dipole interaction; hence we have Ai = At,b for this case. The above expansions have been tested against computer simulation results for potentials of the type u = u()+ua, where u() is either the hard-sphere or the Lennard-Jones A2,6) model, and wa is either a multipolar potential (dipole-dipole, dipole-quadrupole, or quadrupole- quadrupole) or an anisotropic overlap model. A test of the series for g(roj,oi>2) is shown in Fig. 4.5 for the case of a LJ central potential with the addition of a quadrupole-quadrupole potential, B.185). To first order the series for g is obtained from D.12) and D.37) as g(rw, a>2) = g,,(r)[l - /3ua(rw, «2)]. D.40) Also shown in Figure 4.5 is an alternative series obtained by expanding the function yA2) = exp[/3uA2)]gA2) in place of gA2) itself (see D.14)); for the Pople reference system D.14) and D.15) give, to first order e-Bu-(~-^)g()(r). D.41) It is seen that these two expressions for gA2) give similar results at small values of the quadrupole moment. Both expressions fail for Q* values greater than about 0.4 (the values of Q* for F2, N2 and CO2 are roughly 0.3, 0.5 and 0.9, respectively).99-100
'.5 FURTHER DEVELOPMENT OF THE u-EXPANSION 261 rla -hi. t.3. lest of perturbation expansion equations D.40) and D.41) for IJ plus quadrupolc-quadrupolc potential for Q* = Q/(Ecrs)' = 1 (lower set of curves) and Q* = \ (upper set of curves), for the T orientation. Solid lines are cqn D.40), dashed lines are D.41), and points are Monte Carlo results. (From ref. 81.) The series D.12) for g has also been tested to second order101'102 for a nultipole-like ua. The second-order term is given by DA.45) in general, )ut this expression simplifies for multipole-like ua.103 Nevertheless, the ~esulting equation contains fluctuation terms involving three- and four- )ody correlation functions for the reference fluid, which are difficult to evaluate numerically. The difficult terms all occur in the centres correla- ton function part, so that this difficulty can be circumvented by consider- ng only the anisotropic or non-central part gnc(rttIw2) = g(rw,ttJ)-g(r), vhere g(r) = (g(ra),ttJ))o)<i,2 is the centres correlation function. To second >rder we therefore find gncA2) = gnclA2) + gnc2A2) vhere, for multipole-like uaA2), we have gnclA2) = glA2) - gl(r) = -/3goA2)MaA2), = g2A2) - g2(r) = |/32g0A2)[uaA2J- <M goA23)<[uaA3)+ MaB3)]2>w, dr3 D.42) D.43) D.44)
262 PERTURBATION THEORY 4.5 O.81 1.0 rla 1.2 1.4 1.6 -4 - -12 - -16 -20 Flo. 4.6. The harmonic coefficient gB24; r) for a LJ plus quadrupole-quadrupole poten- potential, at p* = 0.75, 7'*= 1.154, O*2 = 0.5. Points are molecular dynamics results [J. M. Hailc, private communication, 1980], and the lines are theoretical calculations; second-order PT; first-order PT; —¦ — — LHNC/SSC/GMF; QHNC; these last two theories are discussed in Chapter 5. (From ref. 101.) In Figs 4.6 and 4.7 are shown results calculated from D.42) for a liquid in which u0 is the LJ model and ua is the quadrupole-quadrupole potential. For the 224 harmonic (and hence the thermodynamic proper- properties) the theory gives excellent results, while the predictions are poorer but still reasonable for the 220 harmonic. For this potential the second- order theory yields the first 18 harmonics of gnc in contrast to the first-order theory, which just yields the harmonics present in ua. The theory has also been tested for other quadrupole strengths,104 for polar fluids,1021 and for non-axial quadrupole fluids.102 It should be noted that D.42) cannot be used to calculate the centres correlation function g(r). However, other theories give this function accurately, as described below. Several authors758'-8298105 have compared the u-expansion for the free energy with computer simulation results. Such a comparison is shown in Fig. 4.8 for the case where u0 is a LJ potential and ua is either a dipole-dipole potential, B.181), or a quadrupole-quadrupole potential,
FURTHER DEVELOPMENT OF THE u-EXPANSION 263 -io. 4. i. me narmonic coefficient gB20; r). Key as in Fig. 4.6. First-order PT gives gB20; r) = 0. (From ref. 101.) Nc -10 - -12 - -14 L Quadrupole ;ig. 4.8. Comparison of the u-expansion with computer simulation data for the configura- tional energy for LJ plus dipole-dipole and LJ plus quadrupole-quadrupole potentials. Dotted lines are the second-order theory, D.45), dashed lines are the third-order theory, D.46), solid lines are the Pade approximant, D.47), and points are computer simulation data.105 Here p* = per3, T* = kT/e, fi* = ^/(ea3^ and Q* = Q/(e<r5)i.
264 PERTURBATION THEORY 4.5 B.185). Included in the figure are the configurational energies corres- corresponding to the second-order theory, D.45) the third-order theory, AC = AO+A2+A3, D.46) and a Pade approximant proposed by Stell et a/.,72108 A=A() + A?{ ). D.47) c ° 2\l-A3/A27 For these two potentials A2 and A3 are negative and positive, respec- respectively, so that the second term in D.47) is always finite; the A2 term is always negative, regardless of the potential model, as shown in Chapter 6. The configurational energies in Fig. 4.8 are obtained from these expres- expressions for Ac by applying the thermodynamic relation D.48) The second-order theory is seen to fail at quite modest values of jx* and Q*. The addition of the third-order term extends the theory somewhat, but this approximation also fails at large values of the moments. The Pade approximation gives good agreement over the whole range of values of jx* and O* studied, however. Verlet and Weis75 have made a similar test for dipoles of strengths up to jx* = 2.3, and find that agreement between the Pade approximation and computer simulation results is still good. The success of the Pade109 is remarkable in view of the small number of terms and the poor series convergence at high strengths; D.47) is clearly just a guess that the complete series A2 + A3 + . . . is geometric, A2A + A3/A2+.. .). However, one can give a rough argument82-3 indicating why D.47) should be at least qualitatively reasonable. We note that for small values of the dimensionless parameter A' = |3 |ua| (i.e. high temperature or weak anisotropic forces), where |wa|«A is a measure of the strength of ua,114D.47) reduces to D.49) N N where O(A'2) = ((S/N)A2, as it should, and for large A' (low temperature or strong anisotropic forces), to
4.5 FURTHER DEVELOPMENT OF THE u-EXPANSION 265 where O(A') = (/3/N)(-A!/A3). D.51) The linear dependence on A' at large A' agrees with what one expects intuitively, since in dense fluids the strong anisotropic forces cause the molecules to become locked into preferred orientations (e.g. T shapes for quadrupoles), and increasing A' now causes no further alignment (i.e. there is "saturation")- There is merely a scaling of the interaction energy with A' once saturation has been reached.115 (If one considers just the A2 and A3 terms in the series, D.47) is the only Pade that preserves this saturation effect.) Although this saturation argument suggests that D.47) is qualitatively reasonable, it is nevertheless surprising that the term O(A') = /3(-A|/A3)/N is quantitatively correct in many cases. In addition to the tests for polar and quadrupolar liquids discussed in this section, On- sager116 and Sullivan et al.U2 have studied classical, rigid polar lattices, for which it is possible to calculate the higher order terms An in the free energy expansion exactly, and also the limiting form of A as A' -^™ (for this case A' in D.49)-D.51) is /3pfi2/3).117 Sullivan et al. find that: (a) the large A' limit given by the Pade agrees remarkably well with the exact results. Thus, for a simple cubic lattice the exact value of the O(A') term in D.50) is 8.028A', whereas the Pade gives 8.040A'; (b) the simple 0/1 Pade of D.47) gives better results than the next higher 1/2 Pade (thus the latter gives O(A') = 7.274A' for the simple cubic lattice). At the present time, then, one regards the Pade as a kind of interpolation formula between the small and large A' limits, eqns D.49) and D.50). In Fig. 4.9 is shown a case where the Pade breaks down at sufficiently large values of A'. The potential is the LJ plus 5-model of B.4); here 8 is an anisotropic strength parameter which determines the non-sphericity of the molecule. When 8 > 0 the molecules are prolate, whereas 8 < 0 corresponds to oblate molecules. In this case the third-order and Pade theories give similar results, and both fail for 8 a 0.35. However, the range —0.2<S<0.35 includes such small molecules as N2, Br2, HC1, CO2, C2H6, etc. In connection with Figs 4.8 and 4.9 we note that there are two ways of constructing a Pade approximant for Uc. The first is to apply D.48) to D.47), which gives Alternatively, we can form a Pade approximant from the series for Uc itself, ) Uc=U0+U2[-———) (U-route). D.53) \1 U3/U2
266 PERTURBATION THEORY 4.5 F-'io. 4.9. Comparison of second-order theory (dashed line), D.45), third-order theory (dotted line), D.46), Pade approximation (solid line), D.47), and computer simulation results' (points) for a liquid in which the intermolecular potential is the LJ plus anisotropic overlap model of B.4). (From ref. 98.) Since D.52) and D.53) disagree, the Pade method is 'thermodynamically inconsistent'.120 Comparisons with computer simulation data show that D.52) is more accurate, and it is this expression that is shown in Figs 4.8 and 4.9. Similar comparisons of the perturbation theory with Monte Carlo results for thermodynamic properties have been carried out for hard spheres with embedded dipoles or quadrupoles, by Verlet and Weis75 and Patey and Valleau82-1. Tables 4.1-4.3 show some of the results of Patey and Valleau. From Table 4.1 it is seen that for dipolar hard spheres the Pade gives good, though not exact, results for the thermodynamic func- functions. For the internal energy the equation derived from the Pade for A is somewhat better than the Pade for U itself. Tables 4.2 and 4.3 show similar results for hard spheres with embedded quadrupoles (where ua is given by B.185)). It is again seen that the perturbation series to A3 is
Table 4.1 Comparison of Fade approximation and mean spherical approximation (MSA) with Monte Carlo (MC) results for hard spheres with added dipolei' at pa3 = 0.8344 (from ref. 107). The MSA results are discussed in Chapter 5 0.056 0.506 1.406 2.755 Pade 0.00304 0.204 1.168 3.24 -AAJMcT (MSA)A§ 0.0018 0.124 0.732 2.12 MC 0.0035 ±0.0003 0.210 ±0.016 1.14 ±0.085 3.19 ±0 23 (Pade),jl- 0.0060 0.372 1.95 5.04 -AUJNkT (Pade)A t 0.0060 0.368 1.87 4.64 (MSA)u§ 0.0036 0.226 1.20 3.21 MC 0.0064 ±0.0004 0.348 ±0.006 1.79 ±0.01 4.80 ±0.04 (Pade) A t 0.0030 0.164 0.70 1.40 -ASJNk (MSA)A§ 0.0018 0.102 0.468 1.09 MC 0.0029 ±0.0005 0.138 ±0.017 0.65 ±0.09 1.61 ±0.23 (Pade)At 0.0045 0.265 1.26 2.92 -Ap/pfeT (MSA)P§ 0.0036 0.226 1.20 3.21 (MSA)A§ 0.0018 0.102 0.468 1.09 MC -0.03 0.58 1.35 3.08 t Here U d § Here subsc Au, AUL= U,-Uth ASC = SC-S(), Ap-p-pn. notes value derived from Pade for U, D.53), while A denotes value derived by differentiating Pade for A, D.52). ripts U, p, A denote the U-, p- and A-routes, respectively (see Chapter 5).
Table 4.2 Comparison of u-expansion and Pade approximation with MC results for the free energy of hard spheres with added quadrupole at per3 = 0.8344 (from ref. 82) 62 = Q2/fc7V 0.02 0.1 0.2 0.4 0.58824 0.71429 0.83333 1.0 1.25 1.66666 -AJNkT 0.00100 0.0249 0.0998 0.0399 0.863 1.273 1.732 2.495 3.898 6.930 -A3/NkT -0.0000061 -0.00076 -0.0061 -0.049 -0.155 -0.277 -0.440 -0.760 -1.484 -3.517 Thermodynamic perturbation theory -(A2+AJINkT 0.00099 0.0242 0.0937 0.351 0.709 0.996 1.293 1.735 2.414 3.413 -(AAc/McT)At 0.00099 0.0242 0.0941 0.356 0.732 1.045 1.382 1.912 2.823 4.597 -(A2 + A3 + A4)/NkTi 0.00099 0.0242 0.0939 0.354 0.723 1.027 1.350 1.853 2.702 4.324 MC -AAJNkT 0.00100±0.00001 0.0242±0.0003 0.0935±0.0011 0.350± 0.004 0.714±0.009 1.014±0.012 1.334±0.016 1.835±0.022 2.687±0.032 4.324±0.052 'From D.47). AAC = AC-AO. : Here A. is fitted from MC data. See text.
lalc ».J Comparison of u-expansion and Pade approximation with MC re*^.,* /< A; -. . . added quadrupole, per3 = 0.8344 {from ref. 82). t From D.52). Al/C= [/„- [/,,. itFrom D.53). § Here [/„ is fitted from MC data. See text. hxll zi ;j llh 62 0.02 0.1 0.2 0.4 0.58824 0.71429 0.83333 1.0 1.25 1.66666 -Ui/NkT 0.00200 0.0499 0.200 0.798 1.726 2.546 3.465 4.989 7.796 13.859 -U3/NkT -0.000018 -0.0023 -0.018 -0.146 -0.464 -0.831 -1.319 -2.279 -4.451 -10.551 Thermodynamic perturbation -(U2+U3)IMT 0.00198 0.0476 0.181 0.652 1.263 1.715 2.146 2.710 3.345 3.308 -(AUJNkT)Ai 0.00198 0.0477 0.183 0.673 1.353 1.904 2.484 3.378 4.868 7.645 theory -(AUJNkT)PU± 0.00198 0.0477 0.183 0.675 1.361 1.919 2.510 3.425 4.962 7.869 u2+u3+u4ymn 0.00198 0.0476 0.182 0.664 1.319 1.838 2.374 3.182 4.497 6.950 MC -AUJNkT 0.00201 ±0.00012 0.0464±0.0009 0.181 ±0.002 0.655 ±0.003 1.301±0.010 1.801 ±0.012 2.333±0.016 3.182±0.014 4.754±0.017 6.987 ±0.027
270 PERTURBATION THEORY 4.5 only valid for moderate Q values, although the convergence is not as poor as in the dipole case; here Q = Q/(kT(r5)*. The Pade again gives good results, although it is somewhat in error at large Q values; for C/c the Pade formed from the Uc series, D.53), is again somewhat worse than that derived from the Pade for A, D.52). Also included in Tables 4.2 and 4.3 are estimates of the sum (A2 + A3 + A4) of the first three non- vanishing terms in the Pople series. For quadrupolar hard spheres this is (at p* = 0.8344) AA —-- = -2.495Q4 + 0.760Q6 + a4Q8+... D.54) NkT where the first two terms are obtained from perturbation theory. If the convergence of D.54) were sufficiently rapid, one additional term, that is Q8, would be sufficient for good results at the values of Q studied. Patey and Valleau82 have therefore studied the function (AAJNkT + 2.495Q4 - 0.760Q6) by computer simulation, and find that it is linear in Q8 within the accuracy of their calculation; from this linear relation they obtain a value for a4. It is also possible to estimate a4 from the Monte Carlo (MC) results for the internal energy, by comparison with the fourth-order series. Both the AAC and AC/C results yield essentially the same values for a4, namely 0.118. It is seen from Tables 4.2 and 4.3 that the fourth-order series obtained in this way gives excellent agreement with the MC results, and that this agreement is better than that of the Pade result. Patey and Valleau82 have also examined dipolar hard spheres in the same way, and find that the A4 term (a4fl8) is again sufficient to give good results, at least up to fI2~2. Most comparisons of the Pade theory with simulation results have been for the internal energy and pressure, since these are the thermodynamic properties most readily calculated in conventional simulation methods. However, Monte Carlo (MC) values of the free energy have been calculated for dipolar hard spheres121 and for LJ plus quadrupole122 models, using the umbrella sampling MC technique. Since the MC studies cover a wide range of temperature and density it is possible to calculate the vapour-liquid coexistence curve, and to compare the MC curve thus generated with theoretical results. Such comparisons123 are shown in Figs 4.10 and 4.11. As seen in Fig. 4.10, for dipolar hard spheres the Pade and MSA theories give coexistence curves that are much broader than the MC result (the MSA theory is discussed in Chapter 5). For the Pade theory the predicted critical density is about 45% too low, while the critical temperature is about 12% too high. Part of this discrepancy between theory and experiment could arise from the use of a hard sphere potential, for which there is no corresponding vapour-liquid region. This supposition is supported by the results for the quadrupolar LJ fluid shown
FURTHER DEVELOPMENT OF THE u-EXPANSION 271 1.28 1.26 0.24 0.22 cnn L ' - 1 V / -i— ¦ i ¦ i ¦ ^ o - \ ; i . i . i i V •iij. t. iu. Liquia-vapour coexistence curves for the dipolar hard sphere fluid from Monte Carlo simulations ( ), the Pade form of perturbation theory ( ), and the MSA (mean spherical approximation) theory ( ). Open circles indicate the critical point, f =feTo-1/|M2, and V= V/Na\ (From ref. 121.) n Fig. 4.11. The coexistence curve is there found to again He above the vIC result and to be too broad, but the differences are much smaller. In his case the predicted critical density and critical temperature are each oo high by 2%. The Pade theory is in better overall agreement with the vIC results than the GMF/LHNC/SSC theory; this latter theory is discus- ,ed in Chapter 5. It should be noted that the Pade theory is in poor 2.5 2.0 - 1.5 - 1.0 TP- 0.5 0.2 0.4 p* 0.6 0.8 7io. 4.It. Liquid-vapour coexistence curves (upper set) for the quadrupolar LJ fluid from MC simulations for 32 and 108 molecule systems ( ), the Pade form of perturbation theory ( ), and the GMF/LHNC/SSC theory ( ). Here T* = kT!r, and p* = pa'. Also shown for comparison is the coexistence curve for the LJ fluid calculated from the equation of state of Nicolas el al.124 (From ref. 122.)
272 PERTURBATION THEORY 4.. agreement with the MC results at low density, as seen from Fig. 4.10 (the poor agreement is less obvious in Fig. 4.11 because of the scale).125 Fo dipolar hard spheres, for example,121 the value of the excess second viria coefficient (relative to hard spheres) predicted by the Pade theory is ir error by 9% at f = 0.67 and by 46% at f = 0.286. These poor results fcr the low-density gas were anticipated earlier (see ref. 115) and are furthe- discussed in Chapter 6. Although the series convergence for the angular pair correlation fum - tion is slow (see Fig. 4.5), considerably better results are obtained for the centres pair correlation function g(r) defined by C.103). The terrr_ g^rw,^) in D.12) vanishes on averaging over the orientations, so tha „,+... D.55 The functions g(r) and go(r) are compared for three different potentia models in Fig. 4.12 (see also Fig. 3.5). The dipolar and quadrupola- forces are seen to have little effect on the correlation of molecula~ centres, despite their large effect on correlation of molecular orientation; (see Fig. 3.4b). Anisotropic overlap forces have a large effect on g(r, however. Madden et af.126'127 have used the Percus-Yevick (PY) anc hypernetted chain (HNC) forms of integral equation theory (see Chapte: 5 for a discussion of these) to estimate the higher order terms g2 and g3 ir_ D.55). They find that a simple Pade approximant for g(r), L g?(r) J D.56 then gives good results. (The modulus sign on the right-hand side o D.56) ensures that g(r) always remains finite). In Fig. 4.13 is shown l comparison of the exact values of g2 with those calculated in the HNC 1 . 3 (a) (b) (c) I;K.. 4.12. Effect of anisotropic intermolecular forces on the centres correlation function Kir) for a fluid of linear molecules with potential u = uo + ua, where u0 is the IJ model, at fKT^ =0.800, kTle. =0.719. Dashed line is for the fluid of isotropic molecules, full line is for the fluid of anisotropic molecules: (a) ua = dipole-dipole interaction B.181), with (j.*2 = (i2/f<t1= 1.4; (b) ua = quadrupole-quadrupole interaction, B.185), with Q*2 = Oz/f<t*= 1; (c) ua = anisotropic overlap model B.4). (From ref. 105.)
4.5 FURTHER DEVELOPMENT OF THE ((-EXPANSION 273 Fig. 4.13. A comparison of the exact and HNC approximation values for g2M, the second-order perturbation term in the expansion of the centres correlation function g(r), for the quadrupolar hard sphere fluid. Here per3 = 0.8344, Q = Q/(fcTcr5)i= 1.291. The solid line gives the exact values of g2, the dashed line gives the HNC approximation to g2, and the points are computer simulation results for (g — gHS), where gIIS is the hard sphere correlation function. (From ref. 127.) approximation for a quadrupolar hard sphere fluid having a large quad- rupole moment. The agreement is not exact, but is quite good, especially at small r values where the perturbation is large. Also shown in this figure are the values of g(r) —gn(r); it is seen that g2(r) alone is not sufficient, and higher order perturbation terms are needed. A test of the Pade approximant D.56) is shown in Fig. 4.14, again for quadrupolar hard 1.4 Fig. 4.14. The difference [g(r)-gIIS(r)J for quadrupolar hard spheres, pa3- 0.8344, O = QI(kTa5)i = 1.291. Points are Monte Carlo results;82 eqn D.55) to g3; —¦ — ¦ — /-expansion D.68); Pade approximant D.56). The Pade approximant is also in good agreement with the MC results at larger r/a values. (From ref. 127.)
274 PERTURBATION THEORY 4.6 spheres with g2 and g3 calculated in the HNC approximation. The agreement with computer simulation results is excellent. Similar results to those shown in Figs 4.13 and 4.14 are obtained for dipolar hard spheres.'26127 Agrafonov et a!.128 have calculated the full angular pair correlation function g(ra),aJ) to second order for dipolar hard spheres, using the HNC approximation to evaluate the perturbation terms. The Pade method described in this section is most useful for the free energy, although we have seen101102111127 that corresponding results for g(ra),(o2) and g(r) can be obtained. It should be noted that it is not required that the Pade yield an accurate g(ro)loJ) in order to yield an accurate A. Since A is more sensitive to particular ws and rs (depending on the nature of u(ru>loj2)), it suffices that g(roIoJ) be accurate in these sensitive regions. Alternatively, it is sufficient that the spherical harmonic coefficients g(/]/20 of g(rajl(iJ) of the same order (M2O as those contained in uitojiCDj) be accurate for the sensitive regions of r. Thus, it is possible for a theory to give a good A and a poor g. The converse can also happen; it is possible for a theory to give a good g but a poor A. An example is the RISM theory for the site-site g^p discussed in § 5.5.1. The reason is essentially the same; when one quantity (A) is an integral over another (g), small changes in the integrand can produce large changes in the integral. 4.6 The /-expansion The u-expansion described in the previous section has the attraction of simplicity, but suffers from several drawbacks. First, all of the anisotropy of the intermolecular potential is included as a perturbation, leading to slow convergence for strongly anisotropic molecules. Second, the u- expansion is not well suited to potentials having a hard non-spherical core. Third, it does not yield the correct expression for the low density limit (e.g. the second virial coefficient). The expansion described in this section, which we call the f-expansionyi29 removes the third objection and goes some way to removing the other two, while still preserving the simplicity of a reference system of spherical molecules. The /-expansion uses a reference potential that includes a contribution from the aniso- anisotropic intermolecular forces, and takes the Mayer /-function as the expan- expansion functional. It was first presented for the free energy by Barker73 in 1957, although he only applied it to fluids of spherical molecules. The expansion for the pair correlation function and applications to molecular liquids are more recent.130 In the /-expansion the reference potential uo(r) is defined by131 exp[-/3uo(r)] = (exp[~(iu(rojioj2)])COiUl7. D.57)
4.6 THE /-EXPANSION 275 This is equivalent to writing uo(r) = <u(rw1a>2)>MlOJ+ Ma(r) D.58) where the first term on the right-hand side of D.58) is the reference potential used in the u-expansion, and 0a is given by exp[-/3ua(r)] = (exp[-/3Ma(rwiaJ)])WlOJ D.59) where ua(roIaJ) is defined as in the u-expansion (see D.22) and D.23)). It should be stressed that the /-expansion uo(r) defined by D.57) differs from the u-expansion uu(r) defined by D.23). We see from D.57) and C.269) that with this definition of u0 the pressure second virial coefficient is the same in both the real and reference systems. We now introduce a potential uxA2) = ux(roIw2) defined by D.60) or, equivalently, U,A2) = u0A2)-^ln[l + A/aA2)] D.61) where HA2) = u(ro)]<o2), woA2) = uo(r), /aA2)=/a(roIoJ), with = exp{-?S[uA2)- uoA2)]}- 1 = exp{-|3[uaA2)-MaA2)]}-l D.62) and uaA2) = Ma(roIaJ), uaA2) = 0a(r). From D.57) and D.62) it follows that </aA2)Liaj2 = 0. D.63) Thus, on using D.61)-D.63) in D.13) we obtain the first-order expan- expansion of the pair correlation function, gA2) = goA2)+ g^l.2), as , D.64) whereas the y-expansion to first order, yA2) = yoA2)+yjA2), obtained from D.14) and D.15), givest gA2) = [1 +/aA2)]{g0A2) + PJ dr3g0A23)</aA3) + /a D.65) tWe note that the term y,A2) in the first-order expansion of yA2) is linear in /a, as it should be. The expression for gA2) = exp[—puA2)]yA2) that corresponds to this is D.65); it is second order in /a due to the presence of the exp[—CuA2)] term. To avoid confusion we shall refer to D.65) and equations derived from it as the first-order f theory.
276 PERTURBATION THEORY 4.6 We shall refer to D.64) and D.65) as the fK and /y expansions for gA2), respectively, in what follows. We note that the /, and /v expansions give different expressions for g(ro)ito2) even at zeroth order; the zeroth-order /,, theory gives gA2) = go(r), whereas from fy theory we have yA2) = y,,(r) or gA2) = A +/a)g,,. From D.19) and D.20) the first two terms in the free energy expansion are± A, = 0, D.66) A2 = -^~ P3} dr, dr2 dr3g0A23)(/aA2)/aA3))OJiaJ2aj,. D.67) x The /-expansion has been tested against computer simulation results for fluids of dipolar132 and quadrupolar127 hard spheres, dipolar135 and quadrupolar135136139 LJ molecules, LJ + S-overlap model of B.4),135 homonuclear diatomic LJ molecules,13JI ">~41) diatomic LJ with embedded quadrupole,13313X repulsive WCA-type reference potentials derived from the diatomic LJ model,1392 homonuclear136139140142"x and heteronuc- lear139142 hard dumbells, linear triatomic BAB models composed of three fused hard spheres,139142 hard spherocylinders,149 and a four-centre tetrahedral LJ model of carbon tetrachloride.150 Several theories have also been proposed151152 which are closely related to the /-expansion. We first briefly consider the results for the structure, and then those for the thermodynamic properties.§ For the centres pair correlation function defined by C.103) the results are generally good, and superior to those from the w-expansion. Both the fK and fy expansions give the simple result g(r) = go(r) D.68) to zeroth order, where g,,(r) is the pair correlation function for the fluid with potential «0(r) given by D.57). At first order the fg expansion, D.64), still yields D.68) since the first-order term in D.64) vanishes on averaging over orientations by virtue of D.63). The first-order fy expan- expansion, D.65), gives g(r) = g«,(r) + 2p J dr3g()A23)</aA2)/aA3))a),^1 D.69) .;: liquation D.67) gives the first non-zero term in the fK expansion for A. The fv expansion for the thermodynamic properties can be obtained by using the f,, expansion for sA2) in the free energy equation for A (see §6.4 and ref. 149), or in the appropriate equations for U or p (see SS6.1, 6.2). § Almost all calculations in which the fii-st order g- or y-expansion, or the second order free energy expansion is used, have used the superposition approximation to evaluate go( 123). I"his is now known to be a poor approximation, and the theory gives better results when computer simulation values of j>(lA23) are used. See: Breitenfeldcr-Manske, H. Mol Phys. 48, 209 A983).
4.6 THE /-EXPANSION 277 For dipolar and quadrupolar hard spheresl27-1S2~4 the zeroth-order theory D.68) gives good agreement with computer simulation studies; a typical result is shown in Fig. 4.14. It should be noted that the u-expansion result, g = gHS, where gHS is the hard sphere function, is in substantial error for these potentials. The zeroth-order /-expansion also gives good results135136 for g(r) for dipolar (up to <x* = fx/(eo-3)i = 1.4) and quad- quadrupolar (up to Q* = Q/(eo-s)i = 0.8) LJ molecules, as well as for the 2.0 1.0 0 - - /  \ I \ /*=0.4 p* = 0.S 1 i 1 . 1.5 2.0 (a) 2.0 1.0 0 / ~f \ \ 1 /*-0.6 p*«0.4 I 1.0 2.0 (b) Fig. 4.15. The centres pair correlation function for the hard dumbell fluid from Monte Carlo simulation151 (solid line), zeroth-order /-expansion, D.68) (dashed line), and the zeroth-order blip function theory (dash-dot line), D.75). described in §4.7. The pair correlation functions needed for the /-expansion and blip-function results are also obtained from computer simulation. Here r* = r/cr, l* = l/cr, p* = po-\ where / is the intersite distance and a is the hard sphere diameter. (From ref. 145.)
278 PERTURBATION THEORY i 1.0 0.5 2.0 Fig. 4.16. The centres pair correlation function for a fluid of hard prolate spnerocynner. from Monte Carlo simulation1'''1'55 (points) and from zeroth-order /-expansion, (solic line); the function g,,(r) needed in D.68) was also obtained by computer simulation. Her; r* — r/a, I* = l/rr where / is the spherocyliner length (excluding the hemispherical caps), r is the diameter, and r\ = pvm where vm is the molecular volume. (From ref. 149.) LJ + S-overlap model (for -0.2 <S=? 0.25); the results are again ar improvement over the u-expansion, particularly in the case of the LJ+< model (see Fig. 4.12). For molecules with shape anisotropy the g(r results are best for moderate elongations and densities; they becomt poorer for very elongated molecules and for high densities. Results fcr dumbells, spherocylinders and diatomic LJ molecules are shown in Fig. 4.15-4.17. For spherocylinders (Fig. 4.16) the results became poor at the highest elongations, particularly at high density. For the diatomic LJ fiuic (Fig. 4.17) the /-expansion curve shows a main peak which is somewha to the left of that for the simulation results; this is because the reference potential used in the /-expansion overemphasizes156 the effect of attrac- attractive orientations on gA2). Also shown in Fig. 4.17 is the g(r) curvt calculated from the first-order /y expansion, which is worse than the zero-order result. The first-order fy expansion is also worse than the zeroth-order result for g(r) for spherocylinders,149 indicating that tht /-expansion is slowly convergent for molecules with significant anise tropy. This conclusion is supported by the results for the orientationa structure, which we discuss below. The superposition approximation, g(ri2)g(rI3)g(r23) D.70
THE /-EXPANSION 279 10 h -10 - L < /) w \X —^ 7 \ \ \ 220) ) T*=2.26 P*=0 524 /*=0.547 i i 1.0 1.2 1.4 1.6 1.8 2.0 7ig. 4.17. Spherical harmonic expansion coefficients of g(r&>i&>2), referred to the inter- molecular frame (see C.146)), for diatomic LJ molecules. Results are from computer simulation ( ), zero-order fy expansion ( ), first-order /B expansion (—¦—), and first-order fy expansion (¦ • ¦). The latter is shown for g@00; r) only. The superposition approximation is used to estimate the first-order terms in the /e and fy expansions. For both g@00; r) and gB20; r) the zeroth- and first-order g-expansion results are identical. The coefficient g@00; r) is related to the centres pair correlation function g(r) by g@00; r) = 4irg(r), as shown in §3.2.1. Here r* = r/a, I* = I/a, T* = fcT/e, p* = per3 with a and e the site parameters. (From ref. 138.) s used for goA23) in the first-order term in the /-expansion in these alculations, and also in the calculations discussed below. Orientational structure can be discussed in terms of the spherical larmonic expansion coefficients of g(raIa>2), the site-site correlation unctions gc,s(r), or the full pair correlation function g(roIoJ). The expressions for the harmonic coefficients are obtained by a harmonic malysis of D.64) and D.65), and are given by Quirke et a/.'33-138 and by ^mith et a(.146149 Typical results for these coefficients157 are shown in Figs J-.17 and 4.18. The zeroth order fy theory reproduces qualitative features )f the harmonics, but is not quantitative. The first-order /R and fy expansions do not offer any significant improvement, again pointing to :iow convergence. In both cases the theory gives harmonics that are too .nort-ranged; for r greater than the range of the angular-dependent part )f wA2) the /g and /y results become identical. Similar results to those ;nown in Fig. 4.18 have been found for spherocylinders.149 The theoreti- :al predictions can be improved146149 by calculating the so-called 're- luced harmonics', g((l/2'r'; r) = g(lil2m; r)/g@00; r). However, this is
280 PERTURBATION THEORY 4.6 /•-0.2 /•-0.4 /•-0.6 5 o  A 1.0 1.2 1.0 1.5 2.0 1.0 1.5 2.0 2.5 F-ifi. 4. IX. Spherical harmonic expansion coefficients for a hard dumbell fluid at p* = 0.5, referred to the intermolecular frame (see C.146)). Results are from computer simula- simulation'51 (points), zeroth-order /v Iheory ( ), first-order fK theory (— ¦ —), first-order /v theory ( ), and first-order blip function (BF) theory ( ). The expression used for the first-order BF term g, is approximate (see text). For gB20;r) the zeroth and first order fe results are identical. At l* = 0.2, the fy and f% first-order results are almost thcS same. At I* = 0.2 and 0.4 the BF results are very similar to those from the first-order fy theory, and so are not shown. At (* = 0.6 representative BF results are shown for gB00;r) and gD00; r). Here r* = r/<r, l* = lla and p* = pcr\ with cr the site diameter. (From ref. 146.) only useful if a theory more accurate than the /-expansion is available for the centres pair correlation function, g(r) = D7r)~' g@00; r). The /y expansion is to be preferred to the /K in general,'46149 because the latter gives unphysical behaviour at small r values. Thus, for r values inside the molecular core g(ra),w2) should be zero, whereas the /s expansion, D.64), usually produces non-zero values. Any attempt to calculate thermodynamic properties from such functions is, of course, doomed to failure. Qualitatively correct behaviour at small r is ensured in the /v expansion by the factor A +/a) that appears on the right-hand side of D.65). Tests of the /-expansion for the full pair correlation function g(ro),aJ) have been carried out,136-'38-'46-149 but the results are usually rather poor
4.6 THE /-EXPANSION 281 2.0 Fig. 4.19. The pair correlation function g(r<o,&>2) = gA2) for hard dumbells in the T- orientation for p* = 0.5. Points are Monte Carlo results: O = sum of first 14 spherical harmonic terms,153 A = direct calculation of gA2) in the simulation.1<i8 Other results are from first-order fv theory ( ), first-order /R theory (— ¦ —), and first-order BF theory ( )¦ The expression used for the first-order BF term g, is approximate (see text, §4.7). At 1* = 0.2 and 0.4 the BF results are almost the same as those from the first-order fy theory. Reducing parameters as in Fig. 4.18. (From ref. 146.) when the molecules are appreciably anisotropic. Results for hard dum- dumbells in the T orientation are shown in Fig. 4.19. The comparison here is somewhat uncertain because the simulation results shown as circles were obtained by summing the first 14 harmonic coefficients obtained in the Monte Carlo (MC) run, and this series is poorly convergent. Nevertheless, a direct MC calculation of gA2) in the simulation for I* =0.6 (triangles in the figure) confirms that the /-expansion is in serious error near contact at the larger elongations. As expected, the first-order /s theory gives un- physical positive values inside the core. The /-expansion has also been used to calculate site-site correlation functions1380147 for a variety of
282 PERTURBATION THEORY 4.6 1.0 Fig. 4.20. The site-site pair correlation function for diatomic LJ molecules, I* = 0.5471, p* —0.524, T* = 2.26. Results are from molecular dynamics simulation (points), 7eroth- order /v theory- ( ), and the non-spherical reference perturbation theory of Tildes- ley"*" ( ) which is discussed in §4.8. Here r* = r/tr, /* = I/a, p* - fxr\ T* = kTle, where f and a are site parameters. (From ref. 140.) 11 10 9 P pkT 8 7 6 5 ¦c. O.I 0.2 0.3 po> i 0.4 0.5 Fie;. 4.21. The pressure for hard dumbells, /* = //<r = 0.6. The results are from computer simulation161 (points), zeroth-order/^ theory ( ), first-order/K theory D.67) (— • —), zeroth-order BF theory ( ), and zeroth-order Bellemans theory, A = Ad ( ). The BF curve is for the u,> reference fluid used in the /-expansion. (From ref. 143.)
4.6 THE /-EXPANSION 283 diatomic and triatomic molecules. A typical result for the diatomic LJ model is shown in Fig. 4.20. The /-expansion gives a first peak that is somewhat too low and is displaced to too large an r* value, it fails to give the shoulder at r*~ 1 + I*, and the range of correlation is too short. The results become worse when the first-order fy term is included. For repulsive potentials such as hard dumbells and repulsive WCA-type reference potentials derived from the diatomic LJ model the results are poorer140 than those shown in Fig. 4.20. For such repulsive potentials a site frame159 /-expansion has been proposed,147139140 and seems to give better results for the site-site correlation functions. The /-expansion usually gives rather poor results for thermodynamic properties, particularly at the higher densities.14(U43'149 Comparisons with simulation data are shown in Fig. 4.21 and Table 4.4 for hard dumbells and diatomic LJ fluids, respectively. For both fluids the predicted pres- pressures are too low, as has also been found to be the case for spherocylin- ders.149 The first-order theory shows a small improvement over zeroth order in the case of the dumbell fluid (Fig. 4.21) and usually makes matters somewhat worse for the diatomic LJ fluid, again indicating poor convergence. In summary, the /-expansion gives a qualitative account of the struc- structure of a variety of fluids composed of anisotropic molecules, while preserving the simplicity of a reference fluid of spherical molecules. For small anisotropy and/or low densities the structural predictions are quan- quantitatively accurate (they are exact in the low density limit), but as molecular anisotropy or density increase the results become poorer. The series is then poorly convergent, and results for the thermodynamic properties become poor. For molecules of non-spherical shape (hard Table 4.4 Configurational energy and pressure for a diatomic LJ fluid (from ref, 140)t (* = 0.329 p* = 0.66 !* = 0.547, p* = 0.524 I* = 0.793, p* = 0.422 Calculation T* = 1.791 T* = 2.26 T*= 1.345 method U* p* U* p* U* p* Zeroth-order/yt -17.9 -3.0 -13.0 -0.1 -10.8 -1.8 First-order /vt -16.6 -3.7 -11.5 -0.8 -9.2 -1.8 NRPT -17.1 +3.1 -12.4 +3.6 -10.5 -0.0 MD -17.64 -0.06 -12.5 +3.31 -10.54 +0.14 + /* = Her, p* = per3, T* = fcT/e, U* = UJNe, p* = per3Is, where e and a are site parameters. % The /v theory results are obtained by using the /v expansion for gA2) in the equations that relate U^ and p to the site-site correlation functions; these equations are given in § 6.7. The NRPT results are from the nonspherical reference perturbation theory of Tildesley'fi" (see §4.8), and MD are the molecular dynamics results.140
284 PERTURBATION THEORY dumbells, diatomic LJ molecules, etc.), the /-expansion is superior to the w-expansion, particularly for fluid structure. For the generalized Stoc." mayer model (u = u,,(r) + wa) the w-expansion is more convenient, sine? the reference potential uo(r) does not vary with the strength or form o" wa, and is usually taken to be some familiar model such as LJ or hart spheres, for which accurate g,,(r) values are readily available. In th^ /-expansion, on the other hand, uo(r) and go(r) must be calculatec (usually numerically162) anew each each time the form or strength of ua 1: varied. The /-expansion has proved useful in studies of properties o_ liquid interfaces, as discussed in Chapter 8. 4.7 Expansions for non-spherical repulsive potentials The M-expansion is not well-suited to molecules with highly non-spherica repulsive cores, because ua(rw,oJ) is then large for some regions o" (ra),aJ); in the limit of a hard core molecule, the perturbation woulc become infinite in certain regions. The /-expansion described in the previous section provides one way of avoiding this problem. In this section we consider two further methods for treating fluids witr a non-spherical repulsive potential w(raIaJ); in both cases the propertiei of the fluid are expanded about those of a fluid of hard spheres ol diameter d. The first approach is due to Sung and Chandler163 and i: called the blip function theory; it can be applied to any purely repulsive- potential. The second approach is that of Bellemans,164 and applies onb to hard non-spherical molecules. The blip function (BF) theory is an extension of the Weeks-Chandler- Andersen18 theory to non-spherical molecules, and uses an exDonentia (j) (b) (c) Fk. 4.22. The function expf-Cu] for the real and reference systems in the blip funcuot theory: (a) the full potential; (b) the hard sphere reference potential; (c) the perturbatior in the exponential term (the 'blip').
4.7 NON-SPHERICAL REPULSIVE POTENTIALS 285 expansion. The parameterization of uxA2) is that of D.60), i.e. exp[-/3^A2)] = exp[-/3udA2)] + A{exp[-0uA2)]--exp[>/3udA2)]} D.71) where ud = ud{r) is the hard sphere potential. The perturbation in exp[-j3n] will often be small even when (u —ud) is large (see Fig. 4.22). The hard sphere diameter d is chosen so as to annul the first-order term in the free energy series Ac = Ad + A, +... D.72) where, assuming pairwise additivity, we have A, = (dAJd\)x=0 = lim ¦?- {-fcT In —^ f dr" do^FI ^"M = lim (-kT)\N{N-1) f drt dr2 dWl dw2 x C- e"^A2)) ePu»A2) f d^-2 dwN-2?^ = -27rNpfeTrdrr2yd(r)<[e^"A2)-e^"-(r)]L,m2. D.73) Thus, if we choose d to satisfy d(r)([e-0"a2)-e-p"..(^]LiUJ = O, D.74) then we have Al = 0, and Ac = Ad+O(A2). The y-expansion D.14) is used for the pair correlation function. In practice this is often truncated at the zeroth-order term, so that g(ra)!aJ) = exp[-/3w(roj1aJ)]yd(r). D.75) The first-order term in the y-expansion can be readily obtained from D.15) together with D.13) and D.71); the resulting expression is more complicated than the /-expansion result, and contains 3- and 4-body terms.165 The BF theory, like the /-expansion, is exact in the low density limit. This can be seen from D.74) and D.75) by noting that yd(r) —> 1 as p —> 0. The BF theory uses the same parameterization (see D.71)) as the /- expansion, but differs from it in the choice of reference potential. The different reference potentials lead to differences in the expressions for the higher order perturbation terms, since D.63) holds for the /-expansion, but not for the BF theory. Extensive tests of BF theory against Monte Carlo data have been made
286 PERTURBATION THEORY 4.7 2 - 1.0 1.4 1.8 rlo Fit;. 4.23. Comparison of the centres correlation function calculated from BF theory fzeroth order), D.75), with Monte Carlo results for hard homonuclear diatomic molecules, [* = ;/„- = 0.6, p* = pa3 = 0.5: BF theory; —•— Monte Carlo. (From ref. 153.) for hard dumbells,143145146153165 and some of these results are shown in Figs 4.15, 4.18, 4.19, 4.21, and 4.23. The results are good at small elongations (e.g. I* = 0.2), but become increasingly poorer as /* increases. Moreover, this deterioration is more rapid than for the /-expansion, so that the latter is superior at the longer bond lengths typical of Br2 or Cl2 (see Figs 4.15, 4.18, 4.19, and 4.21). For the centres pair correlation function (Figs 4.15, 4.23) the zeroth-order BF theory overestimates the first peak, and fails to reproduce the shoulder that occurs to the left of the main peak at high densities and elongations (see Fig. 4.23). The inclusion of the first-order perturbation term (see Figs 4.18 and 4.19) offers little improvement; thus it improves the results for the harmonic coefficient gB00;r), but makes gB20; r) worse.153165 It should be noted that the calculation of the first-order correction is only approximate, since it omits the difficult four-body and fluctuation terms in g, (see ref. 165). At /* = 0.6 the BF expansion gives unphysical negative values for g(r) at small r* values, and also for g(r<o!<o2) in some orientations.146 The BF theory has also been used to study mixtures of hard dumbells166 (with similar results), the repulsive part of the diatomic LJ model163'165167 (i.e. the sum of the site-site terms «"p, where u°p denotes the repulsive part of H,,,,, the site-site LJ potential-see D.100), and a fluid of spherical molecules whose pair potential uo(r) is the reference potential for the /-expansion,137138 D.57). The theory gives poor results in this last case, a consequence of the softness of the u0 potential. The BF theory has also been used to calculate site-site correlation functions for model potentials of the site-site type (see Fig. 5.14 and discussion), and to interpret neutron and X-ray diffraction data for real molecular liquids168 (see Chapter 9). Bellemans164 has considered fluids composed of hard non-spherical molecules. The expansion is made in terms of the anisotropy of the hard core, rather than in terms of a perturbation potential (which would be infinite for some regions), and the reference system is one of hard spheres
4.7 NON-SPHERICAL REPULSIVE POTENTIALS 287 Fig. 4.24. For two hard non-spherical molecules in contact the distance (ftnij^M^) equals the distance between centres, OiO2. of diameter d. The pair potential is = oo, r12<dxA2) = 0, r12>dx A2) D.76) where dK(l2) = dK(wiOJo)l2) is the distance between molecular centres when the molecules touch, their orientations being fixed at a^ and w2 and the orientation of rl2 being fixed at o>12 (Fig. 4.24). When A = 1 D.76) gives the full potential, while A = 0 gives the reference potential. The following parameterization is now introduced: D.77) where the hard sphere diameter d is defined by d-(dA2))MiaJ D.78) where dA2) = dx = 1A2). Thus 7A2) = [dA2)/d]-l and D.79) To calculate the lowest order terms in the perturbation expansion for g(roIoJ) and Ac we need an expression for the first derivative of uxA2) with respect to A. Noting that uK{\2) is a function of (r12 — dk), we get for the first derivative duK{\2) JuK{\2) ddK{\2) 3 0A 04A2) dk ~ dr dk = -TA2)e^«2)-e-^A2) C dr | ^A2) 8(r - dxA2)) D.80) where r = rl2, 8(r~dk(\2)) is the Dirac delta function, and we have used
288 PERTURBATION THEORY 4.7 (B.70). Thus we have ^)=^7A2)e^S(r-d). D.81) dA /op From D.79) and D.81) we have where f(r) is some function of r that remains finite at r = d. From D.13), D.81) and D.82) we see that + 8(r23-d)<7B3)>,J D.83) is the first-order term in the expansion of g(r12a)iaJ). This expression is unsatisfactory because of the singularity at r,2 = d. For spherical molecules with purely repulsive forces the expansion of y(r) gives good results, and one might therefore expect the expansion of y(r12a>,aJ) to give better results than the g(ri2oIoJ) expansion for rigid non-spherical molecules. From D.14), D.15) and D.83), the expansion of y(rl2o>,aJ) gives, to first order ]• D.84) The first-order contribution to the free energy vanishes, as can be seen from D.19) and D.79). The second-order contribution A2 is most easily. obtained by taking the A-derivative of D.18). After using D.81) this gives _ l (a2A\ _ itPN<i2 A2 —— | _.tI — 2\0A2/o C + ^|dr128(r12-d)<7A2)y1A2)>UiA,2 D.85) where the standard property (B.68) of S-functions has been used to obtain the first term on the right-hand side of this equation, and ytA2) is the first-order term in the y-expansion, D.14). The y! term is obtained
4.8 NON-SPHERICAL REFERENCE POTENTIALS 289 from D.13) and D.15), together with D.81), with the result that irpNd2 A2 = p2Nd2 x 5(r13 ~ d)<7(w1w2O(«iw.OL,^. D.86) Bellemans164 has used D.86) to make calculations for prolate ellipsoids. He obtained gd(r12) and g^r^is^) by Monte Carlo simulation for two densities in the dense fluid region and one in the solid region. The second-order perturbation term was found to be Ai 7 = 0.21 ±0.03 for p* = 0.7725 (fluid) NkTe = 0.38±0.08 p* = 0.8369 (fluid) = 4.56 ± 0.09 p* = 1.0000 (solid). Here e = (a — b)/b, where a and b are the principal axes of the ellipsoid {a>b), and p* = per3 is the reduced density. Thus for small e, e.g. e = 0.1, the perturbation term A2/NkT is very small (of the order 0.002) in the fluid region, and can probably be ignored; Ac = Ad is then a good approximation. The perturbation term is larger for solids, however. The zeroth-order Bellemans expansion has also been used to calculate the thermodynamic properties and centres pair correlation function g(r) for hard dumbells.'43 The results for the pressure are very good for Z*=//cr = 0.6 (see Fig. 4.21) and 1.0-much superior to the /-expansion or BF theories. The results for the structure are very poor, however. Thus, the Bellemans expansion at zeroth order, y(r)= yd(r), gives a first peak in g(r) that is much too high (by about 50% at p* = 0.5), and oscillations at larger r that are also much too large; these results are worse than those from the /-expansion or BF theories.143 Thus, the Bellemans expansion apparently gives good results for the ther- thermodynamic properties, but is poor for the structure. The thermodynamic properties of fluids composed of hard convex bodies have also been studied by a perturbation expansion in which the reference system is a hard sphere fluid.169 This expansion is related to that of Bellemans, and makes use of the simplified geometry of convex bodies (see §6.13 and Appendix 6A for a discussion of fluids of convex bodies). 4.8 Non-spherical reference potentials For strongly anisotropic pair potentials it is unlikely that any choice of spherically symmetric reference potential will give good convergence,
290 PERTURBATION THEORY 4.8 because molecular packing will depend strongly on the relative orienta- orientations of neighbouring molecules (see, for example, Figs 3.3 and 3.4). In this section we consider the use of a non-spherical reference potential as a method for obtaining improved convergence.170 We adopt the parameterization ux(r&I&J) = uo(r&),&J) + Au1(raIaJ) D.87) so that the free energy expansion is obtained from D.19) as drr2<u1(r«1«2)g0(r«1«2))u),aJ + ... D.88) \ The expression for gtA2) is obtained by using D.87) in D.13). The reference system should be chosen so that it has a structure similar to that of the real system; accurate methods must also be available for calculating its properties. In view of the success of the WCA theory for atomic fluids (see §4.1), Mo and Gubbins78 proposed a similar method for the molecular case. The intermolecular frame is chosen, and for each set of molecular orientiations (coi<i>2) the potential is divided into repulsive and attractive parts, = 0 , r>rm(cjl(o2), D.89) = —e(co1co2) , r ^rm(co1co2) \ r >/•„,(«! co2) D.90) where eiw^-^ and rm{<olw2) are the magnitude of u(ru>lu>2) and the separation distance, respectively, both evaluated at the minimum of the pair potential (for fixed <olt &>2). The remaining problem is to evaluate Ao and gotrcoto^). There are several possible procedures.78140'141144171 The simplest is to relate these quantities to the corresponding properties for hard spheres, using one of the theories for non-spherical repulsive potentials described in the previ- previous section. Three such procedures might be considered: (a) a single blip function expansion about hard spheres, (b) a blip expansion about hard non-spherical molecules (the potential of eq. D.76)) followed by a second blip expansion about hard spheres, and (c) a blip expansion about hard non-spherical molecules followed by a Bellemans expansion about hard spheres. Method (c) gives the best results for dipolar LJ spheres.78 The blip function expansion about hard non-spherical molecules is carried out as in D.72)-D.75) to give Ao = Ad + second-order terms, D.91) = exp[-/3uo(rco1co2)]yd(rco,w2) + . . . D.92)
4.8 NON-SPHERICAL REFERENCE POTENTIALS 291 where A,, and yd are not the quantities corresponding to hard spheres, but to the system with the non-spherical potential D.76), and d(coico2) has been chosen to satisfy (cf. D.74)) P dir2y<1A2)[e-Bu«<12) -e-B"-(I2)] = 0. D.93) Thus, from D.88), D.91) and D.92) we have, to first order Ac = Ad +27-piVJ d/r2(u1(rcoico2)exp[-/3u0(rw1co2)]yd(rco1co2))<0l<U2. D.94) The quantities Ad and yd =exp[/3ud(rco1co2)]g(r&I&J) can be calculated from the Bellemans theory using D.86) and D.84), respectively. Calculations based on D.94) have been carried out78 for potentials of the type M0(r) + ua(rco1co2), where u0 is the LJ model and ua is either the dipole-dipole, quadrupole-quadrupole, or anisotropic overlap potential (eqns B.181), B.185), and B.4), respectively). For a liquid of dipolar LJ spheres at the reduced conditions per3 = 0.85, fcT/e = 1.15, jx* = fi/(eo-3M= 1.072, D.94) gives (Ac-A0)/NfcT = 0.55, in good agreement with the Monte Carlo value75 of 0.535±0.01. A further test for dipolar LJ spheres is shown in Table 4.5. Similar results are obtained when ua is a quadrupole-quadrupole or anisotropic overlap potential. For very strong dipoles or quadrupoles the Bellemans expansion used to calculate Ad and yd in D.94) is likely to give poor results. Thus, if we define a dimension- less 'eccentricity' e by , , , ,,, ,. „_, e^{dmax-dmin)/dmin D.95) Table 4.5 Comparison of non-spherical reference perturba- perturbation theory with Monte Carlo resu/rs106 for the anisotropic contribution to the internal energy of a dipolar LJ fluid with |x* = f/,(eo-3)^= 1 for various p* = per3 and T* — kT/e values. The estimated accuracy of the Monte Carlo values is ±0.03 (from ref. 78) p* 0.5 0.6 0.7 0.8 0.8 0.8 T* 1.35 1.35 1.35 2.74 1.35 1.15 UJNe Theory -0.50 -0.58 -0.71 -0.61 -0.80 -0.83 MC -0.53 -0.60 -0.70 -0.57 -0.82 -0.86
292 PERTURBATION THEORY 4.8 where dmax and <imin are, respectively, the maximum and minimum values of d(toito2) as a function of the orientations, we find that for dipolar LJ spheres e=0.1 at jx* = 1 but e = 1.44 at fx* = 2.14 (a value used by Verlet and Weis in Monte Carlo studies); in the case of quadrupolar LJ liquids we find e = 0.66 at Q* = Q/(ea5)^= 1. For large values of e (probably above about 0.3) the Bellemans expansion breaks down. Several variations on the above procedure are possible. For many fluids the fy expansion at zeroth order is likely to give better results144 than the BF theory for goA2), as seen from Figs 4.15, 4.18, 4.19, 4.21, and 4.23. The pair correlation function goA2) is then given by D.92), but with YdA2) replaced by yref(r12), the function for a fluid of spherical molecules with pair potential uref(r12) given by (cf. D.57)) exp[-/3uref(r12)] = <exp[-/3uoA2)]Ll(U2. D.96) For molecules with non-spherical shape the results can also be improved by expanding about hard non-spherical molecules of appropriate shape, and avoiding the final expansion about hard spheres that was used to obtain D.94). An accurate equation of state is then needed for the free energy of the reference system of hard non-spherical molecules. Such an approach has been shown144141140171 to be successful for diatomic LJ liquids. The fy expansion is used for goA2), and BF theory is used to expand Ao about Adb, the free energy for a fluid of hard dumbells. The bond length I is kept constant in this expansion, and the diameter cr of the fused hard spheres is adjusted to annul the first-order term in the BF expansion. Thus we have A0 = Adb +second-order terms D.97) with cr determined by the condition (cf. D.74)) f drr2(ydbA2){exp[-/3M0A2)]-exp[-/3udbA2)]})lUl^ = 0. D.98) Thus, the final expression for the configurational free energy is, to first order drr2yref(r)<u1(rco1co2)exp[-/3u()A2)]XOl<U2. D.99) In calculations based on D.99) an equation of state for hard dumbells due to Boublik and Nezbeda172 has been used (see § 6.13 for a discussion of equations of state for hard non-spherical molecules), and yref(r) has been calculated from PY theory143; the function ydbA2) that appears in D.98) has been calculated by the zeroth-order /y expansion.173 Some typical
4.8 NON-SPHERICAL REFERENCE POTENTIALS 293 Table 4.6 The configurational free energy of a diatomic LJ fluid correspond- corresponding to N2 (I* = l/cr — 0.329) from non-spherical reference perturba- perturbation theory and from computer simulation;174 all dimensionless quantities are reduced with site parameters e and cr {from ref. 140) p* 0.6 0.7 0.7 0.7 T* 3.0 3.0 2.0 1.55 Eqn D.99)t -3.53 -3.06 -6.03 -8.75 AJNkT Eqn D.102)* -3.57 -3.11 -6.02 -8.70 Simulation -3.53 -3.07 -6.02 -8.71 t Based on a WCA division of the full molecular potential u(r<o,<o2)-see D.89) and D.90). * Based on a WCA division of the site-site potential uaB(raB)-see D100) and D.101). results of this method are shown in Table 4.6 and Fig. 4.25. Agreement is good except at the lowest temperatures and highest densities. Perturbation expansions based on the splitting of the potential shown in D.89) and D.90) have been studied for fluids interacting with a Kihara potential, B.14), by Boublik and collaborators.783175"8 In this model the molecules contain a hard convex core, and the intermolecular pair poten- potential depends on the shortest distance p = p(ro>ico2) between molecular cores rather than on the distance between molecular centres; for p<0 we have u = °°, while for p > 0 the potential is of LJ form. For such molecules it is possible to use the simple geometry of convex bodies (see Appendix 6A) to simplify the integral that appears in the A! term on the right-hand side of D.88). The resulting expression for Aj involves a correlation function go(p) that depends only on the shortest distance p between cores and is proportional to the probability of finding a molecular pair with distance p between cores, irrespective of molecular orientations or centres positions. The reference properties Ao and go(p) are related to the corresponding values for a fluid of hard convex bodies by expansions of the Barker-Henderson37 or BF types. Quite accurate expressions are available for the free energy of simple hard convex bodies (see §6.13); for the correlation function ghCb(p) approximate expressions have been proposed,175"8 but their accuracy has been tested only for a few simple shapes. This theory has not been tested against computer simulation results,179 but extensive comparisons have been made with experiment783'175"8 for both pure and mixed liquids. These calculations are further discussed in Chapters 6 and 7. An approximate expression has also been given177178 for the second-order term, A2.
294 PERTURBATION THEORY o. 6 T*-2.0 r*—1.5- T*-U t*-o.t 0.4 0.5 P* 0.6 Fig. 4.25. Pressure isotherms for the diatomic LJ fluid, bond length (/<r = 0.63 corres- corresponding to Br2), from the non-spherical reference perturbation theory of D.99) (lines), anc from MD simulation (points). Here p* = pcr'/e, p* = pa3, T* = kT/e, cr and e are the silt parameters, and p, is the triple-point density; the triple and critical temperatures are, ir reduced units, T*~ 1.00, T*~2.20. The MD points shown were obtained by smoothing and interpolating the original data along isochores. (From ref. 141.) For the special case of site-site model fluids a somewhat differen approach to that described above can be used. It is then possible to dividt the site-site potential uaB(r) into reference (u"p) and perturbation (u}aB parts, rather than dividing the full intermolecular potential u(rco1co2). Ir place of D.89) and D.90) we then have = 0 /•>/•„ , r>rc C.m D.100 D.101
4.8 NON-SPHERICAL REFERENCE POTENTIALS 295 where eaB and raBm are the magnitude of uaB and the separation distance, respectively, both evaluated at the minimum of the site-site potential. With this choice the free energy expansion is obtained to first order by using D.19) as180 [ drr2g°aB(r)uiB(r). D.102) Such an approach was first proposed in 1972,79 but numerical calculations have only been carried out more recently.160 The reference fluid proper- properties Ao and g°,e are calculated160 by a BF expansion about hard dumbells having the same bond length I as the diatomic LJ molecules. The dumbell hard sphere diameter daB is obtained by solving the equation f drr2yf3(r){exp[-/3uos(r)]-exp[-/3u*(r)]} = 0 D.103) where yaB(r) = eeu">(r)gaB(r) D.104) and db signifies the function for dumbells. In the usual BF treatment g°p(r) in D.102) would be calculated from the expression y» = y». D.105) However, y^g has a cusp at r = <jafS +1, which should not be present in y°3. To overcome this difficulty Tildesley160 has introduced a 'smooth' y- function, y^g = yae — y^g, where y^B is the part of yaS that gives rise to cusps. This latter function can be calculated in the approximation of RISM theory (see Chapter 5 for a discussion of this theory). Equation D.105) is therefore replaced by y>) = y»- D-106) Tests of D.102) against computer simulation results are shown in Tables 4.4, 4.6, and 4.7; the free energy of the fluid of hard dumbells is obtained from computer simulation data,181 and g°,g is calculated from D.106) with the aid of RISM theory. Agreement between theory and simulation is generally good, particularly at the higher temperatures; there are errors in the calculated pressures at low temperatures and high densities, however (see Table 4.7). This theory can also be used to make approximate calculations of the site-site correlation function ^(r); if the effect of attractive forces is neglected we have and gap(r) can be calculated from D.106). This procedure gives quite good results at high densities for moderate and long bond lengths (see
296 PERTURBATION THEORY 4.8 Table 4.7 Reduced pressures (p* = pcr3/e, where a and e are site parameters) versus reduced density p* = per3 calculated from the non-spherical reference per- perturbation theory based on the reference fluid of eqn D.100), compared with molecular dynamics results,™2 for a diatomic LJ fluid with Her = 0.608. All calculations are at T* = fcT/e = 1.5 {from ref. 160) p* 0.541 0.520 0.512 0.495 0.485 Theory 2.82 1.61 1.21 0.47 0.096 MD 2.51 1.37 1.02 0.40 0.096 Fig. 4.20). The approximation of D.107) is expected to be poor except at high densities, and will be very poor at low densities (cf. Figs 4.2-4.4 for atomic fluids). Moreover, the RISM theory used in calculating g^r) is itself poor at low densities and small elongations. A slightly different approach to site-site model fluids183184 involves the use of a Barker-Henderson potential splitting in place of the WCA splitting of D.100) and D.101), = 0 , r>era&, D.108) = uaB{r\ r>aafi D.109) (see Fig. 4.1 for the corresponding reference potentials in the atomic fluid case). The free energy is again given by D.102), but a somewhat different expansion of Au and g"p about the corresponding dumbell fluid is used. The prescription for the site diameter dafi for the hard dumbells is now {1—exp[— (iuafi (r)]} dr D.110) so that daB depends on temperature but not on density. Equation D.110) gives a much simpler prescription for da& than D.103), which must be
4.8 NON-SPHERICAL REFERENCE POTENTIALS 297 solved by iteration. Moreover, the dafi given by D.103) depends on density as well as temperature. This approach seems to give good results for thermodynanic properties,183184 although tests have so far been limited to rather small bond lengths. So far we have only discussed applications of non-spherical reference perturbation theory to models in which the anisotropy of the intermolecu- lar forces is due primarily to non-spherical shape or to multipolar forces. The situation is more complex when both non-spherical shape and multipolar forces are present. The multipolar forces cannot be easily treated within the framework of the theories that split the site-site potential,160183184 since one must then know the full pair correlation function go('«i«2) for the reference system; the site-site correlation functions g°p(r) are no longer sufficient. The first attempt to tackle this problem was by Sandier;77 he used a potential of the form D.87) with u0 being a hard dumbell model and u, being either a dipole-dipole or quadrupole-quadrupole interaction. The reference properties Ao and goA2) are calculated from the blip function theory (using A0 = Ad and D.75), respectively, with d given by D.74)). Table 4.8 shows calcula- calculations of Ao and Ax for approximate models of nitrogen, chlorine and hydrogen chloride. For N2 and CI2 the perturbation potential is the quadrupole-quadrupole term, while for HC1 it is the dipole-dipole term. The state conditions chosen correspond to the critical, boiling, and freezing points of the fluid. Although the contribution from the multipo- multipolar term is small for N2, it is large for Cl2 and HC1, particularly in the dense liquid region. These conclusions are supported by more recent calculations.185186 Perturbation theory for the pair correlation function g(rco,co2) and its (intermolecular frame) harmonic coefficients g(l,/2m; r) has also been studied'87 for the case where ua is a diatomic LJ model and Uj is a quadrupole-quadrupole interaction. In these calculations exact Monte Carlo values153188 were used for the go(M2m; r) coefficients of the diatomic LJ reference fluid, so that the calculations are 'exact' to first order. The first-order theory is found to give good results for weak quadrupolar forces, but for larger quadrupoles the results are rather poor (see Fig. 4.26), indicating slow convergence. The principal difficulty in the use of non-spherical reference potentials is the evaluation of Ao and particularly gotrcojo^) in D.88). If u0 is a hard diatomic potential and u, includes all attractive forces77186 one may also have to consider higher order perturbation terms in D.88), which are even more difficult to evaluate than the A] term. For such cases a Pade approximant for the A2 and A3 terms has been suggested189 (cf. D.47)), i.e. D.111)
298 PERTURBATION THEORY 4.8 Table 4.8 Values of the perturbation terms in D.88) for the case where u0 is a hard diatomic model and u, is the lowest non-vanishing multipolar term: here I is the bond length (from ref. 77) Species Parameterst State A'0/NkTt A,/N»cT§ I* = 0.367 Q = -1.52xl(T26esu A) pm = 0.311 gem 0.7?43 -0.0018 T= 126.1 K B) Pm = 0.8084gem-3 3.0277 +0.0297 T = 77.4K C) pm = 0.868 gem 3.4469 +0.0446 T = 63.2K Cl2 <rcl = 3.6 A I* = 0.556 O = +6.14xl06esu A) pm = 0.573 gem T = 417.2K B) pm = 1.557gem-3 T = 238.6 K C) Pm= 1.707gem-3 T=171.6K 1.2982 -0.0078 6.956 +0.825 8.513 1.887 HCI (Ta = 3.6 A /* = 0.5333 u. = 1.07xl0~18esu A) Pm = 0.42gem T = 324.6 K B) Pm= 1.193gem-3 T= 188.2 K C) Pm= 1.276gem T= 162.2 K 0.973 +0.0053 4.917 +0.319 5.585 +0.490 t Here a is the site parameter, I* = f/cr, and pm = mass density. For HCI, I* = l/<xH. + Af, = residual part of A,, = A0(iWT)~ Aj,d(iWT), where Aj,d is the ideal gas value; from Sandier,77 using BF theory. § A, = first-order term in D.88), using BF theory for goA2).77 Although the A2 and A3 terms are numerically intractable in general, a rough estimate of their magnitude can be obtained by replacing the true reference correlation functions (g0(rco1co2), etc.) by some effective angle- averaged functions. The simplest approach is to replace the true (angular) reference correlation functions by the corresponding spherically symmet- symmetric functions for hard spheres,186189 using a suitable prescription for the hard sphere diameter. One such prescription186 is to equate the isother- isothermal compressibility of the non-spherical reference system, \0, to that for the effective hard sphere fluid, Xd, Xo = Xd- D.112)
Fig. 4.26. Spherical harmonic expansion coefficients of gA2), referred to the intermolecular frame (see C.146)), for a fluid with potential uQ+ult where u0 is the diatomic LJ model and U[ is a quadrupole-quadrupole term. In these calculations T* = fcT/e = 2.361, p* = per3 — 0.520, (* = l/(T =0.540 (corresponding to Cl2), Q* = Q/(ea5)?= 1.716, and a = 3.679 A, where s and a are site parameters and / is the bond length. Results are from Monte Carlo (MC) simulation for the full system with potential uo+ul (—•—), MC for the reference system with potential u()(A), and first-order perturbation theory with the Ug system as reference ( ). Beyond r = 6-8 A the results for the perturbation theory and reference system are very close, so that only one of them has been drawn. Note that the values shown in (a) are related to the centres correlation function by g@00; r) = 4irg(r). (From ref. 187.)
300 PERTURBATION THEORY 1.0 0.5 0- -0.5 -1.0 1 / / - 'v. \ ^ '\ \ \ \ \ \ I ^-417K ^172 I — - - 417K •N 238.6 0.2 0.3 0.4 n 0.5 Fir,. 4.27. Contributions of the various terms in D.111) to the configurational iree energ' for a model of C\2 in which the pair potential is of the form u0 4- u,, where un is a hare dumbell term and u, is the quadrupole-quadrupole interaction; here <r = 3.6A, O = 6.14 x 10~26 esu, and I* = l/cr = 0.6 (cf. Table 4.8, where all parameters are the same excep /*, which is there 0.556). The results shown are for A'JNkT ( ), AJNkT ( , and [A2A-A.1/A2)~l]/NkT (—• )¦ Here A,r, is the residual free energy for tht reference system, -n = pvm is the reduced density, and um is the volume of the diatomi- moleculcs with site diameter a and bond length /. (From ref. 186.) For the hard dumbell reference fluid the Nezbeda equation of state185 car be used for Xo (sec § 6.13), while for Xd the PY compressibility equatior is a good approximation. Values of the various terms in D.111) are showr in Fig. 4.27 for the model of Cl2 given in Table 4.8. The higher orde- pcrturbation terms make a substantial contribution at all densities, anc are larger in magnitude than the A, term. 4.9 Effective central potentialst When the intermolecular forces are only weakly orientation-dependen. the free energy Ac and centres pair correlation function g(r) can each bt + See also Note added in Proof on p. 340.
4.9 EFFECTIVE CENTRAL POTENTIALS 301 set equal to the corresponding properties for a fluid of spherical molecules with an effective central potential ucn(r). This latter quantity depends on temperature, and so is not a true potential. The use of such effective potentials simplifies calculations, since one can then make use of the simple and accurate theories that have been developed for atomic fluids. The first clear discussion of the use of effective potentials for thermodynamic properties seems to be that by Cook and Rowlinson59 in 1953.190 Here we generalize their derivation191 to encompass arbitrary anisotropic potentials; we also give a treatment of the centres pair correlation function, in addition to the thermodynamic functions. We consider a molecular fluid in which the intermolecular potential is u(r&I&J) = uo(r)+ ua(r&>1 co2) where u0 is the Barker-Pople reference po- potential of D.23) and ua(to)i(o2) is a multipole-like potential. The expan- expansions for the free energy and the centres correlation function are ..., D.113) r) + ... D.114) where A2 is given by D.38) and g2(r) is obtained by averaging DA.45) over the orientations and using the multipole-like property D.36) of ua. Thus, we have dr3g0A23)([uaA3J + MaB3J])(OiW2UK dr3 dr4[g0A234) - g()A2)g0C4)](MaC4J)<UilO4 ~ [p2goA2)][ J dr34g0C4)(MaC4JL3(O4 dr34dr45[g()C45)-goC4)](uaC4J)u),^]} D.H5) where uaA2) = ua(rI2<m,co2), goA2) = go(r,2), etc. We compare these molecular fluid results with those for a fluid of isotropic molecules in which the pair potential is uats(r) = uo(r) + ui"(r), where u0 is again the Barker-Pople reference potential. The free energy and pair correlation function for this central-force system are given by +..., D.116) (r) + ... D.117) with Af and gf* given by D.19) and D.13), respectively; in these expressions (du(r)/5AH = u1"(r). If we now make the choice uf (r12) =^<Ma(r12co1co2J)a,,0J, D.118)
302 PERTURBATION THEORY 4.9 then we see that (a) Af in D.116) is equal to A2 in D.113), and (b) gf in D.117) is equal to g2 in D.114). Thus, to first non-vanishing order in the perturbation, the thermodynamic properties and the centres pair correlation function for the molecular fluid are equivalent to those for the system with the effective isotropic potential, provided that ua is multipole-like. Thus, for molecules with weakly anisotropic forces, per- perturbation methods that are well developed for atomic fluids (e.g. the BH or WCA theories) can be used to estimate these properties. This approach is particularly convenient when the r-dependence of (uaA2J)tOllO2 is the same as that of u0 (or part thereof); with the Lennard- Jones (LJ) model for u0, examples are the dipole-dipole interaction ((ua(l2J)o>i<t>2 oc r~6) and the anisotropic London dispersion interaction ((uaA2J)tOl<U2 <x r~12). In such cases it is possible to rearrange the effective potential into the LJ form, and to calculate properties of the molecular fluid from the corresponding states principle applied to the LJ fluid. For example, if ua is the dipolar interaction, we have \ ^< ?- Combining the coefficients of r~6, we can write this as D.120) where the potential parameters are now temperature-dependent and are given by 16 D.121) ecn = e(l + xJ D.122) where k D-123) The equation of state for such a molecular fluid will be P* = p?j(T*,p*) D.124) where p* = pcreff/eeff, T* = kT/ect,, p* = perl,,, and p^iT*, p*) is the func- function for the LJ fluid. Extensive use has been made of such corresponding states correlations.69072192~4 It should be remembered, however, that the above methods are only valid for cases where the u-expansion can be truncated at the first non-vanishing term. From the discussion in § 4.5
4.9 EFFECTIVE CENTRAL POTENTIALS 303 (see also Fig. 4.8) it is clear that such a truncation can only be made for weakly anisotropic molecules; in the case of dipolar molecules, there will be significant discrepancies if n* = n/iecr3)^ is greater than about 0.5. Nevertheless, the effective potential idea has led to useful empirical equations of state, in which the parameter x that appears in D.120) is replaced by an empirical, state-dependent shape parameter.195 The effective potential method can be generalized72 by writing the configurational phase integral, C.53), as D.125) where <Uen(^) is an N-body effective isotropic potential given by ~. D.126) Clearly, a fluid of central-force molecules with potential energy %eff will have the same free energy and centres correlation function as the molecu- molecular fluid. We can decompose %efl uniquely into a sum of two-body, three-body, etc., terms: %eff= I Weff(r,r,-)+ I ueff(ririrk) + .... D.127) l=si<j==N l<i</<ksN Thus, for an anisotropic potential of the type u(r12co1co2) = Uo(ri2) + ujr12w1w2), the pair term ueff(r12) is (cf. D.57) and D.61)) ue«(rl2)=Ven(i1T2)=u0(r12)~^\n(e-^^))lOi0l2. D.128) Expanding the exponential and then the logarithm in D.128) gives iU'12) = «o(''i2)-^<uaA2J)ll>l@2 ). D.129) The term ueffA23) can be obtained by similar methods as ueff(r1r2r3) = %eff(r!r2r3) - ueffA2) - ue(fA3) - ueffB3) = /32<uaA2)uaA3KB3)L1<1JUK + O(i4). D.130) When these expressions are truncated at the O(u2) terms they reduce to the effective potential of D.118), which reproduces the u-expansion to A2. Including the O(ul) terms will reproduce the series to A3. This gives improved results72'75 but the simplicity of the O(u2) result is lost.
304 PERTURBATION THEORY 4.10 4.10 Perturbation theory for non-additive potentials If the series for the potential energy, ? ..., D.131) can be cut off after the three-body term u(Gfc) = u(rjr,rkcoia)jco|c) it is straightforward66196^8 to extend the earlier results for g(rco1&J) and Ac. In general the three-body potential will be a sum of contributions from dispersion, induction and overlap terms. The three-body overlap con- contribution to thermodynamic properties is poorly understood and is often neglected; for atomic fluids this contribution is claimed to be small.199 This is presumably because the triplet correlation function is small for the short-range configurations where uovA23)^0. The three-body dispersion and induction terms are known to be significant for some molecular fluids, however (see below and §6.11). These three-body potential terms can be included as part of the perturbing potential. For simplicity we consider the u-expansion. When induction forces are present it is convenient to write the potential as * %JrNcuN) = I uo(ri,) + a[*? «?„>„•) + I ua07)+ ? u(ijk)]. D.132) We note that uo(rif) here is not the Barker-Pople reference potential defined by D.23), but is the isotropic potential in the absence of induction interactions; ufncl(rjj) is the isotropic two-body part of the induction interaction and u^=u — u0— u°nd is the anisotropic part of the pair poten- potential. The first-order term in the expansion for Ac no longer vanishes in this case, and is given by (cf. D.18) and D.19)) A,=2P2J dr1dr2go(r12)ufnd(/-12) + |p3 J dri dr2 dr3go(r12rI3r23)<uA23)><1>1<<>2Ml. D.133) The three-body induction term uimlA23) makes no contribution to A,. Thus, for the three-body dipole-induced dipole interaction (cf. B.291)),200 uind(lBK) = -m . T12 . a2 . T23 . m, D.134) the term (Mind(lBK))u>]U2CO3 contains <|x;>^ (for i = 1 and 3) as a factor, which vanishes; the B) in uind(lBK) indicates the molecule that is polarized. The three-body dispersion potential, B.302), will contribute to A\, however. The explicit equation for the dispersion contribution to A, is given in Appendix 6B.
4.10 THEORY FOR NON-ADDITIVE POTENTIALS 305 It is also straightforward, though tedious, to work out the expression for A2 in the presence of three-body forces.196 It will involve integrals over the square of the perturbing potential, and will include terms of the types (we note that terms of the type M°,dua do not contribute by virtue of D.24)): (a) u°nd(ij)u°nd(fc0, (b) uld{ij)u(klm), (c) ua(i/)ua(kl), (d) uJLij)u(klm), (e) u(ijk)u(lmn). Terms of type (c) will be large in general, and have already been included in the A2 expression given in D.29). Of the remaining terms we might expect those of type (d) to be most significant,201 especially when uz(ij) is a multipolar term and u(klm) is the three-body induction term of type D.134). Numerical calculations confirm this expectation (see § 6.11). The expression for the multipolar three-body induction contribution to A2 is readily evaluated by the methods of §§ 4.4 and 4.5 and is ABnon. ¦nuL-ind) = .1^3 J ^ ^ dr3g0A23)(MmultA2)uindA23))OJ,uJM3. D.135) When uindA23) is the dipole-induced dipole interaction, uindA23) = uind((lJ3) + uind(lBK) + uindA2C)), D.136) with the terms uind((lJ3), etc. being of the type D.134), A2 simplifies to A2non-mu"-ind) = -^p3Jdr1dr2dr3g0A23)(MmultA2)uindA2C))><U|(,J(Ui D.137) since the other two terms in D.136) will contain (ni)^ as a factor. The term in D.137) will be appreciable for polar fluids, particularly if ix and/or a are large. Numerical calculations106-197-202-203 for this and other contributions to A2 are discussed in Chapter 6. The influence of induction forces on the thermodynamic properties of molecular liquids has been studied by perturbation theory.66106197202"9 In such studies one must consider, in addition to the two-body dipolar induction term of order a, the effects of (a) higher order terms in a (e.g. the terms in Fig. 2.36(b), (c), (e), (f), etc.), (b) the multibody terms (e.g. the terms in Fig. 2.36(d), (e), (f), (g), etc.), (c) higher order multipole terms, and (d) the anisotropy of the polarizability. In one approximation scheme,106-204 proposed by McDonald,106 one neglects higher order mul- tipoles and higher order terms in a (including multibody terms higher
306 PERTURBATION THEORY 4.10 than three-body), and retains terms in the perturbation series up to A2; thus the two- and three-body induction terms (a) and (d) in Fig. 2.36 are included in this treatment. Since the perturbation series for the O(a)% induction terms converges slowly, a Pade approximant is suggested, = Ac-Au+Ii 1 ^ j D.138) Here Ac = ALJ+(i+a = AU + AA^ +AAa is the total configurational energy, AA^ is the contribution from the rigid dipoles (in the absence of polarizability), AA^ is the additional part arising from the polarizability, A',intl) is the two-body induction part of A, in D.133), and A2mulMnd" = ^ta.n.muii,-inci) arises from the three-body induction potential and is given by D.137). Equation D.138) predicts that the induction contributions are large.62 For a fluid of polarizable dipoles with a central LJ potential at fcT/e = 1.15, f/,/(?<73M= 1, a/<x3 = 0.05, and liquid densities, we find ALV-0.10-0.15, AAJAA^-0.30, and ApJAp^-0.30, where AC/Q, etc. correspond to AA^ and AAa. We shall see below that D.138) in fact underestimates the magnitude of the induction contribu- contributions, so that they are an even larger fraction of the rigid dipole contribu- contributions. The large induction effect may be somewhat surprising at first sight, since the ratio of the induction pair potential energy to the corresponding multipole-multipole energy is of order a/cr3 (see § 2.5) or about 5 per cent in this case. However, this neglects effects due to the angular dependence of the induction and multipolar potentials, and also due to three-body (and higher multi-body) terms. The effect of induction forces on A represents a combination of the effects on internal energy and entropy, since A = U—TS. The induction forces will affect the molecular structure (and hence S) as well as the energy, and these two effects combine to produce an enhanced effect on the free energy. The perturbation calculation based on D.138) omits terms O(a2) and higher (and hence all four-body and higher multi-body induction terms), and does not contain any dependence on the anisotropy of the polariza- polarizability. In order to study these additional aspects of the problem computer simulations have been carried out for polarizable molecules; potentials studied include polarizable dipolar LJ,207 hard sphere,205208 and site-site interaction209 models. Such simulations are time consuming, because of the many-body nature of the induction contribution. The total electrosta- electrostatic energy (for purely dipolar molecules) is given by B.287), %cec = -iI fc • Tf/ . A/ + IIE,. a,. E, D.139)
4.10 THEORY FOR NON-ADDITIVE POTENTIALS 307 where p., =|A; -t-otj . E; (see B.286)) is the total dipole moment, consisting of a permanent and an induced part, T(j is the dipole field tensor given by B.285), and E, is the field at molecule i due to all the other molecules and is given by B.284), E, = lT, .?, D.140) i where ?'/ denotes a sum excluding / = i. The electrostatic energy must be calculated from D.139) and D.140), using an iterative process to obtain E,. In the first step of the iteration the total dipole moment ?.,- on the right-hand side of D.140) is approximated by p,,- = fiy, the permanent moment. The resulting field, E[0), is used to calculate ^l> = Hi+«i -Ei0), which is then used in D.140) to obtain a second approximation, E-n, to the field. For physically realistic a values only a few iterations (typically205-208 3-10) are needed to obtain con- convergence; the electrostatic energy is then calculated by using E|"~n and p.- in D.139). Patey et a/.208 have made such calculations by the Monte Carlo method for polarizable dipolar hard spheres at a reduced density p* = per3 = 0.8344 and a reduced dipole moment (I2 = ix/kTcr3 = 1. Two different choices were made for the polarizabililty. In the first a*x = a*y = a*z = 0.03 (isotropic polarizability, y* = 0) was chosen, while in the second a*x = a*y =0.03, a*z=0.06 (anisotropic polarizability, 7* = a*2 - a*x = 0.03) was used; here z is chosen to lie in the direction of M- and a* = a/cr3. Results for the contribution of the induction forces to the free energy and internal energy are shown in Table 4.9. Even though the polarizability values studied (a* = 0.03 and 0.04) are quite modest, the induction contribution is large, AA^/AA^— 0.29-0.53 and210 A UJAU^ ~ 0.22-0.46. The Pade approximant of D.138) seriously under- underestimates the magnitude of the induction effect, by almost a factor of two for AAa at a* = 0.03, y* = 0. The effect of polarizability is to lower both the free and internal energies. The major contribution to the decrease in the internal energy is seen to arise directly from the average induction energy (%]„<]), but there is also a significant contribution from changes in the average permanent dipole energy (%li); this latter change indicates that the induction forces have a substantial effect on the fluid structure. A direct measure of such structural effects is provided by the changes in the magnitude and direction of the total dipole moment jl associated with a single molecule relative to the permanent moment n. It is useful to introduce a mean total dipole moment n' = (|l)'0, where ( )™ denotes an ensemble average with the molecular orientation w held fixed. Equival- ently, we can obtain n' by choosing the body-fixed axes x, y and z and computing the components (/lx), (/ly), and (jl2), where ( > now denotes an unconditional average. This quantity can be obtained in computer simula- simulations, and some values for the magnitude of \i, (x' = |(i'|, are shown in
Table 4.9 Effects of polarizability on properties for polarizable dipolar hard spheres at per3 = 0.8344, /x2/fcTo-3=l, a^ = a*y =0.03. The free energy and internal energy for the rigid dipole fluid (a = 0) at these conditions are AA(i/NfcT= -0.638±0.005 and M!JNkT= -1.056±0.007, respectively {from ref. 208)i 0.03 0.06 a* 0.03 0.04 7* 0 0.03 -(A,-A,,s)/NfcT 0.820 ±0.007 0.977 ±0.01 MC 0.182 ±0.01 0.339 ±0.01 -^AJNkT MC" 0.173±0.006 0.310±0.0U RT 0.195 0.370 P; 0. ade 104 -<*.VNkT 1.101 ±0.01 1.175±0.01 MC 0.192±0.003 0.370 ±0.004 0. 0. IkT MC" 173±0.006 310±0.011 MC 1.077 ±0.001 1.183±0.002 a- MC 1.074±0.002 1.164 ±0.004 RT 1.083 1.203 + Here a* = altr^, 7* = 7/a3 = a*. — a*x, (I'2 = (i'2/kTcr:\ ja'= magnitude of the average total dipole moment, AAa = AHSt „ —AHS+(ip, where AHS+U<.<> = Ac is the free energy for the system of polarizable dipoles and AHSt;1 is that for the system of rigid dipoles with ot = 0, AA^ = AHS+ll — AHS, where AHS is the value for hard spheres, (AL/^ is defined similarly), (aU^) is the average permanent dipole contribution to the configurational energy, and <%jnd) is the average induction energy; in these latter two quantities the averaging is carried out in the system of polarizable molecules. Note that CUJ) will vary with ot, since the induction forces will influence the fluid structure. Values listed for (Ac— AHS) and (91^) are Monte Carlo (MC) values. MC° = reference fluid MC, eqn D.142); RT = renormalization theory of Wertheim;206 Pade=eqn D.138).
4.10 THEORY FOR NON-ADDITIVE POTENTIALS 309 0.8 1.2 0/radians Fig. 4.28. The probability density f(d) for the angle 8 between the permanent and induced dipole moments for polarizable dipolar hard spheres at per3 = 0.8344, n2/fcT«T3=l, «*x =«*, =0.03. The solid line is for the case a*z =0.06 G* = 0.03), the dashed line is for a*2=0.03 (y* = 0), and the dotted line shows the uncorrelated case, /o@) = jsin0. (From ref. 208.) Table 4.9. It is seen that fx' increases significantly over the permanent dipole moment due to polarizability (further calculations of j/,' are discus- discussed in § 10.1.6). The instantaneous angle 0 between the permanent and induced dipole moments n and Hind gives a measure of the influence of polarizability on orientational structure, and the probability density func- function fF) for this angle is shown in Fig. 4.28. The most probable orientation of the induced dipole is close to that of the permanent one, the values of 0 being about 27° and 10° for the a* = 0.03 and 0.04 cases, respectively. These results do not imply that (tx.;n(i)'° is not parallel to ia; on the contrary, since the molecules considered in Fig. 4.28 are axial, all azimuth angles of |Xj,,d about n are equally likely, so that (Hind)" will be parallel to p.. The computer simulation studies described above are very time con- consuming. When the induction forces are not too large it is possible to use a simpler perturbation approach, which substantially reduces the time needed for simulation (by a factor of about 20, typically). This method has been applied to polarizable, dipolar hard spheres by Patey and Valleau.205208 The potential is written %=^HS+li+^ind D.141)
310 PERTURBATION THEORY 4.10 where %Hs+n 's a sum of the hard sphere and dipole-dipole potentials for all molecular pairs, and °Uina is the induction energy (i.e. all terms in B.291) except the initial dipole-dipole term). Expanding Ac about the value for dipolar hard spheres gives A=A HS-t ./HS+p D.142) where (.. .)Hs+n means an ensemble average over the reference system of dipolar hard spheres. The first-order perturbation term (%ir,d)HS+lt w'" involve g(ra),aJ) and higher order distribution functions for the reference system, all of which are unknown. Patey and Valleau therefore evaluate the first-order term by Monte Carlo simulation; we shall refer to the values obtained by the first two terms in D.142) as MC° (reference fluid MC). From Table 4.9 it is seen that the MC" values lie a little above the exact MC values for AAa but are in good agreement for a*<0.04 G* ?—0.03); this includes most real molecules (see Appendix D). We note that2'1 (cf. §6.8) Ac-AHS+li?(%ind)Hs+^ D.143) so that the MC° values must be an upper bound on the induction free energy; since (%jnd)Hs+M. 's negative, the MC° values are a lower bound on the magnitude of AAa. Results from MC° for the effect of anisotropy of the polarizability on the free energy are shown in Fig. 4.29. It is seen that the induction contribution to the free energy depends strongly on the 0.4 0.2 0.04 0.08 Fig. 4.29. Effect of the anisotropy of the polarizability, 7* = a* — a*, on the free energy for polarizable, dipolar hard spheres, pxr3 = 0.8344, \x2lkT(T^— 1.0, a* = a/<r1 = 0.03, «u = azz, ax = axx = ayy (z along dipoie); AAa is defined in the footnote to Table 4.9. The point at 7* = 0 is from a rigorous MC calculation,208 and the lines are from various approximations: from MC", D.142), including all multi-b.ody terms.to first order in the perturbation (this gives a rigorous lower bound to -AAJNkT); from MC", D.142), but including only the two-body induction terms; from the Pade approxi- mant of D.138); — the renormalization theory (RT) of Wertheim206 (from refs. 205 and 206.)
4.10 THEORY FOR NON-ADDITIVE POTENTIALS 311 anisotropy of the polarizability when multi-body induction potential terms are included, but that this effect is absent in the pairwise additive approximation and also in the Pade approximant of D.138). The pairwise additive approximation is clearly poor except when the polarizability is nearly isotropic. From both Table 4.9 and Fig. 4.29 we see that the Pade approximant of D.138) seriously violates the condition D.143). In view of D.143) the exact values of —AAJNkT are expected to lie somewhat above the MC° values (solid curve) in Fig. 4.29. However, the compari- comparisons of MC° and MC values shown in Table 4.9 suggest that MC° is not in serious error, at least for y* values up to about 0.03. The MC° has been used209 to estimate the contribution of induction interactions to the internal energy of HF and HC1; these calculations are shown in Table 4.10. The potentials used for these molecules consist of atom-atom models with the addition of dipolar and quadrupolar interac- interactions.212 The MC° calculations are based on D.142) with HS+/x now replaced by HS + /X + Q. The induction interaction now contains both dipole-induced-dipole (DID) and quadrupole-induced-dipole (QID) con- contributions. The values shown for —(aUmdH/ NkT provide a lower bound on the magnitude of the polarizability contribution to the free energy, AAJNkT, for these models. For HF the induction contribution is particu- particularly large, approximately 23 per cent of (%H, the total energy of the non-polarizable system, and about half of this amount is contributed by the multi-body forces. The average total dipole moment is about 20 per cent larger in magnitude than the permanent dipole moment. These calculations indicate that a large number of multibody induction Table 4.10 Effect of polarizability on properties of models of HF and HCl from MC". Here (%H and (%eiec)o a^ the average total and electrostatic energy, respectively, for the reference fluid in which a = 0, while (%nd)o *s f^e induction energy averaged over the reference system and is, to first order, an estimate of AAa (from ref. 209)t System HF HCl T/K 278 201 V/cm1 mol ' 19.96 30.74 _mjNkT 9.71 9.89 10.57 3.08 - DID 1.76 A.07) 0.61 @.42) QID 0.50 @.06) 0.24 @.03) kT Total 2.26 A.13) 0.85 @.45) M-7D 2.15 1.29 tFor HF: |i=1.82D, O = 2.6x 10 2<sesu, a, = 0.96A\ a±=0.72A3. For HCl: m. = 1.04D, Q = 3.37x lO^esu, a, = 3.13A', a±=2.39A\ The estimated uncertainties in the calculated values of ("U^Ja/NkT and (%lnd)o/NkT are ±0.15 and ±0.10, respectively. The quantities in brackets are obtained in the pairwise additive approximation.
312 PERTURBATION THEORY 4.10 terms in D.131) may be needed to obtain accurate results. This is impractical using straightforward perturbation theory because reference fluid correlation functions of high order are needed. In order to overcome this problem Wertheim206'2 has developed a form of perturbation theory based on a graphical resummation (renormalization), which accounts for both multibody induction terms and for anisotropy of the polarizability. We refer to this as the renormalization theory (RT); it is further discussed in Chapter 10 in connection with the dielectric constant. In RT the isolated molecule dipole moment |i is replaced by a renormalized value213 , I*' which accounts for multibody induction effects. This renormalized dipole moment depends on the state condition, polarizability and poten- potential parameters. Physically, |i' is simply the mean total dipole moment (ix}u> for fixed orientation u> of the molecule. The effect of polarizability is to make n' larger in magnitude than the permanent dipole moment ji, often by 20 per cent or more (see Tables 4.9 and 4.10 and Chapter 10). In terms of this renormalized dipole moment, the perturbation terms in the expansion of the free energy up to O(/x'6) involve only g()A2) and goA23), and are no more complex than the A2 and A3 terms in the u-expansion. Although the graph-theoretical analysis used in RT is complex, the final results are quite simple and straightforward to use, and we therefore give them here. (In Chapter 7 (Vol. 2) we give a simple non-graphical plausibility argument for the basic equations.) We consider the case of molecules which interact with some central isotropic pair potential uo(r), and possess a permanent dipole moment |i, (and no higher multipoles) and polarizability a. The perturbation terms in the expansion for the free energy are found to involve one or other of two dimensionless state- dependent variables xx and x2, which are given by x2 = pa D.144) where C = 1/fcT, p is the number density, /x' is the magnitude of (*', and a is the mean polarizability. Wertheim chooses to form two separate Pade approximants, P(x,) and P(x2), for these two types of perturbation terms; each Pade is of similar form to the Pade successfully used to resum the u-expansion (see D.47)). The resulting equation for the Helmholtz free energy is206-214 + ^ D.145) where Ao is the free energy of the reference system of spherical molecules with pair potential uo(r). The first term on the right-hand side
4.10 THEORY FOR NON-ADDITIVE POTENTIALS 313 of D.145) gives the contribution to (A — A0)/NkT from the effects of permanent and induced dipolar forces, and the second can be considered to represent the contribution from the polarization of all dipoles from n to the mean moment (*'. (In a mean-field theory, up would in fact be the mean polarization energy per molecule). The Pade approximant P(x) is given by bu <4I46) where I2 and I3 are two- and three-body integrals over correlation functions for the reference system given by215 J2 = jdr12go(r12)T12:T12 = 6jdr12g0(r12K26 D.147) and \ : T23 . T13 , = I dr12 dr,3g0(r12ri3r2; = -24tt2 dr12r7| dr13rr32 dr23r232g0(r12r13r23) J0 J0 J|r12-r13[ x(l + 3 cos al cos a2cos a3). D.148) Here T,, is the dipole field tensor given by B.285), go(r12) and go('"i2'"i3'3) are the pair and triplet correlation functions for the reference system of spherical molecules, and a,, a2 and a3 are the internal angles of the triangle formed by molecules 1, 2 and 3. Note that P(x), 12 and 13 have dimensions of inverse volume. In the limit of rigid dipoles (a = 0) it is easy to show that D.145)-D.148) reduce to the usual Pade approxi- mant of D.47), Ac — Ao = A2(l — A3/A2), where A2 and A3 are the usual perturbation terms for dipoles.216 The mean polarization energy per molecule that appears in D.145) is given rigorously by a mean-field-like expression up = 2-E'.a.E' D.149) where E' is the sum of a particular set of graphs, and is given in RT by E' = |P'(x,)|*' D.150) where dx,
314 PERTURBATION THEORY 4.10 The physical significance of E' is seen from the expression206 EJ = <E,)<ui, i.e. EJ is equal to the mean electric field at molecule 1, which has fixed orientation «,. It remains to determine the renormalized moment ft'. This is given in RT by a] = |i D.152) where 1 is the unit tensor and n is the permanent dipole moment. Equation D.152) follows immediately from \i.[ = p.x + o^ . E',; the latter relation is obtained by averaging the corresponding microscopic relation (see below D.139)). The procedure for numerical calculations based on D.145) is as follows. Equation D.152) is first solved for |i' by iteration, making use of D.151) for P'(xi) and D.144) for x,; in carrying out this solution it is usually true that /x'>/x. Equations D.150) and D.149) are then used to calculate up, and D.146)-D.148) are used for P(x,) and P(x2). The calculations are only slightly more time consuming than for the usual Pade approximant of D.47). The RT has been applied to fluids of polarizable, dipolar hard spheres. The z-direction is chosen to coincide with the direction of ft, and a is taken to be diagonal in the xyz axes,217 D.153) For such molecules D.152) becomes /Xz[l-!P'(*i)azz] = M.z D.154) with /x; = |xv = 0, while D.149) and D.150) together give up = |P'(x1J/x;2a2Z. D.155) We note that although only the zz component of a appears explicitly in D.154), P'(x,) depends on both /x2 and the mean polarizability a, so that /xz itself will depend also on the values of axx and ayy. However, one might expect from D.154) that n' will depend strongly on a2Z, the component in the direction of n, and this is confirmed by the calculations. Some of these RT calculations are shown in Tables 4.9, 4.11, and Fig. 4.29. From Table 4.9 we see that RT is within about 10 per cent of the Monte Carlo value of AAa for the anisotropic polarizability case G* = 0.03), and is even better for the isotropic case. Moreover, the RT values for the mean dipole moment n' are also close to the MC values, providing strong support for the validity of the theory. In Fig. 4.29 is shown the effect of anisotropy of the polarizability on the free energy as calculated by RT, and the various terms that appear in D.145) for (Ac-A0) are
4.11 CONCLUSIONS 315 Table 4.11 Effect of the anisotropy of the polarizability on the free energy of polariza- ble, dipolar hard spheres as calculated from renormalization theory. Here pa3 = 0.8344, /x2/fcT(T3 = 1.0, a* = a/a3 = 0.03, a,, = azz, aL = axx = ayy (z along /x) (from ref. 206)t (Ac-A0)INkT 0 0.03 0.045 0.06 0.075 0.09 < 0.045 0.03 0.0225 0.015 0.0075 0 1 1.0830 1.1362 1.2009 1.2818 1.3864 -P(x,)/p -0.7792 -0.9921 -1.1464 -1.3523 -1.6399 -2.0607 P(x2)/p 0.0077 0.0077 0.0077 0.0077 0.0077 0.0077 0 0.1148 0.2061 0.3363 0.5294 0.8295 RT -0.7715 -0.8696 -0.9326 -1.0083 -1.1028 -1.2235 MC" -0.726 -0.811 -0.860 -0.912 -0.971 -1.035 ^ = AHS+li — AHS is —0.638±0.004 from MC simulation and —0.675 from perturbation theory. The values of AAa from RT and MC° are therefore in somewhat better agreement than the corresponding values of (Ac —A^) shown here (see Fig. 4.29). given in Table 4.11. We note that /x' = ^'z, and hence Xi, depend strongly on a1] = a2z through D.154), and this causes the terms —Pix^/p and f3up to be strongly dependent on a^. These two terms make the major contribution to (Ac-Au). The term P(x2)/p is a constant, independent of a|[. The RT values of AAa lie somewhat below the MC° values, which provide an upper bound. However, the two sets of values agree moder- moderately well, particularly at the smaller values of the anisotropy y. Several other approaches218 to many-body induction effects have been used, in addition to perturbation theory, and some of these are described in other chapters (see particularly § 10.1). 4.11 Conclusions Although no fully satisfactory perturbation theory yet exists for the pair correlation function g(t(olaJ) for molecules with strongly anisotropic forces, several useful approaches are available for the thermodynamic properties. For long-range anisotropic (e.g. multipolar) potentials the Pade resummation of the u-expansion, eqn D.47), gives good results and is simple to use. It has been tested against computer simulation results for dipole-dipole and quadrupole-quadrupole potentials, and has also been compared with experiment for a variety of molecular liquids (see § 6.11). A similar expansion is possible for ionic systems; again, a Pade approxi- mant to the series gives good results when tested against simulation data.111113204 For short-range anisotropic potentials (e.g. overlap) the u-expansion is less satisfactory; it also gives rather poor results for the fluid structure, even for multipolar potentials. At the cost of some
316 PERTURBATION THEORY 4.11 additional numerical work, the /-expansion gives better results for struc- structural properties, and is usually better for molecules that have non- spherical shape forces. However, both the u- and /-expansions rely on a reference fluid of spherical molecules, and this is the main restriction on their range of validity. For highly anisotropic molecules a non-spherical reference potential is called for. Some progress has been made in this direction, particularly in dealing with non-spherical shape forces, but the reference correlation functions that appear in the perturbation terms are now themselves dependent on orientation, so that numerical calculations are usually tedious. Fluids in which the molecules have both short- and long-range forces that are strongly orientation-dependent (e.g. non- spherical shape with strong multipolar forces) are difficult to treat, and no fully satisfactory method is yet available. The effects of multibody forces can often be conveniently treated by perturbation theory. Multibody dispersion forces often have a rather small, though not negligible, effect. For polar fluids multibody induction forces usually have a much larger clfect, typically amounting to 20-40 per cent of the electrostatic contribu- contribution to internal or free energies of simple model polar fluids. Not only must one consider higher n-body induction terms, but the anisotropy of the polarizability also plays an important role. The RT theory is an important step forward in our understanding of these complex effects, and the final equations are simple to use. The discussion of perturbation theory given here is by no means exhaustive. Thus we have omitted any mention of the cluster series approach.75111-219 This method in its various forms (often referred to as LEXP, LIN, L3, QUAD, and LIN + SQ) is a partial resummation of a cluster expansion for the pair correlation function. This apparently gives ' quite good results for dipole-dipole interactions,75220 but may be less accurate for other potentials.221222 We have also omitted any discussion of the van der Waals model,223 which has been extensively used to interpret thermodynamic properties. This model can be derived from perturbation theory by replacing the reference correlation functions g()A2), goA23), etc. by constants-i.e. we assume that terms involving the fluctuations of the g0 functions can be neglected in comparison with those involving the mean value of g0. This should be more accurate the longer the range of the potential. As an example, consider the first-order expression for the free energy using a non-spherical reference potential, D.88). If UiA2) vanishes for r12< cr(aIaJ), corresponding to separations for which go^O, and if we replace go by unity for rJ>cr(aIaJ), then we have A=A() + aNp D.156) and p = -dA/dV is 2 D.157)
4A EXPANSION FOR A GENERAL PROPERTY (B) 317 where a is a constant which depends only on the perturbation potential. When the reference system is a hard sphere fluid, D.157) is the van der Waals equation of state. For non-spherical cores D.157) is a generaliza- generalization of the usual formula, and apparently gives the correct qualitative behaviour for simple molecular liquids.224 Higher terms in the expansion have also been studied by this approach,978 as well as other equations of state.225 Appendix 4A Perturbation expansion for a general property (B) In this appendix we derive general expressions for the first- (§4A.I) and second-order (§ 4A.2) perturbation terms (B)j and (BJ in the expansion of some property (B); both the reference system and method of parameterization are arbitrary at this stage. In §§ 4A.3-4A.5 we consider several applications of these genera! expressions. We use the grand canonical ensemble throughout, since the expressions obtained are then easier to use for calculations than those derived using the canonical ensemble (see ref. 91). We consider a uniform fluid in which the potential energy is allx, where °U^ obeys D.10) and D.11). Any property (B) can now be expanded in a Taylor series about A = 0. This is usually done94 at fixed temperature, density and volume, so that 5@) pVp.A.=0 -i V OA /pVp,A.=0 Taking A = 1 gives (B) for the real system, (B) = {BH + (BI + (BJ + ... DA.2) where (B)o is the reference system value (at the same |3Vp as the real system), (B)i = (a<B)/flA)fJVp.\=o is the first-order perturbation term, etc. 4A.1 The first-order perturbation term(B)i We have already obtained an expression for the grand canonical deriva- derivative (d(B)/dAHVM. in Chapter 3, and this is related to the canonical one (at fixed pVp) by = MB>\ +/a<B>\ MA V = + \ 3A /gvp ^ dk /eV(l V / eV\ pvp The derivatives C<B)/aA)fJVlt and (d<B)/d/x)Pvx are given by C.249) and C.203), respectively. To evaluate (a/x/dAHVp we note that at fixed C and (^) M dA V3A
318 PERTURBATION THEORY 4A and hence we have VdA/0Vp From C.208) we find P X, DA.5) while from C.172) we get (dp\ 1 /d2lnH\ /Aass [—) = I ) • DA.6) Using the definition C.190) of H we have —) DA.7) where ML/d\=d°UJ\. Thus, combining DA.6) and DA.7) gives Using DA.4), DA.5), DA.8) and C.203) we have where, for brevity, {...) = {.. .)K is a grand canonical average and %=%x here. Combining DA.3) with C.249), C.208) and DA.9), gives for the first-order term (we consider the most general case, where B may depend on A): The expression for the first-order term (B)i for any specific case can be obtained from DA.10) by taking226 the limit A —»0. We then get
4A EXPANSION FOR A GENERAL PROPERTY <B) 319 where (.. .H indicates a grand canonical average over the reference system. 4A.2 The second-order perturbation term (BJ Differentiating DA.3) gives, /d2(B)\ = /3_/3<B>\ \ J±(d(B)\ \ /V \ dA2 /0Vp VdA \ dk IpVJ0Vp VdA V d/x /pvi'svp^ The first two terms on the right-hand side of this equation can be put in a more useful form by the use of DA.3), with (B) there set equal to (d(B)/dAHVM. and (d(B>/d/xHVX respectively. The result is 9A /QVo \ dk /f)V,, VM evp N OA ' 3Vn w/v O/X ' 3V WA ' 3Vp 3)\ /^/x\2 /3(B)\ /32/x\ . . _. 2") I—) +( ) \—2) • DA.12) The second derivatives (d2(B)/dA2HVlU {d2{B)/dk 3/xHV and (a2<B>/a/x2Kvx can be evaluated by iteration of C.249) and C.203). This gives DA.15)
320 PERTURBATION THEORY 4A The derivative (d/x/dAHVp has already been evaluated in §4A.l. It is The second derivative of /x with respect to A is obtained from DA.16) as =~vw) w)BV ) w)BVJ w N) 3VP w/v 1<"v ' ' 3Vp DA.17) The various terms in DA.17) can be evaluated with the aid of DA.10) and C.203). The result is The complete expression for the second derivative of (B) with respect to A is given for the general case by substituting DA.13)-DA.18) in DA.12). The second-order perturbation term is then obtained by taking the A —» 0 limit of this equation and multiplying by one half. This will be a long and complicated expression in general. However, considerable sim- simplification occurs in certain special cases, as shown below.
4A EXPANSION FOR A GENERAL PROPERTY <B) 321 4A.3 (B)! and (BJ with the Barker-Pople reference system With the choice of parameterization and reference potential used in the u-expansion (see D.1) and D.2)), the equations given above simplify considerably. In that case we have and 2 O DA.19) = 0 DA.20) where z = exp(/3/x)qqu/AfAr (see §3.3.2). Similarly we have 0, DA.21) lim/iV2 — \ = 0. DA.22) It follows from DA.11) that the first-order perturbation term is given by <BI = -p(B%aH + (^). DA.23) \oa/o From DA.4) and DA.8) we see that (-) =0 and DA.12) simplifies to /d2(B)\ ^ /d2(B)\ | /3<B)\ /3V\ Using the properties DA.19M4A.22) in DA.13), C.203) and DA.18) gives where (.. .}0 denotes an average over the reference system ensemble.
322 PERTURBATION THEORY 4A 4A.4 First-order perturbation contribution to the pair correlation function As a further example of the use of the general expression for (Bt), cq. DA.11), we consider the expansion of the pair correlation function g(ra>,a>2), defined by C.108). We make no specific choice of the reference potential or expansion parameter, and derive the general expression for the first order term gl(ta>1aJ) of D.12). To find the expression for g we use DA.11) with (cf. C.108)) B = I «(r, -O Sfo -Wl) d{i, -r2) 8(a>, -<o2) DA.25) so that from C.198) and C.194) we have g(ro>,a>2). DA.26) We assume pairwise additivity, <MrN<uN) = I u,(riJa,,a,j) DA.27) so that we have Examination of this equation shows that there will be terms of four types: (a) N(N-l) terms with i = k and j = I, (b) N(N-l)(N-2) terms with i = fc, /^l, (c) N(N-l)(N-2) terms with / = fc, iV, and (d) ]MN-l)(N-2)(N-3) terms with i/ fc, ' and /V k, /. Thus, using C.189), we find OTA 3mA2) f pMA3) 3mB3) /A2)+J drdL+ f J dr3 ^4) DA.28) +Mdr3dr4dw3dw4 2 J dA where MA2) = Mx(ri2w1w2), /A2) = /A(r12a>ia>2), etc. By similar methods
4A EXPANSION FOR A GENERAL PROPERTY <B) 323 we get DA.29) /3/A2)\ 3mA2) I-—— =-p \ 3A /3p dA 0 f J d^^ dA +^ f dr, dr2 dr3 dWl d«2 dw3 ^^/A23). DA.30) Z J dA The final form of DA.30) is arrived at by replacing N2 in the first line by N[(N-2) + 2]. Finally, (N) can be written (N)=Jdr1dw1/(l). DA.31) Substituting DA.26) and DA.28)-DA.31) into DA.10) gives f puA3) duB3)l J L dA dA J dr3 dr4 d«3 d«4 ^^ [fA234) - /A2)/C ^ p /a/(i2)\ f f a«C4) + — —— 1 dr3 d»4 dw3 doj4 fC4) V \ dp /3xU dA + \( dr3 dr4 dr5 dw3 d«4 dw5 ^^ [fC45) - 2 J OA DA.32) Taking the A^O limit of DA.32), and using C.194) gives the desired expression, D.13) for g!. 4A.5 The u-expansion for the pair correlation function71221 As a final example of the use of the equations given in §§ 4A.1 and 4A.2, we consider the u-expansion for gA2). In this case B is given by DA.25) and %a(rNa>N) is defined by D.1) and D.2) and is assumed to be a sum of pair potentials ua(riJa>ia),). We have, from D.24), the simplifying feature that the unweighted average of ujjf) over orientations vanishes. Moreover, the correlation functions for the reference system are indepen- independent of orientations for this choice of reference, so that from D.24) and D.13) the expression for the first-order term g! simplifies to D.25), as shown in §4.5.
2 V V dp /oL where f2 is the second order contribution to /(ra),aJ) and 324 PERTURBATION THEORY 4A In this section we obtain the second-order term g2A2) for this series, from the general expression for (BJ. With the present choice of B both dB/dk and d2B/dA2 are zero, so that DA.24) simplifies to DA.33) D A. 34) it Assuming pairwise additivity, %a= I Ua(kl), we have <= Z Z ua(fc/)ua(mn) DA.35) and B%a=Z Z Z b(v)M.(fcOM,(mn) DA.36) where b(ij) = 8(rj — Tj) 8(o)j — o>i) 8(r; — r2) 8(wj — <u2). DA.37) Examination of DA.35) and DA.36) shows that °U2 and B°Ul each involve terms of several types. These are summarized in Table 4A.1. Thus we have = 2p2jdr1dr2goA2)(uaA2J)<ui^ + P3J dr, dr2 dr3goA23)<MaA2)MaA3)Li(U2<U3 DA.38) where we note that the ensemble average of uaA2)uaC4) vanishes because of the property and we have used lim f,A2 ... h) = ^r go(l ...h). DA.39) A
4A EXPANSION FOR A GENERAL PROPERTY <B> 325 Similarly, we get + <N(N-l)(N-2)(N-3 + 3)uaA2)uaA3)>0 = p2 J drx dr2g0A2)<MaA2J>Mia>1 + ip3j dd dr2 dr3g0A23)(uaA2J)CO|<U2 + 3P3J dr, dr2dr3goA23)(uaA2)uaA3)Li(U2a,3 + p4J dr, dr2 dr3 dr4g0A234)<uaA2)uaA3)>u),(O2(O, DA.40) From DA.34), DA.38), and DA.40), together with <N>0 = <N) = pV = pJdri, DA.41) we obtain the second term in DA.33) as [2goC4) + pJ dr5[g0C45)-g0C4)]] dr3 dr4 dr5<MaC4)MaC5)>o>3^lU5 x [3goC45) + PJ dr6[g0C456)-g0C45)]]}. + p3 This expression can be put in a more compact form by using the compressibility hierarchy equation, C.238), drh+1[go(l • • • dp
326 PERTURBATION THEORY 4A Thus, we have 1 P2 (d{B)\ UNq.2^ _/xn /O.2V -1 ~o 77 I ,w ) LV^^a/O VVo\ «aA)J 2 V \ov /0 f dr3 dr4<uaC4J>MVU4 /- [p2g0C4)] 2 J op + Jdr3 dr4 drs<uaC4)uaC5)LjM4M5 ^ [p3g0C45)]} DA.42) where Xo~ P l(dp/dpH is the isothermal compressibility of the reference fluid. The term (B1llH is evaluated by summing the averages of the various terms in Table 4A.1. Thus we get + 2puaA2)Jdr3g0A23)<[uaA3) +1*3 + 2pJ dr3g0A23)<MBA3)MaB3)>^ dr3g0A23)([MaA3J+ uaB3J]L3 y2^ dr3 dr4[g0A234)- g0A2)g0C4)](uaC4JL3OL + P2 J dr3dr4g()A234)<[MaA3)MaA4)+uaB3)uaB4)])<UiOL + 2p2 J dr3 dr4g,)(l234)<[u11A3)uaB4)+ uaA3)uaC4) + uaB3)uaC4)]L3OL + P3J dr3 dr4 dr5[g0A2345)- goA2)goC45)]<MaC4)uaC5)L,W4U>5}. DA.43) Combining DA.42) and DA.43) and noting that
4A EXPANSION FOR A GENERAL PROPERTY <B> Table 4A.1 Classification of terms in "III and Bill 327 Type of term uaA2J uaA2)uaA3) uaA2)uaC4) ftA2)uaA2J feA2)uaA3J feA2)uaB3J feA2)uaA2)uaA3) feA2)uaA2)uaB3) feA2)uaA3)uaB3) feA2)uaC4J feA2)uaA2)uaC4) bA2)uaA3)uaA4) ftA2)uaA3)uaB4) feA2)uaA3)uaC4) feA2)uaB3)uaB4) feA2)uaB3)uaC4) feA2)uaA3)uaD5) bA2)uaB3)uaD5) feA2)uaC4)uaC5) feA2)uaC4)uaE6) Number of terms |N(iV-l) N(N-l)(N-2) |N(N-l)(iV-2)(iV-3) N(N-l) N(N-l)(N-2) N(N-\)(N-2) 2N(iV-l)(N-2) 2N(N-l)(iV-2) 2N(N-l)(N-2) ^N(N-\)(N-2)(N-3) N(N-l)(N-2)(N-3) N(N~\)(N~2){N-3) 2N(N-l)(JV-2)(N-3) 2N(N-l)(JV-2)(N-3) N(N-l)(iV-2)(iV-3) 2JV(N-l)(JV-2)(N-3) N(N - 1 )(N - 2)(N - 3)(JV - 4) JV(N -1 )(N - 2)(N - 3)(N - 4) N(JV - 1)(N - 2)(N - 3)(N - 4) ]jN(N -l)(N- 2)(N - 3)(iV - 4)(N - 5) we have for the second-order perturbation contribution to g(ro>1oJ): + 2pua( 12) J dr3g0( 123)<[mb( 13) + uaB3)]>M, + 2PJ dr3goA23)<MaA3)MaB3)>(U3 + pJdr3g0A23)<[uaA3J+uaB3J]><U3 |P2J dr3 dr4[g0A234)-g0A2)g0C4)](uaC4JL
328 PERTURBATION THEORY + p2Jdr3dr4g(,A234)<[MaA3)MaA4)+MaB3)uaB4)]Ll(U4 + 2p2Jdr3dr4g0A234)<[uaA3)uaB4) + uaA3)uaC4) + uaB3)uaC4)]>(u,oL + P3| dr3dr4dr5[g0A2345)-g0A2)g0C45)]<uaC4)uaC5)L3UL(U5 [p3goC45)])(MaC4)MaC5)L3W4UM]. DA.45) References and notes 1. Here we use the terms 'perturbation theory' or 'thermodynamic perturba- perturbation theory' for the case where °U.U represents a non-trivial reference system. The virial expansion (§ 3.6) and integral equation (Chapter 5) results can be obtained from resummcd perturbation expansions based on the ideal gas (%„ = 0) as reference. 2. Smith, W. R. in Statistical mechanics (ed. K. Singer), Vol. 1, p. 71, Specialist Periodical Reports, Chemical Society, London A973). 3. Gubbins, K. E. AIChE Journal 19, 684 A973). 4. Barker, J. A. and Henderson, D. Rev. mod. Phys. 48, 587 A976). 5. Andersen, H. C, Chandler, D., and Weeks, J. D. Adv. chem. Phys. 34, 105 A976). 6. Ailawadi, N. K. Phys. Rept. 57, 241 A980). 7. Stell, G. and Weis, J. J. Phys. Rev. A21, 645 A980). 8. Boublik, T. Fluid Phase Equilibria 1, 37 A977). 9. Egclstaff, P. A., Gray, C. G., and Gubbins, K. E., in Physical chemistry, series 2, Vol. 2, p. 299 (cd. A. D. Buckingham), MTP international review of science, Buttcrworths, London A975). 10. Gray, C. G., in Statistical mechanics (ed. K. Singer), Vol. 2, p. 300, Specialist Periodical Reports, Chemical Society, London A975). 1 1. Streett, W. B. and Gubbins, K. E. Ann. Rev. phys. Chem. 28, 373 A977). 12. Werthcim. M. S. Ann. Rev. phvs. Chem. 30, 471 A979). 13. van der Waals, J. D. Verslag K. Akad. Wetens Amsterdam 17, 130, 391 A908); Ann. Phys. Beibl. 33, 48, 446 A909). 14. Ornstein, L. S. Proc. K. Akad. Wetens. Amsterdam 11, 116, 526 A908, 1909). 15. Kcesom, W. H., Commun. Phys. Lab. Leiden, Supp. 24b, §6 A912); see
REFERENCES AND NOTES 329 also Hirschfelder, J. O., Curtiss, C. F., and Bird, R. B., Molecular theory of gases and liquids, p. 210, Wiley, New York A954). 16. Falkenhagen, H. Zeit. Phys. 23, 87 A922). 17. Lennard-Jones, J. E. Proc. Roy. Soc. A106, 463 A924). 18. Corner, J. Proc. Roy. Soc. A192, 275 A948). 19. Pople, J. A. Proc. Roy. Soc. A211, 498, 508 A954). 20. Debye, P. Polar molecules, Chemical Catalog Co., New York A929). 21. Van Vleck, J. H. The theory of electric and magnetic susceptibilities, Oxford University Press, Oxford A932). 22. Serber, R. Phys. Rev. 43, 1011 A933). 23. Thin-ing, H. Zeit. Phys. 14, 867 A913); 15, 127, 180 A914). See also Maradudin, A. A., Montroll, E. W., and Weiss, G. H. Theory of lattice dynamics in the harmonic approximation, Academic Press, New York A963). 24. Van Vleck, J. H. J. chem. Phys. 5, 320, 556 A937). 25. Waller, I. Zeit. Phys. 104, 132 A936). 26. Kirkwood, J. J. chem. Phys. 6, 70 A938). 27. Opechowski, W. Physica 4, 181 A937). 28. Kramers, H. and Wannier, G. Phys. Rev. 60, 263 A941). 29. Domb, C. Adv. Phys. 9, 49 A960). 30. Peierls, R. Zeit. Phys. 80, 763 A933); see also Landau, L. D. and Lifschitz, E. M. Statistical physics, 2nd edn, p. 90, Addison-Wesley, Reading A969). 31. Longuet-Higgins, H. C. Proc. Roy. Soc. A205, 247 A951). 32. Leland et al.33 have developed an improved form of conformal solution theory, usually referred to as van der Waals 1 theory, by using different expansion parameters from those of Longuet-Higgins. This gives good results even for quite non-ideal solutions, and is discussed in Chapter 7. 33. Leland, T. W., Rowlinson, J. S., and Sather, G. A. Trans. Faraday Soc. 64, 1447 A968). 34. Zwanzig, R. W. J. chem. Phys. 22, 1420 A954). 35. Rowlinson, J. S. Mol. Phys. 8, 107 A964). 36. McQuarrie, D. A. and Katz, J. L. J. chem. Phys. 44, 2393 A966). 37. Barker, J. A. and Henderson, D. J. chem. Phys. 47, 4714 A967). 38. Weeks, J. D. and Chandler, D. Phys. Rev. Lett. 25, 149 A970); Weeks, J. D., Chandler, D., and Andersen, H. C. J. chem. Phys. 54, 5237 A971). 39. We refer here primarily to LJ and similar rare-gas-like liquids, where the short-range repulsive forces largely determine the structure. [Even in this case the attractive forces have a significant effect on S(k) at small k; see Evans, R. and Sluckin, T. J. J. Phys. C14, 2569 A981).] There are a number of exceptions, particularly for other types of atomic liquids and mixtures, where the attractive forces give rise to important structural effects. Exam- Examples where Coulombic forces are involved include electrolyte solutions (see, e.g. Enderby, J. E. in Microscopic structure and dynamics of liquids, ed. J. Dupuy and A. J. Dianoux, p. 301, Plenum, New York A978)), and liquid metals (where the effective potential has a much softer core: see pp. 262, 263 of ref. 6; Shimoji, M. Liquid metals, p. 72, Academic Press, London, A977); Ashcroft, N. W. and Stroud, D. Sol. St. Phys. 33, 1 A978), especially p. 73). Other exceptions include binary mixtures near liquid/liquid phase separations, in which the strengths of the attractive forces are appreci- appreciably different for the various components (see Sung, S. H., Chandler, D., and Alder, B. J. J. chem. Phys. 61, 932 A974); Henderson, R. L. and Ashcroft, N. W. Phys. Rev. A13, 859 A976)), and structure at interfaces,
330 PERTURBATION THEORY which is generally more sensitive to attractive forces (see Chapter 8 and references therein). 40. The BH d depends only on temperature, whereas the WCA d depends on both density and temperature. At low densities (per3<—0.45) dWCA>dml, whereas for high densities (pa3 > —0.45) dWCA<dim- Both the BH and WCA theories have difficulties in describing liquids at densities above that of the triple point (see, e.g. Ross, M. J. chem. Phys. 71, 1567 A979)). 41. Kirkwood, J. G., Lewinson, V. A., and Alder, B. J. J. chem. Phys. 20, 929 A952). 42. Buff, F. P. and Schindler, F. M. J. chem. Phys. 29, 1075 A958). 43. Smith, W. R., Henderson, D., and Barker, J. A. J. chem. Phys. 55, 4027 A971). 44. Gubbins, K. E., Smith, W. R., Tham, M. K., and Tiepel, E. W. Mol. Phys. 22, 1089 A971). 45. The idea that the structure of a dense atomic fluid is determined mainly by the hard core part of the potential probably originated with van der Waals, although he applied the idea also to the critical point where the approxima- approximation is, in fact, a poor one (see Widom46). Certainly van der Waals' thesis47 seems to be the first clear recognition of the different roles played by the short-range repulsive and long-range attractive forces; the significance of van der Waals' work in the light of modern developments has been . discussed by several authors.4H~^' The idea had been followed up extensively by various workers, without being made precise using perturbation theory (cf., e.g. the ball bearing''2 and gelatin ball" packing models, hard sphere fits to experimental structure factors,54'55 scaled-particle theory results,56'57 im- improved van der Waals type equations of state used in theories of freezing,58 : and computer simulation studies of hard spheres59). 46. Widom, B. Science 157, 375 A967). 47. van der Waals, J. D. Over de Continuiteit van den Gas- en Vloeistoftoes- tand, thesis, Univ. Leiden A873); the English translation is given by Threlfall, R. and Adair, J. F. Phys. Mem. 1, 333 A890). 48. Rigby, M. Rev. Chem. Soc. 24, 416 A970). 49. Rowlinson, J. S. Nature 244, 414 A973). 50. de Boer, J. Physica 73, 1 A974). 51. Lebowitz, J. L. and Waisman, E. M. Physics Today, March 1980, p. 24. 52. Bernal, J. D. Trans Faraday Soc. 33, 27 A937); Nature 183, 147 A959); with S. V. King in Physics of simple liquids, Chapter 6, (ed. H. N. V. Temperley, J. S. Rowlinson, and G. A. Rushbrooke), Wiley, New York A968). 53. Morrell, W. II. and Hildebrand, J. H. Science 80, 125 A934); cf. also Regular solutions, by J. H. Hildebrand and R. L. Scott, Prentice Hall, Englewood Cliffs A962). 54. Ashcroft, N. W. and Lekner, J. Phys. Rev. 145, 83 A966); Ashcroft, N. W. Physica 35, 148 A967). 55. Verlet, L. Phys. Rev. 165, 201 A968). 56. Reiss, H. Adv. chem. Phys. 9, 1 A965). 57. Frisch, H. L. Adv. chem. Phys. 6, 229 A964). 58. Longuet-Higgins, H. C. and Widom, B. Mol. Phys. 8, 549 A964). 59. Alder, B. J. and Wainwright, T. E. J. chem. Phys. 27, 1208 A957). See also the early theoretical work of J. G. Kirkwood and E. M. Boggs [J. chem. Phys. 6, 394 A942); also J. G. Kirkwood, J. chem. Phys. 1, 919 A939)] and O. K. Rice [J. chem. Phys. 12, 1 A944)]. 60. Verlet, L. and Weis, J. J. Phys. Rev. A5, 939 A972).
REFERENCES AND NOTES 331 61. It should be noted that the dispersion forces themselves have a significant effect on the structure, through their role in determining the net repulsive force in the region rsrm (cf. LJ and MK curves in Fig. 4.2). 62. An exception is the 'penetrable hard sphere' model in which the spheres have non-additive diameters; see, for example, Widom, B. and Rowlinson, J. S. J. chem. Phys. 52, 1670 A970); Lebowitz, J. L. and Lieb, E. H. P/iys. Lett. 39A, 98 A972); Ahn, S. and Lebowitz, J. L. J. chem. Phys. 60, 523 A974). For non-spherical molecules the isotropic/nematic liquid crystal transition may be an exception; see Luckhurst, G. R. and Zannoni, C. Nature 267, 412 A977); Boehm, R. E. and Martire, D. E. Mol. Phys. 36, 1 A978). 63. Barker, J. A. J. chem. Phys. 19, 1430 A951). 64. Longuet-Higgins recognized that his conformal solutions theory applied to non-spherical molecules, but only for the rather trivial case in which the molecules of various species were of the same size and shape. 65. Pople, J. A. Proc. Roy. Soc. A215, 67 A952). 66. Barker, J. A. Proc. Roy. Soc. A219, 367 A953). 67. Pople, J. A. Disc. Faraday Soc. 15, 35 A953). 68. See also: Zwanzig, R. W. J. chem. Phys. 23, 1915 A955). In this paper Zwanzig derives the second-order perturbation term for the dipole-dipole interaction. The expression, which is for the pressure, is equivalent to Pople's. Zwanzig appears to have been unaware of Pople's 195367 and 1954|l> papers at the time. 69. Cook, D. and Rowlinson, J. S. Proc. Roy. Soc. A219, 405 A953); this work was extended to mixtures in Sutton, J. R. and Rowlinson, J. S., Proc. Roy. Soc. A229, 271, 396 A955). 70. Rowlinson, J. S. Trans. Faraday Soc. 50, 647 A954); 51, 1317 A955). 71. Gubbins, K. E. and Gray, C. G. Mol. Phys. 23, 187 A972). 72. Stell, G., Rasaiah, J. C, and Narang, H. Mol. Phys. 27, 1393 A974). Pople65 had earlier constructed a Pade-like approximation of this series, based on a cell model. 73. Barker, J. A. Proc. Roy. Soc. A241, 547 A957). 74. Perram, J. W. and White, L. R. Mol. Phys. 28, 527 A974). 75. See, e.g. Verlet, L. and Weis, J. J. Mol. Phys. 28, 665 A974). 76. An infinite order perturbation theory for the direct correlation function, pair correlation function, and thermodynamic properties has also been de- developed. See Wertheim, M. S. Mol. Phys. 26, 1425 A975), and Henderson, R. L. and Gray, C. G., Can. J. Phys. 56, 571 A978). This is more conveniently discussed in the next chapter on integral equations. 77. Sandier, S. I. Mol. Phys. 28, 1207 A974). 78. Mo, K. C. and Gubbins, K. E. Chem. Phys. Lett. 27, 144 A974); J. chem. Phys. 63, 1490 A975). 78a. Boublik, T.Coll. Czech, chem. Commun. 39, 2333 A974). 79. Chandler, D. and Andersen, H. C. J. chem. Phys. 57, 1930 A972); see also ref. 88. 80. Streett, W. B. and Tildesley, D. J. Proc. Roy. Soc. A355, 239 A977). 81. Wang, S. S., Egelstaff, P. A., Gray, C. G., and Gubbins, K. E. Chem. Phys. Lett. 24, 453 A974). 82. Patey, G. N. and Valleau, J. P. J. chem. Phys. 64, 170 A976). 83. For molecular liquids the situation is complicated by the presence of torques, in addition to forces. Although the force can be divided into attractive and repulsive parts, a similar division of the torque is not possible. Thus, if a WCA reference potential is used, the potential and force each go
332 PERTURBATION THEORY smoothly to zero at r = rm, but the torque has a discontinuity at this point. This difficulty does not arise if one uses a site-site LJ potential and a WCA reference potential for each site-site interaction separately (see § 4.8). 84. Gubbins, K. E., Gray, C. G., Egelstaff, P. A., and Ananth, M. S. Mol. Phys. 25, 1353 A973). 85. Lowden, L. J. and Chandler, D. J. chem. Phys. 59, 6587 A973). 86. Lowden, L. J. and Chandler, D. J. chem. Phys. 61, 5228 A974). 87. Ladanyi, B. M. and Chandler, D. J. chem. Phys. 62, 4308 A975). 88. Hsu, C. S., Chandler, D. and Lowden, L. J. Chem. Phys. 14, 213 A976). 89. Sandier, S. I. and Narten, A. H. Mol. Phys. 32, 1543 A976). 90. Agrawal, R., Sandier, S. I. and Narten, A. H. Mol. Phys. 35, 1087 A978). 91. Trie derivation given in §4.2 uses the canonical ensemble for simplicity. Provided that the thermodynamic limit (N, V^=o, N/V fixed) is taken carefully in evaluating the perturbation terms, no difficulty is encountered. In the u-expansion with the Barker-Pople reference system the expressions for A,, A2 and A3 are unchanged on taking this limit, because they do not involve fluctuations. Taking the limit for higher-order perturbation terms is less straightforward because fluctuation quantities occur; thus, in the canoni- canonical ensemble derivation of the u-expansion we find A4 = "(l/24)C3[<%a)o~3<%a)i'i], whereas the grand canonical expression for A4 has additional fluctuation terms. In general it is better to use the grand canonical ensemble to derive perturbation theory, as is done in Appendix 4A and later sections, since the thermodynamic limit has in effect already been taken in that case. For a more detailed discussion of this point see § 3.1.4 and Smith, W. R., ref. 2, pp. 85-8. 92. It is also possible to use a functional Taylor expansion to derive perturbation theory expressions for a general reference system and parameterization. This approach has been developed by Smith93 for the free energy and the pair correlation function. 93. Smith, W. R. Can. J. Phys. 52, 2022 A974). 94. We can also contemplate grand canonical (constant fx; see Kumar, B. and Penrose, O. Mol. Phys. 30, 849 A975)) and constant pressure perturbation theory. 95. In the absence of long-range dipole-dipole forces (see Chapter 10) g is intensive so that (dg/dA.KVo can be written (dg/<5A.)Bp. 96. The series D.16) is expected to converge rapidly if the structure of the real and reference fluids are similar. That this is so can be seen from the free energy equation (see §6.4); when uxA2) = uoA2) + XuiA2) this is 0= f dA-^ = 2plv[ dA[dr Jo OA Jo J Ac-A0= f dA-^ = 2plv[ dA[dr12<u,A2)gxA2)> J A J J If gxA2)~g0A2) for all 0<A<l (which implies the structure of the real and reference fluids are approximately the same) the right-hand side of this equation becomes equal to A, as given by D.19). The first-order perturba- perturbation theory is then very accurate. 97. For polarizable molecules u0 will contain a contribution from induction forces. See §§2.5 and 4.10. 98. For a general anisotropic potential ua the A, term has been evaluated by Gray, C. G., Gubbins, K. E., and Twu, C. H. J. chem. Phys. 69, 182 A978). For "multipole-like" ua the A3 term had previously been evaluated by Stell, G., Rasaiah, J. C, and Narang, H. Mol. Phys. 23, 393 A972); Rasaiah, J. C. and Stell, G. Chem. Phys. Lett. 25, 519 A974); Flytzani-Stephanopoulos,
REFERENCES AND NOTES 333 M., Gubbins, K. E., and Gray, C. G. Mol. Phys. 30, 1649 A975). For the case of non-axial molecules explicit equations have been given for multipo- lar contributions (up to quadrupole-quadrupole) to A2 and A3 by Gubbins, K. E., Gray, C. G., and Machado, J. R. S. Mol. Phys. 42, 817 A981), and for dispersion contributions to A2 by Lobo, L. Q., Staveley, L. A. K., Clancy, P., Gubbins, K. E., and Machado, J. R. S. J. Chem. Soc, Faraday Trans. 2 79, 1399 A983). 99. Equation D.40) suffers from three obvious deficiencies. Firstly, g will become negative for some orientations and r values for sufficiently strong ua. (Such a deficiency is common to many theories for g, and one often tries to modify the theory so that g remains positive everywhere - see Chapter 5). Secondly, D.40) fails to yield the correct low density results, g = exp[— fiu]. Equation D.41), while not suffering from these two deficiencies, usually predicts g values that are in serious error when Cua is large. Thirdly, g(ra),co2) will contain only those spherical harmonics (besides the isotropic term) that are contained in ua(ra>ico2). 100. For large r we note from D.40) that g,A2) —* -CuaA2), provided we assume that go(r)-^l faster than the r~" dependence in uaA2). For polar fluids this result ostensibly does not agree with C.121), hA2)-^> ~(CuaA2)/e)[(e - l)/3y]2; however, it should be noted that in this latter expression e depends on the dipole moment, and hence on ua. If we assume (e — 1)—»3y (see the Debye result E.163)) for small y, we do get consis- consistency between D.40) and C.121). 101. Murad, S., Gubbins, K. E., and Gray, C. G. Chem. Phys. Lett. 65, 187 A979); Chem. Phys. 81, 87 A983). 102. Murad, S. Chem. Phys. Lett. 84, 114 A981); see also Murad, S. Chem. Phys. Lett. 11, 194 A980). 103. The expression for g2 for multipole-like ua is given by Gubbins and Gray.71 104. Murad, S., PhD. dissertation, Cornell University A979). 105. Wang, S. S., Gray, C. G., EgelstafT, P. A., and Gubbins, K. E. Chem. Phys. Lett. 21, 123 A973). Further comparison of the u-expansion with simulation results for the LJ + 8 model are given in Cummings et al., ref. 135. 106. McDonald, I. R. J. Phys. C7, 1225 A974). 107. Patey, G. N. and Valleau, J. P. Chem. Phys. Lett. 21, 297 A973); J. chem. Phys. 61, 534 A974). 108. The Pade approximant n/d = Pn(x)IQd(x) to a function f{x) is the ratio of two polynomials of degrees n and d such that / and PJQd have the same series expansion to order x"+d, i.e. f(x) - Pn(x)IQd(x) + O(xn*d + v). A more detailed discussion of Pade approximants is given in Appendix B. (In the literature the notation [d, n] or [n, d] is often used, rather than the more natural n/d.) 109. H0ye and Stell"" have proposed an expression for g(rco,co2) for dipolar fluids which reproduces the Pade equation for the free energy. This expres- expression requires the dielectric constant as input, and is called the "Pade producing approximation" (PPA). 110. H0ye, J. S. and Stell, G. J". chem. Phys. 63, 5342 A975). 111. Stell, G., in Modern theoretical chemistry, vol. 5. Statistical mechanics, part A. Equilibrium techniques (ed. B. J. Berne), Plenum, New York A977). 112. Sullivan, D. E., Deutch, J. M., and Stell, G. Mol. Phys. 28, 1359 A974). 113. Stell, G. and Wu, K. C. J. chem. Phys. 63, 491 A975). 114. For example, for dipolar fluids or solids one can take A' = 0|x2/cr3 or A' = 2
334 PERTURBATION THEORY 115. It should be noted that this saturation argument (and hence the Pade) will break down for lower densities, and particularly for gases; thus lowering the temperature for a gas then lowers the free energy exponentially as more and more molecules form bound pairs (see §6.10). Computer simulation studies of quadrupolar LJ fluids (J. M. Haile, private communication, 1980) show that the free energy seems to be approaching the saturation limit of eq. D.50) at large Q*2 values (O*2>2 at T* = 1.23, p* = 0.85); however, solidification usually occurs before the saturation limit is reached. 116. Onsager, L. J. phys. Chem. 43, 189 A939). 117. There appears to be no general study of rigid quadrupolar lattices available, although Felsteiner'18 has discussed the special case of a face-centred cubic lattice. 118. Felsteiner, J. Phys. Rev. Lett. 15, 1025 A965). 119. Haile, J. M., private communication A978). 120. A theory is said to be thermodynamically consistent if all possible routes to the calculation of a given property (e.g. pressure or energy) give identical results. Thus thermodynamic perturbation theory to some order k is ther- thermodynamically consistent, but the Pade discussed here is not. The integral equation methods given in Chapter 5 are also inconsistent; for example, the pressure calculated from the virial and compressibility equations of state give different results in PY theory. 121. Ng. K. C, Valleau, J. P., Torrie, G. M, and Patey, G. N. Mol. Phys. 38, 781 A979). 122. Shing, K. S. and Gubbins, K. E. Mol. Phys. 45, 129 A982). 123. The question of whether the perturbation series D.46) or the Pade D.47) for Ak encompass phase transitions of various types is difficult to answer in general. Fluid-fluid phase transitions are known to occur (see Figs 4.10, 4.11, and Chapter 7), which is not surprising for a van der Waals-type theory. On the other hand, fluid-solid and orientational transitions in solids apparently do not occur. 124. Nicolas, J. J., Gubbins, K. E., Streett, W. B., and Tildesley, D. J. Mol. Phys. 37, 1429 A979). 125. To correct the Pade at low density, one can add terms which make A correct to the second virial level —see the similar approximation of Larsen, B., Stell, G., and Wu, K. C. J. chem. Phys. 67, 530 A977), which has been applied to ionic systems. 126. Madden, W. G. and Fitts, D. D. Chem. Phys. Lett. 28, 427 A974); Mol. Phys. 31, 1923 A976). 127. Madden, W. G., Fitts, D. D., and Smith, W. R. Mol. Phys. 35, 1017 A978). 128. Agrafonov, Y. V., Martinov, G. A., and Sarkisov, G. N. Mol. Phys. 39, 963 A980). 129. This expansion has also been referred to as the RAM (for 'reference- averaged Mayer-function') expansion. 130. The /-expansion for molecular fluids as given here was put forward indepen- independently by Perram and White74 and by Smith93 in 1974. 131. The use of un(r) defined by D.57) as an effective central potential in the statistical mechanics of polar and other molecular liquids has a long history. The first clear discussion seems to be that of Cook and Rowlinson69, who started from the assumption that the properties of the molecular liquid with potential u(raIaJ) can be approximated by that of the reference fluid with potential u,,(r) - i.e. a zeroth order /-expansion. However, these au- authors expand the exponential term on the right-hand side of D.57), thus obtaining a u-expansion, and only keep terms to O(CuaJ. A more general
REFERENCES AND NOTES 335 discussion of the use of temperature-dependent effective potentials in statis- statistical mechanics is given by G. S. Rushbrooke [Trans. Faraday Soc. 36, 1055 A940)]. The effective isotropic potential defined by D.57) has the virtue that it gives the pressure second virial coefficient exactly (see C.272)), while for dense fluids it provides a zeroth-order approximation to thermodynamic properties that is useful if the intermolecular forces are only weakly depen- dependent on orientation (see § 4.9 for a detailed discussion of effective isotropic potentials). Some early workers [e.g. Hirschfelder, J. O., Curtiss, C. F., and Bird, R. B. Molecular theory of gases and liquids, pp. 985-8. Wiley, New York A954); Balescu, R. Bull. Acad. Belg. Classe Sci. 41, 1242 A955); also Prigogine, I. Molecular theory of solutions, Chapter 14. North-Holland, Amsterdam A957)] used a different effective isotropic potential, obtained by averaging the pair potential uA2) weighted by the Boltzmann factor exp[-CuA2)] over orientations. This isotropic potential does not give the pressure second virial coefficient correctly, nor does it provide a useful starting point for a perturbation theory of dense fluids, and it has therefore been criticised (see Rowlinson, J. S. Mol. Phys. 1, 414 A958)).t 132. Smith, W. R., Madden, W. G., and Fitts, D. D. Chem. Phys. Lett. 36, 195 A975). 133. Quirke, N. Faraday Disc. Chem. Soc. 66, 166 A978). 134. de Leeuw, S. W., Perram, J. W., Quirke, N., and Smith, E. R. Mol. Phys. 42, 1197 A981). 135. Smith, W. R. Chem. Phys. Lett. 40, 313 A976). The /-expansion for the LJ + S-overlap model of B.4) has also been tested for the harmonic coeffi- coefficients of gA2), in addition to g for molecular centres [Cummings, P. T., Ram, J., Barker, R., Gray, C. G., and Wertheim, M. S. Mol. Phys. 48, 1177 A983)], and the results have been compared with those from the SSC/LHNC, SSCF, and PY integral equation theories; see Chapter 5 for a discussion of these latter theories, especially § 5.4.9a. The /-expansion and SSCF theories are of about equal quality except for the harmonics contained in uaA2), i.e. (, l2l — 202 and 022; for the latter harmonics SSCF is superior. 136. Smith, W. R. Faraday Disc. Chem. Soc. 66, 130, 165 A978). 137. Fischer, J. and Quirke, N. Mol. Phys. 38, 1703 A979). 138. Quirke, N., Perram, J. W. and Jacucci, G. Mol. Phys. 39, 1311 A980). 139. Nezbeda, I. and Smith, W. R. Mol. Phys. 45, 681 A982); Melnyk, T. W., Smith, W. R., and Nezbeda, I. Mol. Phys. 46, 629 A982). 140. Quirke, N. and Tildesley, D. J. J. phys. Chem., 87, 1972 A983). 141. Fischer, J. J. chem. Phys. 72, 5371 A980). 142. Nezbeda, I. and Smith, W. R. J. chem. Phys. 75, 4060 A981). 143. Kohler, F., Marius, W., Quirke, N., Perram, J. W., Hoheisel, C, and Breitenfelder-Manske, H. Mol. Phys. 38, 2057 A979). 144. Kohler, F., Quirke, N., and Perram, J. W. J. chem. Phys. 71, 4128 A979). 145. Nezbeda, I. and Smith, W. R. Chem. Phys. Lett. 64, 146 A979). 146. Melnyk, T. W. and Smith, W. R. Mol. Phys. 40, 317 A980). 147. Nezbeda, I. and Smith, W. R. Chem. Phys. Lett. 81, 79 A981). 148. Smith, W. R. and Nezbeda, I. Chem. Phys. Lett. 82, 96 A981). 149. Smith, W. R. and Nezbeda, I. Mol. Phys. 44, 347 A981). 150. Steinhauser, O. and Bertagnolli, H. Zeit. physikal. Chem., neue Folge 124, S33 A981). 151. Lee, L. L., Assad, E., Kwong, H. A., Chung, T. H., and Haile, J. M. t See also Note Added in Proof, p. 340.
336 PERTURBATION THEORY Physica 110A, 235 A982); Lee, L. L. and Chung, T. H. J. chem. Phys. 78, 4712 A983). 152. Wertheim, M. S. J. chem. Phys., 78, 4619 A983). The approach used here bears some similarity to the /-expansion, but involves graphical and ulti- ultimately integral equation methods for the case of hard dumbells. 153. Streett, W. B. and Tildesley, D. J. Proc. Roy. Soc. A348, 485 A976). 154. Monson, P. A. and Rigby, M. Chem. Phys. Lett. 58, 122 A978). 155. Nezbeda, I. Czech. J. Phys. B30, 601 A980). 156. Better results for g(r) can be obtained by neglecting the attractive forces in the full potential uA2) when calculating ufi(r) and go(r), as shown in ref. 137. 157. Many authors138146'49 have reported calculations for intcrmolecular har- harmonic coefficients gir,,,(r); these are related to those used here by girm(r) = 4irg(//'m; r). 158. Streett, W. B. and Tildesley, D. J., private communication A979). 159. In the usual centre frame /-expansion the averaging over orientations in D.57) is carried out at fixed centre-centre distance r, whereas in the site frame /-expansion the orientational averaging is at constant site-site dis- distance roB (see ref. 94f of Chapter 5). The remaining equations used in deriving the /-expansion are formally the same in the two frames.13gl447 For the diatomic LJ fluid the two frames seem to give similar results, but for hard dumbells the site frame gives better results13914"; at low densities these results are superior to the RISM theory (see Chapter 5). 160. Tildesley, D. J. Mol. Phys. 41, 341 A980). 161. Freasier, B. C. Chem. Phys. Lett. 35, 280 A975). 162. Several integral equation theories have been used to calculate the reference fluid pair correlation function go(r) needed in the /-expansion theory.141148 Both the py""-I4K and various modifications of the HNC theory148 have proved successful. 163. Sung, S. and Chandler, D. J. chem. Phys. 56, 4989 A972). 164. Bellemans, A. Phys. Rev. Lett. 21, 527 A968). 165. Steele, W. A. and Sandier, S. I. J. chem. Phys. 61, 1315 A974). These authors given an approximate expression for g,A2), the first-order term in the expansion of gA2) in the blip function theory, which consists of the 3-body term only (the second term on the right-hand side of D.13)). They ignore the other 3- and 4-body terms present in D.13). 166. Aviram, I. and Tildesley, D. J. Mol. Phys. 35, 365 A978). 167. The BF results for this model have not been compared with simulation results for the same model, but instead with results for the full diatomic LJ model.'63165 The agreement is rather poor. Such tests are therefore ambigu- ambiguous, since they implicitly assume that the attractive forces have a negligible effect on the liquid structure. Such comparisons test both this assumption and the BF theory itself. 168. Das Gupta, A., Sandier, S. I., and Steele, W. A. J. chem. Phys. 62, 1769 A975). 169. Nezbeda, I. and Leland, T. W. J. Chem. Soc, Faraday Trans. II 75, 193 A979). 170. The first perturbation theory calculations to employ a non-spherical refer- reference potential were made independently by Sandier,77 Mo and Gubbins,78 and by Boublik783 in 1974. At that time the use of such expansions was hampered by the lack of accurate equations of state for suitable reference
REFERENCES AND NOTES 337 fluids. For the special case of molecules interacting with site-site inter- molecular potentials, Chandler and Andersen79 suggested a non-spherical reference perturbation scheme in 1972, but no numerical calculations were made until much later, by Tildesley160. Non-spherical reference perturbation theory has been applied to liquid crystals by Singh and Singh (Singh, S. and Singh, Y. Mol. Cryst. Liq. Cryst. 87, 211 A982). 171. Kohler, F. and Quirke, N., in Molecular-based study of fluids (ed. J. M. Haile and G. A. Mansoori), A. C. S. Adv. Chem. Series 204, Amer. Chem. Soc. Washington, D.C., A983), p. 209. 172. Boublik, T. and Nezbeda, I. Chem. phys. Lett. 46, 315 A977). 173. Kohler et al.144 use the zeroth-order /s expansion for ydbA2), whereas Fischer141 replaces ydbA2) by yref(r), where yref(r) is the function for the fluid with potential uTQf(r) given by D.96). The two methods give essentially the same results.171 174. Quirke, N. and Jacucci, G. Mol. Phys. 45, 823 A982). 175. Boublik, T. Mol. Phys. 32, 1737 A976). 176. Boublik, T. Fluid Phase Equilibria, 3, 85 A979). 177. Boublik, T. Coll. Czech, chem. Commun. 46, 1355 A981). 178. Boublik, T. Fluid Phase Equilibria 7, 1, 15 A981). 179. Although convenient for theoretical calculations, the Kihara model does not easily lend itself to computer simulation studies except for very simple shapes, because the distance p between cores cannot be expressed as a simple analytic function of (ra>ico2) >n general. 180. The derivation of D.102) is similar to that of the internal energy equation in §6.7. 181. Tildesley, D. J. and Streett, W. B. Mol. Phys. 41, 85 A980). 182. Singer, K., Taylor, A., and Singer, J. V. L. Mol. Phys. 33, 1757 A977). 183. Lombardero, J., Abascal, L. F., and Lago, S. Mol. Phys. 42, 999 A981). This work is extended to multicomponent fluids in: Enciso, E. and Lombar- Lombardero, M. Mol. Phys. 44, 725 A981). 184. Lombardero, M. and Abascal, J. L. F. Chem. Phys. Lett. 85, 117 A982). 185. Martina et al.™6 have calculated Ao and A, for the models of N2 and Cl2 of Table 4.8 using methods that should be more accurate than the BF theory used by Sandier.77 For Ao Martina et al. use the accurate Nezbeda equation of state for convex bodies (Nezbeda, I. Mol. Phys. 33, 1287 A977)), with the shape factor determined by equating the second virial coefficient for dum- bells to the B2 expression for convex bodies (see § 6.13 for a discussion of this equation of state and its application to dumbells). This is known to be in good agreement with computer simulation data except at the higher den- densities, and is more accurate than BF theory. However, the differences in the A(, values from the two methods do not change the qualitative conclusions to be drawn from Table 4.8. Martina et al. calculate A, using Monte Carlo data153 for the harmonic coefficients g,,m(r) of g(roI(o2). They obtain A, values that are of similar order of magnitude of the BF values given in Table 4.8, but the values so obtained are highly sensitive to errors in the gt(•„, (for example, glrm from computer simulations for 256 and 500 molecules give A, values that differ in sign for /* = 0.4 and p* = 0.6). 186. Martina, E. Stell, G., and Deutch, J. M. J. chem. Phys. 70, 5751 A979). For errata to the equations and numerical calculations in this paper see J. chem. Phys. 74, 3636 A981). In addition to errors listed there, note also that the values listed in Table III for y, eq. B.7) and blip are incorrect, as are the numerical results shown in Fig. 1 for Ao and A^
338 PERTURBATION THEORY 187. Valderrama, J. O., Sandier, S. I., and Fligner, M. Mol. Phys. 42, 1041 A981); 49, 925 A983). 188. Fligner, M, M.Ch.E. Thesis, Univ. of Delaware A979). 189. Rasaiah, J. C, Larsen, B., and Stell, G. J. chem. Phys. 63, 722 A975). 190. Barker's'11 1951 paper on perturbation theory for polar liquids implies that an effective spherical potential can be used to calculate the free energy, although this is not brought out explicitly. 191. The treatment of Cook and Rowlinson69 starts from a zeroth-order /- expansion, as discussed in ref. 131, but is equivalent to the derivation given here. 192. Leland, T. W. and Chappelear, P. S. Ind. Eng. Chem. 60, 17 A968). 193. Rowlinson, J. S. and Swinton, F. L. Liquids and liquid mixtures, 3rd edn, * pp. 230-43. Butterworth, London A982). 194. Reed, T. M. and Gubbins, K. E. Applied statistical mechanics, pp. 157-62 and 323-7, McGraw-Hill, New York A973). 195. See, for example, Leach, J. W., Chappelear, P. S., and Leland, T. W. Proc. Am. Petrol. Inst. 46, 223 A966); Rowlinson, J. S. and Watson, I. D. Chem. Eng. Sci. 24, 1565 A969); Watson, I. D. and Rowlinson, J. S. Chem. Eng. Sci. 24, 1575 A969); Gunning, A. J. and Rowlinson, J. S. Chem. Eng. Sci. 28, 521 A973); Teja, A. S. and Rowlinson, J. S. Chem. Eng. Sci. 28, 529 A973); Murad, S. and Gubbins, K. E. Chem. Eng. Sci. 32, 499 A977). 196. Singh, Y. Mol. Phys. 29, 155 A975). 197. Shukla, K. P., Ram, J., and Singh, Y. Mol. Phys. 31, 873 A976). 198. Clancy, P. and Gubbins, K. E. Mol. Phys. 44, 581 A976). 199. Barker, J. A. in Rare gas solids, Vol. 1, pp. 212-64, (ed. M. L. Klein and J. A. Venables), Academic Press, New York A976). 200. The term uindA23) in D.134) is the sum of the two equal terms, -j[l«.i .T12 .ct2-T23 .iit + ii.3 .Ti2-c^'T2i .Hi] which occur in the O(a) term in B.291). Note that the total uindA23) for the 123 molecular triplet is u,ndA23) = u,nd(lBK) + u,nd((lJ3) + uindA2C)), where u,nd(lBK) is the term given in D.134) and (i) indicates the molecule that is polarized. 201. Terms of type (a) should be small since they are O(a2); type (b) terms also involve higher orders of a and should be small; terms of type (e) should be small since they involve a product of two three-body terms. An early treatment of the dipole-induction term of type (d) was given by Barker66 in 1953. 202. Shukla, K. P., Pandey, L., and Singh, Y. J. Phys. C12, 4151 A979). 203. Shukla, K. P., Singh, S., and Singh, Y. J. chem. Phys. 70, 3086 A979); Shukla, K. P. and Singh, Y. J. chem. Phys. 72, 2719 A980). 204. Larsen, B., Rasaiah, J. C. and Stell, G. Mol. Phys. 33, 987 A977). 205. Patey, G. N. and Valleau, J. P. Chem. Phys. Lett. 42, 407 A976); erratum, 58, 157 A978). 206. Wertheim, M. S. Mol. Phys. 37, 83 A979). 207. Vesely, F. J. Chem. Phys. Lett. 56, 390 A978). In this paper the pressure and configurational energy (but not the free energy) are calculated for molecules with isotropic polarizability. 208. Patey, G. N., Torrie, G. M. and Valleau, J. P. J. chem. Phys. 71, 96 A979). 209. Patey, G. N., Klein, M. L., and McDonald, I. R. Chem. Phys. Lett. 73, 375 A980). 210. Note that AUa = ?/HS+ll+et- C/Hs+m- contains a contribution from <%ind) and also one from <%u), due to the effect of induction on the fluid structure.
REFERENCES AND NOTES 339 Thus AUa =[(aUja-(aU^)n,0] + (%„<,), where <...)„ and <...>„_<, mean av- averaging over the real system of polarizable molecules and over the system of rigid dipolcs (with « = 0), respectively. 211. Isihara, A. J. Phys. A, Ser. 2, 1, 539 A968). 212. Klein, M. L. and McDonald, I. R. J. chem. Phys. 71, 298 A979); McDonald, I. R., O'Shea, S. F., Bounds, D. G., and Klein, M. L., J. chem. Phys. 72, 5170 A980). The potentials proposed for HCl and HF in these papers use point charges to represent the electrostatic interactions. In the calculations of Patey et al209 shown in Table 4.10 these point charge interactions are replaced by the leading dipolar and quadrupolar terms. 213. The theory here referred to as RT is Wertheim's 1-R theory. Wertheim2 considers several successive levels of renormalization, which he calls 0-R, 1-R and 2-R. At the 0-R level the parameters ft and « remain simply the single molecule parameters; the 0-R level.corresponds closely to the treat- treatment of induction effects by McDonald106-see D.138). At the 1-R level |j. is replaced by the renormalized moment (j.' but a is unchanged. At the 2-R level |j. is replaced by |j.', and a is replaced by either «' or a", depending on the class of graphs considered; here a' and a" are two different renormalized polarizabilities, for which explicit equations are given by the theory (see also § 10.1.5). The 2-R results for the free energy are only slightly different from the 1-R results, and the latter are a little closer to the simulation results.2"8 The extension of RT to mixtures is given by Venkatasubramanian, V., Gubbins, K. E., Gray, C. G., and Joslin, C. G. Mol. Phys. in press A984). For an extension to include quadrupoles see Gray, C. G., Joslin, C. G., Gubbins, K. E., and Venkatasubramian, V. Mol. Phys. submitted A984). These papers also give a simplified, non-graphical derivation of RT. 214. Wertheim, M. S. Mol. Phys. 34, 1109 A977). 215. The integrals I2 and J3 are simply related to the integrals /„ and K(/!'/"; nn'n") that arise in the u-expansion for polar fluids (see Flytzani- Stephanopoulos et al.9S and Appendix 6B), The relations are where J6=J dr*r*~4g0(r*), f.rt,2 f " " J|r?2-r?,l x go(r*2rtV*3)(l +3 cos d, cos d2 cos a3) where r* = rjcr. 216. When a = 0 we have up = 0, x, =pfrx2/3, x2 = Q, and's A (Jp/V?iVcr-3KB22; 333) where J6 and 1CB22; 333) are defined in note 215 above.
340 PERTURBATION THEORY 217. The simpler equations D.154) and D.155) will hold whenever a takes the form D.153), with z being the direction of ij., and so will include linear and other molecules with axial symmetry (e.g. H2O-like symmetrical molecules, etc.), where n lies along one of the principal a directions. We see from D.154) that in these cases |j.' will lie in the same direction as ft. 218. We mention in particular the 'polarization models' [see Stillinger, F. H. J. chetn. Phys. 71, 1647 A979)], which, like RT, introduce effective pairwise potentials and correlation functions which include the multibody effects. One can also include polarizability via the SSC integral equation (Chapter 5). For numerical results see Carnie, S. L. and Patey, G. N. Mol. Phys. 47, 1129 A982) and Stell, G., Patey, G. N., and H0ye, J. S. Adv. chem. Phys. 48, 183 A981). 219. Stell, G. in Phase transitions and critical phenomena, Vol. 5B (ed. C. Domb and M. S. Green). Academic Press, London A976). 220. Stell, G. and Weis, J. J. Phys. Rev. A16, 757 A977). 221. Martina, E. and Deutch, J. M. Chem. Phys. 27, 183 A978). 222. Patey, G. N. Mol. Phys. 35, 1413 A978). 223. This idea goes back to van der Waals' thesis.47 He understood that his equation of state rested on the separation of effects of long-range attractive and short-range repulsive forces. The idea that the attractive forces must be weak but long-range was understood more clearly by Rayleigh [Lord Rayleigh, Nature 45, 80 A891); Sci. Papers 3, 469 A902)] and Boltzmann [Boltzmann, L. Lectures on gas theory, trans, by S. G. Brush, pp.220, 375. Berkeley, 1964]. For the modern statistical mechanical derivation of the van der Waals equation of state see Kac and Uhlenbeck [Kac, M. Phys. Fluids 2, 8 A959); Kac, M., Uhlenbeck, G. E., and Hemmer, P. C. J. math. Phys. 4, 216, 229 A963); 5, 60 A964); Lebowitz, J. L. and Penrose, O. J. Math. Phys. 7, 98 A966)], The historical development of such van der Waals models (or 'mean-field theories' in modern parlance) is described in more detail by Rowlinson49 and others.48'"" 224. Rigby, M. J. phys. Chem. 76, 2014 A972). 225. Henderson, D., in Equations of state in engineering and research, (ed. K. C. Chao and R. L. Robinson, Jr.), Adv. in Chem. Ser., No. 182, p. 1 A979). 226. The term (&(B)ldp)BVK can be written in terms of dynamical quantities by using C.210). However, in the applications to follow it is often convenient to leave this term as a density derivative. 227. Gray, C. G., Gubbins, K. E., and Twu, C. H. J. chem. Phys 69, 182 A978). Note added in proof. Recently a temperature-independent effective central potential un,cd(r) lias been proposed, where uInc,j(r) is the median potential al fixed r, as contrasted with the mean potential D.23); see Shaw. M S., Johnson, J. D. and Holian. B. 1.., Phys. Rev. Lett. 50, 1141 A983), and Lebowitz, J. L. and Percus, J. K., J. diem. Phys. 79, 443 A983). Accurate results are obtained for the thermodynamic properties at high densities for repul see G qualit i(incJ ive site-site type potentials (I.e.), and also at low densities (second virial coefficients); ay. C. G. and Joslin, C. G., Chem. Phys. Lett., 101, 248 A983), where also a simple tive explanation for the .success of unKii(r) is given. For muhipole potentials, however, ) fails completely (Gray and Joslin, I.e.). For structural properties, e.g. the centres m,|('l is less accurate even for site-site potentials; see Joslin, C. G., Gray, C. G., and Wojcik. M, submitted A984).
5 INTEGRAL EQUATION METHODS Two functions are introduced, one relating to the direct interaction of molecules, the other to the mutual influence of two elements of volume. An integral equation gives the relation between the two functions. Leonard S. Ornstein and Fritz Zernike, Proc. Akad. Sci. (Amst.) 17, 793 A914). In this chapter we describe some of the integral equation methods which have been devised for calculating the angular pair correlation function g(r&>i&>2) and the site-site pair correlation function gcpM for molecular liquids. These methods are in the main natural extensions of methods devised for calculating the pair correlation function g(r) for atomic liquids.1013 They can be derived from infinite-order perturbation theory (an example is given in § 5.4.8), whereby one partially sums the perturba- perturbation series of Chapter 4 to infinite order10c usually with the help of diagrams, or graphs, but alternative methods of derivation are also available, e.g. functional expansions. The original integral equation theories are in a certain sense more complete than perturbation theories, in that the full correlation function g (°r gap) 's calculated, whereas in perturbation theory one calculates the correction g — g0 to the reference fluid value gA. On the other hand the perturbation theory approximations are controlled; one can estimate the error by calculating the next term. It is extremely difficultrod to estimate a priori the error in integral equation approximations, since certain terms are neglected almost ad hoc. Their validity must therefore be a posteriori, according to agreement with computer simulation results (or, less satisfac- satisfactorily (cf. Fig. 1.1), with experiment). Of particular interest are theories which are a combination^'¦'•13-13b-15 of perturbation theory and an integral equation, which tend to have some of the advantages'53 of both ap- approaches (see also §5.3.1). An example41 is the GMF/SSC theory of §5.4.7. The structure of the integral equation approach for calculating g(rft),ftJ) is as follows. One starts with the Ornstein-Zernike (OZ) in- integral equation C.117) between the total correlation function h — g— 1 and the direct correlation function c, which we write here schematically1511 as h = c + pch E.1)
342 INTEGRAL EQUATION METHODS or, even more schematically, as h --= h[c], (OZ) E.2) where h[c] denotes a functional of c. The relation E.2) is exact. The second relation is the so-called closure relation c=c[h], (closure) E.3) and is an approximation. We now have two equations in two unknowns. The various integral equation theories to be described below (PY, HNC, MSA, GMF) differ in their closure relations E.3). From E.2) h depends on c, and from E.3) c depends on h; thus the unknown h depends on itself and must be determined self-consistently. This (self-consistency requiring, or integral equation) structure is characteristic of all many- body problems. n-1^19b We note that E.2) and E.3) can be rewritten as a non-linear integral equation for h, so that solving for h will be difficult in general. The solution is usually carried out by iteration of E.2) and E.3), although other methods4'l9c e.g. variational,lyce perturbational (see § 5.3.1), factorization (Appendix 5A), etc. have also been devised, and analytic solutions exist in a few cases (see §§ 5.1 and 5.4.2). Thus, beginning with an initial guess /i(" for h, one calculates the corresponding cA> from E.3), cm = c[hm], and then the new value of h, hm = h[cn)] from E.2). One then repeats the cycle and continues until h(n + i) = h(n) to the desired accuracy.19' In practice one often uses standard tricks419c to speed up the convergence, such as using some average of h(n) and h<n~x) as input to calculate c<rl) from E.3). Two19" of the classic integral equation approximations for atomic liquids are the PY (Percus-Yevick1013-20-21) and the HNC (hyper- netted chain 1~10b-22~6) approximations h-c = y-l, (PY) E.4) h-c = )ny, (HNC) E.5) where y is the indirect correlation function (see C.99)), defined by kA2) = cxp(—/3wA2))y(l2), with wA2) the pair potential. We assume pairwise additivity throughout this chapter; see refs. 1, 77, 79 for exten- extensions to multibody potentials. The closures E.4) and E.5) can be written in the form E.3) c = g(l-e<3"), (PY) E.6) c = ft-|3u-lng, (HNC) E.7) which also display explicitly how the pair potential wA2) enters the problem.263 Since the derivations of E.4) and E.5) are somewhat lengthy and are
INTEGRAL EQUATION METHODS 343 given in many other places, 1~lob we give here26b a heuristic argument. We have introduced (see Chapter 3) two measures of indirect correlation between molecules 1 and 2, viz. yA2), the indirect part of gA2), and h(\2)~cA2), the indirect part of h(\2). Assuming the indirect part of h = g — 1 is y — 1, we equate these two measures of indirect correlation in h to obtain the PY approximation E.4). If y is close to unity,26c we can write In y = y — 1. In the HNC approximation one employs In y as in E.5) rather than y — 1, even if y is not close to unity. As mentioned above (see ref. 10c), the PY and HNC approximations can be derived by partially summing the density series26d for c,101Oa cA2) = Here O denotes a fixed point A or 2), • denotes an integrated point C, 4,...) [i.e. • denotes multiplication by p, integration over r3, and averaging over co3], and denotes a Mayer /-bond, /A2) = exp(-|3wA2))- 1, where /A2)=/(ri2<<>i<<>2)- For example we have pJdr =/A2)pJdr3(/A3)/C2))u, etc. where (.. .)„ = J (dw/H)... (see B.17)). In E.8) the two-body term is independent of density p, the three-body term linear in p, etc. The PY and HNC approximations both keep the first two rows of E.8), but drop terms313 from the third and succeeding rows. Thus, for the four-body terms (third row), one drops the last term in the HNC approximation, and the last four terms in the PY approximation: + 2 In spite of the fact that the HNC theory sums more diagrams, it is not necessarily a superior theory, since there can be cancellation among the last four diagrams in E.8) for certain potentials. Thus, for atomic li- liquids513131'27 the PY theory does better for steep repulsive pair poten- potentials, e.g. hard spheres, whereas the HNC theory does better when
344 INTEGRAL EQUATION METHODS 5.1 attractive forces are present, e.g. Lennard-Jones, square well, and Coulomb potentials. One expects that the same situation might hold true for molecular liquids, although there have as yet been very few tests of this hypothesis.31b We discuss below the theories for g(raIa>2) and for the site-site g^r) that have been developed and applied for molecular liquids. For simplic- simplicity we discuss only pure liquids. Three sections containing background material precede this discussion. The reader who is mainly interested in results, and not in the details of the derivations, should read only §§ 5.4.1, 5.4.2, 5.4.7, 5.5.1, and 5.5.4. In fact, many other readers might want to start with these sections. 5.1 The PY approximation for hard spheres We briefly discuss the solution to the PY approximation for the hard sphere atomic fluid. Besides its historical interest as an analytical solu- solution3233'35 to a classical many-body problem, it also forms a basis for other theories, e.g., in molecular fluids the MSA and RISM theories to be discussed below, and the non-spherical reference perturbation theory (§§ 4.7, 4.8). The importance of the hard sphere fluid as a proper reference for atomic fluids was discussed in §4.1. For atomic fluids the OZ equation C.117) simplifies to h{rl2) - c(rl2) + pjdr.1c(r,3)h(r32). E.9) Choosing the origin at molecule 1 one can also rewrite E.9) as h(r) = c(r) + pjdr'c(r')h(|r-r'|) E.10) or, in k-space since the convolution term factorizes, (see C.156) or 8 3B.4) = c(k) + pc(k)h(k). E.11) We note for future reference two alternative forms of E.11): T—rr, E.12) l-pc(fc) l. E.13) For hard spheres one obviously has g(r) = 0 for r<a, where a is the hard sphere diameter, since there is zero probability of overlap for two hard spheres. Thus we have the exact condition h(r) = -l, r<a. E.14)
5.1 THE PY APPROXIMATION FOR HARD SPHERES 345 The PY approximate condition E.6) reduces to c(r)=0, r>a E.15) for hard spheres. The problem is thus35a to solve the OZ equation E.10) for c{r) for r<a and h(r) for r>a, given h(r) for r<a and c(r) for r>a. The simplest method of solution is based on the Wiener-Hopf factorization technique (see Appendix 5A). The result for c(r) is a cubic polynomial (see EA.29)) = co + c,(r/cr) + c3(r/crK = 0 where the (dimensionless) constants cx = Cj(t|) are given by with c3 = 2f)C0. Here = (tt/6)po-3 E.16) E.17) E.18) is the 'packing fraction',35b which is proportional to the reduced density per3. Because of the discontinuity in c(r) at r = a, the result13 for h(r) is only piecewise analytic for the regions @, a), {a, 2a), etc.35c However, as we shall see, E.16) is sufficient to obtain the equation of state. In Fig. 5.1 we compare the theoretical results for g(r) with Monte — PY • Monte Carlo V.. 1.5 2 (r/o) 2.5 Fig. 5.1. Comparison of the 'exact' Monte Carlo and PY theory results for the pair correlation function g(r) for hard spheres at density tj =0.49. (From ref. 5.)
346 INTEGRAL EQUATION METHODS 5.1 Carlo values, for the high density tj = 0.49, which is close to freezing. The PY results are seen to be qualitatively good, but there are two principal deficiencies: (a) the first peak height, at contact, r = a, is about 20 percent too low-this is important since (cf. E.24)) the pressure depends on g at contact; (b) for larger r values, not shown in Fig. 5.1, the PY g(r) peaks are slightly too high and out of phase-this gives rise to a principal peak height in the structure factor S(k) (see Chapter 9) which is about 5 per cent too high. The hard sphere system solidifies in reality5 for r\ S 0.49. PY theory, on the other hand, continues to predict fluid phase g(r)s for 0.49 Stj <0.62. For tj>0.62 PY theory gives unphysical results, g(r)<0. The PY results for c(r) and y(r) for r<cr are reviewed and assessed in refs. 13, 13a. (They are related by E.25) in PY.) The y(r) for r<a is of interest since in perturbation theories for liquids based on a hard sphere reference fluid (see Chapter 4) one needs ynsM f°r r<a, where yHs(r) "s the indirect correlation function for the hard sphere fluid. It turns out that, for dense fluids, the PY c(r) is quite good, but the y(r) is poor. Better theories for y(r<cr) are discussed in refs. 13 and 35c'. The equation of state can be obtained in two ways. The first method is based on the compressibility relations [see C.113), C.123), or Appendix 3E] E.19) = [l-pjdrc(r)] E.20) where x = P~'(dp/3p)T is the isothermal compressibility. From E.16) and E.20) we get35d fcT- = ^^. E.21) Integration of E.21) with respect to density, with boundary condition (p/pfcT)p_0 —> 1, gives the 'compressibility equation of state'35e (x-route). E.22) For the second route to the equation of state one uses the virial relation [see Appendix E], J) E.23)
5.1 THE PY APPROXIMATION FOR HARD SPHERES 347 which for hard spheres reduces to35f P/pkT=l+§7rpcr3g(o-+) E.24) where g(cr+) denotes the limit of g(r) as r approaches a from above. Using the fact that y(r) is a continuous function513 at r = a, and is given in PY theory (cf. E.4)) by () g(r), r>a • = -c(r), r<a E.25) we see that g(cr+) = — c(a~). Using this result together with E.24) and E.16) gives the 'pressure equation of state' (p-route). E.26) A-T,J We first note that the results E.22) and E.26) differ, so that the PY theory is 'thermodynamically inconsistent' (cf. ref. 120 of Ch. 4). In Fig. 5.2 the two PY equations of state are compared with computer simulation data (and also with the HNC theory results, which are seen to be in worse agreement with the data and to have errors of opposite sign from PY). The two PY curves bracket the correct one, with the compressibility curve being closer to the correct one. One expects the compressibility equation of state to be better than the pressure equation of state, since the latter 12 10 CS • • Monte Carlo Fig. 5.2. Equation of state for the hard sphere fluid according to the PY and HNC theories; \ and p denote the compressibility and pressure equations of state respectively. The points are the computer simulation data and the dashed line is the Carnahan-Starling (CS) equation of state. (Adapted from refs. 5, 13.)
348 INTEGRAL EQUATION METHODS 5.2 depends on only g(r) at contact (cf. E.24)) which is considerably in error in PY theory (Fig. 5.1), whereas the former depends on the integral E.19) over the whole g(r) curve. From E.22) and E.26) we see that the pressure is predicted to be continuous and analytic up to the singularity at tj = 1 (where space is completely filled). Thus, the theory fails to predict the freezing transition which is known from computer simulation513 to occur at -rjr-0.49. This failure is perhaps not surprising since p and g in the OZ equation have been assumed homogeneous and isotropic. If we average the two PY equations of state, p = (§)p(x-route) + (g)p(p- route), we obtain the phenomenological Carnahan-Starling (CS) equation of state,51338 ^X7^3' (cs) E-27) which is seen (Fig. 5.2) to agree well with the simulation data for the stable fluid density range (tj =30.49); for the metastable range small discrepancies occur.36 Phenomenological fits to the simulation g(r) also exist,3Aa and improved closures3* have been devised, based on ther- modynamic consistency, which agree with the simulation g(r) to within 1 per cent. From E.22), E.26), or E.27) we can also obtain closed-form expressions for the free-energy and chemical potential (ref. 13 and Chapter 6). The theory is readily extended to mixtures (see ref. 13, Appendix 5A and Chapter 6). 5.2 Spherical harmonic form of the OZ equation In practice the integral equations for g(io>1o}2) are almost always solved using spherical harmonic expansions. This is because the basic form C.117) of the OZ relation contains too many variables to be handled efficiently. We remind the reader (§ 3.2.2 and Appendix 3B) of the spherical harmonic form of the OZ equation for linear and general shape molecules; we restrict ourselves here to linear molecules. We have seen (Chapters 2, 3, and Appendix A) that a rotationally invariant pair quantity /(ro>,<x>2) such as the pair potential u(ra>,<x>2), pair correlation functions g(raI<o2), h(T(ii^u>2), c(iu>loJ), Mayer /-bond f(ru>iu>2), etc. can be expanded in terms of products of three spherical harmonics with respect to space-fixed axes as /(rco,oJ)= Y. fiW; r)<Pw(oi0J<*>r) E.28) I,!2I where ^ + B+! = even (from parity; see argument following B.27)), and
5.2 SPHERICAL HARMONIC FORM OF THE OZ EQUATION 349 ill2i is the rotational invariant = Z C(lll2l;m1m2m)YUmi(o}l)Y,2m2((o2)Ylm(<o)* E.29) with C( ) a Clebsch-Gordan (CG) coefficient. The space-fixed axes expansion coefficients filial', r) are related to the intermolecular axes (the 'r-frame') expansion coefficients f{lj2m;r), defined by C.146), by the CG relations C.147), C.148). The quantity f(iu>xo>2) can be Fourier transformed to k-space using C.154) and C.155) and is denoted by f(ka)xw2). We can similarly expand /(kft>i&>2) in spherical harmonics = Z E.30) where i//(|,2,(oj1aJa)k) is the same rotational invariant E.29) with <o = ojk. The two sets of space-fixed coefficients, f(lj2l; r) and f(Ul2l', k), are mutual /( Hankel transforms, CB.43) and CB.44). Finally, /(ka>ja>2) can be referred to the 'k-frame', i.e. polar axis along k (cf. CB.20)), with corresponding expansion coefficients f(lil2m; k). The k-space coefficients f(lj2l; k) and f(lil2m; k) are related in the same way as are f(lj2l; r) and f{lxl2rn;r), i.e. by the CG relations CB.21), CB.22). In Fig. 5.3 we summarize the Clebsch-Gordan (C) and Hankel (jt) relations between the four expansion coefficients f(lj2l;r), f{lxl2m;r), f(lj2l;k), f(lj2m; k). (Also shown are two auxiliary coefficients introduced in §5.4.3.) r-space k-space Fig. 5.3. The spherical harmonic expansion coefficients (in boxes) in r-space and k-space showing the Clebsch-Gordan (C) and Hankel transform (/i) relations between them. [Also shown are two auxiliary coefficients /""(/,l2l', r) and /""(f^m; r) introduced in §5.4.3, eqns E.77), E.78).] The transformation relations f(l,Bm; r)-»f(l,!2l; r)-> /(I, I2l; fc) -»/(*! l2m; k) are defined precisely by C.148), CB.43), and CB.22), respectively. [The inverse relations f(;,!2m; fc)-*/((,B/; fc)->/((,B/; r)-^/(!,/2m; r) are given by CB.21), CB.44), and C.147), respectively.]
35A INTEGRAL EQUATION METHODS 5.3 It is most convenient (see §§5.4.1, 5.4.2, 5.4.7, 5.4.9a, and Appendix 5B) to employ the OZ equation in terms of the k-space harmonic coefficients f(/,/2/;fc) and f(ltl2rn;k), see C.160) and C.158). (The factorized form analogous to E.13) is EB.12).) In the PY, MSA, and GMF theories discussed below in terms of the harmonics of g^a^a^), the expansions E.28) and E.30) are truncated after a finite number of terms. The validity of the truncations rests on the rate of convergence of the harmonic series, which depends in turn on the degree of anisotropy in the intermolecular potential (see Chapter 3, and also Chapter 2 for comments on the convergence of the harmonic series for g(roIo>2) and for the potential M(ro>1o>2)). If we truncate the harmonic series for g^a^a^) too soon, the rigorous condition g(rai|(iJJ0 may be violated. For example,41 consider a fluid with Lennard-Jones (LJ) iso- tropic potential and quadrupole-quadrupole (QQ) anisotropic potential, and let us truncate the g(ra>io>2) series E.28) keeping only the /,/2/ = 000, 220, 222, and 224 harmonics; we assume these are the dominant har- harmonics in g, since the potential contains only 000 and 224 harmonics. We put g(ro>,oJ) = g(r) + ga(ro>1oJ), where g(r) is essentially (see C.145)) the 000 term, and ga comprises the 220, 222, and 224 terms. With only these three harmonics in ga, it is easy to show that ga(—) = -2ga(T), where — and T denote the 'end-to-end' and 'T' configurations of the two molecules, respectively. The 'T' configuration is the most probable for QQ interactions, and the first peak in g(T) becomes larger and larger41 as the quadrupole moment Q increases. Hence for large Q the truncated series383 will give g(—) = g(r) + ga(—) < 0 in the region of the main peak in g(T). Thus, even if the theory gives accurate values for the four harmonics 000, 220, 222, and 224 of g(ro),aJ), the sum will not be a good representation of g for large Q. On the other hand certain properties of the fluid depend only on the lower-order harmonics. For the fluid in question (LJ + QQ) the energy can be determined from g@00; r) and gB24; r) (cf. Chapter 6), and the Kerr constant angular correlation parameter depends on gB20; r) (cf. Chapter 11). If the theory yields accurate values for these harmonics it will still be useful. Computer simulation yields directly the individual harmonics of g (§ 3.2.1), so that it is easy to test the theoretical values. 5.3 Separation of the OZ equation into reference and perturbation parts One often divides the intermolecular pair potential uA2) into two parts, e.g. reference and perturbation parts, isotropic and anisotropic parts, long- and short-range parts, etc. It is then of interest to derive the
5.3 SEPARATION OF THE OZ EQUATION 351 corresponding decomposition of the OZ equation.18a'3441'4344 For defin- itehcss we consider our standard decomposition [B.15), D.22)] of the pair potential into reference and perturbation parts, ub(t<o1<x}2), E.31) where M0(r) = {M(ra>1aJ))u,l<U2 is isotropic, and the anisotropic perturbation Ma(roIa>2) satisfies (uB{iwl<o2))atai = 0. E.32) We have defined multipole-like potentials, cf. B.19), D.36), as those satisfying the stronger condition <Ma(ro),aJ))Ml = («a(iwi&J))^ = 0. E.33) Corresponding to E.31) we now decompose h and c into reference and perturbation, or correction, parts E.34) E.35) At this stage we do not assume that ha and ca are small; in §§ 5.3.1 and 5.4.7 we develop /ia and ca in perturbation series in wa. The real fluid pair h, c satisfy the 'molecular' OZ C.117), and the reference fluid pair h0, c,, satisfy the 'atomic' OZ E.9). It must be possible to combine the real and reference fluid OZs to obtain an equation for the correction terms ha, ca. Now in general h.d and ca do not satisfy an angular condition of the type E.32), let alone the stronger condition E.33). Therefore, in general the centre-to-centre pair correlation function h(r) = (/i(raIco2))<ul<o2 differs from the reference fluid value ho(r). In Fig. 4.12 are compared Monte Carlo results42 for h(r) and ho(r), for (i) LJ plus multipolar, and (ii) LJ plus overlap potentials. In case (i) the shift of h(r) from h()(r) is small, whereas in case (ii) it is large. A similar result is shown in Fig. 5.11 for a quadrupolar hard sphere fluid. In the MSA and GMF theories discussed below, and applied to multipolar fluids, the difference between h(r) and ho(r) is neglected. It will be seen that in these theories the strong angular condition <ca(ra)l0J)>o)i =<ca(raIa>2)>^ = 0. E.36) is satisfied if the potential ua satisfies E.33). It then follows from iteration of the OZ C.117) and the use of E.34) that in MSA and GMF /ia also satisfies the strong angular condition, (K(rwlw2))oll = <ha(ra>,a>2)>^ = 0. E.37)
352 INTEGRAL EQUATION METHODS 5.3 It now follows from substitution of E.34) and E.35) into the OZ C.117) the use of E.36) and E.37), and the removal of the reference OZ that ha and ca satisfy38b an OZ equation: J E.38) Thus, when E.36) holds a decoupling occurs; the reference fluid values h(), c() and corrections ha, ca satisfy independent OZs. In the more general case one gets Ur^w^J = ca(T12wl(o2) + P\ dt3c0{rx2){hjj32oj3oj2))^ 4- pi dr3<ca(r13a),aK)>(U3/i0(r32) + pi dr3<ca(r13aIa»3)fia(r32a»3toJ)L,, E.39) so that the reference fluid values h0, c0 appear explicitly. Since h0, c0 satisfy the reference OZ, it must be possible to eliminate c0, say, from E.39). This can be done in a systematic way4344 as follows. We use the symbolic, operator, or matrix notation E.1), E.40) which, in coordinate space,44" stands for (cf. C.117)) Jdr3<cA3)hC2)L, E.41) where h{\2) = h{ti<x)x, r2a>2) = /i(r,2a),aJ) for homogeneous fluids. Simi- Similarly, a multiple product like D = ABC, which occurs below, stands for DA2) =|dr3 dr4<AA3)BC4)CD2)>UlWl, E.42) etc. Corresponding to E.38) the reference fluid values h0, c0 satisfy the OZ E.43) The OZ E.40) can be rewritten (compare E.13)) as (l-pc)(l + ph) = l E.44)
5.3 SEPARATION OF THE OZ EQATION 353 where 1 is the unit operator, with 'matrix elements' IA2) = fi8A2) = ftS(r, -r2)S(o), - (o2), so that 1 . A = A according to the multiplication rule of E.40), E.41). Multiplying E.44) from the left with (l-pc) gives44" (l-pc)-1. E.45) We now introduce the obvious (operator) identity440 A1 = B1 + B\B-A)A~1 E.46) and put A = 1 — pc and B = 1 — pc0. This gives A - pc)^ = A - pee) + A - pcT Wl - pc)-\ E.47) or, using E.45) and the corresponding relation between h0, c0, A + ph) = A + pfc0) + A + pfto)pca(l + ph). E.48) Putting h = ho+ha in E.48) finally gives K = A + P^o)Ca(l + pho) + P(l + P^o)Ca^a- E.49) The relation E.49L40 is the desired result; it is the symbolic version of E.39) with c0 eliminated. Using E.42) we see that E.49) reduces to E.38) when Jdoj!CaA2) = 0 (cf. E.36)). The relation E.49) will be needed whenever441 Jda>jCaA2) ^0. In particular, it is valid for a non-spherical reference potential (§4.8). Equation E.49) has been applied43 to the calculation of the dielectric constant of rigid polar fluids, and is also useful for deriving the GMF approximation for general anisotropic potentials (§ 5.4.9), and for deriving the RISM equation in a simple way (ref. 83h). 5.3.1 Perturbation theory solution of integral equations134145 In 'grafting together' perturbation theory and an integral equation theory, the object may either be to (i) use perturbation theory to solve the integral equation, or (ii) use the integral equation to sum the perturbation series. We are concerned with use (i) here; an example of (ii) is given § 5.4.7. To solve the OZ equation E.40) with the help of perturbation theory, we expand h and c in powers of the perturbation potential wa in E.31): h = ho + h1 + h2 + ..., E.50) c=co + c1 + c2 + ... E.51) where h0, c0 are the reference fluid values corresponding to m0 in E.31). Here h, is linear in wa, h2 is quadratic, etc. Substituting E.50) and E.51) in E.40), and equating terms of the same
354 INTEGRAL EQUATION METHODS 5.4 order in ua gives h() = co+pcohit, E.52) hl=cl + p{clh() + cohx), E.53) h2 = c2+p(coh2 + clhl + c2ho), E.54) etc. Equation E.52) is just the reference fluid OZ E.43), and E.53), E.54), etc. relate hn to cn and the lower order h{ and c{. If the set {cn} is known, E.53), etc. then determine the {hn}, and conversely. In Chapter 4 we have worked out hx and h2 explicitly, so that c^ and c2 can be obtained from E.53) and E.54). For the special case where E.36) (or E.37)) holds, E.53) and E.54) reduce to fc,=c,, E.55) 1 E.56) since c() and h0 are assumed to be isotropic. The result C] = hx is used in § 5.4.7. 5.4 Theories for the angular pair correlation function g(ro>1o>2) We discuss now the PY, MSA, and GMF/SSC/LHNC approximations for the angular pair correlation function g(ro>1o>2) for molecular fluids. 5.4.1 The Percus-Yevick (PY) approximation As discussed earlier (see introduction and § 5.1) the PY theory is based on the OZ equation C.117) together with the closure relation E.6). Because of the large number of variables in h(ro>,a>2) ar|d c(ra),a>2), the solution is most easily carried out using the spherical harmonic form C.160) of the OZ relation. The PY closure E.6) can also be written directly in terms of spherical harmonic components (see Appendix A, eqns (A.305) and (A.306c)). We thus have two algebraic relations for the harmonic coefficients of h and c. In practice4647'4711 the harmonic series for h and c must be truncated (somewhat arbitrarily), so that the two algebraic relations referred to become finite sets of coupled equations. Chen and Steele46 applied the theory to a relatively low density symmetric diatomic fused-hard-sphere fluid (sphere diameter a, bond length /), for reduced bond lengths I* = I/a = 0.3 and 0.6. The harmonics /,/2 = 00, 20, 02 were kept. The results for the harmonic coefficients g(/,/2/; r) and the pressure are in good agreement with computer simula- simulation482 and the virial series.46 One expects good results at low densities for the exact PY theory (see Chapter 5, introduction). The good agree- agreement found here suggests that truncation of the harmonic series for can give reasonable results.
5.4 ANGULAR PAIR CORRELATION FUNCTION 355 Morrison47 applied the PY theory to calculate the structure factor S(k) for liquid Cl2. He used an atom-atom LJ potential, and kept the har- harmonics /,/2 = 00, 20, 02, 22, 40, 04. Because of the usual473 difficulties in comparing theoretical and experimental S(k) curves, and also because of the lack of accurate data for liquid Cl2, no conclusions could be drawn from this study about the validity of the theory. Cummings et al.41h have tested the theory at high density against computer simulation; their method of solution takes full advantage of the harmonic forms C.160) and (A.306c) of the OZ and closure relations discussed above. They employed a LJ central potential, with 5-model B.248) short-range anisotropic overlap forces, with S2 ranging from 0 to 0.3. Solution is by iteration, and is extremely slow to converge. The theoretical and simulated harmonics, for lj2l = 000, 202, 022, 220, 222, 224, are in reasonable qualitative agreement, but with some quan- quantitative discrepancies. Because the harmonics of c and h were truncated at (, l2 = 22, it is not possible to ascribe the discrepancies totally to the PY approximation. These preliminary results show that the PY approximation appears to be potentially useful for short-range anisotropic forces, and methods of improving it have been proposed-see ref. 79d' of §5.4.9. 5.4.2 The mean spherical approximation (MSA) The MSA theory for fluids originated48 as the extension to continuum fluids of the spherical model for lattice gases. Besides its intrinsic interest for the qualitative insights and simple results it provides, and its historical interest as an analytic solution50 to an approximation scheme for a semi-realistic49a molecular liquid model (dipolar hard spheres), it is a basis for later improved theories, e.g. GMF (§ 5.4.7), GMSA ('generalized' MSA),1334 CMSA15 = ORPA ('corrected' MSA = 'optimized random phase approximation'), and others.65 Also many of the techniques introduced in MSA such as the form E.75) for the OZ equation, and the transformation E.85) to short-range functions, are useful in other theories. For molecular liquids with non-central intermolecular forces the MSA does not, with a few exceptions, yield quantitatively accurate structural and thermodynamic properties. The MSA is usually applied to potentials with spherical hard cores, although extensions to soft core1313b53a and non-spherical core13c poten- potentials have been discussed. We adopt a simplified model and assume the pair potential u(iwi<o2) = M0(r) + Ma(roIa>2) has a spherical hard core part uo(r) = uHs(r) of diameter cr: r>a E.57)
35fi INTEGRAL EQUATION METHODS 5.4 where ua(ra>,to2) is the longer-range part of the potential which is assumed anisotropic.52" We shall discuss explicitly the case of a pure52b fluid with wa the dipole-dipole potential B.159a), and indicate briefly the extensions to other uas.57~9 We summarize the results in this section, and defer the technical details to §§5.4.3-5.4.6. The MSA theory is based on the OZ relation C.117) together with the closure -1, r<a, E.58) c(rw](xJ) = — /3Ma(ra>i<x>2)» r>a. E.59) The condition E.58) is exact for hard core potentials E.57), since g(rto,a>2) = 0 for r<cr. The condition E.59) is the approximation. We recall (see C.119) and Appendix 10A) that asymptotically (large r) c(rto,co2) is equal to -|3Ma(ro>1aJ); in MSA one assumes E.59) for all r (r>cr). We thus expect the approximation to be worst near the hard core. We also note that the theory is not exact at low density (except for a pure hard sphere potential, where MSA = PY, see below), where c = h=f = exp(-|3u)—1. This is not too important since good theories exist for low densities, and we are therefore mainly interested here in high densities. For Ma = () or C-> 0 the MSA E.59) reduces to the hard sphere PY approximation. For a = 0 the MSA reduces to an RPA or Debye-Hiickel type approximation.n-4S-49-52 The solution is most easily accomplished using the spherical harmonic components /(/,/2/;r) [defined by E.28)], of h and c. For dipolar fluids the alternative Cartesian tensor expressions50 for some of the lower-order harmonic invariants in E.28) are often employed:593 . E.60) where A(to,to2a>) = n, . n2 E.61) (and is in fact independent of to) where n( is a unit vector along o>;, the direction591' of molecule (', and D(a>,a>2a>) is the angular dependence of the dipole-dipole interaction wa (cf. B.159a)) D(o>1o>2o)) = 3(n:. n)(n2 .n)-n! .n2 E.62) with n = rlr a unit vector in the direction to. Comparison of E.28) and E.60) gives (cf. (A.246), (A.248)) A(o>!to2to) = -D7rM(gJi^no(a>ia>2to), = D<n\)i(t|)i(//112(to1to2to)
5.4 ANGULAR PAIR CORRELATION FUNCTION 357 so that we have h(r) = -Dir)-iC)i/A10; r), E.64) /D(r) = Dir)-S(?)l/(l 12; r). As we shall see shortly only the three harmonics lj2l = 000,110,112 enter in MSA for dipolar hard spheres. We now consider the case where wa is a dipole-dipole interaction. The pair potential E.57) contains an isotropic (/1/2( = 000) term, and an anisotropic term (dipole-dipole) with just one non-vanishing harmonic, ltl2l = 112 (see B.172) or (A.247)); the coefficient of the latter is uaA12; r) = -Dir)'(?)*(fA2/r3). E-65) Since c(ra>ia>2) is proportional to ua(ro>ia>2) for r>cr (cf. E.57)), we see that for r>cr, c(ro>1co2) (ar>d therefore h(ro>1a»2)-see E.37)) satisfies the strong angular condition E.36) characteristic of multipole interactions (cf. E.33)). Because of this fact and the fact that the hard core boundary is spherically symmetric, we know from the discussion of § 5.3 that the OZ equation [C.117) or C.158)] decouples into an OZ equation for the isotropic parts (l\l2l =000) of c and h, and an equation for the anisotropic parts (^y/000) of c and h. The isotropic OZ equation, involving c0, h0, is just that corresponding to the reference590 (hard sphere) fluid. With the closure ho(r) = -l, r<a, and co(r) = 0, r>cr, (cf. E.58), E.59)) we see that c0, h0 are just the solutions to the PY hard sphere problem, discussed in § 5.1. The isotropic harmonic coefficients /s(r) are thus given by cs(r) = CpyOO, hs(r) = hPY(r) E.66) where fPY is an abbreviation for /PYhs> the PY hard sphere value. We also see immediately from E.59), E.65) and the harmonic form of the OZ equation [C.160) or C.158)] that only the ^y = 112 and 110 anisotropic harmonics of c and h will be non-vanishing. As we show in § 5.4.3, these two harmonics can also be expressed in terms of the PYHS functions. We find59dc for c±(r) ca(/-) = 2k[ci>y('';2kp)-c,>y('", ~Kp)] E.66) where CpY(rlP) iS the PYHS c(r) at density p, and k is a scaling parameter defined by qPYB/<T})~qPY(-KT}) = 3y E.67) with t} the packing fraction E.18), y the dimensionless combination of density, temperature, and dipole moment, y = D7r/9)Cpfx2 E.68)
358 INTEGRAL EQUATION METHODS and qt>y(ri) = (pkTxPY) ' the dimensionless PYHS inverse compressibilit1 defined by E.21), i.e. . A + 2-nJ QpyKV ) = ~7l ^3" ¦ (J .09 A-T)L The harmonic coefficient cD(r) contains a long-range59' r~3 component - 3r~3 f'drW^V) Jo cD(r) = cg>(r) where c?JV) is a short-range function. The value of the latter is E.70 E.71 (so that, in fact,59* cJSV) = 0 for r>a). Expressions of the forms E.66 and E.70) also hold for hA(r) and hu(r). Thus, all the harmonic coefficients (/s, fA, ft3) of c and h are given ir terms of the known PYHS values. The cA, cD results are therefore analytic (see E.16)) and the hA, hD results piecewise analytic. In Fig. 5.^ we compare the MSA harmonic coefficients hs (=gs-l), hD, and hA witr computer simulation results13'60 at density p* = pa3 = 0.9, and dipole strength (I2 = Cju,2/o-3 = 1. As expected, the MSA coefficients are reasona- reasonable at long range, but poor near contact. From the three MSA harmonic: one can reconstruct the complete g(rco1oJ);61 however, one should not^ the precaution given in § 5.2 regarding theories containing a finite number of harmonics. As discussed in §5.4.3 the MSA can also be solved for quadruDola- 4 - 2 - MSA oand» Computer simulation 1.0 1.2 1.0 F-Ki. 5.4. Comparison of the theoretical MSA harmonic coefficients tor uipoiar *,,.._ wilh computer simulation values, for (I2= 1 (O) and jI2 = 0 (¦). The density is p*-( (F-Yom ref. 13.)
5.4 ANGULAR PAIR CORRELATION FUNCTION 359 1.0 0.4 0.3 0.2 0.1 0 0.1 _ i " \\ r \\ \\ 2 1 i i \ - v .0 \ \\ \\ \ ¦^.__ 1 —^^ • vx l MSA LHNC-GMF QHNC Monte Carlo 1.2 1.3 1 1.5 2.0 2.5 3.0 (rlo) Fig. 5.5. Comparison of theoretical MSA and Monte Carlo _results for the harmonic coefficient hp B24; r) for quadrupolar spheres at p* = 0.83, Q2= 1.25. Also shown are the LHNC and QHNC theoretical curves. [hpB24; r) is proportional to hB24; r), sec ref. 61a.] (From ref. 62.) spheres,58 although the results are less explicit. The pair potential u = mHs+uoq contains lj2l = 000 and 224 harmonics. This leads, in MSA, to 000, 224, 222, 220 harmonics in h and c. In Fig. 5.5 are compared613 the MSA and the computer simulation values of hB24; r) for p* = 0.83 and Q2 = (iQ2/a5= 1.25. The MSA value is seen to be inaccurate. (The LHNC = GMF and QHNC values (see § 5.4.7) are seen to be much more accurate.) The MSA thermodynamics is worked out in § 5.4.4, and the results are also analytic. For dipolar spheres the anisotropic potential contribution to the thermodynamics is determined by the g(Zj/2Z; r) = gA12; r), or go(rX component of the pair correlation function, since Ma(ro>1o>2) contains only a 112 harmonic (see E.131)). Since the MSA goM is inaccurate (Fig. 5.4), especially near contact, we expect to obtain inaccurate ther- modynamic results. As an example,63 we compare in Fig. 5.6 the MSA and Monte Carlo values for A A = A — Ao, the additional free energy due to the dipole-dipole potential. The dipole strength is /I2 = 0.5. The value obtained from the Pade approximant to the Pople series, D.47), is also shown and is again seen to be accurate. As expected, the MSA value is
360 INTEGRAL EQUATION METHODS -0.05 -0.1 -0.15 -0.2 -0.2S -0.3 - MSA PADE • Monle Carlo 0.2 0.4 0.6 0.8 F[(i. 5.6. Comparison of MSA, Monte Carlo and Padc values for the dipolar irec cnerg1 (AA/NfcT) as a function of density p*, for (I2 = 0.5. (From refs. 5, 63.) not in good agreement with the 'exact' Monte Carlo values. MS/ pressures (Ap), energies (At/), and free energies (AA) are compared U Monte Carlo values for dipolar spheres in Table 4.1, for various dipok strengths, and again the MSA values are seen to be inaccurate althougr they do follow the correct trend with dipole strength. The MSA is alst seen (Table 4.1) to be thermodynamically inconsistent; the Ap = p —p( values calculated from the pressure and free-energy routes are seen tt disagree (see also § 5.4.4). Detailed phase diagrams for dipolar hare spheres according to MSA have also been plotted.6170 Numerical result: for mixtures have also been worked out.70 Similar results are obtained fcr quadrupolar spheres (cf. Table 5.2). The MSA theory of the dielectric constant e (for rigid dipolar spheres is worked out in § 5.4.5. The result is again analytic for e = e(y): e =- where ? = ktj is determined by y using E.67), i.e. = 3y. E.72) E.73)
ANGULAR PAIR CORRELATION FUNCTION 361 MSA Onsager • Monte Carlo 200 160 120 40 2.0 2.5 3.0 •ig. ,V7. Comparison of theoretical and Monte Carlo values of the dielectric constant s for p* = 0.8 and various dipole strengths /I. (From refs. 63, 64, 65a'.) \s we shall see e is determined by an integral over the harmonic :oefficient h(\\0; r) or hA(r). In contrast to the thermodynamic properties his integral receives significant contributions from the large r values, e.g. lp to6464a r~10cr for /I2 = 2.0, and the MSA e is thus aacurate to ibout 10 per cent for jl2ssl (cf. Fig. 5.7). In Fig. 5.763-64 we give the vlSA e, and also the values for the Debye, Onsager (§ 5.4.5) and ^MF^LHNC (§ 5.4.7) theories for p* = 0.8; it is convenient to compare vlSA with the other theories here, but the reader may wish to postpone a ;erious study of the results of the other theories until later. (In GMF e lepends on p as well as y.) Based on the limited and somewhat uncertain lirect Monte Carlo data that is available64-64'653' (i.e. 3 points in Fig. 5.7), ;nd also on indirect Monte Carlo values of e inferred from the SCOZA65 self-consistent OZ approximation), it appears that the GMF values of e ire the most accurate of the theories shown in Fig. 5.7 for small /I; lowever, GMF appears to be seriously in error at large /I. More recent ind more extensive simulation tests65" for dipolar spheres and dipolar LJ nolecules confirm that the GMF e is accurate for small dipole strengths, )ut seriously overestimates e at large dipole strengths. This work is liscussed in detail in Chapter 10. This completes the discussion of results for the MSA theory. Generally, or multipolar fluids, it is not a quantitatively accurate theory. For central "(otentials, including electrolyte solutions, it has been more success- uj_i3.i3b,34,53.s3a Nevertheless, as mentioned in the introduction, the de- ails of the solution are of interest in their own right, as well as being
362 INTEGRAL EQUATION METHODS 5.4 useful in suggesting improved approximation schemes and introducing general theoretical techniques. The reader not interested in the technical details can skip §§5.4.3-5.4.6. 5.4.3 Solution of the MSA In this section we derive E.66) and E.70) of the preceding section. The procedure will be to carry out the transformations f(lil2l; r)—* fmdil2l; r)->/@)(/,Bm; r) on c and h, which are illustrated in Fig. 5.3. In terms of the /<0)('i'2m; 0s (which will be introduced shortly in E.77)) we shall see that the MSA problem reduces to the PYHS problem, and is therefore trivially soluble in terms of the solutions of the latter problem. As discussed in the preceding section, because ua contains only an /,B' = 112 harmonic E.65), the only non-vanishing harmonics of c and h correspond to lrl2l = 000, 110, 112, (i.e. the s, A, and D harmonics of the other notation). The OZ C.158) (or C.160)) decouples. The c and h solutions for the /,/2'-000 harmonics are just the PYHS values E.66). To find the 110 and 112 harmonics, it is easiest to use the k-frame harmonics /(llm; fc), (m =0, ±1), and corresponding OZ C.160). Since (see CB.22a)) ;fc) E.74) (where m =— m) we focus attention on the 110 and 111 harmonics. Since the non-vanishing harmonics of c and h correspond to IJ2 — OO and 11, the OZ C.160) also decouples. For the lll2 = \\ case, since coeffi- coefficients such as cA0m; k), etc. vanish in the product term, we have h(llm; fc) = c(llm; k) + Dir)-'P(-)mc(llm; k)h(Um; k). E.75) We note that the equations for the l^m —110 and 111 harmonics are themselves uncoupled in E.75) (whereas the 11[2! = 112 and 110 har- harmonics are coupled in C.158)). Thus E.75) is of the same form as for atomic liquids, and can be solved immediately for h(\\m; k) in terms of c(llm;/c), (cf. E.12)). We must now transform E.75) directly to r-space, since the closure relations for c and h will be given in terms of the f(lll; r)s (see E.92)). (We do not use the other possible route to r-space, f(llm; k)—» f(l\l;k)-*f(\ll;r), (see Fig. 5.3), since the OZ equations become coupled by this set of transformations (cf. C.158)). The r-space form of E.75) is (cf. E.11) and E.10)) E.76) where65b /<0)(llm;r) [which differs from /(llm;r), see Fig. 5.3] is the
5.4 ANGULAR PAIR CORRELATION FUNCTION 363 Fourier transform (see C.154), C.155)) -*-Tf{lxl2m;k) E.77') ;/0(/cr)/(ii;2m;/c) E.77) where jo(x) = sin(x)/x is a spherical Bessel function; the angular integral in E.77') is simply done since f(.lj2l;k) depends only on fc = |k|. We similarly define a Fourier transform of the space-fixed k-space harmonic coefficients f(lil2l; k), 4tt BttK The inverse transformation to E.78) is Jfc2dfcj0(fcr)/(M2J;fc). E.78) drj0(kr)f°\lll2l;r) E.79) and similarly the inverse of E.77) can be written down. The harmonic coefficients f(l\l2l; k) and f{lj2m; k) are related by the Clebsch-Gordan relations CB.21), CB.22). Because of E.77) and E.78) the same relations hold between fm(lil2l; r) and fi0\lil2m; r): rihhh r) = I t^fcdJJ; mmO)f@\lJ2m; r), E.80) fm{lj2m; r) = I (^-^felxy; mipO/^dWjI; r). E.81) \ 4tt / i 4tt To relate the auxiliary quantities fm(l}l2l;r) to the space-fixed har- harmonic coefficients f(lj2l; r) of interest, we substitute the Hankel transform CB.43) for f(lj2l; k) into the Fourier transform relation E.78). This gives {Ztt) where the kernel Kt(r, r') is given by \r'2dr'Kt(r, r')/(i,BI; r') E.82) J Kt(r, r') = i'|/c2 d/c/0(/cr)/,(/cr'). E.83)
364 INTEGRAL EQUATION METHODS 5.4 Similarly, we find the inverse relation from E.79) and CB.44) f{Uhl;r) = {-)l~~ fr'2dr'K1(r',r)/@)(M2Z;r'). E.84) Btt)- J The kernel K,(r, r') is evaluated in §5.4.6, and is given by eqn E.179). Using this result in E.83) and E.84) gives for / even65c the explicit *¦- relations l;r)-- f dr'PJ(r/r')/(/1l2I;r'), E.85) 'dr'P\(r'lr)f@\l1l2l;r'), E.86) where P\(x) = (d/dx)P[(x), and Pt(x) is the Legendre polynomial. In addition to the original four spherical harmonic expansion coeffi- coefficients f(lj2l;r), f(l\lzm;r), f(Ul2l; k), f(lj2m;k), we have now intro- introduced two auxiliary6'" coefficients, fia\lxl2l; r) and fm{lxl2m; r). To assist the reader in keeping track of these coefficients and the various relations between them we have listed the coefficients and their relations in Fig. 5.3. It turns out that the f(m(lj2m; r) have two remarkable properties:650 (i) they decouple the OZ equation (we have seen this already, eqn E.76)), (ii) the MSA closure reduces to a PY closure in terms of these coeffi- coefficients. As a result of these two properties we can immediately (and analytically) solve the MSA problem in terms of the known solution to the PY problem. A preliminary result, needed to prove assertion (ii) above, is that the transformation E.85) annuls functions of the type r~(i+1); i.e. we have if f(hl2l;r) = Ar-"+l\ { ' This is proved for general I in § 5.4.6. For the dipole case, we have cA10;r) = 0 and c(l 12; r) = Ar^ (see E.59), E.65)) for r>a. Using P',,(x) = 0, and P'2(x) = 3x in E.85) and E.86), we get the explicit relations E.88) ;r)- f ^-/A12;r'), E.89a) 12;r)-^ [ r'2 dr'/@)A12; r'). E.89b) r Jo A simple calculation now gives c<0)A12; r) = 0 for r>a. In essence, then,
5.4 ANGULAR PAIR CORRELATION FUNCTION 365 the transformation E.85) transforms awayt the interaction outside the core. Hence because of the relation E.81), we see that the c@)(fi/2m; r)s satisfy the closure property (ii) mentioned in the preceding paragraph. We now fill in the details. The explicit forms of E.80), E.81) for the dipole-dipole case are needed. The /@)(ll(; r)s are given by (note E.74)) ; r) = D7r)KM/@)Ull; r) + /@)A10; r)], )] Conversely65* the /<0)(llm; r)s are given by ; r) = (^-f{lfu(O)(n0; r) + (§)^0)(H2; r)], / i\viv ; r) = (^JEJ[-/<0)A10; r) + A0)*/<0>A12; r)] The OZ E.76) is in terms of the /<0)(llm; r)s. To formulate the closure relations in terms of these quantities we carry out the transformations f(.ni;r)^f°Xni;r)^>fm(Um;r), (see Fig. 5.3). In terms of the /A11; r)s the closure relations are (see E.58), E.59), E.65)) h(U0;r) = 0, r<a, E.92a) cA10;r) = 0, rXr, E.92b) r<a, E.92c) r><r. E.92d) Transforming to the /(O)A1Z; r)s via E.89a) gives (note E.87)) h@)(HO;r)=0, r<(T, E.93a) c(o)A10;r) = 0, r>a, E.93b) h@)A12;r) =-3D71)^)^^ r<(T, E.93c) c(mA12;r) = 0, r>a, E.93d) where k is a dimensionless constant, ¦r). E.94) t Because the transformation E.85) annuls the long-range multipole interactions, the coefficient /@1((,BJ; r) can be read as 'the short-range part of /((,B'; r)\ or. f°r ' even (see E.77)) as 'the ;,, transform of /(I,I2I; k)\ or more simply as (see E.77) or E.168), E.169)) 'the Fourier transform of f(I,l2i;fc)'. The quantity f(lxl2l;r), on the other hand, is (see CB.44)) 'the (/,) Hankel transform of fil^k)'.
36<S INTEGRAL EQUATION METHODS 5.4 Transforming to the /""(llm; r)s via E.91) gives finally r<<r, E.95a) r>a; E.95b) , r<a, E.95c) c(o)A10;r) = 0, r><7. E.95d) The relations E.95) are essentially PYHS closures h(r) = -\ for r<a, c(r) = 0 for r>o-. Consider, for example, the OZ E.75) for m = 1, E.96) or schematically, h = c + Dir)-\-p)ch, E.97) with closure E.95a,b). We denote the PYHS solutions at density p by /iPY(r;p) and cPY(r; p). The (schematicN5* OZ for these quantities is hPY(p) = cPY(p) + pcpY(p)hPY(p). E.98) Multiplying E.98) by 4ttk, i.e. E.99) shows that 47TKhpY(p), 4-7rKCPY(p) are solutions of an OZ-like equation; the closures are 4ttk/ipy = -Attk for r<cr, 4ttkcpy = 0 for r>a. Scaling the density p by —k in E.99) then gives E.100) The quantities 47r«hpY(-Kp) and 4-7rKCPY(-xp) in E.100) satisfy the same OZ-like equation as the quantities in E.97), and the same closure relations. Hence the solutions of E.97) (or E.96)) are /@)(lll;r) = 47n</pY(r;-/<p) E.101) where /"" denotes c(m or h<0\ with /PY the corresponding PY value of c or h. Similarly, for m =0 we get f'"A10; r) = 47rBK)/PY(r; 2Kp) E.102) We must now carry out the reverse transformations /A11; r) ^/<0)(li;; r) «-/@)(llm; r) E.103) to obtain the space-fixed harmonics of interest. From E.90) and E.101),
5.4 ANGULAR PAIR CORRELATION FUNCTION 367 E.102) we obtain the /<O)A1Z; r)s /(n)(H0; r) = D-7rMCM2K[/PY(r;-/<p)-/pY(r;2Kp)], E.104) Finally, using E.89b) we get the /(Hi; r)s /A10; r) = D7r)^2«[/pY(r;-Kp)-/pY(r;2Kp)], E.106) /A12;r) = /@)A12;r)-3r-3f r'2dr'/@)A12; r') E.107) Jo with the /@)(lli;r)s given by65h E.104) and E.105). The relations E.106), E.107) were given in the preceding section (cf. E.66), E.70)) in the other notation E.64). There remains only to evaluate the scaling parameter k, which is defined implicitly65' by E.94). A more explicit expression can be obtained by equating the two expressions E.92d) and E.107) for cA12; r) for r>a; since c<0)A12; r) = 0 for r>cr we have DirM(AIPfA2r3 = -3r-3 {"radr1 c<0\\\2; r'). E.108) Jo Substituting E.105) in E.108) gives 4-7T 2 f f — Cpju, = —Kp|drcpY(r; —«p) — 2«p JdrcPY(r; 2kp) or 4-7T -—Cpfx =qPYBKT)) —qPY(—ktj) E.109) where f •cPY(r;p) E.110) is the dimensionless inverse PYHS compressibility E.20), which is ex- explicitly given by E.69). The relation E.109) gives k as a function of y E.68) and tj E.18). From E.69) one shows that qPYB?) — qPY(— ?) is a monotonic function of ? in the interval 0 <?<i so that from E.109) as y =Dtt/9)Cpju,2 varies from 0 to oo, ? varies from 0 to \. Solution for other multipoles We briefly indicate the solution to the MSA for quadrupoles and other multipoles.58 For the pair potential u(t(xIw2) = MHS(r) + MQO(ro>1a>2), in MSA the non-vanishing harmonics of c and h are lxl2l = 000, 224, 222,
V,,S INTEGRAL EQUATION METHODS 5.4 220 since the only non-vanishing harmonic in the anisotropic potential is i(,,B24; r) (cf. B.172); we consider linear molecules), E.111) Since for r>cr we have in MSA c(rwiw2) = — CMa(ra),aJ), the only non- vahishing lxl2l harmonic of c outside the core is cB24; r), analogous to E.92). To carry out the transformations (and inverses) fB2l;r)—* /""B2/;r)^/'O)B2m;r), we need the analogues of E.88), E.89) and E.90), E.91). Analogous to E.88), E.89) we find /(<"B20;r)=/B20;r), /""B22;r) = /B22;r)-3| -4/B22; r'), E.112a) 35 ~,f°°dr' 15 f°°dr' /@>B24;r) = /B24;r) r2 — /B24;r')+— — /B24; r') 2 Jr r'3 2 Jr r' and conversely /B20;r) = /<mB20;r), /B22; r) =/(A)B22; r)-\ [ r'2 dr'/@)B22; r'), E.112b) /B24; r) = /(n)B24; r) 5 r'4 dr'f())B24; r') Jo p)B24; r'). Analogous to E.90) we have for the /(O)B2/;r)s in terms of the /""B2m; r)s /(O)B24; r)= (|^J[/(o)B20; r) + |/(O)B21; r) + ^/(O)B22; r)], /"l>B22; r) = (—J[-/(n)B20; r) + /@'B21; r) + 2/@)B22; r)], E.113) /""B20; r) = (yV[f(O)B20; r)-2/A1)B21; r) + 2/@)B22; r)].
5.4 ANGULAR PAIR CORRELATION FUNCTION 369 Conversely, analogous to E.91) we have for the /<0>B2m; r)s r>B22; ^)= (^)'[/(mB20; r) + 5 (~fr B22; x/@)B24;r)], x/'0)B24; ; r) = (^[ ^f ^f x/(wB24;r)l. From E.112a) we obtain the closure relations in terms of the fmB2l; r)s, analogous to E.93): h@)B20;r) = 0, r<a, E.115a) c<0)B20;r) = 0, r>a, E.115b) h@)B22;r) = -3J %-h{222;r'), r<a, E.115c) c(o'B22;r) = 0, r>a, E.115d) h<0)B24; r) = ~ r2 f^ hB24; r') + ^ f"^ hB24; r'), E.115e) C@)B24;r) = 0, r>o-. E.115f) The first non-trivial difference from the dipolar case appears here; in E.115e) there occurs the form a + br2 on the right-hand side, rather than a constant as occurs in E.93c). In transforming E.115) to the hmB2m; r)s (analogous to E.93) -> E.94)) using E.114), there will occur closures of the type h@)B2m; r) = polynomial in r, r<cr, E.116a) c@)B2m;r) = 0, r>a. E.116b) The OZ equation analogous to E.76) for the /<0)B2m; r)s together with the closure E.116) can be solved58 using the Wiener-Hopf factorization method of Appendix 5A (see ref. 119). The final results are not as explicit as in the dipole case, owing to the fact that three constants of the type E.94) occur in E.115) and must be determined self-consistently.
370 INTEGRAL EQUATION METHODS 5.4 A further complication57"9 arises for some unlike multipole interac- interactions (e.g. dipole-quadrupole, charge-dipole), and for induction, disper- dispersion, and overlap potentials, due to the presence of harmonics f(l\l2l', r) with / odd. The transforms E.77), E.78) must be replaced by cosine transforms (see § 5.4.6), and as a result the OZ E.76) takes a more complicated form. A relatively simple example involving the charge- dipole interaction, which occurs for electrolyte solutions with polar sol- solvents, has been worked out explicitly.4357'57" Finally, we mention that a variational method of solution of the MSA equations has also been developed.55 5.4.4 Thermodynamics of the MSA63-610 From the MSA pair correlation function and the theoretical expressions for the thermodynamic quantities given in Chapter 6 we can calculate A (free energy), U (internal energy), p (pressure) and x (isothermal com- compressibility). The numerical results have been discussed in § 5.4.2. The results can all be obtained in closed form, and also illustrate nicely the problem of 'thermodynamic inconsistency'; i.e. different routes to the thermodynamic quantities give rise to different results, because the theory is approximate. For a pair potential of the form uK = uQ + Aua (where here uQ = uHS and ua = u^), the additional free energy AA = A — Ao due to the anisotropic potential ua is given by (see § 6.4O0a =4pj3f Jo j3AA//V=4pj3f dA fdrd^rco^M™,^))^ E.117) where hK = ho + ha is the total correlation function for the fluid with pair potential uK. In terms of the spherical harmonic components of ua and hx E.117) becomes (see §6.4) J3AA/N = !p/3D7r)-3 f dA fdr ? B1 + l)ua(M2l; r)hk(hl2l; r). Jo J i,i2i E.118) Here only uaA12;r) is non-vanishing, and is given by E.65). Thus, we. have ^j3p,i,(^VD7rrf dA f -hAA12;r). E.119) 2 \15/ J J f dA f - But from E.94) we see that the r-integral in E.119) is proportional to ) E.120)
5.4 ANGULAR PAIR CORRELATION FUNCTION 371 where kk is determined from (cf. E.109)) 4*77 y Kx-n). E.121) Hence, from E.119) and E.120) we have j3AA/N = -3y[ dkxK E.122) [ Jo l2 where y = Dtt/9)j3p/ll2. Changing integration variable in E.122) to y' = Ay, and introducing ?' = KKr\, we have j3AA/N = --/(y) E.123) where fV E.124) df E.125) where we have integrated by parts, and introduced ? = ktj. The quantity y' is determined by ?' from E.121), i.e. E-126) Substituting y' from E.126) in E.125) and carrying out the integration gives where ? is determined by y in the same way E.126) as ?' is determined by y' (see E.73)). The (dimensionless) free energy change is thus given70b in the closed form E.123). For future use we note the relation obtained by differentiat- differentiating E.124) ~Ky) = 4- E.128) The internal energy change700 AU=U-U0 will be calculated here in two of the four possible ways. First, by differentiating the free energy E.123) according to j3At//N = j3O/3j3)(j3AA/JV) we get . E.129) dy
372 INTEGRAL EQUATION METHODS 5.4 Because of E.128), E.129) gives j3A[//N = -3/<y, (A-route), E.130) where (A-route) indicates the value derived from the free energy route. In the second method we use the relation for Uc in terms of gA2) and mA2), or rather the spherical harmonic form (see §§6.1, 6.6). Since only uA12; r)^0 for r>a, and since in MSA E.33) and E.37) are satisfied we therefore have J3AL//N = ij3pD77-r3jdrh(l 12; r)u(l 12; r). E.131) Again using E.65) and E.94) we find E.131) becomes t3Mf/N =-3xy, ([/-route), E.132) which agrees70" with E.130). We now calculate the pressure in four ways. First, by differentiating the free energy E.123) according to |3Ap/p = p(d/dp)(KAA/iV), using again E.128), we get CAp/p=-/(y)-3i<y. E.133) V Comparing E.133), E.123), and E.130) we see that C\p!p=(iAUIN-CAA/N E.134) which, in view of the thermodynamic relation A = U — TS, where S is the entropy, can also be written as j3Ap/p = AS/Nk, (A-route). E.135) In the second method of calculating p, we use the virial relation [see § 6.2 or Appendix E] between p and gA2). Because the dipole-dipole poten- potential varies as r 3, it satisfies where u^ttu.Jrlr, so that by comparison with the energy equation and the use of E.33) and E.37) we see 0Ap/p = /3A[//N, (p-route) E.136) where A.U is given by E.132). In the third route to the pressure we use the compressibility relation C.13) -l] E.137) where g(r) = gs(r) is the centres pair correlation function. Since in MSA
5.4 ANGULAR PAIR CORRELATION FUNCTION 373 gs(r) = goM. the PY hard sphere value, we have x — Xo- Since this is true for all values of the density p, integration of the inverse of x — P~' dp/dp with respect to density will give p = p0, or Ap = p - p0 = 0. Thus, from the compressibility route we get j3Ap/p = 0, (x-route). E.138) The fourth route to the pressure is based on the energy equation (§ 6.1) [or see C.107) and discussion]. We integrate the thermodynamic relation CAU/N = j3C/3j3)@AA/iV) with respect to |3 to obtain j3AA/N, and then differentiate with respect to density using {ih.plp = pC/dp)(j3AA/N) to obtain Ap. However, because of the equality of E.130) and E.132), the result of this route will be the same as that of the A-route, E.135), so that we have j3Ap/p = AS/Nk, ([/-route). E.139) Finally we discuss the compressibility. If we use the compressibility equation E.137) to calculate x, we get pAx/j3=0, (x-route) E.140) for the reasons already discussed following E.137). This is equivalent to E.138). Non-vanishing, and therefore inconsistent, expressions for Ax can also be derived from the A-, p-, and [/-routes (equivalent to E.135), E.136), and E.139), respectively). We have listed above the incremental quantities Ap, A[/, etc. To obtain the total quantity, we simply add the PYHS value, which can be obtained from §5.1. We have seen that in MSA there is inconsistency in the thermodynamic quantities. These inconsistencies are substantial in practice. For example, at p* = per3 = 0.59 the two values E.135) and E.136) of Ap can differ by a factor of 2 or more (see Table 4.1). This thermodynamic inconsistency can be made the basis of an improved theory,13'34 GMSA (generalized MSA), wherein one introduces additional tail terms in c(roj1aj2) which contain adjustable parameters, and then fixes the values of the parame- parameters by forcing agreement700 between the inconsistent quantities. 5.4.5 Dielectric constant of the MSA5063-67-71 Here we give the derivation of the dielectric constant e for rigid dipolar spheres in MSA, eq. E.72). The relation E.72) for e = e(y) can also be written as e=qPYB!)/qPY(-?) E.141) as we see from the explicit relation E.69) for qPY(T|). Here ? = ?(y) is determined by E.73), with y = Dtt/9)|3pil<,2.
374 INTEGRAL EQUATION METHODS 5.4 As discussed in Chapter 10, there are various routes from g(ra>|i<j2) to the dielectric constant. As we shall see, these routes all give rise to the same expression E.141), so that the MSA theory is 'dielectrically consis- consistent', in contrast to the 'thermodynamic inconsistency' found in the preceding section. In two of the routes we shall obtain e directly from g(rw1aJ) [in fact from the harmonic coefficients gA(r) and gD(r)] and in the third route from c{io}lo}2) [using cA(r)]. The dielectric constant is a system property which is believed to be intensive, independent713 of the size and shape of the sample. Due to the long-range (r~3) nature of the dipolar field, however, the relation between e and gOra^o^) does depend on sample size and shape, which compen- compensates for the fact that gOra^o^) for dipolar systems depends on size and shape (see Chapter 10). In what follows we write down expressions for e which are valid for an infinite sample.7 The first expression we use is the famous Kirkwood exact relation710 (see § 10.1.4), (e - l)Be + U = l+ [drfliOraMtoJcos YizL,^ E.142) 9ey J where y12 is the angle between the two dipoles of molecules 1 and 2. Since we shall consider only linear molecules, y12 is also the angle between the two symmetry axes of the molecules. The relation E.142) can be re-expressed (eq. A0.41)) in terms of the l^l k-space harmonic coefficient 6A10; fc), or hA(k),t for k = 0: — =l+|p6A@). E.143) The y0 Hankel transformation7" CB.43) then gives (e-l)Be = l+?p drhA(r). E.144) 9ey Substituting the analogous relation to E.66) for hA(r) in E.144) gives (e-: 9ey 3 3 + ?[l + (-Kp)Jdr/iPY(r;-KP)j. E.145) Noting the compressibility relations E.69) and E.19) we get71e +§qPY(-gr1. E.146) We use f(k) for a k-space quantity here so that f@) will be unambiguous.
5.4 ANGULAR PAIR CORRELATION FUNCTION 375 Using the relation E.67) 3y = q+-q_ E.147) where q+ = qPYB?) and q- = qPY(~?), we rewrite E.146) as ), E.148) or alternatively as (e - l)(e + 2) = (q+/q_ - l)(q_/q+ + 2) E.149) from which we derive the desired relation E.141), e =q+/q_. The second route to e from g{iwloJ) is based on the formally exact asymptotic relation (see C.121), and for derivation (§ 10A)) E.150) which in turn follows from the formally exact relation E.151) where ullLIL(toilw2) is the anisotropic r 3 dipole-dipole potential. Since u^ contains only an lxl2l = 112 or D harmonic (see E.65)), h(r<uioj2) will fall off at large r as r~3, and contain only a D harmonic. The relation E.150) can be re-expressed (see § 10.1.4) in terms of this D harmonic in H0ye- Stell form O-1J ^-^ = -pM0). E.152) 3ey Using the Fourier7" transform relation E.79) we write E.152) as ^-=^ = -pfdrhg)(r) E.153) 3ey J where /iSV) is the Fourier transform of hD(k). Substituting the h{o analogue of E.71) in E.153) and again using the compressibility relations E.69) and E.19) gives ^^-^ E.154) 3ey which we rewrite using E.147) as (e - l)(e-1 - 1) = (qjq_ - l)(qJq+ - 1). E.155) We see again from E.155) that e =q+/q_. In the third route to e we make use of the exact Ramshaw expression
376 INTEGRAL EQUATION METHODS 5.4 (§ 10.1.4) ^= I l-pjd'<c(raIaJ)cos712>o)lW2J E.156) which is derived from E.151). We transform this expression in the same way E.142) was transformed to E.143) and E.144): ^ = [1-^@)]-' E.157) y e +2 [J] . E.158) Substituting E.66) in E.158) and using the compressibility relations E.69) and E.20) gives = 3(q+ + 2q_)~1 E.159) or, using E.147), e~l ' w . _ ,_, E.160) = I +r E-161) From E.161) we see again71' e =q+/q_. Thus, all three routes lead to the same expression E.141) for e in MSA.71s We note that e depends only on ? and therefore only on y « j3p)x2, and is independent7111 of o\ For small711 y, ? is given by ? = sy-256y +•••¦ E.162) It is interesting to compare the MSA result for e with the results of the early theories of Debye72 (D) and Onsager73 (O): "~1=v, (D) E.163) = y. (O) E.164) e+2 (e -1X26 + 1) 9e The Debye and Onsager approximations arise from mean field theories in which the angular correlations between the molecules [represented by h(raj,a>2) in E.142)] are neglected (see Chapter 10). The unphysical singularity in the Debye theory (e—»c° for y —* 1) is avoided in t|je Onsager theory by the use of an average 'reaction field'.
5.4 ANGULAR PAIR CORRELATION FUNCTION 377 It is convenient to compare the Clausius-Mossotti function (e - l)/(e + 2) for the three theories. If this function is expanded in powers of y, for small y, the nearly exact result is74 ?Zi=y-iiy3 + -.. • (Jepsen) E.165) e + 2 The Debye theory gives the linear term E.163) for the Clausius-Mossotti function, whereas the Onsager approximation E.164) gives ?zi=y_2y2+... (O) E.166) e +2 and the MSA E.160) gives e — 1 ,, , e+2 (MSA) E.167) Thus the MSA theory is nearly correct to the first two orders in y, which is an improvement over the Debye and Onsager theories. The values of e for larger values of y = Dtt/9)p*jI2 are given for all three theories in Fig. 5.7, and are also compared to the available Monte Carlo data. Also shown is the GMF^LHNC theory (§5.4.7) result. Discussion of these results was given in § 5.4.2; further results are given in Chapter 10. In conclusion, MSA for the dielectric constant is quantitatively correct for small y, and gives an analytical result for larger y which has at least a qualitatively correct trend. Further insight into the reasons for the relative success of MSA as regards the dielectric constant can be found in the papers of Stell and coworkers.13c65-74a 5.4.6 Evaluation of the kernel K{(r, r') The kernel K,(r, r') defined by E.83) leads to a simple explicit expression only5658 for even /. For odd /, we therefore modify the transform E.78) (and E.77)) in such a way as to lead to a more convenient kernel K,. For even I we use E.78) and E.79) (/even). E.168) For odd I, we can either58 replace jQ(kr) in E.78) and E.79) by ji(kr), or57 replace sin kr in E.168) by cos kr. Since the latter procedure gives rise to
378 INTEGRAL EQUATION METHODS 5.4 a simpler result we follow it here. We therefore use I odd). E.169) The kernel E.83) then becomes K,(r, r') = i'J fc2 d/c^-/,(fcr'), I even E.170) c2dfcC°,S rh(kr'), kr and can now be evaluated.56'57 We write E.170) as E.171) which is valid for all /. Using the integral representation75 duetauP,(u) E.172) where P[(u) is the Legendre polynomial, and introducing a convergence factor _ lim e into the k integral in E.171), we obtain for E.171) K,(r, r') = Re— duP,(u) k dfc eilc(ur'r+ie) E.173) 2r J J where it is understood that the limit e —* 0 is to be taken after the integrations are carried out. We can write E.173) as K,(r, /•') = Re(—) — fdMP,(u) [dfc eilt("r"r+i<!) E.174) \2rI drJ J and then do the fc integral in E.174). This gives K,(r,r') = Re(^i)f fdnP,(n) , \. . E.175) \2r / dr J ur-r + ie Using the relation76 (see (B.73)) ()iTr8(x) E.176)
5.4 ANGULAR PAIR CORRELATION FUNCTION 379 where P( ) denotes the principal value, we get J = (r')~1 E.177) Using 8(ur'-r) = (r')~18(u -r/r') (see (B.71)) and then doing the u in- integral in E.177) gives Kt(r, r') = - (~) f [P,(/-//H(r'" r)] E.178) \Zrr I or where Q(r' — r) is the unit step function, 0 or 1 according as r'<r or r'>r, respectively. Carrying out the differentiation in E.178), using (d/dxH(x) = 8(x) (see (B.70)) gives finally ] E.179) where P[(u) = (d/du)P[(w). The result E.179) is valid for both even and odd I. Substitution of E.179) in E.82) leads to the explicit transformation E.85). To prove the property E.87) of this transformation, we first integrate E.85) by parts, f°\r) = -\\ r' dr'P((r/r')[2/(/) + r'/'(r')] E.180) where we have used the facts that Pj(l) = 1 and /(<») = 0; we also use for brevity the notation f(lll2l; r)=f(r) here. Putting/(/•) = Ar~('+1) in E.180) now gives Changing integration variable to x = r/r' in E.181), we get ^2. E.182) But x'~2 can be written as a linear combination of Legendre polynomials Pj(x), where I' runs over 1 — 2,1 — 4,.... Because of this fact, the even- evenness of the integrand, and the orthogonality relation (A.9b) for Legendre polynomials, the integral E.182) vanishes. 5.4.7 Generalized mean field (GMF) approximation The GMF (generalized mean fieldO63 approximation was originally de- derived77 in the general context of polar and polarizable fluids, where it was christened773 the SSC (single superchain) approximation. A simplified
380 INTEGRAL EQUATION METHODS 5.4 diagrammatic rederivation41 (see § 5.4.8) was later given in which molecular polarizability is assumed absent; the approximation can be applied for arbitrary anisotropic pair potentials (see §5.4.9). Another rederivation64 has been given based on the HNC approximation E.7), wherein the approximation was called LHNC (linearized HNC-see § 5.4.9). For purposes of derivation we shall refer to the GMF approxi- approximation. Later we shall often refer to the SSC, LHNC, SSC/LHNC, etc. approximation. We assume7715 the pair potential u(xtx>l<iJ) has the decomposition E.31) into an isotropic reference part uQ(r) and an anisotropic perturbation ujx<ti1o}2) satisfying E.32). For simplicity we shall also consider here only multipole-like potentials satisfying E.33), although the theory can be applied for general was (see §5.4.9, and ref. 47b for application to anisotropic overlap forces). The theory is based on the OZ equation together with the closure E.191), where ca and ha are defined by E.34) and E.35). Since in the GMF theory ca will be seen to satisfy the strong angular condition E.36), the correction terms ha and ca themselves satisfy an OZ relation E.38) which is not coupled to h0 and c0. GMF is an extension and improvement upon MSA in the following ways: (i) the theory is applicable to soft core (e.g. LJ) potentials, (ii) the exact values of h0 and c0 are used, rather than PY approximations to these quantities, (iii) the GMF ca contains the correct first-order term in ua (for multipole-like ua), and also contains higher order terms (whereas MSA has neither of these attributes), and (iv) extensions of the theory can be given to account for the difference between g(r) and go(r) (see §§5.4.9, 5.4.9a, and discussion p. 351). Because of (ii) and (iii) GMF is expected to be more accurate than MSA. The disadvantage is that (so far) the results have not been worked out analytically, although as we shall see, some analytic steps can be carried out. In common with MSA, in GMF, g(r<u,oJ) automatically contains only a finite number of harmonics (if ua does), which simplifies the theory,77c and satisfies the core condition g(roj]&J) —» 0 as r —* 0. The theory also gives the correct asymptotic (large r) results for c and g. It does not, however, become exact at low densities; cf. § 5.4.2 for comments concerning why this defect is relatively unimpor- unimportant. We start with the perturbation theory solution of the OZ equation derived in §5.3.1. The perturbation corrections hx in E.50) have been worked out in Chapter 4. In particular, for multipole-like ua, ht is given byD'37) h,(rw1«U2) = -pg0(r)iia(rw1<a2), E.183) and clearly satisfies the strong angular condition E.37). Hence, from E.55), C] is given by E.184)
5.4 ANGULAR PAIR CORRELATION FUNCTION 381 The approximation ca = cu = -/3go(r)u11(wB1<a2) E.185) or, c = co + cu cMF(rwlto2) = Co(r)-pgo(r)M.(r<a1<a2), E.186) is designated the mean field (MF) approximation,77 by analogy with mean field and RPA theories of linear response.51116 We wish to solve the OZ E.38) for h^*, with c™? given explicitly by E.185). For concreteness we take an LJ + QQ pair potential uo(r) = Mu(r), Ma(roIoJ) = Moq^oj,^). E.187) The anisotropic potential WqqOnuiwJ contains a single lj2l space-fixed harmonic coefficient uaB24; r), given by E.111). From E.185) we see that c™f(to}1o}2) will also contain a single harmonic, with coefficient ; r) = -D7r)iA4/45)igo(r)(j3Q2/r5). E.188) For the reasons discussed in §5.4.2 (p. 357, 367) this leads to three non-vanishing harmonics ha, corresponding to lxl2l = 224, 222, and 220. The coupled OZ equation C.158) for these harmonics reduces to a 3x3 system in this case, and is readily solved explicitly41 for the hJ22l; r)s, in terms of the known caB24; r). Alternatively,41 and even more simply, we can use the OZ C.160) in terms of the k-frame harmonics, /B2m; fc), where m=2,1,0, (with /B2m; fc) = /B2m; fc)-see CB.22a)). In this frame, the OZ relation decouples, haB2m; fc) = caB2m; k) + D7r)-\-)mpcaB2m; k)KB2m; fc), E.189) and so is easily solved, ^g^ and GMF7-). h.B2m;fc) [4^g^w;fc), E.190) Transformation of this result using CB.21) [which is explicitly written out in E.113) for the IJ2 = 22 case] gives the h™FB2l;k)s. Either of the methods just described for solving the OZ equation thus yields a closed form for the k-space harmonic coefficients h™F{22l; fc). Hankel transfor- transformation77^778 of these coefficients using CB.44) yields the desired r-space coefficients h%iFB2l; r). The reference fluid go(r) values for a LJ fluid, needed in E.188), are known over a wide range of density and tempera- temperature from computer simulation studies.7711 For small Q (Q*<i where Q* = Q/(ecr5)J) the results41 of the MF
382 INTEGRAL EQUATION METHODS 5.4 theory are found to be only marginally better than the results of first- order perturbation theory (§4.5), based on ha = -pgo(r)Ma(raj1&J). For larger Q values the theory fails completely, in that the harmonic coeffi- coefficients haAFB2l; r) do not vanish in the core region. This will lead to unphysical values of the thermodynamic properties. Although of no intrinsic interest then, the MF theory is of value in being a direct antecedent of a useful generalization, the GMF (generalized MF) approx- approximation. The closure E.185) is replaced by E.191) Ca = -j3go"a + ho(ha-ea), (GMF). E.191) This result is derived in § 5.4.8, by summing the perturbation series E.51) for ca, ca = Ci + c2 + ..., selectively to infinite order. Two properties of E.191), which are in fact built into the derivation, are evident. First, for small r where ho = — 1, E.191) reduces to ha = 0, so that g = go+ga(=ha) vanishes in the core region. Second, because the anisotropic quantities (ua, ca, Jia) occur linearly in E.191), only the IJ2 harmonics which occur in ua will arise in ca and ha. Thus, for QQ interaction, only i1J2 = 22 occurs in ua so that in GMF ca and ha will contain only IJ2 = 22 harmonics. In more detail, ua contains only ^y = 224; the coupling in the OZ equation C.158) [for ca, ha; see also E.38)] results in the three specific harmonics I J2l = 224, 222, 220 for ca and ha. Expressed equivalently, the lxl2m =222, 221, 220 harmonics of ca and ha arise in C.160), just as in the MF E.189). Since E.189) is uncoupled for the different ms, the solution is again E.190). In terms of harmonic coefficients, then, the GMF theory consists771 of the OZ E.190) in terms of the /B2m; fc)s, together with the closure E.191) which can be written in terms of the /B2Z; r)s, caB2f ;/•) = -(igo(r)uaB2l; r) + ho(r)[haB2l; r) - ca{22l; r)] E.192) where uaB2l; r)^0 only for 1 = 4 (see E.111)). This relation can be solved explicitly for caB2i; r) in terms of fiaB2Z; r) so that E.191) and E.192) become a harmonic version of the standard forms E.2) and E.3). The solution is by iteration (cf. ref. I9f, or ref. 41, for a rapid method). The results for the harmonics haB2l; r) are shown in Figs 5.8-5.10 for the LJ + QQ interaction, at density p* = per3 = 0.75, temperature T* = fcT/e = 1.07 and quadrupole strength Q*2 = Q2/ecr5 = 0.5. Also shown in the figures are Monte Carlo values and, for haB24; r), the first-order u-expansion perturbation theory (FT) result (cf. Chapter 4, eqn D.37)). GMF represents a significant improvement over first-order PT for haB24; r); for the other two harmonics, GMF is seen to be in qualitative agreement with the data (first-order FT gives zero for these harmonics). Comparisons with second -order PT have been made for Q*2 = 0.5 (see ,
5.4 ANGULAR PAIR CORRELATION FUNCTION 383 Or -4 -12 -16 -20 — GMF ... PT • Monte Carlo 1.0 1.2 1.4 1.6 (rlo) Fig. 5.8. The harmonic coefficient fia B24; r) for a quadrupolar LJ fluid with p* = 0.75, T*= 1.07, Q*2 = 0.5. The values shown correspond to Monte Carlo, generalized mean field (GMF), and first-order perturbation theory (PT). (From ref. 41.) 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 Fig. 5.9. The harmonic coefficient Ha B22; r) for a quadrupolar LJ fluid. (From ref. 41.)
384 INTEGRAL EQUATION METHODS 5.4 -2 C2 -4 -6 — GMF • Monte Carlo 1.0 1.2 1.4 1.6 1.8 2.0 2.2 (r/a) Fig. 5.10. The harmonic coefficient ha B20; r) for a quadrupolar LJ fluid. (From ref. 41.) Figs 4.6, 4.7, and ref. 77j). Somewhat surprisingly, second-order FT does better than GMF for gB24; r) and gB20; r); it does less well for gB22; r). Tests at larger Q* and other state conditions are obviously needed. Similar results have been found for dipolar hard spheres.77k Comparisons of a modified form of the /-expansion PT (PTF) with a modified GMF approximation have also been carried out,47c as well as comparisons47b of PTF with LHNCF (see below). Similar calculations have been carried out for dipolar64 and quadrupo- quadrupolar62 hard spheres, with similar results. An example for the latter case was shown in Fig. 5.5. The GMF = SSC = LHNC theory can be applied to non-multipole potentials as discussed in § 5.4.9. When applied47*5 to the short-range anisotropic overlap S-model B.248), the theory does poorly, as antici-\ pated for HNC-type theories. Also, convergence of the iteration method of solution is very slow for these short-range potentials, whereas it is fast for the long-range multipole potentials. A modified version LHNCF = SSCF (i.e. SSC based on an /-expansion -see §5.4.9) is much more accurate,4711 and is only slightly less accurate than the PY theory (§ 5.4.1). As mentioned earlier, and as will be discussed in § 5.4.9, the GMF approximation is equivalent to the LHNC approximation E.210), ob- obtained by linearizing the HNC closure E.7). A natural extension of the GMF = LHNC approximation is the QHNC approximation E.211), in which one keeps the linear and quadratic terms from HNC. As seen from the example in Fig. 5.5, QHNC represents a further improvement. Another feature771 of QHNC is that g(r) = <g(raIaJ)><Ol^^ go(r), i.e. the difference between the centres and reference correlation functions is accounted for (which is not so in LHNC = GMF). The quadratic terms
5.4 ANGULAR PAIR CORRELATION FUNCTION 385 2.5 fe 1.5 05 i - V 6 4 2 IT , I , , l.i / , i Hard sphere QHNC Monte Carlo 1 1.0 2.0 2.5 (rlo) Fig. 5.11. The centres pair correlation function g(r) = (g(rtu,tu2))o)|<(>2 for a quadrupolar hard sphere fluid at density p* = 0.834 and Q2 = 1.25. The dashed line gives the reference fluid (hard sphere) value. (From ref. 62.) give rise to coupling between the lj2l = 000 and the lJ2l^000 har- harmonics. One must return to the general (coupled) OZ relations C.117), C.158), C.160); the decoupled E.38), and harmonic versions thereof, cannot be used in QHNC. An example77 of the results for quadrupolar hard spheres is shown in Fig. 5.11, for Q2 = Q2/(kT)a5 = 1.25, p* = 0.834. It is seen that QHNC accounts nicely for the observed shift of g(r) from go(r). QHNC and second-order PT have been compared,77' for the LJ + QQ potential with Q*2 = 0.5, and appear equally accurate on balance, with particular harmonics given better in the one or other theory. GMF thermodynamics is derived in § 5.4.9. In common with other integral equation approximations, the theory is thermodynamically incon- inconsistent (see also Table 5.1). As an example,78 for the LJ + QQ case discussed earlier, we show in Table 5.1 the values of AA (=A — Ao), At/, and Ap compared with values from molecular dynamics (MD) and from the Pade approximant to the free energy perturbation series (Chapter 4). The thermodynamic quantities are determined mainly by hB24; r) (see § 5.4.9), which is given reasonably accurately in GMF (see Fig. 5.8), so Table 5.1 Thermodynamic quantities for a quadrupolar LJ fluid GMFt PADE MD pAAIN PAU/N pAp/p -0.419 (A) -0.676 (A) -0.798A7) -0.670 (A) -1.32 (p) -0.379 -0.692 -0.678 — -0.637 ±0.03 — t Here (A), (U) and (p) indicate that the quantity is determined by the A-, U- or p-route, respectively (see §5.4.9).
386 INTEGRAL EQUATION METHODS 5.4 Table 5.2 Energy -j3AL//iV of a quadrupolar hard sphere fluid, calculated by the energy route E.212), for p* = 0.834 (from ref. 62) Q2 MSA Pade LHNC QHNC MC 0.4 1.0 1.25 1.6666 0.320 1.811 2.703 4.450 0.673 3.378 4.868 7.645 0.697 3.396 4.830 7.464 0.663 3.228 4.629 7.272 0.655 ±0.003 3.182±0.014 4.574 ±0.017 6.987 ±0.027 that we expect the GMF thermodynamic quantities to be given fairly accurately. This is borne out in the table. For the quadrupolar hard sphere fluid (Table 5.2) the LHNC = GMF approximation does about as well as the Pade; QHNC does even better. (The density and temperature derivatives of go(r) needed for the A-route values of Ap and At/ were obtained78 by numerical differentiation of the simulated go(r) values./ Liquid-vapour coexistence curves and critical points obtained from GMF have been obtained (Fig. 4.11). The Kerr constant and depolarized light scattering angular correlation parameter G2 (see E.245a) or Appendix 3E and Chapter 10) is given by G2 = Dir)-iE)-lpJ r2drhB20;r) E.193) and has the value G2 = —0.11 for the quadrupolar LJ fluid in GMF at the state condition given above. As G2 is notoriously difficult to simulate there are no simulation data with which to compare this value. In view of Fig. 5.10, the GMF values is not expected to be accurate. The dielectric constant for dipolar systems is given by E.144), and the results for dipolar spheres were discussed in § 5.4.2; further results are given in Chapter 10. GMF appears to give accurate values for the dielectric constant for not too large dipole strengths. The reason that GMF (and, to a lesser extent, MSA) give good values for e is presumably that the angular correlations between the molecules occurring at longer range783 contribute significantly [through h(\ 10; r)~hA(r) in E.144)] to the effect. The GMF closure E.191) is accurate at longer range (as is MSA). Finally we note that GMF = SSC can also be generalized to include multi-body induction potentials,7779 non-electrolyte and electrolyte mix- mixtures,78*' and non-axial multipole effects.780
5.4 ANGULAR PAIR CORRELATION FUNCTION 387 5.4.8 Derivation of the GMF approximation We saw in the preceding section that the mean field (MF) approximation E.185) failed in that g(r<u1oj2) did not vanish in the core region. We now modify the theory to remedy this defect. We also want to retain the simplicity of MF, in that only79a IJ2 = 22 harmonics of ca and ha are non-vanishing, so that the OZ C.160) automatically reduces to a finite system of equations. We use a graphical argument here; an alternative derivation based on HNC theory is given in the next section. In the standard graphical treatment103183-3940 (see introduction to this chapter) one expands h (and c) in powers of / = exp(— j3m)-1. Corres- Corresponding to the decomposition u = M0+"a of the potential into reference and perturbation terms, we have / = /0 + /a+/0/a E.194) where /0 = exp(-/3M0)-l and /a = exp(-j3ua)-1. If we substitute E.194) into the h expansion just mentioned, a complicated expression results in general.34'44 We shall consider here only a simpler case where (i) /a is linearized to — j3ua (cf. § 5.4.9 where /a is kept), and (ii) the f0 bonds are resummed79b everywhere to ho= go~ 1. Our simplified graphical rules are then taken to be: 1. O, fixed point A or 2), 2. •, integrated point C,4,...) [see eqn E.197) for exact prescrip- prescription], 3. = h0, 4. = "|3go«a where we have included a g0 in for convenience. The MF approxi- approximation E.185) is thus cmf=o_-_o E.195) = -j3go(r12)"a('l2WiOJ). E.196) Iteration of the OZ E.38) then yields the MF value of ha, h^FA2) = o—-o+/*\ +?""¦? + ... o o o 6 E.197) - c^A2) + PJd3crFA3)ca4FC2) +... where A2) = (r12ii>ii<J2) and d3 = dr3 daj3/fl. We wish now to go beyond MF E.197) and make a selection from the other graphs which are allowed in ha, such as
388 INTEGRAL EQUATION METHODS 5.4 Note that two points cannot be connected by more than one line, as in Xi b, b b, (unallowed) since such graphs would introduce (g0J between two points, which is forbidden. Graphs such as \ d'---b (neglected) will be neglected in order not to introduce harmonics additional to the ones occurring in MF, Z,/2 = 22. It is clear that if only one ua line terminates on each fixed point, the graph (G) is proportional to ^2m,(<jJi)^'2m2('t'2)» and therefore satisfies (G(ri2(Oico2))OJl — \G(ri2(O](o2))(U2 = 0. E.198) The conditions necessary (cf. E.36), E.37)) for the decoupling of the OZ into a reference part (involving only h0, c0) and a perturbational part (involving only ha, ca) are therefore satisfied in GMF. As mentioned above, the MF graphs in E.197) [beyond the first, which contains goA2)] do not vanish in the core region (r^>0). We can compensate the second graph in E.197) by adding another graph to it, in such a way that the sum vanishes in the core region; thus d * * r .• i ¦ \ + / \ = go ,' \ o o—o L° oj clearly vanishes for r —* 0. Similarly, we can compensate the other graphs in E.197) by adding corresponding terms with an h0 line across the fixed points. The result is that h™F-c™F is augmented with ho(h™F-c™F). But these latter graphs are irreducible.790 We therefore add them to c™F instead of to h™F-c™F; use of the OZ relation then generates, as always, a series for ha contain- containing the irreducible ca terms plus reducible combinations of ca terms. Thus, our second approximation for ca is cL2) = cr+MhLn-O E.199) E.200) where ca" = c!^F = -j3goua, and h*,1) = h^F. Use of the OZ relation then
5.4 ANGULAR PAIR CORRELATION FUNCTION 389 generates from E.200) , ,21 ha = 0--- a / \ + • : +... O O 6 O o—b 6—A + .... E.201) The graphs of the last line in E.201) do not vanish in the core region, so that we add the set <?>)] E.202) where the last equality follows from inspection of E.201) and E.197). Since the graphs E.202) are irreducible we add them to ca, giving as our next approximation c? = c?+hQ(h™-c?)-K{h^-c?) = caMF+ho(hi2)-c<2)) E.203) where we have used E.199). We can continue this procedure. The general term will be c(rl) = -j3gowa+ K{h^-c^) E.204) where h^ and c^ are related by the OZ equation. Evidently the limit of this iterative process is achieved with functions ca and ha which satisfy the OZ and the closure c. = -pgou.+ ho(ha-cj E.205) which is the GMF approximation. From the above iterative scheme for constructing ca, ca is given by in which in each term an h0 line connects the two fixed points, the ua lines appear only in a single chain,79d and the remaining h0 lines do not cross. Although derived above for multipole-like ua, we show in the next section that the closure E.205) is in fact valid for arbitrary ua. In the most general case (where ua is such that E.206) is not satisfied), the general form E.39) (or E.49)) of the OZ relation must be used. (Alternatively, one can return to the complete OZ; see also next section).
390 INTEGRAL EQUATION METHODS 5.4 5.4.9 Equivalence of the GMF and LHNC approximations15-78 If we put u = uo+Ma, c = co + ca, h = ho+ha, and g = go+ga in the HNC closure E.7), and rearrange, we get c(, + ca = (h0-j3M0-lngo)-i3Ma+^-ln(l + ga/g0). E.207) If the reference fluid is also treated in HNC approximation, then E.207) glVCS co=ho-|3Ko-Ingo, E-208) c, = -/3«a+ha-ln(l + ga/g0). (HNC) E.209) If we now expand the logarithm in E.209), and retain only the linear term, we get ca = -?ua+Mo/go, (LHNC) E.210) where we have used the relation ga = ha. This is the LHNC (linearized HNC) approximation.54 If we retain two terms in the expansion of the logarithm, we get ca = -?Ma+h>o/go + ^a/goJ, (QHNC) E.211) which is the QHNC (quadratic HNC) approximation.62 We can improve E.209) by using the exact reference fluid values hu, c0, instead of the HNC values E.208). If we use these exact values in E.209), we get a 'corrected' HNC (i.e. CHNC) theory,1215 (also called RHNC-renormalized or reference HNC). 'Corrected' PY theories can be derived similarly.79d In order for E.210) to be equivalent to GMF E.205), we first must use the exact h0 and g0 values in E.210) (not the HNC values). Then, if we write E.205) as ca = -j3gnWa+ KK- Kca, put ho= g0— 1 in the last term, and divide by g0, we obtain E,210). Thus GMF is really equivalent to a CLHNC (corrected LHNC) theory. We shall, however, continue to use the briefer designation LHNC. Earlier we derived the GMF closure E.205) only for multipole-like was and spherical reference potentials uu, whereas LHNC E.210) is valid without these restrictions. The derivation of GMF can be extended78 to remove both these restrictions; we use the form E.49) of the OZ relation, and graph theory similar to that used above. This derivation shows that the graphs in caA2) now include the following types in addition to those in E.206): Hence LHNC and GMF/SSC can be regarded as equivalent in general.
5.4 ANGULAR PAIR CORRELATION FUNCTION 391 We should note two small provisos for the equivalence of GMF/SSC and LHNC. First, in deriving E.210) we have divided by g0, thus tacitly assuming go^0. Thus E.210) is not valid in the core region of the potential where go = 0. Thus, in LHNC for hard sphere u0 for example, one would use E.210) outside the core, and augment this with the condition ga = 0 inside the core. This is equivalent to using GMF E.205) everywhere; the validity of E.205) in the core region was in fact built into the derivation. Second, we could set gA2) = gc(rj2) + gncA2) in the HNC closure and linearize in gncA2). Here gc(r12) is the central (or centres) and gncA2) the non-central (or anisotropic) part of gA2). In terms of spheri- spherical harmonics, they comprise the lxl2l = 000 and IJ^l^ 000 terms, respec- respectively. This version of LHNC would thus in general differ from one based on gA2) = go(r12) + gaA2), where g0 is the reference fluid g and ga the correction. For multipole-like ua, however, we saw that in LHNC (gaXu,^ = 0, so that in this case gc = g0 and gnc = ga. We also remind the reader that in GMF for non-multipole potentials we cannot use the simple decoupled form E.38) of OZ equation for ca and /ia; we must use one of the more complicated forms E.39) or E.49), or the complete OZ equation E.40). Equivalently, the (j^I = 000 and ^y^OOO harmonics are coupled in general. In most theories (PY, QHNC, LHNCF) this is the case. Only for MSA and GMF with multipolar forces do we get a decoupling. There are still other ways47bc to linearize the HNC equation, as we now discuss. 5.4.9a Beyond GMF (=SSC = LHNCL7b We recall the identity E.194), which we rewrite as l+/=(l+/0)(l+/a) E.211a) which is obviously equivalent to e = eoea, with e = exp(—|3h), e0 = exp(—(lu0), ea = exp(— (luj, where M = uo + ua and e = l+/, eo=l+/o, We keep the reference (u0) and perturbation (ua) parts of u arbitrary for the time being. If we replace (—fiu) in the HNC closure E.7) by ln(l+/), and use E.211a), c = co + ca, and g = go+ga, we have -)¦ ?o/ f— . E.211b) go/ If we assume the reference parts c0, f0, hQ obey the HNC relation for the reference fluid and subtract this from E.211b) we have go
392 INTEGRAL EQUATION METHODS 5.4 Linearizing in /a and hjho we get c* = L+ha-hJg0 E.211c) or, rearranging, c. = /.go+ho(h.-c.). (LHNCF-SSCF) E.211d) As in LHNC above, we replace the HNC values g0, h0 in E.21 Id) by the exact values. Equation E.21 Id) is then really a 'corrected' SSCF approxi- approximation. Comparing E.21 Id) with E.205), we see that LHNC is obtained from LHNCF by linearizing /a = exp(— (iua) — 1 to (—0ua). At short range, where ua is not necessarily small compared to u0, it can happen47b that /a is small compared to /0, so that LHNCF may be more reasonable in some cases than LHNC. Various choices for «„ and it,, and the corresponding gn and /io=go— 1, are possible. Consider for concreteness the total potential wA2) = Mu(ri2) + MooA2). Choosing u0 = uu and Ma=MoQ gives a version of LHNCF analogous to u-expansion PT (§ 4.5). On the other hand, choos- choosing «0(r,2) defined by exp(-«,,(r12)) =(exp(-j3MA2)>A)](O2 and ua=u-u0 gives a version of LHNCF analogous to /-expansion perturbation theory (i.e. PTF). [Advantages of PTF are discussed generally in § 4.6. Here we simply recall that defining u0 by exp(-/3M0) = (exp(—0u))<U|a>2 builds some of the anisotropy of uA2) into uo(rl2), and that the isotropic part of /A2) is the reference value fo(r12) (equivalently /a is purely anisotropic, </aA2))o)]O>2 = 0), so that solution of the integral equation by spherical harmonic analysis is simplified a bit; we have /a@00; r) = 0.] Tests47b of the latter version for the potential uA2)= uLJ(r,2) + usA2), where usA2) is the short-range anisotropic overlap model B.247), shows that LHNCF is superior to LHNC for anisotropic short-range perturbations. As men- mentioned, this is expected since /a remains relatively small even when wa is large. The PY approximation (§ 5.4.1), however, seems to be even better than LHNCF for these short-range Mas. We note that to derive E.2lid) graphically in the spirit of § 5.4.8, and which we would then denote as GMFF (=SSCF), we merely replace (-(iuj everywhere by /a; i.e. we let now denote /ag0 instead of (-j3«a)g0 as in rule 4, p. 387. We have already noted in the preceeding section the extension of LHNC to (truncated - see below) QHNC; applications to the calculation of the dielectric constant are discussed in Chapter 10. Solution of the full HNC E.7) seems prohibitive at this time, even by spherical harmonic methods, since one would need to do spherical harmonic analysis inte- integrals of In g at each iteration (in contrast to PY E.6), for example, where harmonic analysis of A —expOu)) is done once at the beginning). An
5.4 ANGULAR PAIR CORRELATION FUNCTION 393 approximate solution of a truncated CHNC (see below E.211)) approxi- approximation (in which harmonics up to / = 4 only are kept) has, however, been carried out79' for fused hard spheres for various values of bond length ( and equivalent diameter d (see E.245)). The harmonic coefficients g(/i/2w;r) are in qualitative agreement with simulation, and the ther- modynamic properties agree within a few per cent if /* = I/a is not too large (I*<0.6). Just as for MSA and LHNC/SSC/GMF, the A-integration in the A-route E.117) can be carried out analytically in HNC (see Appendix 5B); it can be carried out approximately79'' in CHNC. 5.4.10 Thermodynamics of the GMF approximation7879 From the GMF pair correlation function g(i(Oi(o2) we can evaluate the thermodynamic quantities using the relations of Chapter 6. There are four routes linking g(rcolco2) to the thermodynamic quantities (the A-, U-, p-, ^-routes), and, as with MSA, the theory is found to be thermodynam- ically inconsistent. For concreteness we again take wa(raIaJ) to be a quadrupole- quadrupole (QQ) interaction. Then, in terms of the harmonic coefficients of ua and h (see § 6.6) we have for the incremental energy AU = U- Uo (since (hX,^ = 0 in GMF) PAU/N = ^PDtt)~3 (drhB24; r)uB24; r), (G-route) 1 E.212) with uB24; r)= uaB24; r) given by E.111). We can also calculate At/ from the A-route, /3AL//N = ?—OAA/N). E.213) dp We show later that this gives dgoto PAUIN=(I3AUIN)U -|Dt7)Pp ? B1 +1) fdr- x[hB2l;r)-cB2l;r)-l3uB2l;r)Y, (A-route) E.214) where (pAU/N)^, denotes the value E.212) obtained by the L/-route. We can calculate the pressure from the p-route (virial relation) [see § 6.2 or Appendix E], and since for the QQ interaction (which varies as r~5) we have rwa = —5wa, we find (since in GMF (hj^^ = (ga)OJlQ,2 = 0) (p-route) E.215) Calculated by the A-route, -^03AA/N), E.216) dp
394 INTEGRAL EQUATION METHODS 5.4 we show later that Ap is given by PAp/p = OALT/AOu - OAA/N)-iDir)-3p2 I B1 + 1) fdr ^~ x[hB2/;r)-cB2/;r)-?uB2/;r)]2, (A-route) E.217) where AA is given by E.228). From the compressibility route we get for Ap ?Ap/p = 0, (x-route) E.218) since g(r) = go(r) for all values of p in GMF theory (see E.137) and . discussion following). We could also obtain Ap from the t/-route using E.213) and E.216). To calculate the free energy increment AA = A — Ao, we start with E.117). As we now show, it is possible to do the A-integration analyti- analytically. Furthermore,790 the result can be analytically differentiated using E.213) and E.216), leading to E.214) and E.217), which contain only reference fluid go(r) derivatives. Thus we do not have to solve the GMF equations over a range of state conditions to obtain Ap and At/ at a given state condition. In E.117) hk can be replaced by ha, since h0 is assumed isotropic. We now manipulate the GMF closure Ca = -?go(A-«a) + ho(ha-Ca) E.219) to find a more convenient form for uaha in E.117). We write the last term in E.219) as ho/ta — hoc^, put ho = go— 1 in hoc^, and obtain go(ca + 0Aua)= hoha. E.220) Differentiation with respect to A gives °f ¦ ¦<522I) Cross-multiplying E.220) and E.221), and dividing79f by goho gives ^(^ + e«a)=^(c^Aua). E.222) \9A / dA We rearrange E.222) 2 dA dA and integrate between A = 0 and A = 1 to give f1 t1 dcK j3 dAh>a = i/ta(ca + ?ua)- d\K-f E.223) Jo Jo "A where ha = h^ = 1.
5.4 ANGULAR PAIR CORRELATION FUNCTION 395 We now substitute E.223) into E.117) (with hx replaced by h$) to give = \p\ di(ha(iw1w2)[ca{iw1w2) + ?ua(r«1«2)])<Oia>2 ^ . E.224) The integrals in E.224) can be transformed using the generalized Parse- val theorem (B.88) to a form involving the harmonic coefficients f(lj2m; fc) [=/B2m; fc) here] used in § 5.4.7. The second term in E.224) gives (j3AA/NJ = -2-p [ dAB-rr)-3 [dk ? Dir)~2h*B2m ;fc) — c*B2m; k) Jo J d\ E.225) where we have used the fact that the harmonic coefficients /B2m; k) are real (cf. CB.22a)). In GMF we have (cf. E.190)) E-226) From E.226) we can do the A integration in E.225): E.227) Combining E.227), E.225), and E.224) we get the final result for AA: j3AA/N = —^— X fc2 dfchaB2m; fc)[caB2m; fc) + ?uaB2m; fc)] o(ZiT) m Jo +^T Z } fc2 dfc[D7r)^(-)mpcaB2m; fc) + ln(l-D7r)-1(-)mpcaB2m; fc))]. E.228) To calculate the pressure E.216) and energy E.213), we need the
396 INTEGRAL EQUATION METHODS ^-derivative of E.228), where v = p or v = fi. We get 4tt p „ J 9v L 1-Dtt) (-) pc, E.229) where we have used the abbreviation fa = /aB2m; fc). The last term in E.229) can be transformed using the OZ relation E.190) to so that, with (dldv)(pca) = 8vpca+ p(dcjdv), and E.228), we have f (dAA/N) = -f- I f e dfcff (ha(ca + (iua)) - 2ha ^ flv 8Btt) m J l3i^ 9i^ E.230) We can now transform E.230) back to r-space expressions using the generalized Parseval theorem (B.88), f (fSAA/N) = ^ fdr(^ [ha(ca + ?ua)] - 2ha ^) UeL E.231) where here ha = /ia(ro),oJ), etc. and (. . .)„ =(. . .)<Ol<O2. The GMF closure E.205) can now be manipulated as follows to eliminate the ^-derivative of ha and ca in E.231). With A = 1 in E.220), we differentiate with respect to v giving <3g<), . q \. aCa. d (r, X , dll*,L ag° /com —- (ca + j3«J + g() — + g0 — (j3ua) = h0 — + ha — . E.232) c)v ov dv ov ov
5.5 THE SITE-SITE PAIR CORRELATION FUNCTION 397 Cross-multiplying E.220) [A = 1] with (.5.232) and rearranging gives ¦ a a 9i-> J a dv a 9^ E.233) An inconvenient expression for the first term in E.231) would result from E.233) since hogo would occur in the denominator. We invoke the closure again to re-express the last term in E.233). For A = 1 we substitute go=ho+l in E.220) to get ca+0ua=ho(ha-ca-0ua) E.234) whereas substituting ho = go—1 in E.220) gives ha= go(^a~Ca~ PUa)- E.235) Substituting E.234) and E.235) in E.233), and cancelling hogo, gives dv a a dv dv a a "ftp E.236) When we substitute E.236) in E.231) we obtain the final result l Jdr<haua)<u. E.237) The working expressions E.214) and E.217) are obtained by putting v = f3 and v = p, and transforming E.237) to space-fixed harmonic com- components fB2l; r) using the generalized Parseval theorem (B.84). 5.5 Theories for the site-site pair correlation function The site-site pair correlation functions ga0(ra0) have been introduced in § 3.1.6. Since they depend only on the radial variables rae (between sites), they are naturally simpler than gfj-o^a^). They contain less information than g(ra>iOJ) but sufficient to calculate the compressibility x-> the struc- structure factor S(fc) (Chapter 9), and the dielectric constant and angular correlation parameters G, (Chapter 10) quite generally. For site-site model potentials, other thermodynamic properties (i.e. U, A, but not p) can also be expressed in terms of the ga0s (Chapter 6). The theories for ga0 fall naturally into two categories, based on site-site or particle- particle OZ equations, and are discussed in §§5.5.1 and 5.5.4, respec- respectively. Various closures can be used with either category. By analogy with
398 INTEGRAL EQUATION METHODS 5.5 the atomic liquid case (cf. introduction) one expects PY-type closures to work better for short-range repulsive potentials, and HNC-type closures to be more accurate for the longer-range part of the potential. As we shall see, on the limited available evidence this naive expectation appears to be valid for the short-range structure, but the simple PY and HNC- type closures both have serious defects as regards the long-range (or long-wavelength) structure. 5.5.1 Site-site OZ equation methods (RISM) The 'reference interaction site model', or RISM,79e is a theory designed to calculate the site-site correlation functions ga^ir^) for the case that the intermolecular pair potential u(r<wi«2) is modelled by the site-site form B.5), (see §§2.12 and 2.8). The original intuitive derivation80 of RISM is based on exploring the possibility of decomposing g(r«i«2) a'so in the form B.5), i.e. as a sum of site-site gaB(r)s. Functional expansion81 and graphical82'83 derivations were subsequently given. (Among the graphs summed in RISM are some unallowed ones83" - see below.) In this section we state the RISM approximation, and give some results. A derivation along the original lines is given in § 5.5.2, and in § 5.5.3 a simplified derivation based on atomic (or particle) fluid mixture theory is given for the diatomic case, which sheds further light on the nature of the approximation. The RISM theory consists of (i) an OZ-like relation between the set {fiaB}83c (where hal3(r) = ga0(r)— 1) and corresponding set {ca&} of direct correlation functions, and (ii) a PY-type or other type of closure. The OZ-like relation is a matrix one; in k-space it readst (see E.265)) h = wc(x> + pojch, E.238) or,83f in full, E.239) Here a)aa-(k) is an intramolecular correlation function which, for rigid83* molecules, is given by «w« = ^~ E.240) act' where /„„¦ is the intramolecular site-site distance between sites a and a' in a single molecule. The self-terms (a = a') in E.240) are simply a)m(k)=l. E.241) t We use the notation cscin this section (see §5.5.2).
5.5 THE SITE-SITE PAIR CORRELATION FUNCTION 399 The r-space form of E.240) is (cf. CB.2), CB.3)) ,rl8(r- laa.) E.242) and is proportional to the probability density of finding the a site at position r from the a site. Equation E.238) can be regarded as a definition of the {c^} and is therefore exact. The approximation enters later with the closure. The equation E.238) (or E.239)) can be interpreted physically in the usual way, i.e. by iteration. A typical p2 term found is (note that both self {a = a') and distinct (a^a1) <w terms arise) 2 which we represent pictorially in Fig. 5.12. We show a diatomic fluid for simplicity. The correlation between sites a and (i of molecules 1 and 2 occurs via first an intramolecular correlation from site a of molecule 1 to site a' of molecule 1, then an intermolecular correlation to y' of 3, then an intermolecular correlation to 5' of 4, then an intramolecular correla- correlation to 5" of 4, and finally an intermolecular correlation to site fi of molecule 2. The complete series obtained from E.239) is a sum over all possible such chains of intra- and intermolecular correlations. For the special case of a homonuclear diatomic AA fluid, which has only one distinct li^ and Caa, E.238) can be solved explicitly8311 hAA(k) = -—^ ,..r'"AA —-. E.243) 1 — 2pcAA(fc)U + Waa'WJ where (oAA.(k) = sin(fcJAA.)/fc/AA., with ZAA. the bond length. For /AA. —* 0, E.243) reduces to the atomic case E.12), apart from a factor of 4 in the definition of Caa. For general site-site potentials wa0(r) one begins by trying8081831 standard atomic fluid type closures (e.g. MSA (co0 = -j3ua(i), PY83i' (cafi = /apVcp) or HNC (cf. §5.5.4)). Applications have largely focussed on the fused hard sphere model (wc>e(r)=<» for r<crafi, uafi(r) = 0 for r>o-a(i), 2 Fig. 5.12. Correlation of site a of molecule 1 with site fi of molecule 2, via the sites a', 7', 8' and 8". The solid lines denote intramolecular correlation functions (w), and the dashed lines intermolecular correlation functions (c).
400 INTEGRAL EQUATION METHODS 5.5 using a PY-type closure (E.14), E.15)) ha(s(r) = -l, r<<rafi, E.244a) c«0(r) = O, r>aa0. E.244b) Equation E.244a) is exact, and expresses the obvious fact that gc,0(r) = O for r<<ja&. Equation E.244b) is the approximation. We shall see that RISM [E.238) and E.244)] gives similar results to those of hard sphere PY theory, although quantitatively worse. There are some qualitative differences also (e.g. RISM is not exact8284 at low densities, whereas PY is exact; see also below and § 5.5.3a). Applications have also been carried out for a variety of other potentials, with various closures.83l~" Also, soft core and attractive tails of the potential have been treated by modifying* the hard core RISM results with 'blip-function-type' (§ 4.7) perturbation theory.8290 Attempts have been made to incorporate non-site-site poten- potentials.88 The extensions to mixtures,83" fluid/solid interfaces83" (Chapter 8), and non-rigid molecules (see ref. 172 of Chapter 1) have also been given. We shall restrict our quantitative discussion mainly to the case of hard sphere sites with PY closure E.244). To solve the RISM, one seeks functions ca0 (r) for r < cra& such that when substituted into the OZ relation they yield the result haP(r) = —l for r<o-a0. Having found the ca0(r)s, one then finds the hal3(r)s from the OZ relation. In practice a variational method,84'85 Gillan's method,86cl9c and the factorization method (Appendix 5A.)83p-86-86a give rapid solutions for the hard sphere site potentials. For more general potentials, the latter two methods and iteration methods (both ordinary83 and 'chain recur- recurrence'8611 types) have all been employed, with Gillan's method perhaps the most rapid. In Fig. 5.13 we show the results89 of the RISM approximation for a homonuclear AA diatomic fluid, at density pd3 = 0.9 and l/cr = 0.6. Here d is the diameter of the sphere having the same volume as that of the molecule, d[ll(/)i(//K] (l) E.245) Also shown is the result for gAA when an 'auxiliary site' is placed at the centre of the molecule. This site has a diameter small enough so that it is enveloped by the two fused A spheres, and therefore contributes nothing to the intermolecular potential. If RISM were an exact theory, no effect would result in gAA due to the auxiliary site.89'89c The fact that the effect is observable in Fig. 5.13 shows that RISM is not internally consistent; some measure of the error inherent in RISM is given by the discrepancy between the two RISM curves of Fig. 5.13. (The authors of ref. 89 do not specify the diameter chosen for the auxiliary site; presumably it is chosen to give best agreement with simulation - see ref. 89c.)
5.5 THE SITE-SITE PAIR CORRELATION FUNCTION 401 3.0 1.0 - RISM (no auxiliary site) RISM (auxiliary site at center) o o o Monte Carlo 0.5 1.0 1.5 2.0 {rla) 2.5 3.0 Fig. 5.13. Comparison of the RISM and Monte Carlo site-site pair correlation functions SaaW- (From ref. 89.) The RISM gAAs are in qualitative agreement with the Monte Carlo data. As with hard sphere PY theory, there is a large error at contact. An interesting feature is the ability of RISM to give the cusp at r = cr+l = 1.6a. One intuitively expects91 such a cusp since r = cr-f- I is a characteris- characteristic site-site distance; when site A is at distance o- from site a (i.e. two spheres are at contact), another intermolecular site must be within cr+l from a due to the bonding of a third site to C. In Fig. 5.14 we show the results90 for a LJ site-site model92 of liquid N2. The hard core RISM results of the type shown in Fig. 5.13 have been 'softened' using the 'blip function' method (cf. § 4.7). The agreement with molecular dynamics results92 is again qualitatively correct. Similar re- results83' are obtained by solving RISM for the complete LJ potential. Results have also been obtained for heteronuclear diatomic89 and polyatomic89a-b86c molecules (see also Chapter 9). In Fig. 5.15 are com- compared the RISM and Mont© Carlo results for the triatomic BAB liquid (with parameters chosen to model CS2). Again the gross qualitative features are given correctly by RISM. However, we note that the RISM value of gBB at contact is 20 per cent too high (it was too low in the diatomic case of Fig. 5.13) and that RISM fails to predict the small secondary peaks on the small r side of the main peaks of gAB and gAA- Qualitative (and quantitative) errors are similarly found for tetrahedral molecules ;R6c RISM with PY closure gives a second peak height lower
402 INTEGRAL EQUATION METHODS Fir,. 5.14. The NN pair correlation function for the diatomic LJ model determined d> RISM and by molecular dynamics (MD) at pa-3 = 0.696 and fcT/s = 1.83. (From ref. 90.) than the first, opposite to the behaviour observed in simulation (Fi<_ 5.15A). Short-range structural results from RISM will be discussed further lr Chapter 9 on the structure factor S(k). We simply note here that it i: difficult to draw conclusions about the validity of a theory by comparisons with experimental structure factors, since S(k) is fairly insensitive to tht theory used, and also intermolecular potential parameters are often fittec The pressure p/pkT can be obtained from the compressibility (x) routt (§6.7.1), or by the free-energy (A) route (§6.7.4). The results for tht AA diatomic fused hard sphere fluid at I* = I/a = 0.3 and 0.6 are showr in Fig. 5.16. The results are somewhat similar to the hard sphere PY result: (Fig. 5.2), but with a larger discrepancy with simulation, and between tht X- and A-route values here than occur between the x~ and p-routt values there.93 Also there is a qualitative feature13 which is incorrect. "Wt see that for a given molecular volume (~d3), the x-route value of r increases with increasing I/a (as one expects intuitively, and from viria coefficients,13 scaled particle theory,13 and simulation data13-9187 (Fij. 5.16)-see also §6.13), but the A-route value of p decreases witr increasing l/cr. In ref. 87 the RISM equation of state is compared with t number of other, superior, 'hard-object' equations of state. The latter art discussed in §6.13.2. Since RISM is primarily a qualitative theory o~ short-range structure, it is not too surprising that the thermodynamk properties are not given accurately. For LJ sites, the RISM pressure \i similarly poor, but the internal energy is given accurately.860 This u
5.5 THE SITE-SITE PAIR CORRELATION FUNCTION 403 RISM Monte Carlo (rlan) Fig 5.15. The site-site correlation functions g^M for the triatomic BAB fused hard sphere model, according to RISM and Monte Carlo simulation, for l/crH = 0.897, p* = F/7r)pD111 = 0.897 (where u,n is the molecular volume), and orA/o-B = 0.857. D indicates a discontinuity and C a cusp. (From ref. 89a.) similar to PY for LJ atomic liquids, where the energy route gives the best thermodynamics.l3 The most serious deficiencies of RISM arise when we calculate the angular correlation and dielectric properties. This is discussed in detail in Chapter 10 and we give only a brief summary here.
404 INTEGRAL EQUATION METHODS RISM MD simulation ~~= 0.90 - Fk;. 5.ISA. Site-site pair correlation function gB8(r) for tetrahedral B4 (four-site) fluic with U sites, at pa^ = 0.2453, kT/eBB = 2.54 and IbbAtbb = 0.85. (From ref. 86c.) We consider linear molecules for simplicity. The angular correlatior parameters Gi are defined by G, = N<P,(cos 7J)> = PJdr12<gA2)P,(cos y12)L1&J E.245a where yl2 is the angle between the symmetry axes of molecules 1 and 1 16 - 8 - 4 - RISM (/*-0.3) RISM (/*=0.6) Monte Carlo (/*-0.3) Monte Carlo (/*-0.6) F"i(i. 5.16. The equation of state for the fused hard sphere fluid for I* = 0.3 and l* = 0.6. The RISM values are obtained from the compressibility (x) and free-energy (A) routes. (From refs. 84, 87 (Tildesley and Streett).)
5.5 THE SITE-SITE PAIR CORRELATION FUNCTION 405 G, « (cos -y12) is important for the dielectric constant (see also E.142) and ref. 71e) and G2 a C cos2-y12— 1) arises in the theory of the Kerr constant (Chapter 10) and depolarized light scattering (Chapter 11). The G|S can be expressed in terms of the hal3s or ca0s in either r- or k-space (see refs. 93a, 93b, Appendix 10E). For example, Gt is related to the second moment of the ha0(r)s, or equivalently to the k-space second-power expansion coefficients (d2/ia3(fc)/dk2)|C=0. When G; is evaluated using the RISM approximation for the ha0s, one obtains zero,93c instead of a correct non-zero value arising from angular correlations between molecules. This is true for the usual fused hard sphere PY-type RISM, and more generally for any closure and intermolecular potential which has the asymptotic (large r) property93d caC(r)-^ — Pua0(r) built in, e.g. PY, MSA, and HNC closures. Similarly, for charged sites (together with any shorter-range site in- interactions), when we express the dielectric constant e in terms of the hafss [see § 10.1.4] and evaluate e using RISM hafis we find93e the trivial ideal gas-type result e = l+3y, where y is defined in E.68). This holds for molecules of arbitrary symmetry, arbitrary site-site forces and arbitrary closures which have the asymptotic behaviour cal3(r)-^ -pn,B(r). The RISM theory is thus inadequate for the calculation of these long- range93' structural properties. Suggestions for modifying the theory are discussed in § 5.5.3a. Useful theories for calculating e and G, are dis- discussed briefly in § 5.4.2 and in detail in Chapter 10. 5.5.2 Derivation of the site-site OZ equation We wish to derive the RISM site-site OZ relation E.238). We follow here more or less the original argument;80 in the next section a simplified derivation is given for the case of an AB diatomic liquid. The site-site correlation function hctB(raB) = ^(r^) —1 is defined in r-space by C.126), and is given alternatively in k-space as C.133), hal3(k) = (Ji(kWla>2) e~ik-r- e'k-r«»><1>lW2 E.246) where rcla is the position of the site a with respect to the centre of molecule 1. We define cal3(ral3) in terms of c(raI<w2) analogously to C.126) Ccptae) = (c(rci« +ra0 -rc20, w,^)),^ E.247) or, equivalently, in k-space as ca0(k) = (c(kWlw2)e-ik-'- eik*->MlM2. E.248) We now explore the possibility of decomposing c(r<W!<w2) into a sum of
406 INTEGRAL EQUATION METHODS 5.5 site-site terms ca0(rafj), c(ra>lM2) = Y,cal3(rali), E.249) where we expect cal3^cal3 in general. The expression E.249) seems plausible at first sight. For example, if the potential M(ra),aJ) is of short-range site-site form B.5) [e.g. fused hard spheres], then E.249) satisfies the approximate PY relation943 c(raIoJ) = 0 (no overlap) E.250) if the caB(ra0)s satisfy a PY relation caB(raB) = 0 for rae>acc&. The relation E.249) can also be written in k-space form; Fourier transforming E.249) according to CB.3) we have = X [dreik-rc«B(r,,B). E.251) c<0 J The left-hand side of E.251) is c(k<w1oJ). The integral on the right-hand side is for fixed a^ and co2, i.e. fixed icla and rC20- Transforming the integration variable to taB using (cf. Fig. 3.8) ra0 = r + rC2B ~'ci<*, we get dr = dra0 so that c(koIoJ) = ? caP(k) eik-r- e-k-'-» E.252) where cal3(k) is the Fourier transform of ca0(ra0). We now work out the relation between the two types of site-site cs, ca0 and 4,0. Substituting E.252) in E.248) we get ca0(k)= Z 4-e-(k)<e'k-(r--r-)L1<e-|k-('--'^)>M2. E.253) a'0' We define the real and symmetric matrix waa(k) by «««(fe) = <eiM'--r-))lUl E.254) which for rigid molecules is easily evaluated9415 as sin kla.a o>a-a(k)=— E.255) where /„.„ =|rcla--rcla| is the intramolecular aa' separation. Thus, from E.253) and E.254) we have or, in matrix notation E.256)
5.5 THE SITE-SITE PAIR CORRELATION FUNCTION 407 Next we show from the OZ relation that if c(r<w1<w2) possesses the site-site decomposition E.249), so does h(ioIoJ), i-e. fi(t«#2) = Z^(^) E-257) or equivalents (cf. E.249) and E.252)) = ? hae(k) dk-r-e-'k-*-». E.258) We substitute E.252) into the iterated form of the molecular OZ equa- equation C.118), to get = I caP(k) e'k-r- e-ik-r-3 a/3 + P Z <c^(fc)eik-'-e yp(fc) eik*~. e^^L, + • • • • E.259) Carrying out the averaging over <w3 in E.259) using E.254) gives E.26 We see that indeed ^(kwiwj has a site-site decomposition E.258), with or, in matrix notation h = c + pcwc + ... E.261) or h=c+pcwh E.262) where iteration of E.262) generates E.261). Corresponding to E.256), we find the relation between the two types of site-site hs, h=wha). E.263) We now have two sets of site-site cs, and two sets of site-site /is. The relation E.262) is an OZ-like relation between the pair (h, c). We can derive three more such relations [between (h, c), (h, c), and (h, c)] by eliminating from E.262) c, h, or both. Thus multiplying E.262) from the
408 INTEGRAL EQUATION METHODS 5.5 left and from the right with <w, and using E.263) and E.256), we get94c h E.264) where co1 is the inverse matrix to <w. If we eliminate c from E.264) using E.256) we get h + h (RISM) E.265) where we use the designation (RISM) to indicate that this is the RISM OZ relation actually used in practice (see remarks below). Finally, eliminating h from E.264) using E.263) gives, for completeness /^w-W+poT'c/t. E.266) The quantities of experimental interest are the haBs, so that we focus attention on E.264) and E.265). As mentioned above (cf. E.250) and discussion) the ca0s have desirable physical properties, so that E.265) appears the best candidate for a site-site OZ relation. Further arguments for choosing E.265) are given in the next section. There is, however, a fundamental difficulty. The relation E.257) (and therefore E.249) on which it is based) is in fact invalid, since it violates the fundamental physical requirement94*1 (see Fig. 5.17) h(rw1w2) = -l, overlap. E.267) We can now adopt one of two points of view. In the first, we regard E.265) as an approximation, based on the inexact relations E.257) and E.249). In the second viewpoint, we divorce ourselves from the invalid relations E.257) and E.249), and define cafi by E.265). The relation E.265) is then exact. Whether this procedure is useful then depends on whether we can invent approximate closures which generate accurate ha0s from E.265). Finally we note that E.265) can be solved formally, E.268) or, since a>c commutes with a function of <wc, h = o)c(l-po}c)-lo), E.269) which should be compared to the atomic liquid case E.12). The form of Fig. 5.17. Two overlapping fused hard spheres. Of the four site-site terms (ac, ad, be, bd) in E.257), the last three give rise to a violation of E.267).
5.5 THE SITE-SITE PAIR CORRELATION FUNCTION 409 E.265) corresponding to E.13) or E.44) is / E.270) where I is the unit matrix. 5.5.3 Simplified derivation of the site-site OZ equation79*1'831 To shed further light on E.265) we give a simplified derivation from atomic mixture theory for the special case of a diatomic AB liquid.946 As a by-product the intermediate results are useful in § 5.5.4. We consider a 50-50 mixture of A atoms and B atoms (pA = pB). We shall eventually fuse together each A with one B, so that we end up with a diatomic AB liquid of number density p = pA = pB. The mixture OZ equation in k-space is95-95a Hae(k) = CaB(k) + pY,Cay(k)HyB(k) E.271) y where p = pa = Pb- Here a and (i label the species (A or B). We use capital letters since later h and c will be needed for the site-site functions. The {H^} are explicitly {Haa, Hbb, Hab}. In matrix notation E.271) is H=C + pCH E.272) which can be rewritten as (cf. E.13) or E.44)) (I-pC)(I + Pm = I E.273) where I is the unit matrix. After fusing together the As with the Bs, we shall now show that the following relations are valid for the two factors in E.273): I + pH = o) + ph, E.274) I-pC = w~l-pc E.275) where <oaa'(k) is the intramolecular correlation function E.240). Once E.274) and E.275) are established, we have then proved E.270), and therefore the desired relation E.265). The validity of E.274) is clear from the following physical argument. The total pair correlation function of the mixture, consisting of a self- term [a delta-function in r-space, or the 'constant' I in k-space] plus a distinct interatomic term pH, is decomposed into an intra molecular part a> plus an intermolecular part ph, thereby defining h. (On the right-hand side of E.274) the self-term is included in <w (see E.241).) The relation E.275) has a similar interpretation: I — pC consists of an intramolecular plus an intermolecular part. To show that these parts
410 INTEGRAL EQUATION METHODS 5.5 correspond to the two terms on the right-hand side of E.275), we let J-pc = cin(ra+cinler ^QMra-pc, E.276) thereby defining c. To find Cintra in E.276) we substitute E.276) and E.274) in E.273), (C,ntTa-pc)(w + ph) = I. E.277) Since95b this must be true for all values of the density p, we take the case p —¦> 0. This gives Cintra = aT1, thereby establishing E.275), and hence the relation to be proved. 5.5.3a Beyond RISM Despite some qualitative successes, some failures of RISM have been noted, so that amendments are required. The three deficiencies which are most helpful in pointing the way to improved theories are: (i) the trivial values predicted for e and Gh (ii) the dependence of physical properties (e.g. the 'real' site g^s) on 'auxiliary' (non-interacting) sites, and (iii) the theory is graphically unsound;95b the RISM OZ equation will generate unallowed graphs in h if c is assumed to consist only of the subset of irreducible h graphs. Defect (i) can be traced to the property of standard RISM closures that asymptotically the site-site c obeys caB(r)-^> — /3ua0(r) for r—»°°. This appears reasonable at first sight by analogy with the formally exact result C.119) for the molecular cA2). The trivial results for e and Gu which depend on moments of the ha0{r), or k —»0 behaviour of haa{k), suggest that ca0{r) does not in fact approach ~j3uctB(r) asymptotically. We can, of », course, work backwards7911 (see Chapter 10), and obtain the asymptotic behaviour of cati(r) by demanding that RISM yield correct Gt and e values. Alternatively, we can critically re-examine the derivation of the preceding sections to see more directly where the flaw in RISM lies. For cxample,79h in defining C,,ncr^-pc by E.276), with C^^^oj'1, there is no guarantee that Cinter or c is, in fact, a normal type of direct correlation function, with the usual properties of being short-range (see Fig. 3.7) and asymptotically equal to the potential. In fact, if we solve the RISM OZ equation E.270) for the defined c(fc), pc(fc) = co(fc)^1-[a;(k) + ph(fc)r1, E.277a) we can evaluate the k -»0 behaviour of cae(k) from the known95c behaviours of coa0(k) and hae(k). Due to the inverse matrices in E.277a) it is plausible that c{k) will contain singular [O(&T")] fc-space behaviour, since <o(k) and h(k) are regular95d (at least up to k2) at small fc, and
5.5 THE SITE-SITE PAIR CORRELATION FUNCTION 411 therefore long-range behaviour in r-space. As examples ,79h>95d for non- polar (uncharged sites) AB diatomics we find for small k, coie(k)«- [G1;/A + Gi)]fc~2; for polar (charged sites) diatomics Gt is replaced by a function of e. This corresponds to the large-r behaviour ca0 (r)« [G1/A + G1)]r~l. For polar and non-polar ACB linear triatomics, the corresponding asymptotic results are caS(fc) <x[G2/(l + G2)]fc~4 and coib(>'HC[G2/A + G2)>. Thus, we see that cae(k) is singular956 at k=0, (except for the AA diatomic case, where Gx = 0) and this corresponds to long-range r-space behaviour, even for short-range (e.g. hard-sphere) ua&(r)- It is therefore not surprising that short-range-type closure approx- approximations (e.g. E.244b)) will lead to inconsistent and erroneous results for the long-wavelength (k —> 0) properties. Defect (ii), the dependence on auxiliary sites, is also manifest in the above examples. If we take site C in the ACB molecule to be auxiliary, and the AB distance the same for both, then the gAB, g^, and gBB should be the same in the two cases. They are, in fact, quite different asymptotically. This difference is particularly great in the symmetric cases of AA and ACA. Here cAA(r) is short-ranged (since G1 = 0) for the AA fluid, but varies as r at large r for the ACA fluid. These considerations also demonstrate that the RISM ca0(r), unlike ha0(r), is not an intrinsi- intrinsically intermolecular quantity. Two possible ways of remedying the situation suggest themselves: (i) modify the usual RISM closures, and (ii) modify the usual RISM OZ equation. The simplest way95' to accomplish (i) is to add terms to ca0(r) such that the rigorous asymptotic result for cat3(r) is satisfied. Another method is to invoke closures in terms of the modified direct correlation function c' = u>cu>, where c is the RISM direct correlation function (cf. E.256) and note the notational change c-^c, c—>c'). However, in the applications9569511 to date where PY-, HNC-, and MSA-type closures were applied to c', results worse than RISM were obtained for the hap(r)s, even though a priori there appeared to be reasons8194 for expecting better results! More work along these lines might be worth- worthwhile. In inventing new closures generally, it may be necessary to treat the real and auxiliary sites on different footings. In the second basic approach to improving RISM, modifying the RISM OZ equation, various possibilities suggest themselves. For example, one might try one of the three other site-site OZ equations derived in § 5.5.2. For example, using the modified OZ E.264) and a PY-type closure on c' (denoted as c in E.264)) is equivalent to the procedure of using the old (RISM) OZ equation and the PY closure for c' discussed in the preceding paragraph. The latter was unsuccessful, but other possibilities can be explored. Another proposal94' is to derive a site-site /-expansion refer- reference potential (§4.6) u°B(r) from the full potential uA2), and use this in
412 INTEGRAL EQUATION METHODS 5.5 standard atomic liquid integral equations and closures, thereby yielding directly a site-site g(r). This avoids using u> (and the RISM OZ equation). The above methods are all admittedly somewhat ad hoc. A more sys- systematic approach, based on defect (iii) above, is to derive a new site-site OZ equation which is graphically sound, i.e. c defined by the OZ equation is obtained by topological reduction, and has the usual property of consisting of the subset of irreducible h-graphs. This has been done83d but the result is more complicated than in the original RISM; one is forced to introduce four new topologically distinct cs which are related to each other and to h through four coupled OZ-like equations. Closures can now be introduced in the usual way of omitting graphs. Numerical results using this approach are beginning to be explored.95' Finally, it is possible to return to the particle-particle approach, as we * discuss in the next section. 5.5.4 Particle-particle OZ equation methods (central force model) For longer-range site-site model potentials, theories of the HNC type for the site-site g«g(r)s are expected to be more accurate than those of the PY type (§5.5.1), at least for the short-range structure. Thus one can use83k the exact site-site OZ relation E.265) together with an HNC-type closure E.7) between ha0(r) and cae(r). For H2O, for example, three-site (H, H, and O) RISM/HNC calculations831* give results which capture some of the qualitative features of gHH> 8oh> ar|d goo seen in simulation. Alternatively, we can return to the particle-particle approach96963 of § 5.5.3; the molecular liquid is envisioned as an atomic liquid mixture, of species A, B, C,.... Then we use the atomic mixture OZ relation E.271) [generalized to a general mixture95'953] for the Hae(k), Ca0(k), l,yyyek) E.278) y together with the closure (cf. E.7)) ff(r)-ln Gae(r) E.279) where Uae(r) is the total particle-particle potential (including the in- infra molecular bonding contribution; the molecules are allowed to be non-rigid and dissociatable, and chemical saturation arises automatically from the force model) and Ga& =Ha0 + l. The Hat3s (and Ca0s) contain both the infra and intermolecular contributions (cf. §5.5.3). Equations E.278) and E.279) have been applied to liquid H2O for a particular atom-atom plus charge-charge model96 for the Uaes (i.e. Uoo, UHH, and UOh)- The results for the three H»3(r)s (i.e. Hoo, Whh, HOH)
.5 THE SITE-SITE PAIR CORRELATION FUNCTION 413 .vj. _,. iu. ¦ aniLit-particle correlation functions Ga^(r) = Ha0(r) + 1 for H2O according to the HNC approximation (at mass density 0.32 g/cm3, T= 1900 K). (From ref. 96.) ;re shown in Fig. 5.18. Equations E.278) and E.279) were solved by teration96b (see the introduction to this chapter). Due to numerical :onvergence difficulties96" the solution was obtained only for low density, ligh temperature state conditions (Fig. 5.18 corresponds to a mass lensity of 0.32 g/cm3, and a temperature T= 1900 K). There are significant intramolecular contributions in Fig. 5.18. Thus, as ¦xpected, GOH(r) is dominated by the intramolecular peak at r = 0.98 A, he OH bond length. The intramolecular peak in GHH(r) occurs at ¦= 1.46 A, which compares with 1.52 A found in the free molecule. This ;hift of about 4 per cent agrees with that found from computer simula- ion.97 The intermolecular peak in GHH(r) at r = 2.32 A is consistent with he usual linear hydrogen bond model of liquid H2O. An interesting feature of the HNC solutions is that they satisfy the lgorous zeroth and second moment conditions98'9811 for charged fluids.
414 INTEGRAL EQUATION METHODS 5.6 For the H2O case we have Po|dr[GOH(r) - G00W] = 1, E.280) (r)-GHH(r)]=l, E-281) Jdrr2[2GOH(r) - Goo(r) - GHH(r)] = CkT/8Trp2oq2), E-282) where po = jpH = p, and q is the charge on the H particles. These conditions express the overall neutrality of the charged-particle system, and the local neutrality around each ion due to screening by ions of opposite sign. The areas under the first peaks in GOH(r) and GHH(r) are also consis- consistent with the facts that in each H2O molecule two Hs surround each O, and another H is near each H. 5.6 Conclusions For general intermolecular potentials u(rw1w2) where one needs the complete angular pair correlation function g(r<o1a>2) the theories MSA—* GMF (=LHNC = SSC)-*QHNC or LHNCF-»HNC or PY represent a natural progression toward increasing complexity and accuracy. The re- results seem potentially more accurate for g(rw1w2) than the various perturbation series of Chapter 4, but in practice98b47b it is still an open question for a general potential whether a particular perturbation theory or integral equation theory is the more accurate for the structure. The elementary perturbation theory (with Pade) has a definite advantage in its simplicity and accuracy for thermodynamic properties. Because of the mathematical complexity of these integral equations, spherical harmonic decomposition is necessary in order to obtain a solution. For a potential ujl^l) containing a single anisotropic harmonic each of these theories yields only a finite number of harmonic coefficients g(/,/2/), either because of approximations used in the closure of the OZ equation or because of truncation (by necessity) of the harmonic series; for example, if ua(/,/2/) = uaB24) then the MSA and GMF theories yield the ga(/i/20 for lj2l =220, 222, and 224 only. In most applications these three harmonics are sufficient; in others (e.g. S(k) calculations) more of the harmonics of the full pair correlation function g(rco,a;2) may be needed. In theories where the harmonic series has been truncated arbit- arbitrarily (by necessity), e.g., PY, HNC, QHNC, LHNCF, the harmonics retained have some error due to truncation. In principle this error can be estimated by retaining a few more harmonics, and observing if the lower-order ones change. Tests of this sort should be carried out.
5A FACTORIZATION METHOD FOR OZ EQUATION 415 For site-site models of the intermolecular potential, the site-site (RISM) and particle-particle (central force model) integral equation theories for the site-site correlation functions g«B(raB) appear to be at least qualitatively accurate for the short-range structure. For the long- wavelength angular correlation structure, further developments of the theories are needed. As pointed out in Chapters 3, 4, 9, and 10, but not discussed explicitly here, the short- and long-range orientational and site-site structures found in many of the cited references depend on both the long-range and short-range forces, in contrast to dense Ar-type atomic fluids where only the centres correlation function g(r) exists and which usually depends mainly on the short-range forces, at least for pure fluids. On the limited available evidence, one can hypothesize that the PY- type theories for gA2) or g«B(r) are generally speaking the most success- successful of present theories in handling the short-range anisotropic potentials, whereas the HNC-type theories are the best for dealing with the longer- range anisotropic potentials, in agreement with the overall picture in atomic fluids.13 Further tests of this hypothesis are certainly required. As yet, quantum effects" have not been taken into account for molecu- molecular liquid gA2)s or g»p(r)s. Appendix 5Af Factorization method of solution of the OZ equation The equations for radiation equilibrium in the stars belong to a type now known by Eberhard Hopf s name and mine. They are closely related to other equations which arise when two different physical regimes are joined across a sharp edge or a boundary .... Norbert Wiener, I am a mathematician A956) The use of the Wiener-Hopf factorization method100 for solving the OZ equation was introduced by Wertheim32'35 via Laplace transformation of the OZ equation. Baxter subsequently introduced two new derivations and reformulations, the first101 also based on Laplace transforms (and which gives a more complicated factorized form), and the second102 based on Fourier transforms. Wertheim113 has also derived the factorized form of the OZ equation using combined Fourier-Laplace methods, and has extended the result in various ways (see below). We shall follow Baxter's second derivation since it is pedagogically simple. Both the Fourier and t In preparing this appendix the authors are indebted to Dr. D. Henderson for the use of his unpublished notes.
416 INTEGRAL EQUATION METHODS 5A Laplace transform versions have been applied widely and extended in various ways. Here we can only note a few highlights,103 develop the basic equations for pure atomic fluids, and then apply the factorization method to the PY problem for hard spheres (§ 5.1), originally solved analytically and in closed form by Wertheim32 and Thiele33 by other methods. The method has been used primarily to obtain analytic104* solutions to the OZ equation, but there have been a few purely numerical410411105 applications. The original pure atomic fluid theory has been extended to atomic fluid mixtures,10*"9 fluids at walls,11011 and to the triplet correla- correlation function.112 For molecular fluids, the factorization method has been applied to the angular pair correlation function, with58 (see E.116)) and without spherical harmonic expansions, both in bulk and at walls, and also to the site-site863-114 and particle-particle79h correlation func- functions, again both in bulk and at walls83p (see Chapter 8). We restrict ourselves here to a discussion of the simplest case, pure atomic fluids; this illustrates the method and is adequate for the purpose at hand, to solve the hard sphere PY problem. We shall see that the factorization method solves analytically the hard sphere PY problem extremely simply. We begin by noting three equivalent forms for the Fourier transform CB.3) of a spherically symmetric function f(r), (e.g. h(r) or c(r)), r/(r) E A.I) dr^f(r) EA.2) = 4tt[ dr cos(fcr)F(r), EA.3) where F(r)= dr'r'f(r'). EA.4) liquation EA.2) is derived by choosing the polar axis along k in EA.I) and then doing the angular integrations, or alternatively by using the Rayleigh expansion (B.89), the fact that JdtorY,m(wr) = 0 for (^0, and the relation /0(x) = sin(x)/x. Equation EA.3) is obtained from EA.2) by integration by parts. The advantage of EA.3) is that it is a simple one-dimensional Fourier transform. The OZ relation E.13) for an atomic fluid can be written as = tl-pc(Jk)]-1. EA.5) It is well known (see Chapter 9 or ref. 5) that l + ph(k) in EA.5) is the
5A FACTORIZATION METHOD FOR OZ EQUATION 417 structure factor S(fc) for the fluid, and is real and positive definite with no singularities.115 Hence 1 — pc(k) must be real and positive definite with no zeros. It is then plausible116 that l-pc(fc) can be written as the absolute square of some function, l-pc(k) = \q(k)\2^q(k)q(kT = q(k)q(-k) EA.6) where q(fc) has no zeros, and corresponds to some real r-space function q(r). The relation EA.6) is referred to as a Wiener-Hopf factorization.100 A rigorous proof that the factorization EA.6) is possible has been given by Baxter,102 who also shows using complex variable analysis that q(k) can be defined in such a way as to have the following properties. A) q(k) — 1 possesses a Fourier transform q(r), defined by q(fc) = l-2npf dreikrq(r), EA.7) Jo where q(r) is a real function, so that q(k)* = q(-k). EA.8) B) For the case that c{r) satisfies117 c(r) = 0, r>R EA.9) q(k) can be analytically continued in such a way that q(r) = 0, r<0, EA.10a) q(r) = 0, r>R. EA.10b) (The upper limit in EA.7) can therefore be taken as R, cf. EA.47).) C) q(k) is analytic and has no zeros in the upper half complex k plane. A simplified proof of properties (l)-C) of q(k) is given in § 5A.2. We wish now to transform EA.6) to r-space. Using EA.3) and EA.8) and the fact that C(r) = 0 for r>R (cf. EA.4) and EA.9)) we rewrite EA.6) in the form •r 1-4ttP\ dr cos fcrC(r) = q(fc)q(-fc). EA.11) We now invert the transforms in EA.11) to relate C(r) and q(r). Substituting EA.7) in EA.11), operating on both sides with J!L dk e~'kr..., using 2 cos kr = eikr + e-'kr, and (see (B.72)) S(r)=-^- [ dke-ikr, EA.12)
418 INTEGRAL EQUATION METHODS 5A and the relations EA. 10), we find \ dr'q(r')q(r'-r) EA.13) for 0<r<R. We note from EA. 13) that q(i?) = 0 (since C(R) = 0), so that from EA. 10b) we see that q{r) is continuous at r = R. The relation EA. 13) is the r-space version of the factorization EA.6). We can also relate q(r) to h(r) through the OZ relation EA.5), which we rewrite using EA.6) as or q(k)[l + ph(k)]-l=q(-k)-1-l. EA.14) The relation EA. 14) can be converted to r-space by operating on both sides with JT^ dk e~ikr.... We evaluate the resulting integral on the right-hand side of EA.14) by contour integration, closing the contour with a large semicircle in the lower half k plane. There is no contribution from this semicircle since (cf. EA.7)) q{—k)—* 1 for Im(fc)—* —<=°, so that the integrand e~lkr[q(—k) — 1] vanishes for Im(fc) —» —co. Since q(—k)~l has no poles in the lower half plane (see property 3 above for q(k)), the integral on the right-hand side of EA.14) therefore vanishes by the pole-residue theorem. On the left-hand side, before integrating over k, we substitute EA.7) for q{k), and EA.3) for h{k), which since H{\r\) is an even function of r we can rewrite as dr eikrH(|r|). EA.15) Integrating now with respect to k and using EA. 10b) and EA. 12) again we find [ dr'q(r')H(\r~r'\) EA.16) for all r>0. This is the OZ relation with c(r) eliminated in favour of q(r). By differentiating EA.13) and EA.16) with respect to r, and using q(R) = 0, we can write these relations directly in terms of c(r) and h(r): R dr'q'(r')q(r'-r) EA.17) for Ossrssi?, where q'(r) = (d/dr)q(r), and rh(r) = -q'(r) + 27rp\ dr'q{r'){r~r')h(\r-r'\) EA.18) Jo for all r>0.
5A FACTORIZATION METHOD FOR OZ EQUATION 419 The relations EA.17) and EA.18) are the desired reformulation118 of the OZ equation. The great advantage of EA.17) and EA.18) over the original OZ E.10), is that the one-dimensional convolution-type integrals in EA.17), EA.18) are over the reduced range @, i?), whereas in the original three-dimensional equation the convolution integral involves the r range @, 2R). As we discuss in detail in the next section, this reduction of range35.101102113 property of the transformation reduces the solution of the hard sphere PY problem for c(r) for r<R to simple quadratures, because h(r) is known119 in the range @, R). Also noteworthy (cf. E.20) and EA.6)) is the fact that the compressi- compressibility x (and hence the equation of state) can be obtained from q using EA.19) where q@) = q(k =0) is real and given in terms of q(r) by EA.7). 5A. 1 Solution of the PY approximation for hard spheres We begin with the factorized form EA.17), EA.18) of the OZ relation, together with the closure E.14), E.15). Putting ^=0- in EA.17) and EA.18), and introducing the dimensionless quantities x = r/cr and q(x) = q(r)/<r2, we have xc(x) = -q'(x) + 12tj [ dx'q'(x')q{x' - x) EA.20) for 0<x<l, where r\ = (tt/6)pct3, and dx'q(x')(x-x')h(\x-x'\) EA.21) for all x>0, where q'(x)^(d/dx)q(x). We also have the closure E.14), E.15) h{x) = -\, x<l, c(x) = O, x>l. Setting h(x) = -\ for x<l in EA.21) gives q'(x) = ax + b, (x<l), EA.22a) EA.22b) EA.23) where a = 1 - 12tjj dxq(x), EA.24) | dxq(x)x. EA.25) o
420 INTEGRAL EQUATION METHODS 5A The linear relation EA.23) is easily solved for q(x), q{x) = \a{x2-l) + b{x-\), x=sl EA.26) = 0, I>1, where we have used the boundary condition q(l) = 0 (cf. discussion following EA. 13)). The parameters a and b are found (self-consistently) by substituting EA.26) in EA.24) and EA.25). This gives the two equations a = l+4va + 6rib, ' EA.27) b = — 5-rja—2t)?> with solution 1+2t) a = A-T)J' EA.28) 3 b 2A-t,J- Having determined q(x), we now obtain c(x) by substituting EA.26) in EA.20). After a number of elementary integrations and the use of EA.28) we get 3, x<\, EA.29) = 0, x>\. Equation EA.29) can be rewritten in the form E.16). An analytic solution EA.29) has thus been found for c(x), from which one can obtain the thermodynamic functions in closed form (see §5.1). The correlation function h(x) for x>l can be obtained from EA.21) but the result13'32'33120 is only piecewise analytic over the x ranges A,2), B, 3), etc. 5A.2 Properties of q(k) The properties of q(k) given on p. 417 have been proved rigorously by Baxter.102 We give here a heuristic argument due to Sullivan.1" Because of the OZ relation EA.5) the structure factor S(fc) = 1 + ph(k) of the fluid is related to c(fc) by EA.30) so that the relation to be proved EA.6) is S(fe)-1=|q(fc)|2 EA.31) where q(k) has the three properties given on p. 417.
5A FACTORIZATION METHOD FOR OZ EQUATION 421 The relation EA.31) defines the modulus |q(fc)|. We show that the phase of q(k) can be chosen in such a way that properties (l)-C) hold. If S(fc) is to have a product form EA.31), its logarithm will be a sum. We therefore introduce In S(k) and its Fourier transform121 L{r), In Siky1 ={ dr eilcrL(r) EA.32) where L f dke~ikr In S(k)-1. EA.33) 2 tt .Lo The physical values of fc are real and positive. We define S(fc)~! for real negative k by EA.30) and EA.3), so that S(fc) is a real and even function of k. Since S(k)~l is also positive definite, In S(fc) is a real even function of fc. From EA.33) it follows that L(r) is a real and even function of r. The transforms EA.32) and EA.33) can therefore be reduced to one-sided ones lnS(fcr1=2| dr cos(fcr)L(r), EA.34) L{r) = - I dfc cos(kr)ln S(fc)-1. EA.35) IX Jo Since 2 cos kr = exp(ifcr) + exp(—ikr), we get from EA.34) In S(fc) = !(fc) + /(fc)* EA.36) where l(k)=\ dreckrL(r). EA.37) As expected In S in EA.36) is a sum of two terms, so that we have j(k) — (,e )(c ) ¦ pa.js) We now adopt as our definition of q(fc) e1(@ EA.39) so that S(fc) takes the desired form EA.31). Since /(fc) is completely defined in terms of S{k) (or c(fc)) by EA.37) and EA.35), EA.39) defines completely (modulus and phase) q(k). We now show that with this definition q{k) has the desired properties. Up to now fc has been real. It is useful to define the functions c(fc), S(fc), /(fc), and qik) for complex fc. We define c(fc), and hence S(fc)^1
422 INTEGRAL EQUATION METHODS 5A EA.30) for complex fc by EA.3), dr cos(fcr)C(r) EA.40) where we have used EA.9). Since the integral EA.40) is over a finite interval, c(k), and hence S(fc), is analytic throughout the complex fc plane. The function /(fc) defined by EA.37) is clearly analytic in the upper half plane (UHP). Hence q(k) defined by EA.39) is analytic in the UHP, and also (because of the exponential) has no zeros there. For y = Im(fc) —* o°, we have /(fc)—*0 and therefore q(k)—>1. In the lower half plane (LHP) /(fc) and q(k) are analytic for finite fc, but will diverge for y —* — °°. The exact behaviour can be obtained from (cf. EA.31)) q(-fc)-1 EA.41) which extends to complex fc the real fc relation based on q(k)* = q(—k). Since for y -» -oo we have q(-k) -> 1 and (cf. EA.30), EA.40)) S(fc) -> l + O(elv|R), we therefore have q(fc)-^l+O(elv|R), (y->-oo). EA.42) Since (cf. EA.39), EA.37)) for real fc-»°°wehave q(k)-*l, q(fc)-l will possess a Fourier transform q(k)- 1 = -27rp[ dr eikrq(r) EA.43) where -2irpq(r)=-M dfc e-'kr[q(fc)- 1]. EA.44) 2tt J_«, For r<0 the integrand in EA.44) is analytic in the UHP and vanishes for Im(fc) = y—>o°. Closing the contour integral by a large semicircle in the UHP we therefore have q(r) = 0, r<0. EA.45) For r>i? we close the contour in EA.44) by a large semicircle in the LHP. The integrand is analytic in the LHP, and for y —»—oo e"lkr decays as O(eHvlr)whereasq(fc)~l grows as (cf.EA.42))O(elvlR).Thus for r>R the integrand vanishes exponentially on the semicircle. Hence, because there is no contribution from the semicircle and because of the absence of poles in the LHP we have q(r) = 0, r>R EA.46) which is EA. 10). Because of EA.45) and EA.46) one can rewrite
5B THERMODYNAMICS OF THE HNC APPROXIMATION 423 EA.43) as r 2Tpl Jo R drelhrq(r). EA.47) Appendix 5B Thermodynamics of the HNC approximation From the harmonic coefficients h{lxl2l; r) of hA2) we can calculate the thermodynamic quantities from the U-, p-, and ^-routes in the usual straightforward ways (cf. §§5.4.10 and 5.4.4) in HNC. The A-route E.118) would appear to present problems because of the A-integration, but just as in MSA and LHNC/SSC/GMF, we can carry out the A- integration analytically121 in HNC. Also analogous to MSA and LHNC, we can analytically differentiate the resulting AA expression with respect to density and temperature to obtain the pressure and energy in HNC. In contrast, the LHNCF/SSCF and PY closures do not possess these proper- properties. 5B. 1 Free-energy We begin by rewriting the free-energy equation E.117) as where uA2) = u(rw1w2) is the complete potential, so that AA is the residual free energy, i.e. the excess over the ideal gas value, gxA2), is the pair correlation function corresponding to the fluid pair potential Am (('/), (all i,j), and A is integrated from 0 to 1. The HNC closure E.7) for pair potential Am can be rewritten as gx=e-»u+(hK-cM EB2) so that we have ^ + g^(^-Cx). EB.3) 3A dA For the quantity j3ugx occurring in EB.1) we therefore have ?+ g(hc) oA dA = ^[-gx + gx(^-cx)]-~(^~cA). EB.4) dA dA The last quantity in EB.4) can be rewritten as ~(l/2)(d/dA)(hxJ + (dgx/3A)c\ so that we have ?"gx=-^-Q(kxJ+hx-gx0xx-<:x)] + c^ EB.5) 5A 5A where we have replaced the term gx in EB.4) following (d/dA) by h\
424 INTEGRAL EQUATION METHODS 5B If we substitute EB.5) in EB.1) we get (j3AA/NJ, where the first term arises from the d/dA term in EB.5). The A-integration in EB.1) can be carried out immediately for this term giving OAA/N), = -ip|dr<ihA2J+hA2)-gA2)[hA2)-cA2)])MlUj EB.6) where h = hx = 1, etc. The expression h2l2+h can be written a bit more simply as (g2—l)/2. This term can be expressed immediately in terms of r- or k-space harmonic coefficients of hA2) and cA2) using the generalized Parseval relations of Appendix B (§ B.4). The second term (?SAA/NJ is given by ^^) EB.7) where we have used dgxld\ =dhKld\. We express this in terms of the k-frame harmonic coefficients f(lJ2m\k) using the generalized Parseval relation (B.88), dA EB.8) \ Using the symmetry properties CB.22a) we rewrite the last part of this expression as v f , a/, , ,, dhx(/iBm; k) v f f . , dhx(m;k)] 1 dAc^/^mjk) = 2. dATr cx(m;k) i,i2m J 5A m J L 5A J EB.9) where we have introduced matrix notation (Appendix B); the matrices cK(m; k) and hx(m; fc) have elements c^il^m; k) and hx(?iBm; k), and Tr[AB] denotes the trace of the matrix product AB. Thus, we have ipB7r)-3|dkDw)-2 I jdA EB.10) The k-frame OZ equation C.160) can be written in matrix notation as or equivalent^ as (cf. E.13) and E.270)) -1p(~)mhx(m; k)] = [/~D7r)-'pHmcx(m; k)V EB.12)
5B THERMODYNAMICS OF THE HNC APPROXIMATION 425 where I is the unit matrix. We note the identity122 I EB.13) where A is any non-singular matrix which depends on a parameter A. From EB.13) and EB.12) we have ^ In det[/ + Dtt)-VR-V] - Tr{[/ D7r)pHc]D7rr V() ^ 5A I dA where hK = hK(m; k), etc. Hence the expression Tr[cx d?ix/dA] in EB.10) is given by ]}. EB.14) With the help of this expression the A integration in EB.10) can now be carried out immediately. We get (/3AA/NJ = --!- B77-r3 I dk X {In det[I + D-ir)-1 p(-)mh(m; k)] 2p J EB.15) With this expression and EB.6) we achieve our objective, to express AA directly in terms of the harmonic coefficients of hA2) and cA2). Whereas in EB.6) we can use any of the four types of harmonic coefficients discussed in § B.4, in EB.15) we must use the k-frame harmonic coeffi- coefficients f{lxl2m; k). In terms of the latter harmonics we can write EB.6) as (/3AA/N), = -^B77)-3JdkD77) ?TrSh(m; kf + h(m; k)B7rKD-77-)S(k)S,il2m,O0O -g(m;fc)(fi(m;fc)-c(m;fc))] EB.16) where e>lli2m?Ooo = 1 for lj2m = 000, and zero otherwise. In principle det[ ] and Tr[ ] in EB.15) and EB.16) are to be evaluated for infinite dimen- dimensional matrices. In practice we truncate the ltl2 harmonic series, usually with lu /2<4 or Z1; B<2, so that det and Tr are evaluated for finite dimensional matrices. If h and c are assumed diagonal in IJ2 (i.e. we neglect coupling between different harmonics (, and l2) EB.6) plus EB.15) reduces to the form E.228) derived previously for LHNC/SSC/GMF. The superficial differences between [EB.6) + EB.15)] and E.228) arise from the fact that in EB.5) we expressed |3ugx in terms
426 INTEGRAL EQUATION METHODS of cA dgx/dA =cx dhx/5A; we could just as easily have expressed /3ugx in terms of hx dcx/d\, and this would lead to a final expression involving Tr[c] and In det[/-D7r)-1pHmc], as in E.228). (We also took /,/2 = 22 for purposes of illustration in E.228).) For atomic fluids EB.6) and EB.15) reduce to -c(r))], EB.17) (/3AA/NJ = ~ B77-r3 fdk[ln(l + ph(k)) - ph(k)] EB.18) 2p J where h(k) is the Fourier transform of h{r) (see CB.3) or EA.1)). Equation EB.18) follows from EB.15) on noting that the only non- vanishing element of h(!,/2m; k) for atomic fluids occurs in the m=0 matrix, and is /i@00; k) = 4irh(k). 5B.2 Pressure and energy The A-route residual (or configurational) energy (CAU/N) and pressure (/3Ap/p) are obtained from (/3AA/N) using E.213) and E.216). Hence we must calculate the ^-derivative of [EB.6) + EB.15)], where v = |3 or v = p. The calculation is tedious but fortunately can be avoided by recalling the result26 established by functional methods for atomic fluids (and which presumably readily extends to molecular fluids) that the free-energy, pressure and energy are thermodynamically consistent in HNC theory. Hence the A-route energy (/3AL///V) will equal that calcu- calculated from the [/-route, and the A-route pressure (|3Ap/p) will equal that calculated by the p-route (virial expression). Thus, of the four traditional routes, the only inconsistency in HNC theory arises from the x-route as compared to the other three routes. We note the contrast with LHNC theory (§ 5.4.10), where all four routes are generally inconsistent. References and notes 1. Rushbrooke, G. S. in Physics of simple liquids (ed. H. N. V. Temperley, J. S. Rowlinson, G. S. Rushbrooke), p. 25, North-Holland, Amsterdam A968). 2. Henderson, D. and Davison, S. G. in Physical chemistry, an advanced treatise, (ed. H. Eyring, D. Henderson, W. Jost), p. 339, Vol. 2, Academic Press, New York A967). 3. Rowlinson, J. S. and Swinton, F. L. Liquids and liquid mixtures, 3rd edn, pp. 251-253, Butterworth, London A982); see also Rept. Prog. Phys. 28, 169 A965); Physics of simple liquids (ed. H. N. V. Temperley, J. S. Rowlinson, G. S. Rushbrooke), p. 59, North-Holland, Amsterdam A968). 4. Watts, R. O. Statistical mechanics, Specialist Periodical Report (ed. K. Singer), Vol. 1, p. 1, Chem. Soc. London A973) 5. Hansen, J. P. and McDonald, I. R. Theory of simple liquids, Academic Press, New York A976).
REFERENCES AND NOTES 427 6. Stell, G. in Equilibrium theory of classical fluids (ed. H. L. Frisch and J. L. Lebowitz), pp. II, 171-266, Benjamin, New York A964). 7. Rice, S. A. and Gray, P. G. The statistical mechanics of simple liquids, Wiley, New York A965). 8. Baxter, R. J. in Physical chemistry, an advanced treatise, (ed. H. Eyring, D. Henderson, W. Jost), Vol. 8A, p. 268, Academic Press, New York A971). 9. Cole, G. H. A. The statistical theory of classical dense fluids, Pergamon, New York A967). 10. Croxton, C. A. Liquid state physics, a statistical mechanical introduction, Cambridge University Press, Cambridge A974); cf. also Liquid state physics, Wiley, New York A975). 10a. McQuarrie, D. A. Statistical mechanics, Harper and Row, New York A976). 10b. March, N. H. and Tosi, M. P. Atomic dynamics in liquids, Macmillan, London A976). 10c. The orthodox approach, in fact, is to sum partially the density expansion (cf. E.7)); however, the density expansion can be derived" as a rear- rearranged perturbation expansion, with the ideal gas as reference, and the full pair potential as perturbation. lOd. Correction terms can be calculated4 in principle from the so-called PY II and HNC II approximations. For more recent work based on HNC, see Rosenfeld, Y. and Ashcroft, N. W., Phys. Rev. A20, 1208 A979) (cf. refs. 12). For some of the simpler integral equations, corrections can be worked out more readily (see §§5.4.2, 5.4.9a, 5.6). 11. Brout, R. and Carruthers, P., Lectures on the many-electron problem, Gordon & Breach, New York A963). 12. Lado, F. J. chem. Phys. 60, 1686 A974); Phys. Rev. A8, 2548 A973); A125, 135, 1013 A964); Phys. Lett. 89A, 196 A982). For alternative derivations of CHNC [§ 5.4.9] see refs. lOd, 15, and McGowan, D. Phys. Lett. 76A, 264 A980). 13. Barker, J. A. and Henderson, D. Rev. mod. Phys. 48, 587 A976). 13a. Torrie, G. and Patey, G. N. Mol. Phys. 34, 1623 A977). 13b. Andersen, H. C. Ann. Rev. phys. Chem. 24, 145 A975). 13c. H0ye, J. S. and Stell, G. J. chem. Phys. 64, 1952 A976). 14. Madden, W. G. and Fitts, D. D. Mol. Phys. 31, 1923 A976). 14a. Madden, W. G., Fitts, D. D., and Smith, W. R. Mol. Phys. 35, 1017 A978). 15. Smith, W. R. and Henderson, D. J. chem. Phys. 69, 319 A978). 15 a. The advantage of perturbation theory is its simplicity. The integral equa- equations are more powerful in that they go to infinitely high order in the pertrubation potential. 15b. For atomic liquids ch is a convolution in r-space, or a simple product in k-space [see E.10), E.11)]. For molecular liquids, ch can be written in terms of radial and angular convolution (see C.117)), or as a matrix product (using spherical harmonic indices see C.158), C.160)), EB.11) (see also § 5.3). 16. Brout, R. Phase transitions, Benjamin, New York A965). 17. Abrikosov, A. A., Gorkov, 1,. P., and Dzyaloshinskii, I. Ye. Quantum field theoretical methods in statistical physics, Pergamon, New York A965). 18. Fetter, A. L. and Walecka, J. D. Quantum theory of many-particle systems, McGraw-Hill, New York A971).
428 INTEGRAL EQUATION METHODS 18a. Stell, G. in Phase transitions and critical phenomena, (ed. C. Domb and M. S. Green), Vol. 5B, p. 205, Academic Press, London A976). 19. Mattuck, R. D. A guide to Feynman diagrams in the many-body problem, 2nd edn, McGraw-Hill, New York A975). 19a. The structure of the OZ equation E.1) is, in fact, similar to that of the Dys*on equation179 of many-body theory (cf. also § 5.3). The structure of the Lippmann-Schwinger integral equation of scattering theory'*' is also similar. 19b. Other examples of this many-body structure are given in §§ 2.10 and 10.1. 19c. Dale, W. D. T. and Friedman, H. L. J. chem. Phys. 68, 3391 A978). Other methods devised to speed up convergence of the iteration method include: Rossky, P. J. and Dale, W. D. T. J. chem. Phys. 73, 2457 A980) ['chain recurrence', cf. § 5.4.8]; Gillan, M. J. Mol. Phys. 38, 1781 A979) [Newton-Raphson aspect incorporated]; Ceperley, D. M. and Chester, G. V. Phys. Rev. A15, 755 A977) [recast as eigenvalue problem to optimize the mixing of iterates]; Leribaux, H. R. and Miller, L. F. J. chem. Phys. 61, 3327 A974) [optimized mixing]; Larsen, B. J. chem. Phys. 68, 4511 A978) [solves for the small and smooth difference c-cGMSA]; Hall, D. S. and Conkie, W. R. Mol. Phys. 40, 907 A980) [iterates on G, c) pair, where y = h~c; and utilizes various numerical methods techniques]. See also refs. 19f, 96b, and 104b. Other methods are reviewed by Watts.4 19d. Rodberg, L. S. and Thaler, R. M. Introduction to quantum theory of scattering, Academic Press, New York A967). 19e. Olivares, W. and McQuarrie, D. A. J. chem. Phys. 65, 3604 A976); Vicsek, T. Physica A102, 523 A980); MacCarthy, J. E., Kozak, J. J., Green, K. A., and Luks, K. D. Mol. Phys. 44, 17 A981). 19f. In some cases convergence is faster by starting with c<n, calculating hm from E.2), then c<2) from E.3), etc. and stopping when C(" + 1) = C(>1) (see §5.4.7). 19g. The third classic method1'*'13 has a somewhat different basis. One uses the BBGKY28~31 hierarchy C.245) to relate the pair correlation function gA2) to the triplet function gA23), and then 'closes' with the superposi- superposition approximation gA23) = gA2)gA3)gB3). This theory is generally less successful for atomic liquids,513 and has not been applied to molecular liquids. Closely related is the theory based on the Kirkwood hierarchy410313 where A is a potential parameter, such that A = 0 decouples molecule 1 from the system, and A = 1 corresponds to the full interaction of all N molecules, i.e. The MSA (and closely related ORPA - optimized random phase approxi- approximation) theory has also been applied to atomic liquids.'313hl5'53a 20. Percus, J. K. and Yevick, G. J. Phys. Rev. 110, 1 A958). 21. Percus, J. K. in Equilibrium theory of classical fluids, (ed. H. L. Frisch and J. L. Lebowitz) p. 11-33, Benjamin, New York A964). 22. Rushbrooke, G. S. Physica 26, 259 A960).
REFERENCES AND NOTES 429 23. Verlet, L. Nuovo Cimento 18, 77 A960). 23a. Green, M. S., Hughes Aircraft Rept. (Sept. 1959); J. chem. Phys. 33, 1403 A960); Percus, J. K., ONR Rept. A956), also published in App. B. of ref. 21. 24. Meeron, E. J. math. Phys. 1, 192 A960). 25. van Leeuwen, J. M. J., Groeneveld, J., and de Boer, J. Physica 25, 792 A959). 25a. Abe, R. Prog, theoret. Phys. 19, 57, 407 A958). 26. Morita, T. Prog, theoret. Phys. 23, 829 (I960); Morita, T. and Hiroike, K. Prog, theoret. Phys. 23, 1003 A960). 26a. It is also seen that the PY and HNC approximations satisfy the formally exact asymptotic condition cA2)-»—/3uA2), (r12-»«) (see C.119)). 26b. A linearized version of the HNC theory is derived in § 5.4.9. 26c. We recall that y = 1 at low densities (binary collision regime) where 26d. Note that cA2) contains only irreducible diagrams, whereas the corres- corresponding hA2) series103 contains also reducible diagrams. A reducible diagram is one which contains one or more integrated points such that when cut at such a point the diagram falls into disconnected pieces, thereby separating the fixed points. An example is I I . See refs. l-10a, 18a, 39, 40, 40a for a precise classification of diagrams. 27. Madden, W. G. and Fitts, D. D. J. chem. Phys. 61, 5475 A974). 28. Bogoliubov, N. N. in Studies in statistical mechanics, (ed. J. de Boer and G. E. Uhlenbeck), Vol. 1, p. 1, North-Holland, Amsterdam A962). 29. Born, M. and Green, H. S. Proc. Roy. Soc. A188, 10 A946). 30. Kirkwood, J. G. J. chem. Phys. 3, 300 A935). 31. Yvon, J., La theorie statistique des fluides et Vequation d'etat, Act. Sci. et Indust. No. 203, Hermann et Cie, Paris A935). 31a. The PY and HNC cA2) functions thus err in the p2 terms (and above), which leads to errors in (p/p/cT) at the p3 terms; i.e. the PY and HNC theories give the correct second and third virial coefficients, but err in the higher coefficients (see ref. 5, p. 118). 31b. So far only linearized and/or truncated (finite spherical harmonic expan- expansions) versions of the PY and HNC approximations have been solved. 32. Wertheim, M. S. Phys. Rev. Lett. 10, 321 A963). 33. Thiele, E. J. chem. Phys. 39, 474 A963). 34. Stell, G., in Modern theoretical chemistry (ed. B. Berne), Statistical mechanics, Vol. 5, Chapter 2, Plenum, New York A977). 35. Wertheim, M. S. J. math. Phys. 5, 643 A964). 35a. One can show34 quite generally that the OZ equation together with h(r) for r<r0 and c(r) for r>r0 are sufficient to determine h(r) and c(r) for all r. The result is proved most simply using the factorized form of the OZ equation given in Appendix 5A; see ref. 119. 35b. This is the fraction of space occupied by the spheres. The value t) = (tt/6)V2 = 0.74 corresponds to a close-packed crystal, t) ~ 0.64 corres- corresponds to random close packing (glass; metastable), t) —0.545 corres- corresponds to the melting density, and tj ~ (tt/6JV2/3 = 0.49 corresponds to the freezing density.5'36:'6a 35c. The structure factor (see Appendix 5A and Chapter 9) S(k) = A - pc(fc))~'
430 INTEGRAL EQUATION METHODS can, however, be written in closed form since the Fourier transform of E.16) is easily carried out (N. W. Ashcroft and J. Lekner, Phys. Rev. 145, 83 A966)); setting fccr = fc* we have ) = -24T)fc*-2[c0(sinfc*-fc*cosfc*) + c1fc*"'Bfc*sinfc* -(fc*2-2)cos fc*-2) + c3fc*-3(Dfc*3-24fc*)sin fc* -(fc*4-12fc*2 + 24)cosfc* where the q are defined in E.17). 35c'. Stell, G. Physica 29, 517 A963). 35d. Ekiuation E.21) is also easily obtained from EA.19). 35e. The result E.22) was obtained earlier37 from scaled particle theory (SPT), see Ch. 6; however, the SPT and PY theories are not13 in general equivalent. 35f. To derive E.24) we substitute g(r) = exp(—pu(r))y(r) into E.23), note the relations -f}u' exp(-Cn) = (<)/dr)e.xp(-(}u(r)) = 8(r — a), and use the fact that y(r) is continuous5 at r = a (see also Chapter 6). 36. Reed, T. M. and Gubbins, K. E. Applied statistical mechanics, Chapter 8, McGraw-Hill, New York A973); de Llano, M. J. chem. Phys. 65, 501 A976). 36a. Verlet, L. and Weis, J. J. Phys. Rev. 5, 939 A972); for the hard sphere solid g(r), see Kincaid, J. M. and Weis, J. J. Mol. Phys. 34, 931 A977). For a review of random close-packing of spheres, see R. Zallen, in Studies in statistical mechanics (ed. E. W. Montroll and J. L. Lebowitz), North- Holland, Amsterdam 7, 177 A979). 36b. See Verlet, L. Mol. Phys. 41, 183 A980); 42, 1291 A981), and references therein. 37. Reiss, H., Frisch, H. L., and Lebowitz, J. L. J. chem. Phys. 31, 369 A959). 38. Carnahan, N. F. and Starling, K. E. J. chem. Phys. 51, 635 A969); 53, 600 A970). 38a. Similar problems will arise, for example, for a dipolar fluid if one truncates the harmonic series with (,B/ = 000, 110, 112. 38b. Because E.38) has the OZ form C.117), the OZ relations C.158) and C.160) for the harmonic coefficients, ha(lil2l; fc), c,(lil2l; fc) and /ia(/,f2m; fc), cJJxl2m; fc), respectively, will also be valid. In terms of the complete harmonics h(lxl2l; fc), c(Ul2l; fc) this means that [when E.36) and E.37) hold] the OZ C.158) decouples into two relations, one for the harmonic /,/2( =000 and another for the harmonics U l2l^ 000. In general, however, this decoupling does not occur. 39. Uhlenbeck, G. E. and Ford, G. W. Studies in statistical mechanics, (ed. J. de Boer and G. E. Uhlenbeck), North-Holland, Amsterdam Vol. 1, p. 119 A962). 40. Andersen, H. C. in Modern theoretical chemistry, Vol. 5, (ed. B. Berne), Statistical mechanics, Chapter 1, Plenum, New York A977). 4()a. McDonald, I. R. and O'Gorman, S. P. Phys. Chem. Liq. 8, 57 A978). 41. Henderson, R. L. and Gray, C. G. Can. J. Phys. 56, 571 A978). 42. Wang, S. S., Gray, C. G., Egelstaff, P. A., and Gubbins, K. E. Chem. Phys. Lett. 21, 123 A973); 24, 453 A974). 43. Adelman, S. A. and Deutch, J. M. Adv. chem. Phys. 31, 103 A975). 44. Lebowitz, J. L., Stell, G., and Baer, J. J. math. Phys. 6, 1282 A964). 44a. See also note 15b relevant to E.1). 44b. If need be the matrix elements of (l-pc)~' can be found explicitly from
REFERENCES AND NOTES 431 the series representation (l-pc) = l + pc + p2c2+... together with the multiplication rules E.41), E.42). 44c. The use of such identities is common in scattering theory.44'3 44d. See ref. 19d, p. 178. 44e. The relations E.48) and E.49) also have the simple Dyson equation structure (cf. ref. 19a). 44f. The relation Jda>,cA2) = 0 is only satisfied in simple theories (cf. discus- discussion following E.35)). 45. Kirkwood, J. G., Lewinson, V. A., and Alder, B. G. J. chem. Phys., 20, 929 A952). 46. Chen, Y. D. and Steele, W. A. J. chem. Phys. 50, 1428 A969); 52, 5284 A970). 47. Morrison, P. F., Ph.D. thesis, University of California A972). See also Morrison, P. F. and Pings, C. J. J. chem. Phys. 60, 2323 A974). 47a. Comparison of theory and experiment tests jointly the potential model and the theory. Also S(fc) is relatively insensitive to the type of theory used (cf. also remarks in Chapters 1, 4, and 9). 47b. Cummings, P. T., Ram, J., Barker, R., Gray, C. G., and Wertheim, M. S. Mol. Phys. 48, 1177 A983). 47c. Lee, L. L. and Chung, T. H., J. chem. Phys. 78, 4712 A983). These authors further approximate ca in E.210) by a PY-type approximation, and term the resulting approximation MLHNC (modified LHNC). 48. Lebowitz, J. L. and Percus, J. K. Phys. Rev. 144, 251 A966). 48a. Streett, W. B. and Tildesley, D. J. Proc. Roy. Soc. A348, 485 A976). 49. Jepsen, D. W. and Friedman, H. L. J. chem. Phys. 38, 846 A963). 49a. The 'dipolar Debye-Huckel approximation', or dipolar random phase approximation (RPA),11 in which one sets c(ra)]OJ) = -fiu^fra),^) for all r (where m^ is the dipole-dipole potential), was solved somewhat earlier,491, (as was the MSA for charged hard spheres52). 50. Wertheim, M. S. /. chem. Phys. 55, 4291 A971). 51. Ramshaw, J. D. J. chem. Phys. 64, 3666 A976). 52. Waisman, E. and Lebowitz, J. L. J. chem. Phys. 52, 4307 A970); 56, 3086, 3093 A972). 52a. Assessment of the MSA for monatomic potentials (Coulomb, square well, Yukawa, LJ) is given elsewhere.311-34-53-53" 52b. The MSA for dipolar hard sphere mixtures has also been solved.547 53. Smith, W. R., Henderson, D., and Tago, Y. J. chem. Phys. 67, 5308 A977). 53a. Rosenfeld, Y. and Ashcroft, N. W. Phys. Rev. A20, 2162 A979); Madden, W. G. and Rice, S. A. J. chem. Phys. 72, 4208 A979). 54. Isbister, D. and Bearman, R. J. Mol. Phys. 28, 1297 A974); Freasier, B. C. and Isbister, D. J. Mol. Phys. 38, 81 A979). 55. Martina, E. and Deutch, J. M. Chem. Phys. 27, 183 A978). 56. Adelman, S. A. and Deutch, J. M. J. chem. Phys. 59, 3971 A973). 57. Adelman, S. A. and Deutch, J. M. J. chem. Phys. 60, 3935 A974). 57a. Blum, L. J. chem. Phys. 61, 2129 A974); Chem. Phys. Lett. 26, 200 A974); Perez-Hernandez, W. and Blum, L. J. stat. Phys. 24, 451 A981). 58. Blum, L. J. chem. Phys. 57, 1862 A972); 58, 3295 A973). 59. Maclnnes, D. A. and Farquhar, I. E. Mol. Phys. 30, 457, 889 A975). 59a. Here . .. indicates the remaining harmonics in E.28), i.e. lj2l = 101, 011, 202, 022, 220, etc. In simple theories (e.g. MSA, GMF) the harmonic series is truncated (cf. §5.2). In particular, for u = «(ref.) + «(dipole-
432 INTEGRAL EQUATION METHODS dipole), the MSA and GMF theories keep only the three harmonics written explicitly in E.60). 59b. We take co, to be defined by n-,, the dipole of molecule ('. 59c. It is perhaps worth pointing out again that in general the following two decompositions of h (or c) do not coincide: (a) h = h (reference)-!- h (correction), (b) h = h(isotropic) + h(anisotropic). How- However, when h(correction) [or c(correction)] satisfies the strong E.33) or weak E.32) angular condition, (a) and (b) do coincide. (We assume the reference fluid potential uo(r) is isotropic; see also the discussion in § 5.3.) 59d. There is no difficulty with the negative density term, see E.16). 59e. It is plausible that cs(r), cA(r), cD(r) are expressible in terms of cPY(r), since apart from an r~3 tail in cD(r), E.57) is a PY closure for cs, ca, cD. 59f. The 112 or D harmonic of the potential is long-ranged (cf. E.65)), so one expects cD(r) and hD(r) to depend on r asymptotically as r 3. 59g. The analogous quantity h'o(r) is also a short-range function (i.e. contains no r term at large r), but does not vanish for r>cr, since hPY(r)^0 for r>cr. 60. Verlet, L. and Weis, J. J. Mol. Phys. 28, 665 A974). 61. Quirke, N. and Perram, J. W. Chem. Phys. 26, 191 A977). 61a. The quantity plotted hPB24; r) is in Patey's convention,52 and is related to our coefficient by h"B24; r) = Dir)-iG)(n)'hB24; r). 62. Patey, G. N. Mol. Phys. 35, 1413 A978). 63. Rushbrooke, G. S., Stell, G., and H0ye, J. S. Mol. Phys. 26, 1199 A973). 64. Patey, G. N. Mol. Phys. 34, 427 A977). 64a. The reason for such a large range is not clear intuitively. It may be that this is simply analogous to calculating the compressibility from h(r) using E.19); the latter is always difficult because of cancellations arising from the oscillating integrand. Alternatively the system may be approaching an orientational phase transition analogous to a liquid crystal. A third possi- possibility is that it is an artifact of the theory. 64b. Levesque, D., Patey, G. N., and Weis, J. J. Mol. Phys. 34, 1077 A977). 65. Stell, G. and Weis, J. J. Phys. Ret;. A16, 757 A977). 65a. Pollack, E. L. and Alder, B. J. Physica 102A, 1 A980); de Leeuw, S. W., Perram, J. W., and Smith, E. R. Proc. Roy. Soc. A273, 27 A980); and further references in Chapter 10. 65a'. Adams, D. J. Mol. Phys. 40, 1261 A980). 65b. We give E.77) for general l^lj. The result E.76) is also valid for, for example, quadrupole-quadrupole interaction if (,/2 = 22 is used in place of 11; The quadrupole case is discussed briefly later. For cross multipole interactions (e.g. IJ2= 12; dipole-quadrupole) the transformation E.77) is not useful and is replaced by another one (see ref. 65c). 65c. It turns out that E.83) can be put in useful form only for even I. For odd I (which arises when (some) unlike multipole interactions are present, e.g. dipole—quadrupole, dipole-charge, and also for some non-multipolar in- interactions) the definition E.83) is modified in such a way that the results E.87), E.85), and E.86) are in fact valid for all I (see § 5.4.6). 65d. These two quantities are not expansion coefficients. 65e. It would be useful to have further insight into the reasons these properties hold. 65f. Since m takes the values m = 1, 0, and ( the values / = 2, 0, the meaning of /""A10; r) will be clear from the context, i.e. whether it is a member of the set {/""A12; r),/""A10; r)} or of {/""A11; r), /""A10; r)}.
REFERENCES AND NOTES 433 65g. See E.1) or E.40). 65h. As expected (see ref. 59f) the /A12; r)s contain a long-range r~3 part for r—*co. The /l<))A12; r)s are the short-range parts (see footnote p. 365) of the /A12;r)s, and are quite generally given by E.89a). In MSA the c(m(ll/;r)s in fact vanish for r>a (see E.104), E.105)). 65i. It will be seen in § 5.4.4 (eqn E.132)) that k is proportional to the (dimensionless) energy per molecule of the fluid. 66. Andersen, H. C. and Chandler, D. J. chem. Phys. 57, 1918 A972). 67. Nienhuis, G. and Deutch, J. M. J. chem. Phys. 56, 5511 A972). 68. H0ye, J. S., Lebowitz, J. L. and Stell, G. J. chem. Phys. 61, 3253 A974). 69. H0ye, J. S. and Stell, G. J. chem. Phys. 67, 439 A977). 70. Sutherland, J. W. H., Nienhuis, G., and Deutch, J. M. Mol. Phys. 27, 721 A974). 70a. Note that gx in E.117) can be replaced by hK since ua satisfies E.32). 70b. Alternative expressions for AA can be obtained by integrating the ther- modynamic relations for Ap, A U, and Ax, which are in turn given in terms of gA2) later. Since the corresponding inconsistent expressions are given for Ap later, we do not give these alternative expressions. 70c. For hard spheres Uo = (|)NfcT is entirely kinetic. 70d. The A- and l/-routes for AU in MSA do not always agree for central potentials.69 For the dipolar case, inequivalent expressions for AJJ derived from the p- and x-routes can also be written down (see ref. 70b). 70e. Similar self-consistency requirements can be imposed on other13b (e.g. PY and HNC) theories. 71. Ramshaw, J. D. J. chem. Phys. 57, 2684 A972). 71a. References to the extensive literature on this point are given in Chapter 10. 71b. MSA finite volume corrections (i.e. allowing for V<c°) are discussed in ref. 50. 71c. h or g can be used in E.142), since (cos 7i2)«w = 0. 71d. The Fourier transformation E.79) gives the same result, since these transformations coincide for I = 0. 71e. The result E.142) is often written (e - 1)Be + 1)/9e = g,y, where g, is the Kirkwood angular correlation parameter. Equation E.146) then yields the MSA result for g,, gl = &(qV + 2qZl). 71f. The j2 Hankel transform CB.43) cannot be directly utilized for fc —» 0 (see ref. 43 of Chapter 10). 71f. Other routes discussed in Chapter 10, e.g. the Wertheim exact relation A0.48), also give e =q4/q_. 71g. The reason is that the MSA cA2) satisfies the formally exact asymptotic condition C.119). 71h. e will depend indirectly on a through p if p is taken as an independent thermodynamic variable rather than p. 71i. From the numerical examples63 (a) ? = 0.2, e = 144, y = 8.275, and (b) f = 0.25, e = 625, y =21.299, we see that, in MSA, the physically interest- interesting ranges of f and y are ?Sy, y<8; the complete ranges allowed by the MSA solution (?:=j, y<co) are not utilized. 72. Debye, P. Polar molecules, Chemical Catalog Co., New York A929) (Dover reprint). 73. Onsager, L. J. Am. Chem. Soc. 58, 1486 A936). 74. Jepsen, D. W. J. chem. Phys. 44, 774 A966); 45, 709 A966); see also Rushbrooke, G. S. Mol. Phys. 37, 761 A979). The coefficient a3 of y3
434 INTEGRAL EQUATION METHODS differs from -15/16 by a few per cent; the exact result is a3(p*) = ~15/16 + 0.0314p* + .... In GMF the coefficient is a^(p*) = -15/16-0.0181p* + . . ., again nearly equal to -15/16. Here p* = jxr\ 74a. Martina, E. and Stell, G. J. chem. Phys. 69, 931 A978). 75. Morse, P. M. and Feshbach, H., Methods of theoretical physics, p. 622, McGraw Hill, New York A953). 76. Merzbacher, E. Quantum mechanics, 2 edn, Wiley, New York A970) p. 85. 76a. The reason for this name will be apparent following E.185) and E.191). 77. Wertheim, M. S. Mol. Phys. 26, 1425 A973). 77a. The aptness of this name will be seen in § 5.4.8. 77b. Some of the results of §§ 5.3 and 5.3.1 will be used in this section. 77c. See also the cautionary remarks on p. 350 concerning theories with a finite number of harmonics. 77d. Equation E.186) reduces to the usual mean field result c-~pu for an ideal gas reference system (ref. 16, p. 16). For a hard core reference potential, MF E.186) (and also the GMF E.191)) reduces to MSA E.59) outside the core if c0 is neglected and g0 is approximated by its low density limit (a step function). 77e. For future use, we anticipate in E.190) that /ta and ca also satisfy this relation in the GMF theory. The extension41 of E.190) to arbitrary like-multipole (/i = B) potentials is immediate. 77f. The jo, j2, and /4 transforms are readily assembled from separate sine and cosine transform routines, making use of the power law form of ;'i(x) at small x. As a check on the numerical routines caB24; fe) can be calculated from caB24; r) in the approximation where go(r) in E.188) is replaced by a step function, using ? r2 drj,(kr)r'" + l) = k'^i^k^KkaI''. (For ct ^ 0 the right-hand side becomes fc'~2/Bf -1)!! where x!!=x(x—2) (x —4) . . . A).) Another check on the numerical Hankel transform routines is provided by transforming a given test function, and then back- transforming the result. 77g. It may be useful numerically in some cases62 to carry out the Hankel transformation by the indirect route f(lj2l; k)-* f(a\lj2l;r)-~^f(l]l2l;r), which uses a Fourier transform in the first step (cf. Fig. 5.3 and § 5.4.3). 77h. Verlet, L. Phys. Rev. 165, 201 A968). More extensive LJ g,,(r)s from computer simulation are given by Nicolas, J. J., Gubbins, K. E., Streett, W. B., and Tildesley, D. J. Mol. Phys. 37, 1429 A979). An empirical fit to these results is given in Goldman, S. J. phys. Chem. 83, 3033 A979). 77i. The closure E.191) is derived first only for multipole-Iike «as. We can, however, extend the derivation to general uas (see § 5.4.9). In the most general case one needs the more general form E.39) (or E.49)) of the OZ relation. 77j. Murad, S., Gubbins, K. E., and Gray, C. G. Chem. Phys. Lett. 65, 187 A979); Chem. Phys. 81, 87 A983). 77k. Agrofonov, Yu. V., Martinov, G. A., and Sarkisov, G. N. Mol. Phys. 39, 963 A980). 771. Alternative methods of calculating the shift, based on the effective central potential method (§ 4.9) and using HNC theory to evaluate the perturba- perturbation series for g(r) (Chapter 4), have also been developed.14" 77m. Actually the full QHNC was not solved in ref. 62; the harmonic expan- expansions of c and h were truncated, keeping only the l,l2l = 000, 224, 222,% 220 terms (i.e. the ones that occur naturally in LHNC). 78. Gray, C. G. and Henderson, R. L. Can. J. Phys. 57, 1605 A979).
REFERENCES AND NOTES 435 78a. There is no r~3 weighting of the integrand in E.144), as occurs for the thermodynamics (cf. E.131)). 78b. Levesque, D., Weis, J. J., andPatey, G. N. Mol. Phys. 38, 535 A979); 72, 1887 A980); Carnie, S. L., Chan, D. Y. C, and Walker, G. R. Mol. Phys. 43, 1115 A981). 78c. Carnie et a!;78b Carnie, S. L. and Patey, G. N. Mol. Phys. 47, 1129 A982). 79. Wertheim, M. S. Mol. Phys. 33, 95 A977). 79a. For concreteness the discussion is given for QQ interactions, where the three specific harmonics of ca and ha correspond to ltl2l — 224, 222, 220. The extension to arbitrary multipole interactions is obvious. The extension to general «a is discussed at the end of this section. 79b. As a result, only reference pair functions goA2) occur in GMF. This is equivalent to using the superposition approximation for goA23), etc. which occur in the exact perturbation series for g (see Chapter 4). 79c. See Chapter 5 introduction, ref. 26d; note that c contains only irreducible graphs, whereas h contains both irreducible and reducible graphs because of the OZ relation. 79d. The origin of the name SSC (single super chain) approximation [§ 5.4.7] is now apparent. The name is less apt for non-multipole potentials, however; see E.211a'). 79d'. The 'corrected' PY closure1215 ca = /a[go + (l+/oHia-';a)] + /o(ha-ca), (CPY) is obtained by substituting c^Cq + c^, / = /o + /a(l+/o), and y = yo+ya in the PY closure c = fy, and then using the forms c0 = foyo and ya = ha - ca of the PY closure. We also use the exact g0 in the final expression. A similar closure47b PYX (i.e. PY 'excess') c» = /a[go + god, - c,)] + ho(ha - ca), (PYX) can be derived graphically. CPY can be obtained from PYX by approx- approximating h0 by /0 in the two (ha-cj terms. LHNCF/SSCF E.211d) is obtained from PYX by dropping the term go/,(ha~ca). Linearized versions of CPY and PYX, where /„ is approximated by — CMa> can also t>e contemplated by analogy with LHNC. The CPY and PYX approximations have been tested for the S-model anisotropic overlap model potential B.247), and found to be about equally good, with both somewhat better than PY and SSCF; see Cummings, P. T., Ram, J., Gray, C. G., and Wertheim, M. S. Mol. Phys. 50, 1183 A983). 79e. These remarkable properties of GMF were first pointed out by Wer- Wertheim,79 (see also ref. 78). Similar properties arise in MSA (§ 5.4.4) and in HNC (Appendix 5B). 79f. It is not difficult to show that E.222) also gives correct results [of the form 0 = 0] for small r (where gu = 0) and large r (where ho = 0). 79/'. Lado, F. Mol. Phys. 47, 283, 299 A982). 79g. We have followed the historical nomenclature but note that Cummings and Stell7911 have suggested the following sensible changes: A) ISM (in- (interaction site model) to denote any site-site model potential (see §2.1.2). B) RISM (reference ISM) to denote the fused-hard-sphere site-site model potential, where uafl(r) = 0 for r>craB. C) ISA (interaction site approxima- approximation), where one assumes caB(r) = -fiuali(r) for r>aaS (with uaB forr>aoS the tail of the potential outside a hard core). D) RISA (reference ISA), where one assumes caS(r) = 0 for r>craB. Thus new RISA = old RISM.
436 INTEGRAL EQUATION METHODS 79h. Cummings, P. T. and Stell, G. Mol. Phys. 46, 383 A982). 80. Chandler, D. and Andersen, H. C. J. chem. Phys. 57, 1930 A972). 81. Chandler, D. J. chem. Phys. 59, 2742 A973). 82. Ladanyi, B. M. and Chandler, D. J. chem. Phys. 62, 4308 A975). 83. Chandler, D. Mol. Phys. 31, 1213 A976). 83a. This statement refers to the RISM OZ equation. When an approximate closure is introduced (e.g. PY-like), some graphs are then dropped, as is usual for closures. See refs. 83, 83b, 83c, 83d. 83b. Cummings, P. T., Gray, C. G., and Sullivan, D. E. /. Phys. A14, 1483 A981). 83c. Pratt, L. R. Mol. Phys. 43, 1163 A981). 83d. Chandler, D., Silbey, R., and Ladanyi, B. M, Mol. Phys. 46, 1335 A982); Chandler, D., Joslin, C. G., and Deutch, J. M. Mol. Phys. 47, 871 A982). The latter authors verify explicitly that this theory can yield a non-trivial e using non-Coulomb parts of their new direct correlation functions which are short-ranged. However, to date, just as in amended RISM-type theories7911 which have the correct asymptotic behavior of c built in, non-trivial es have not been truly predicted; rather, closures have been written down with a prescribed e essentially incorporated into the closure. 83e. For an AB diatomic fluid for example, {haii} = {hAA, hlils, /tAE1}. 83f. Naturally E.238) or E.239) can also be written in r-space as convolutions (cf. E.10), E.11)). 83g. For extensions to non-rigid molecules, see ref. 172 of Chapter 1. 83h. The scalar RISM OZ equation E.243) can be written in various alterna- alternative forms, analogous to the general RISM OZ equation, e.g. G))ch, (A) l (B) where h=hAA(k), ai = oiaa^/c), c=cAA(k). A scalar (non-matrix) equation can be derived7911 whenever all sites are equivalent. The form (A) closely resembles the general form E.49)-with good reason. Equation E.49) results whenever h and c are divided into two pieces in any manner, corresponding to the division of the potential u intoi two pieces. Here the original atomic fluid atom-atom potential can be regarded as separated into an intra (bonding) piece plus an inter (site—site) piece. (There arc subtleties connected with this division of the potential - see ref. 79h). The form (B) is the natural one to which to apply the factorization method of solution-see ref. 86a and Appendix 5A. 83i. H0yc, J. S. and Stell, G. SUNY CEAS Report 307 (Dec. 1977); See also Stcll, G., Patey, G. N., and H0ye, J. S. Adv. chem. Phys. 48, 183 A981). 83i'. The PY closure E.6) can be written alternatively as c = fy. 83j. Kojima, K. and Arakawa, K. Bull. Chem. Soc. Japan 51, 1977 A978); Johnson, E. and Hazoume, R. P. J. chem. Phys. 70, 1599 A979); Steinhauser, O. and Bertagnolli, H. Zeit. f. phys. Chem. neue Folge 124, 33 A981). [LJ sites, PY and HNC closures]. 83k. Hirata, F. and Rossky, P. J. Chem. Phys. Lett. 83, 329 A981); Hirata, F., Pettitt, M., and Rossky, P. J. J. chem. Phys 77, 509 A982). [LJ plus charged sites, PY and HNC closures; renormalization (topological reduc- reduction) techniques used to treat the long-range Coulomb tail.] Pettitt, B. M. Rossky, P. J. J. chem. Phys. 77, 1451 A982). [RISM with HNC closure, applied to H2O-Iike case].
REFERENCES AND NOTES 437 831. Cummings, P. T., Morriss, G. P., and Wright, C. C. Mol. Phys. 43, 1299 A981) [hard sphere sites, GMSA (Yukawa) closure]. 83m. Thompson, S. M, Gubbins, K. E., Sullivan, D. E., and Gray, C. G., Mol. Phys., 51, 21 A984) [full LJ and repulsive-only LJ sites, PY closure; for bulk fluid and fluid/wall interface]. 83n. Cummings, P. T., Morriss, G. P., and Stell, G. J. phys. Chem. 86, 1696 A982) [central HS + charged sites, modified MSA-type closure]; Morriss, G. P. and Perram, J. W. Mol. Phys. 43, 669 A981) [hard sphere plus charged sites, MSA closure]. 83o. Lombardero, M. and Enciso, E. J. chem. Phys. 74, 1357 A981); Pratt, L. R., Hsu, C. S., and Chandler, D. J. chem. Phys. 68, 4202 A978); see also ref. 83p. 83p. Sullivan, D. E., Barker, R., Gray, C. G., Streett, W. B., and Gubbins, K. E. Mol. Phys. 44, 597 A981). 84. Lowden, L. J. and Chandler, D. J. chem. Phys. 59, 6587 A973), erratum 62, 4246 A975). 85. Lowden, L. J. and Chandler, D. J. chem. Phys. 61, 5228 A974). 86. Maclnnes, D. A. and Farquhar, I. E. /. chem. Phys. 64, 1481 A976). 86a. Morriss, G. P., Perram, J. W., and Smith, E. R. Mol. Phys. 38, 465 A979); see also ref. 79h and refs. therein. 86b. See ref. 83k and also Pratt, L. R., Rosenberg, R. O., Berne, B. J., and Chandler, D. /. chem. Phys. 73, 1U02 A980); the general method is described in ref. 19c. 86c. Monson, P. A., Mol. Phys. 47, 435 A982) [LJ sites]. 87. Freasier, B. C. Chem. Phys. Lett. 35, 280 A975); Jolly, D., Freasier, B. C, and Bearman, R. J. Chem. Phys. Lett. 46, 75 A977); Freasier, B. C. Mol. Phys. 39, 1273 A980); Tildesley, D. J. and Streett, W. B. Mol. Phys. 41, 85 A980). 88. Johnson, E. J. chem. Phys. 67, 3194 A977). 89. Chandler, D., Hsu, C. S., and Streett, W. B., J. chem. Phys. 66, 5231 A977). 89a. Streett, W. B. and Tildesley, D. J. Faraday Disc. Chem. Soc. 66, 27 A978). 89b. Narten, A. H., Sandier, S. I., and Rensi, T. Faraday Disc. 66, 39 A978). 89c. Ref. 83b studies in detail the dependence on the size and position of the auxiliary site. It is found, for example, that if the auxiliary site C is placed at the centre of an AA molecule, no effect in gAA occurs if the diameter <rc of site C is small enough to be contained in the overlapping volume (co-sphere) of the two A spheres. Also, the effect on gcc is considerable; unphysical (gCc(r)<0) gccS result unless <rcc is sufficiently large. These defects are not surprising, since auxiliary and real (interacting) sites are treated on the same footing in RISM. 90. Hsu, C. S., Chandler, D., and Lowden, L. J. Chem. Phys. 14, 213 A976). 91. Streett, W. B. and Tildesley, D. J. J. chem. Phys. 68, 1275 A978). 92. Barojas, J., Levesque, D., and Quentrec, B. Phys. Rev. A7, 1092 A973). 93. The p-route cannot be used here since RISM does not yield the full g(ro)ioJ), which is needed in the p-equation (see §6.7.3) to calculate <ri2cos YaeXw However, for (atomic) hard spheres, the A-route (§6.7.4) gives the same value as the p-route. The L/-route (§ 6.7.2) gives trivial results of course for hard spheres. 93a. Sullivan, D. E. and Gray, C. G. Mol. Phys. 42, 443 A981). 93b. H0ye, J. S. and Stell, G. J. chem. Phys. 66, 795 A977). 93c. This was stated without proof by D. Chandler [Farad. Disc. Chem. Soc.
438 INTEGRAL EQUATION METHODS 66, 74 A978)] for the case of G2 for a symmetric (ACA) linear triatomic. General proofs for G, and G2 are given in ref. 93a. See also App. IOC. 93d. This might appear to be correct by analogy with the formally exact result C.119) for the molecular cA2). However, as we discuss later, this relation is in fact incorrect. 93e. See ref. 93a; other references and derivation are given in Chapter 10. 93f. Since d and e depend on integrals over all r of haB(r), we should perhaps refer to them as 'long-wavelength' (fc —» 0) properties. 94. Topol, R. and Claverie, P. Mol. Phys. 35, 1753 A978). 94a. Equation E.249) leads to unphysical results in the overlap region, how- however, as shown later in this section. 94b. See for example, the Rayleigh expansion (B.89); alternatively choose the Z axis in E.254) along k. 94c. The relation E.264) and others below containing a> ' are somewhat formal, since oj~' does not exist for fc = 0. (The nature of the divergence of a) ' can be found, and exploited; see § 5.5.3a.) 94d. Equation E.257) is also invalid on other grounds. For a homonuclear AA diatomic liquid, for example, E.257) implies that a function of four variables fi(ra>iCiJ) is determined by a single function hAA(r) of one variable, which seems unreasonable. Equation E.257) is an additive SSA (site superposition approximation); multiplicative SSAs have also been proposed (see C.127)). 94e. For the diatomic AA case, the result follows simply from the remarks in ref. 83h. 94f. Nezbeda, I. and Smith, W. R. Chem. Phys. Lett. 82, 96 A981); Naumann, K.-H. and Lippert, E. Ber. Bunsenges. phys. Chem. 85, 650 A981); Naumann, K.-H. Ber. Bunsenges. phys. Chem. 86, 519 A982). (Related methods are also discussed in the last two references.) The site-site /¦-expansion reference potential u"B(r) is defined by exp[-Cu^B(rnS)] = (exp[-CuA2)]X,,loB, where (. . .)„,„,, is done with fixed ap distance r,>B; i.e. the two molecular centres are shifted to the two sites, and {. . .)am is then carried out. We can also define a site-site u'lB(r) as in u-expansion perturbation theory, uaB(rae) = (,u(\2)),,,lO>2; however, the former procedure would be expected to be superior for strongly anisotropic cores. 95. Ref. 5, eqn E.106). 95a. Pearson, F. J. and Rushbrooke, G. S. Proc. Roy. Soc. Edin. A64, 305 A957); Munster, A. Statistical mechanics, Vol. 2, Springer, Berlin, p. 716 A974). 95b. Alternatively,7911 we can decompose the particle-particle potential in the atomic fluid UaB into intra (bonding) and inter (site-site) parts, UnB(r) = U"a(r) + Uaij(r). Instead of varying the density, we can vary the strength of K,p(r) (cf. ref. 83h). 95b'. This defect is not on the same footing as the other two, in that an orthodox graphical expansion (or any graphical expansion for c for that matter) is not a rigorous requirement. However, it is convenient to have such an expansion. 95c. See eqn A0.69) for the asymptotic behaviour of haB(r). 95d. Cummings, P. T. and Sullivan, D. E. Mol. Phys. 46, 665 A982). 95e. For charged sites, on the naive assumption caB(r) —> ~(iuaB(r), we expect cal3(k) to contain a term proportional to qaqBk~2. The result in the text shows that, for diatomics for example, the coefficient of fc is different from the naive expectation qa%.
REFERENCES AND NOTES 439 95f. See Cummings, Morriss, and Stell83" and ref. 79h. 95g. Kojima, K. and Arakawa, K. Bull. Chem. Soc. Japan. 53, 1795 A980). 95h. In ref. 83p the wall-RISM analogue of c' is introduced, and closures worked out. 95i. For fluid/solid interfaces, the new theory with PY-type closure gives results for the short-range structure of about the same overall quality as in RISM83p for a hard diatomic A A fluid at a hard wall (see Chapter 8). Also, for an AB diatomic fluid there are no difficulties in solving the equations analogous to those found in RISM;83p Sullivan, D. E., Thurtell, J., and Gray, C. G. unpublished. 96. Lemberg, H. L. and Stillinger, F. H. Mol. Phys. 32, 353 A976). This is also called the 'central force mod".!'. 96a. The relation between the particle-particle and site-site approaches is explored graphically in ref. 83c. 96b. To assist in the solution, the transformation to short-range components Or), fQr) [cf. E.85)], and variational methods, are available.96 96c. More recently the solution has been obtained at 1 g/cm3 and 302 K; see Thuraisingham, R. A. and Friedman, H. L., /. chem. Phys. 78, 5772 A983). 97. Rahman, A., Stillinger, F. H., and Lemberg, H. L. J. chem. Phys. 63, 5223 A978). 98. Stillinger, F. H. and Lovett, R. J. chem. Phys. 48, 3858 A968); Mitchell, D. J., McQuarrie, D. A., Szabo, A., and Groeneveld, J. J. stat. Phys. 17, 15 A977); van Beijeren, H. and Felderhof, B. U. Mol. Phys. 38, 1179 A979). 98a. The relations between these particle-particle moment conditions for GaB(r) of non-rigid molecules and the moment conditions for the site-site hae,{r) of rigid molecules (the latter are mentioned briefly in the last section and discussed in detail in Chapter 10) are discussed by H0ye, J. S. and Stell, G. J. chem. Phys. 67, 1776 A977). 98b. Murad, S., Gubbins, K. E., and Gray, C. G., Chem. Phys., 81, 87 A983). 99. In principle one can apply 'quantum corrections' to the gs calculated by classical integral equations. Alternatively, one can devise quantum ver- versions of the integral equations - see ref. 137 in Chapter 1, ref. 4 of this chapter and Ripka, G. Phys. Reports 56, 1 A979). 100. Noble, B. Methods based on the Wiener-Hopf technique for the solution of partial differential equations, Pergamon, New York A958); Roos, B. W. Analytic functions and distributions in physics and engineering, Wiley, New York A969). 101. Baxter, R. J. Phys. Rev. 154, 170 A967). 102. Baxter, R. J. Aust. J. Phys. 21, 563 A968). 103. For reviews of applications to various atomic fluid problems, see ref. 4, and Cummings, P. T., Ph.D. Thesis, University of Melbourne A979). For representative recent work see, e.g. H0ye, J. S. and Blum, L. J. stat. Phys. 16, 399 A977); Cummings, P. T. and Smith, E. R. Chem. Phys. 42, 241 A979); Niizek, K. Mol. Phys. 43, 251 A981). 104a. By this we mean the problem is reduced to algebraic equations and quadratures. In some special cases (e.g. PY for hard spheres) the solution , can be reduced to a closed form expression. 104b. Kohler, F., Perram, J. W., and White, L. R. Chem. Phys. Lett. 32, 42 A975) [iterative/perturbationa! solution of factorized OZ equation].
440 INTEGRAL EQUATION METHODS 105. Dixon, M. and Hutchinson, P. Mol. Phys. 33, 1663 A977); Jolly, D., Freasier, B. C, and Bearman, R. J. Chem. Phys. 15, 237 A976) [used to extend simulation data for h(r)]. 106. Lebowitz, J. L. Phys. Rev. A133, 1 A964). 107. Wertheim, M. S. ref. 6, p. 11-282. 108. Baxter, R. J. J. chem. Phys. 52, 4559 A970). 109. Hiroike, K. and Fujui, Y. Prog, theoret. Phys. 43, 660 A970); Hiroike, K. J. Phys. Soc. Japan 27, 1415 A969); Prog, theoret. Phys. 62, 91 A979). 110. Blum, L. and Stell, G. J. stat. Phys. 15, 439 A976). 111. Sullivan, D. E., Levesque, D., and Weis, J. J. J. chem. Phys. 72, 1170 A980). 112. Parrinello, M. and Giaquinta, P. V. J. chem. Phys. 74, 1990 A981). 113. Wertheim, M. S. J. chem. Phys. 73, 1398 A980); 74, 2466 A981). 114. Maclnnes, D. A. and Farquhar, I. E. J. chem. Phys. 64, 1481 A976). 1 15. We assume that the fluid is not at the critical point. 1 16. In stochastic theory, for example, the (real) power spectrum of a random variable is essentially the square of the Fourier component of the (real) random variable. 1 17. For potentials u(r) of finite range R, EA.9) is satisfied in PY and MSA approximations. In general one expects c(r) to be of short range if u(r) is (cf. § 3.1.5). In any case, by taking R sufficiently large, the error in c(r) (and h(r)) can be made arbitrarily small. [It therefore seems plausible that the final results EA.17), EA.18) can be generalized to the case that EA.9) is replaced by a weaker requirement,'"9 that c(r) fall off 'suffi- 'sufficiently fast' at large r.] 118. If q(r) is eliminated between EA.17) and EA.18) the original form E.10) of the OZ relation can be recovered.109 This shows that the OZ E.10) and EA.17), EA.18) are equivalent. [Those readers who do not wish to follow the details of the derivation of EA.17), EA.18) can therefore justify these relations by eliminating q(r).] From either10'9 E.10) or2 EA.17), EA.18) one can also derive the alternative1"' but more compli- complicated Baxter 'factorized' or 'reduced range' form of the OZ relation. 119. The method is a general one; whenever h{r) is known for r<R, EA.18) gives q(r) for r<R, and EA.17) then gives c(r) for r<R. An application to the molecular MSA for higher multipoles is mentioned following E.116). These remarks apply to the case c(r) = 0 for r>R. More generally (e.g., atomic MSA with Yukawa tail) if c(r) is known for r>r0 and h(r) is known for r<r0, EA.18) and EA.17) (with R—*°°) determine q(r) and then c(r) and h(r) for all r [G. Stell and P. T. Cummings, personal communications]. 120. Smith, W. R. and Henderson, D. Mol. Phys. 19, 411 A970); Perram, J. W. Mol. Phys. 30, 1505 A975). 121. Since for large fc, In S(k)'1 =ln[l-pc(k)] =-pc(k), the integral EA.33) is well defined. 121 a. See ref. 79f; this generalizes the atomic fluid results of M. S. Green23" and T. Morita and K. Hiroike,26 and the LHNC results of § 5.4.10. 122. The identity EB.13) follows from differentiating the identity lndet(A) = Trln(A). The latter identity is easily proved using the representation in which A is diagonal, with eigenvalues A,, A2 . . . along the diagonal. The LHS is then !n(A,A2 . . .), and the RHS In A,+ ln A2+. . . (since/(A) has eigenvalues /(A)). The two sides are obviously equal.
APPENDIX A SPHERICAL HARMONICS AND RELATED QUANTITIES ... in the early days .. . there was much resistance to the adoption of invariance principles as everyday working tools. This resistance disappeared in the course of the years and is not even understood by the new generation of physicists. Eugene P. Wigner, Rev. mod. Phys. 37, 595 A965). The theory of spherical harmonics can be approached from various viewpoints,25 e.g. special function theory,273 potential theory,346 quan- quantum theory of angular momentum,13'5"8105 or the representation theory of the rotation group.216926 The rotation group approach is perhaps most relevant in fluid theory because we are constantly dealing with physical quantities (tensors) which are either invariant under rotations37 (e.g. intermolecular potentials, pair correlation functions, Mayer /-bonds, etc.), or transform under rotations in some other simple way (e.g. dipole moments, quadrupole moments, polarizabilities, etc.). Invariance or symmetry properties lead to two important simplifications in the theoretical work: (a) They assist in deriving (spherical harmonic) expressions for various physical quantities, e.g. the pair potential (§ 2.3,2.4), the multipole moments (§ 2.4.3), the polarizabilities (Appendix C), the pair correlation function (Chapter 3), etc. (b) They assist in the calculation of various statistical mechanical averages, e.g. thermodynamic (Chapter 6), polarization and alignment (Chapter 10), etc. by means of standard properties available for the spherical harmonics. These properties are derived in the representation theory itself, and lead to useful and well documented5"8 auxiliary quan- quantities such as the Clebsch-Gordan coefficient (§ A.3), the rotation matrix (§ A.2), and the / symbols (§ A.5). Sectionst A.1-A.3 give a compendium of properties for the spherical harmonics, the rotation matrices, and the Clebsch-Gordan coefficients. For proofs of these relations we refer for the most part to the standard sources listed at the end of this Appendix. A discussion of the / symbols, which is to a large extent self-contained, is given in § A.5 from the point of view most relevant to irreducible spherical tensors. Irreducible spheri- spherical tensors themselves, which are physical quantities transforming under rotations like spherical harmonics, are discussed in § A.4. This section also considers reducible spherical tensors, and Cartesian tensors (both t A detailed Table of Contents for this Appendix is given on p. x.
442 APPENDIX A A.I reducible and irreducible), together with the relations between Cartesian and spherical tensors. A.I Spherical harmonics Definition We consider a right-handed coordinate system (Fig. A.I) and the stan- standard polar angles (x> = 6(t> [where O<0<7r, 0<<?<2tt] defined by x = r sin 6 cos 4>, y = r sin 6 sin <p, (A.I) z = r cos d. ""> (xyz) or (re<f>) Fig. A.I. Polar coordinates rd<f>. For m>0 the complex spherical harmonics Ylm(d4>)= Ylm(<i>) [/ = 0, 1, 2, ...; m = -/,...,+/] are defined by \ 4tt 7 \(l + m)\J xP|m(cos0)eim<(', (m>0) where i=-J~ 1, and for m<0 one uses (A.2) (A.3) where m=—m. For a definite / there are 21 + 1 independent functions. We note the phase (—)m in (A.2), which is the standard3510 Condon and Shortley phase convention.! A consistent set of phases for the t'rhe phase (—)'" holds for positive m only. The phase factor can be written i*1^ which because of (A.3), is valid for all m.
A.I SPHERICAL HARMONICS 443 spherical harmonics, rotation matrices and Clebsch-Gordan (CG) coeffi- coefficients must be followed; the Condon and Shortley convention is the most commonly used. Ref. 38 (p. 17) lists other conventions used in the literature. In (A.2) Plm(cos 6) is the associated Legendre function ,(x) (A.4) where P((x) is the Legendre polynomial, defined for x in the range (—1,1), and is given by ^()' (A5) B1-1)!! n L B)B/-i) |x- + ...] (A.7) where {21- 1)!! = B0!/2'i! = {21- l)Bi - 3)... C)A). In (A.6) the sum over fc runs over the integers fc = 0, 1, 2,. .. . The upper limit in (A.6) can be put equal to [112] where [Q] denotes the integer part of Q (e.g. [§] = 2, [3] = 3), since (? —2fc)!=c° for larger values of k. The polynomial (A.7) contains even powers for / even (and therefore ends with x°), and odd powers for I odd (and then ends with x1). P((x) is a polynomial of degree /, parity (—)' (i.e. even function for I even, odd function for I odd), and has I zeros in the range (— 1,+1). The explicit values of the lowest order polynomials are P0(x)=l, Pi(x) = x, P2(x) = K3x2-l), (A.8) and so on. Higher order polynomials can be obtained from (A.7), or from the recurrence relation . (A.9)
444 APPENDIX A A.I The generating function for the P|S is (l-2rCos0 + r2)-2= ? p,(Cos0)f(, (|r|<l), (A.9a) 1 = 0 and they satisfy the orthogonality relation I d(cos 0)P,(cos 0)Pr(cos 0) = 2B1 + I) 5,r. (A.9b) J-i From (A.4) and (A.8) we find Poo(cos0) = l, P,(,(cos 0) = cos 0, P,, (cos 0) = sin0, P20(cos 0) = 1C cos20 - 1), P2i(cos 0) = 3 sin 0 cos d, P22(cos 0) = 3 sin20, P30(cos 6) = |E cos30 - 3 cos 0), (A. 10) P31(cos 0) = | sin 0E cos2 0 - 1), P32(cos 0) = 15 sin20 cos 6, P33(cos 0) = 15 sin30, P40(cos 0) = |C5 cos40 - 30 cos20 + 3), P41(cos 0) = | sin 0G cos30 - 3 cos 0), P42(cos 0) = ^ sin2eG cos2e -1), P43(cos 0) = 105 sin30 cos 0, P44(cos 0) = 105 sin40 etc. Also, we have P,[(cos0) = B/~l)!!sin'0 Pl[_!(cOS 0) =COt 0P,,(COS 0). In the subsections below we summarize some of the properties of the spherical harmonics. For a detailed treatment the reader should consult Rose5 or Edmonds,6 for example. Other notations Ylm(d<t>) = YIm(o>) = Y,m(r) = Y,m(r) where f = r/r is a unit vector in the direction of r.
A.I SPHERICAL HARMONICS 445 The Racah spherical harmonics C|m@<?) are defined by4'8 Ylm(e<t>) = (^^fclm{e<t>). (A.ii) The Y|m(co) are normalized to unity with respect to the argument u> (i.e. Jdco | Y,m(co)|2= 1), and unnormalized with respect to index m (Sm l^im('t))|2 = B1 + 1)/4tt). C|m(co) has the opposite properties. Eigenfunctions5'10 The Ylm(d4>) are simultaneous eigenfunctions of the differential operators lz and I2, where I = —i(rxV) [the angular momentum operator, or generator of rotations]. The factor of —i is included to agree with quantum theory (apart from an h factor used there); it makes I hermitian (i.e. Jdtt)A*lB =Jdco(lA)*B). The operators I2 and lz commute, so that simultaneous eigenfunctions are possible.10 The operator I is given in polar coordinates by L = i( sin d> hcot d cos / = —), =i(-cos<? — + cote sin 4>—), (A. 12) \ dd d4>/ so that f = ll+ly + ll is given by sinfl We have (A. 14) Operators of the form exp(—ian.l), where n is a unit vector in an arbitrary direction, are called rotation operators,510 and n. 1 the generator of rotations about n. For example, note that exp(-ialz)f(<t>) = A - ialz +. . .)/(<?) = /(<?-«) (A. 15) where (A. 12) and Taylor's expansion are used. Putting exp(—ia^)/ = /', and <p + a = <p', we can write this as which has the obvious interpretation that the value of the new function at
446 APPENDIX A A.I Fig. A.2a. Active rotation through angle a. Fig. A.2b. Passive rotation through angle a. the new point equals the value of the old function at the old point. This defines the (active) rotation of the function / into /'. (The rotation is called active since the space points (or body) are rotated, keeping the coordinate system fixed (see Fig. A.2a)). Thus lz generates rotations about z, etc. In a passivet rotation, on the other hand, the space points (or body) remain fixed and the axes are rotated (Fig. A.2b). In a passive rotation (A. 16) is valid, with <?' = <?-a (cf. Fig. A.2b). Thus a clockwise passive rotation is equivalent to a counter clockwise active rotation, as is also clear geometrically. In the general case, where exp(-jan .1) performs the rotation of angle a about axis n, the resulting relation between a> = 04> (or 4>6x) and a>' = d'4>' (or 4>'d'x') is more complicated. Since we shall usually consider rotations about one (or more) of the axes (X, Y, or Z) we do not discuss + Throughout this book we use passive transformations (rotations, inversion, translations).
A.I SPHERICAL HARMONICS 447 the general case37a explicitly here. The relation between w and u>' can also be expressed in terms of the Euler angles characterizing the rotation, but again we shall need only the existence of the relation [see (A.45) and discussion] and not the explicit relation. For small angles a of rotation, the small change 5/ of a function f(a>) is given by the leading term of the Taylor expansion of exp(—ian . 1)/, i.e. 8f = f'-f=-Uxn.lf=-an.VJ (A. 16a) where in the last step we have introduced the angular gradient operator V<u = A (see next section). One may also need the transformation law for the Cartesian compo- components (x, y), instead of for the polar coordinates (r, 4>) of point P in Fig. A.2b. (We later consider the three-dimensional case.) The components in the rotated frame, (x1, y'), are given by x' = (cos a)x + (sin a)y, y' = —(sin a)x + (cos a)y. (A.17) The transformation (A.17) is passive for a >0 (counterclockwise). An active clockwise rotation of (x, y) gives the same formal result as (A. 17). In Fig. A.3 we illustrate schematically active and passive rotations in three dimensions. Later (see Fig. A.6) we discuss three-dimensional (pas- (passive) rotations quantitatively. We also note the commutation relations among the la [(„/,] = ilz (andcycl.) (A. 18) where [A, B] = AB —BA denotes the commutator. These follow algebrai- algebraically from (A. 12), or geometrically391 from the fact that the product of Z' (a) Y (b) Fig. A.3. Active (a) and passive (b) rotations in three dimensions. In (b), a rotation fi = <f>0x (see Sec. A.2 for precise definition of Euler angles) brings the old coordinate system XYZ into coincidence with the new one X'Y'Z'. The point r has orientation to with respect to XYZ and o>' with respect to X'Y'Z'.
448 APPENDIX A A.I the four infinitesimal rotations in the order (a), +e about X, (b), +e about Y, (c), -e about X, (d), ~e about Y, is a second-order rotation of -e2 about Z, i.e. Ry(-e)Rx(-e)Ry(e)Rx(e) = Rz(~e2). Using jRx(e) = exp(-i8/x) = (l-ie/x-382/2), etc. and equating coefficients of g2 gives (A. 18). Angular gradient operator We introduce here the angular gradient operator for a point, or a linear molecule (with rotational degrees of freedom d<p). On p. 462 we generalize to a rigid-body of general shape (with rotational degrees of freedom 4>6x)- The gradient operator V = id/dx+j d/dy + kd/dz can be decomposed into radial and tangential parts, V = f -fxl dr r AIALA (A.18a) dr rdd rsin0d<? The angular gradient operator V^ is denned by l = -iVO) = -irxV (A. 18b) so that Vu = rxV. Hence we have 1*1 ¦*. ^ = -6 + <{> — . (A.18c) sin 6d<t> 90 The § and 4> components of VM are given by (A.18c); the space-fixed X, Y and Z components are given by Vu = i\ and (A.12). We also have Vl = -l2. (A.18d) Note in particular that V^ is not equal to the tangential part of V given in (A. 18a). However, Vf, is essentially equal to the angular part of V2 (see next section). Potential functions34 The operator I2 is essentially the angular part of the Laplacian V2 =
A.I SPHERICAL HARMONICS 449 where r2 dr dr _<?_ 2d_ ~2 " rdr2' (A.19) \dr r i and V2 = —I2 is given by V2=-^— — sin0" + -r^—^-5, (A.20) The functions designated solid harmonics, are homogeneous harmonic polynomials38 in x, y, z; i.e. B/lm satisfies Laplace's equation V2B/jm =0, and each term in ¦2/^ is of degree I. Examples: f\azxy). tJ 2 (Further examples are given by (A.63)). The second set34 of fundamental solutions of Laplace's equation is YlmF<t>)lr'+\ (r^O). (A.23) The solutions (A.21) and (A.23) are often referred to as the regular and irregular solutions respectively. It is easy to verify (A.21) and (A.23) satisfy Laplace's equation. Thus
450 APPENDIX A A.I from (A.19), (A.18d) and (A.14) we get V2[r" Ylm] = (V2r") YIm + rn~2(VlYlm) = [n(n+l)~l(l+l)]r"~2Ylm (A.23a) which clearly vanishes for n = ( and n = — (!+ 1). An arbitrary solution of Laplace's equation can be written32 as a linear combination of the fundamental solutions (A.21) and (A.23). Recurrence relation LYlm=[l(l+l)-m(m±l)$Ylm±l (A.24) where l±= lx±it,. (The positive square root is to be taken in (A.24) which is consistent with the Condon and Shortley phase convention.) From (A.12)weget () (A.25) The quantities (± are step up and step down operators, respectively. The relations (A. 14b) and (A.24) can also be combined into the single relation5 lYlm = (-y[l(l+l)fC(lll; m + vvm) YIm+v (A.26) where {„ denotes the spherical components (see (A. 167)) of I: lo=lz,l±x = ^2'Kl±ily),, and C denotes a Clebsch-Gordan coefficient (§A.3). Orthogonality Jdco YIm(a))* Yrm.(a>) = 5,rSmm. (A.27) where Sn- is the Kronecker delta, and jdco^j d<t>\ d(cose) = J d<t>\ d0sin0 = 47r. (A.28) Completeness One can expand an arbitrary function of the angles f(a>) in spherical harmonics " /M = I/lmYlm(o,), (A.29) lm where the expansion coefficients are given by flm = | dtuYIm(«)*/(*)). (A.30)
A.I SPHERICAL HARMONICS 451 Substituting (A.30) in (A.29) we see that we must have (A.31) where (A.32) is the angular Dirac delta function. Equation (A.31) is the completeness relation. In this book we often write the expansion (A.29) as /(w) = Zim/im^tm(w)> which is then in manifestly rotationally invariant form- (see § A.4.2). Addition theorem5 Ylm(a,)* Y!m(co') = 7) (A.33) where y is the angle between u> and co' (Fig. A.4). Equation (A.33) is proved later (see (A. 181)). For the case / = 1 (A.33) becomes cos 7 = cos 6 cos 6' + sin d sin 6' cos(<? — < i.e. the law of cosines of spherical trigonometry. For w' = co (A.33) becomes v-, , >,-, 21+1 m +/^" which is sometimes referred to as Unsold's theorem. z (A.34) (A.35) Fig. A.4. The angle y between the directions cu = (A.34)). and w' = 8'4>' (see (A.33) and
452 APPENDIX A A.I Product rule5 x C(!,/2I; OOOJCd^al; m1m2m)Ylm(a)) (A.36) where C(lil2l', mlm2m) is a Clebsch-Gordan coefficient (see § A.3). The inverse of (A.36) is obtained using (A. 141) to be rBf1 + DBi2+i)i| I D^B1+1) JC«.M;000)YIm(a,) D^B1+1) = I C(M2l;m1m2m)YllMl(a,)YIjm2(a,). (A.37) mim2 Note that the arguments of the spherical harmonics in (A.36) and (A.37) are all the same. Integrals J dco Ylm (co) = Dir)l 8,o Sm0, (A.38) * Ylim,(co) = S,,,2 Smim2, (A.39) Jda>Yl,m»*Ylim»YIimi(a>) " hhl3; mim2m3) (A.40) The relation (A.40) is sometimes referred to as Gaunt's formula. Integrals over products of more than three spherical harmonics can be evaluated using the product rule (A.36) and (A.39). 5^0 Transformation under rotations and the inverse transformation YIm(o)) = Z D'm,m(n~l) Y,m.(o)') (A.42) where a> and a>' are the orientation of r with respect to XYZ and X'Y'Z', respectively, and ?1 denotes the rotation carrying XYZ into coincidence with X'Y'Z'-see Fig. A.3(b) and § A.2 (we use the passive interpre- interpretation (see p. 446) of rotations here). Here D'mm<(?l) is the rotation matrix
A.I SPHERICAL HARMONICS 453 (§ A.2) exactly as definedt by Rose or Messiah (see (A.64)). Using D(fT!) = D\Q) = DT(fi)* [see p. 464 and eqn (A.99)], where DT is the transpose of D, we write (A.42) as VImM = lDL,.(n)*Ylm.(o>) (A.43) The interpretation of (A.41) is that under rotations the Ylm of the same I transform among themselves. The precise meaning of this statement is as follows. We recall (eqn (A. 16)) that a function /(o>) is rotated into a new function f'(u>') using /(co) = /V) (A.44) where a>' is a definite function of co co' = co'((o) (A.45) and conversely co = w(co') (A.46) The transformation relation (A.46) also depends on the particular rota- rotation Cl, and a more precise notation is &> = co(co'; Cl). We substitute114 this value of co into Yim(w), thereby generating a function of co' (and f2). The result can be put in the form (A.43), where the D'mm.(Ci)* are the coefficients of the Y,m-(co'). In the language of vector spaces, one says that the B1 + 1) functions Ylm span a space which is invariant under rotations. This space has the further important property that it is irreducible (i.e. it contains no invariant subspace). Thus, no subset of Yims, or linear combinations, can be found which transform under rotations only among themselves. The rotational transformation relation for spherical harmonics can also be pictured geometrically. For example, consider the (active) rotation of Yoo. This spherically symmetric function obviously transforms into itself under rotations. As another example, consider the three / = 1 functions Yn, Y10, Yli. These are equivalent (by linear combinations) to the functions x/r, y/r, z/r, which are figure-eight-like lobes along the X, Y, and Z axes, respectively (i.e. the familiar px, py, pz orbitals). If one of the lobes is rotated in some arbitrary way, the result is intuitively seen to be a linear combination of the three original lobes: no higher or lower har- harmonics are mixed in. The / — 1 space is therefore invariant under rota- rotations. The irreducibility property (with respect to arbitrary rotations) is t One must be careful to note the differences in the D-matrices of various authors. Brink and Satchler8 give a number of comparisons. We repeat that, in our convention for (A.41) and (A.43) we use the passive interpretation of rotations, and the explicit D-matrices of Rose5 and Messiah10 (see eqn (A.64)).
454 APPENDIX A A.I also clear intuitively in the latter case since an arbitrary rotation of one lobe will be a linear combination of all three original lobes. (By contrast, the / = 1 space is reducible with respect to rotations about the Z-axis, since z/r is invariant under these rotations, and (jc/r, y/r) transform among themselves; thus the three-dimensional / = 1 space is reducible into two invariant subspaces with respect to these restricted rotations). Inversion (parity) Inversion in the origin is defined actively and passively in Figs. A.5(a) and A.5(b), respectively. The polar coordinates ((o) = F4>) of a point trans- transform into (-w) = (tt-6, 4> + tt). We have Ylm(-a,) = (-)'Y,mM. (A.47) Thus, even (odd) order spherical harmonics have even (odd) parity. (a) -Z (b) Fig. A.5. (a) Active inversion: the point r transforms into r' = — r. Thus the Cartesian coordinates of the new point are (x'y'z') = {—x, -y, — z). (b) Passive inversion: the axes transform into X' = —X, Y' = — Y, Z' = —Z. Thus the new Cartesian coordinates of a point are (x'y'z') = (-x, -y, -z). The gradient formula57'8 We denote by V,, the spherical components (see (A.167)) of the gradient operator V;V,, = V2, V±1 = =F2^(VX ±iVy). Then V,,(/(r)Y,m(w)) = C(lll + l; mvm + v) - <-> x/(r)Y,_lm+v(a>) where f(r) is an arbitrary function of r. (A.48)
A.I SPHERICAL HARMONICS 455 Corollaries >'Y,mM) =-C(/l(-1; mwn + k) l-lY^m+A<o\ (A.49) ,,(YlmM/rI+1) = -C(lll + 1; mwn + v)Bl +1) (A.50) Example: Proof of (A. 50) This relation is used a few times (see, e.g., Appendix C and Chapter 11) so that a short derivation is perhaps of interest. We expand Vv(Ylm(co)/r1+1) in spherical harmonics where AIT'lM are the unknown expansion coefficients. Since Vv trans- transforms as D\ and Y,m as D1, the left-hand side (of (A.51)) transforms under rotations according to D^D1, and therefore the expansion coeffi- coefficient must contain the Clebsch-Gordan coefficient C(llL; vmM) (cf. (A.182), and analogous arguments in §§2.3, 2.4.3, and 2.4.5): Arrtf(r) = CA1L; vmM)AUL(r) (A.52) where the factor AUL{r) is independent of the ms. Since V2 commutes with V the quantity Vv(Yim(co)/r'+1) is a potential function (cf. (A.21), (A.23)) and the r-dependence of A\T™(r) must be /-~<L+1); no terms in r1' are allowed as V^(Y(m(o))/rl+1) vanishes for r—»oo. Also the dimension of Vv(V,m/rl+1) is (inverse length)i+2, so that only L = l+1 contributes. Hence (A.51) becomes V,(Ylm(co)/r'+1) = A,C(lll + 1; vmv + m) xYl+lv,+m(co)/r1+2 (A.53) where M=v+m is used, and A, is some remaining constant. To find A,, take the special case v = m = 0, and a> = @0). Since r is along z, we can put V0 = d/dz =d/dr, so that we have l;000) x Y1+10@0)/r1+2. The values of Y,o@0) and CAH + 1;OOO) are given by (A.55) and
456 APPENDIX A A.I (A. 162). Hence we find which completes the derivation of (A.50). Special cases For co = 0<? we have For raaO we have f, o) . M>" 2 / (A.55) (A.55a) m = 0, f + modd (A.55b) with jc!!=jc(jc-2) ... D)B) for x even, and ) (A.55c) (A.56) P,,(cos0) = B/-l)!!sin'0, (A.57) Pu-iicos 6) = B1- l)V. cos 6 sin'^tf (A.58) where B/ —1)!! is defined on p. 443. Pim(l) = Plm(-l) = 0, m^O, (A.58a) P,()(l) = l, P10(-l) = (-)', (A.58b) = 0, /-m = 2s + l, (A.58c) P,(D = 1, (A.59) Pi(-l) = (-)', (A.60) P,@) = 0, I odd
A.I SPHERICAL HARMONICS 457 The lower order harmonics are given explicitly by loo = (A-62) G \i i — 2-Ecos30-3cos0), Air; 2 Y40 = {-^-f - C5 cos40 - 30 cos2© + 3). \4ijv 8 In terms of the Cartesian coordinates (A.I) the above relations can be written, using r cos 0 = z, r sin 6 e'* = x + iy, (A.62a)
458 APPENDIX A A.2 as (A.63) V 30z V A.2 Rotation matrices510 Definition The rotation matrices (also called the representation coefficients of the rotation group) D'mn(Cl) [where I = 0, 1, 2,. . . ; m, « = -(,...,+/] have been defined in eqn (A.41) as the transformation law for the spherical harmonics Ylm(w) under rotations. The Ds are functions of the rotation Cl which carries the initial frame XYZ into coincidence with the final frame xyz. One usually choosest to represent ft by the three Euler angles <j>dx (see Fig. A.6). + Other choices of three parameters to label a rotation are possible, e.g. Caley-Klein parameters (or quaternion parameters - see footnote p. 463), axis of rotation42 (two parameters) together with angle of rotation, direction cosines (see (A.122); because of (A. 125) only three of the nine direction cosines are independent), etc.
A.2 ROTATION MATRICES 459 Fig. A.6. The Euler angles <p8x- The space-fixed axes are OXYZ and the body-fixed axes are Oxyz. Hie line OV (the intersection of the XY and xy planes) is called the line of nodes. Note that x = X'", V = Y", y = Y'", z = Z" = Z'", Z = Z', Z. YOY' = <?, LX'OX" = Euler angles The rotation fl = <?#\- can be carried out in two equivalent ways: Method I (i) rotate positively by <f> @<<?<2it) about the Z axis, thereby bringing XYZ into position X'Y'Z', (ii) rotate by 0(O<0<it) about Y', thereby bringing X'Y'Z' into position X"Y"Z", (iii) rotate by x@<x<27r) about Z", thereby bringing X"Y"Z" into the final position X'"Y'"Z'"=xyz. A positive rotation about an axis is one such that the positive direction of the axis advances in the direction in which a right-handed screw advances. In the above method the rotations are carried out in the body-fixed or moving frame of reference; in applications discussed in this book, the xyz frame will be attached in some definite way to a rigid body (molecule). On can show,5 algebraically or geometrically, that the same overall rotation is effected by carrying out all three rotations in the original space-fixed axes XYZ, provided they are carried out in reverse order, i.e. (i) rotate by x about Z, (ii) rotate by 6 about Y, (iii) rotate by <f> about Z. We note in passing (Fig. A.6) that the polar coordinates of z in XYZ are F, 4>) and the polar coordinates of Z in xyz are F, tt-x)- We also note that the inverse of ?l = (<f>,6,x) is H = {-y, -6, ~<t>) = (tt — x, 6, Tr — <f>), where the first form is strictly speaking incorrect, as we have defined <j>, 0, x to be positive. For the special case of linear molecules (N2, HC1, CO2, C2H2, etc.),
460 APPENDIX A A.2 with the z axis along the line of molecular symmetry, only two angles @, <f>) arc necessary; in this case the third Euler angle x >s redundant and can be put equal to zero. For an intuitive picture of the Euler angles, see Fig. 2.5 The notation a(iy=<f>dx is also used for the Euler angles (see, e.g., (A.120, 123)). Definition of Dlmn The D'mnD>dx) are given explicitly by D'mnD>dX) = e-'"">-d'mn(d) e-*, (A.64) where the real function dlmnF) = D'm,,@d0), which represents a rotation of 6 about Y, is given byt (-)fc(cos0/2J'+m-"-2fc(sin0/2Jk~m+" . , . (A.65) f (/ + m - k)\ (I- n - k)\ kl (k - m + n)\ The ds can also be expressed in terms of hypergeometric functions or Jacobi polynomials.6-7-27 Symmetries5 Dllnn(<fNx) = {~~)m+"Dlmn(<fNx)* (A.66) where m=—m. <*!„„(») = (-)m+BdL@). ,AA^ (A.67) '(e) dl(e) Eigenfunctions2''8 The functions Dlmn(<f>6x)* are simultaneous eigenfunctions of the differen- differential operators Lz, Lz and L2 LzD'mn(Q,)* = mD'mn(Q,)*, (A.68) f In (A.65) the sum over k can be taken over values such that the arguments of the factorials in the denominator are positive, since (—x)!=°°. Hence d'mn(8) contains a finite number of terms (see (A.114), (A.115) for the 1=1 and \ = 2 cases).
A.2 ROTATION MATRICES 461 where43 L has the following space-fixed and body-fixed axes components respectively: / d d d\ Lx = —i\ -cos d> cot d sin d> hcos d> cosec d— I, \ dd> dd dx/ I d d d\ LY = —i\ —sin d> cot d hcos<? 1-sm d> cosec d— , (A.69) \ dd> dd dx' d and / a a d\ Lx = — ilcos x cot d hsin y cos y cosec 6— , \ dx dd dd>) Ly =-((-sinxcot 6 hcosx~: +sin x cosec 0—-), (A.70) \ 9x 9^ dd>J so that L2 = LX+Ly+L| = L2 + L2 + Lj is given by L2 r i a5inQa , i (d2 , d2 Lsin 6 dd dd sin2 d \dd>2 dx2 (A.71) The operators (Lx, LY, Lz) are the generators of rotations of a rigid body about the space-fixed axes X, Y, Z respectively. They satisfy the usual (cf. (A. 18), valid for a point or particle) commutation relations [Lx, LY]= iLz, etc. The operators (Lx, Ly, Lz) are the generators of rota- rotations about the body-fixed axes x, y, z. They satisfy the 'anomalous' commutation relations [Lx, !*,,] =—iLz, etc. [One can derive the 'anomal- 'anomalous' commutation relations algebraically44^7 from (A.70), or geometri- geometrically from the fact that the product of the four rotations in the order (i), +s about x, (ii), +s about y, (iii), —e about x, (iv), —e about y, is a net rotation of +s2 about z (cf. analogous derivation of (A.18)).] Since rotations about the space-fixed axes commute with those about body- fixed axes, one also has [Lx, L,,] = 0, etc. The eigenvalue problem (A.68) arises in quantum mechanics for the spherical top.21843 For a freely rotating top the Hamiltonian is H = L2/2J, where I is the moment of inertia. Hence (A.68) provides an immediate solution to the Schrodinger equation (see CD.48) and follow- following discussion). Similarly, the Ylm(8<t>) are the solutions of the
462 APPENDIX A A.2 Schrodinger equation for a freely rotating dumbell in three dimensions (cf. CD.42)), and the functions eim<* are the solutions of the Schrodinger equation for a freely rotating dumbell in two dimensions. Just as for a point or linear molecule (cf. p. 448) one defines an angular gradient operator Vn and angular Laplacian V^ by L = -iVn, L2 = -V^ (A.71a) with L and L2 defined by (A.69) and (A.71). Equations (A.69) and (A.70) together with Vn = i'L, give the Cartesian components of Vn, analogous to (A. 12) in the linear (or point) case. It is not possible to introduce in a simple way an analogue of (A. 18c), the angular components in the linear case, since «j> and 0 vary from point to point in a rigid body. As for the linear, or point, case the operator exp(— ian.L.) rotates a function f(w) = fD>6x)- In particular, for small angles a of rotation (about axis n), the change 8f = f'~ f in a function /(co) is given by (cf. (A. 16a)) 8f=-ian-Lf=-an-Vnf, (A.71b) Note on notation The alternative notation J is often used for L and/or I (see, e.g. Appendix 3D). Also, when no confusion can arise (i.e. when body-fixed components (lx, /y), A^, Ly, Lz) etc. are not usually being considered), the notation (lx, ly, lz), and (Lx,Ly,Lz), (Ax,Ay,Az), and (Txx, Txy,...), etc. are usually used for simplicity for the space-fixed components (/x, lY, lz), (Lx, Ly, Lz), (Ax, Ay, Az), (Txx, TXY,. ..) [see, e.g. earlier in the appen- appendix, and also Appendix B, Chapters 2, 10 etc.]. [The body-fixed (usually principal axes) components are then denoted by (ix., Jy-), (Lx-, Lv., Lz.), (Ax., Ay-,AZ), (Tx-X., TyV,. . .) if they need be introduced (see, e.g. Chapter 2 and (A.213b)) at all.] Generalized (four-dimensional) spherical harmonics1*8" The functions D'mn(<f>dx)* furnish a set of spherical harmonics in four dimensions; hence it is not surprising the Ds and Ys have analogous properties. For fixed /, the B1+ IJ functions Dlmn(<f>dx)* form an or- orthogonal set of spherical harmonics of order 21. To prove the above assertions one transforms from the four- dimensional Cartesian coordinates (xix2x3xA) to the polar coordinates
A.2 ROTATION MATRICES 463 usingt x2 = rsin@/2)cos[D>-x)/2], x3 = r cos@/2)sin[(<? + x)/2], (A.72) x4 = r cos@/2)cos[D> + *)/2]. The Laplacian V2 = ?¦ 32/3x? becomes 7 4 , V2 = V2 + -V^ (A.73) where la a l / a2 a2 „ a2 \ sin 0— + —y~ —5H 5-2cos d sin 6 86 86 sin26 \8<t>2 dx 3<A &x' so that Vn = —L2, where L2 is the operator defined in (A.71). Hence, from (A.68) we have ^DL(n)* = -i(i+i)DL(nr (a.76) showing that the Dlmn(Sl)* are indeed four-dimensional spherical har- harmonics. The four-dimensional potential functions are r2lDlmn(nr, DL(fl)*/r2l+2 (A.77) which are analogous to (A.21) and (A.23). In (A.76, 77) we have 1=0, 5, 1, §,...; nowhere else in this book do we require half-integral ( values. In particular the expansion (A.82) given below, when applied to a rigid body in three dimensions, requires only 1 = 0,1,2,.... Recurrence relations (Lx ± iLv)DL(n)* = W + 1) - '"(m ± l)$Dlm±Un(Cl)*, (A.78) (I, ±i^D'^Q.)* = [Id + 1)- n(n =F l)fD^nT1(nr. (A.79) Orthogonality | (^) 8h. Smm. 8m. (A.80) t Various other transformations to polar coordinates are possible in four dimensions, and in fact more common. Biedenharn51 discusses the more common cases, and the corresponding forms of the