INFRARED AND RAMAN
SELECTION RULES
FOR MOLECULAR AND
LATTICE VIBRATIONS
The Correlation Method

Infrared and Raman
Selection Rules
for Molecular and
Lattice Vibrations:
The Correlation Method
William G. Fateley and Francis R. Dollish
MELLON INSTITUTE
PITTSBURGH, PENNSYLVANIA
Neil T. McDevitt and Freeman F. Bentley
AIR FORGE MATERIALS LABORATORY ()
WRIGHT-PATTERSON AIR FORGE BASE, OHIO
WILEY-INTERSGIENCE, A DIVISION OF JOHN WILEY & SONS, ING.
NEW YORK ' LONDON • SYDNEY . TORONTO

Copyright © 1972, by John Wiley & Sons, Inc.
All rights reserved. Published simultaneously in Canada.
No part of this book may be reproduced by any means, nor
transmitted, nor translated into a machine language
without the written permission of the publisher.
Library of Congress Cataloging in Publication Data:
Main entry under title:
Infrared and Raman selection rules for molecular and lattice vibrations.
1. Crystal lattices. 2. Vibrational spectra.
3. Representations of groups. I. Fateley, William G.
QD921.148 548'.81 72-4120
ISBN 0-471-25620-X
Printed in the United States of America
10-9 8 7 6 5 4 3 2 1

INFRARED AND RAMAN
SELECTION RULES
FOR MOLECULAR AND
LATTICE VIBRATIONS
The Correlation Method

PREFACE
The application and development of the correlation theorem
presented in this book grew out of conversations with Prof^or
James R. Durig at a NATO Meeting on the Solid Sta/^ ^id m
Delft, Holland, in 1968. At that time we were only frnidly interested
in the derivation of selection rules for molecul?jr vibrations in solids.
However, we found the application of the correlation theorem of
group theory, as suggested in the original papers by Halford and
Hornig, quite an enjoyable J&sk and certainly a time-saving,
foolproof method for generating selection rules for crystals.
A seminar on this subject given that fall encouraged us to write
down some practical rvtfes for the application of this method to
various crystals. As w,^ became more adventurous in our studies of
crystals, we found, xo our amazement, that the application of the
correlation method was incorrectly applied to a number of crystals
reported in the literature. The main difficulty seemed to be the
improper choice of a correlation table relating the site group to
the factor group. Wishing to remove this obstacle, we constructed
a table that now provides the correct correlation for each site
whenever choices exist.
We have attempted to present adequate examples so that the
reader- may better understand this method and to provide a number
of shortcuts that reduce this compilation to only a few minutes
work.

vi Preface
An outgrowth of this effort was the construction of general tables
for the vibrational selection rules in molecules. Certainly Herzberg's
excellent book, Molecular Spectra and Molecular Structure, Vol. II,
Infrared and Raman Spectra of Polyatomic Molecules, provided many of
these tables. Herzberg does not provide tables for all the molecular
point groups that are physically possible; therefore we decided to
extend them. In the tables presented here we have identified the
sets of equivalent atoms present in molecules by their site symmetries.
This approach is consistent with the method used in crystals.
We have not attempted to predict the selection rules for nonrigid
molecules and for crystals beyond the boundary k — 0 of the
Brillouin zone, but, we are sure that the reader can make this
extension with the proper application of the more expanded double
groups and group theory.
Even though we are only applying well-known principles of the
correlation theorem in group theory according to the rules
provided by Halford and Hornig, we continue to find new enjoyment
inlhJ '?c1xT'^vat^on °f these selection rules. We hope you will.
We should like to acknowledge many helpful discussions with
Dr. Gerald L. NG5rJson? Mr. Peter Larsen, Professor William
White, Professor Coln7, Farmer, and Professor F. A. Miller. We
wish to thank Carolyn Ke*1' f°r ner patience in translating and
typing our illegible chicken ti^cks into a manuscript. One of us,
William G. Fateley, wishes to acknowledge partial support from an
unrestricted grant provided by the Gulf ^search and Development
Foundation. We also greatly appreciate the partial financial
support received from the Mellon Institute\of Science of Carnegie-
Mellon University and from Air Force Contract No. F 33 615-70-
1382.
Willia^. G. Fateley*
Francis R.. Dollish
Neil T. McDevitt
Freeman F. Bentley
Pittsburgh, Pennsylvania
Wright-Patterson Air Force Base, Ohio
February 1972
* Present address: Head, Department of Chemistry,
Kansas State University, Manhattan, Kansas 66502

CONTENTS
CHAPTER 1 PRACTICAL METHODS FOR SELECTION RULES 1
CHAPTER 2 SITE SYMMETRY AND CORRELATION TABLES 35
CHAPTER 3 THE BHAGAVANTAM AND VENKATARAYUDU
METHOD 53
CHAPTER 4 MOLECULAR SELECTION RULES 65
CHAPTER 5 APPLICATION AND SPECIAL CASES 117
APPENDIX I SITE SYMMETRY TABLE FOR THE BRAVAIS
SPACE CELL 171
APPENDIX II CHARACTER TABLES 181
APPENDIX III CORRELATION TABLES 201
INDEX 217
vii

CHAPTER ONE
PRACTICAL METHODS FOR
SELECTION RULES
With the recent increase in interest in infrared and Raman
spectra of crystals, it has become important to determine which
vibrational modes are optically active. Hornig [1], Winston and
Halford [2], and Bhagavantam and Venkatarayudu [3]
pioneered in developing methods for solid-state selection rules.
Heretofore the application of these rules has been a laborious
procedure fraught with difficulty and with many points of
indecision. Among the latter is the choice of the primitive cell and
the correct site symmetry of each atom. What is needed is a
short, straightforward, foolproof method. In this book we have
outlined some practical rules for the use of the correlation
Pages 1 to 42 of this book originally appeared as "Infrared and Raman
Selection Rules for Lattice Vibrations: The Correlation Method," by W. G.
Fateley, Neil T. McDevitt, and Freeman F. Bentley, Applied Spectroscopy, 25,
No. 2, 155-173 (March/April, 1971).

2 Practical Methods for Selection Rules
method to derive the vibrational selection rules for both crystals
and molecules. These practical rules reduce the calculation to
only a few minutes work. The correlation procedure which has
already been discussed in several papers and books dealing with
lattice vibrations [2-5], is explained in detail by the use of
numerous examples. We have chosen not to review the theory
but to proceed directly to a demonstration of its use, and that of
the correlation tables, to obtain the vibrational selection rules
for solids.
An orderly procedure is outlined for the step-by-step
calculation of selection rules that will predict infrared and Raman
activity. The reader is warned that there may be slight
variations in some of the steps used in this method, but intelligent
reasoning will help him to overcome them.
1. CRYSTAL STRUCTURE
The crystal structure of the sample must be known. Alternatively,
a structure can be assumed and predictions made for the
vibrations, which can then be compared with observations to
prove or disprove the assumed structure. It is far better,
however, to know the structure in advance.
Crystallographic information may be obtained from Refs.
7 and 8 or from the original literature. Examples of the data
needed are given in Table 1.
2- MOLECULES PER BRAVAIS SPACE CELL
The Bravais space cell is used by molecular spectroscopists to
obtain the irreducible representation for the lattice vibrations.
The crystallographic unit cell may be identical with the
Bravais cell or it may be larger by some simple multiple. This
can be ascertained from the capital letter in the x-ray symbol
for the crystal structure. For all crystal structures designated by

2. Molecules Per Bravais Space Cell
TABLE 1
Crystallographic Information for Several Examples
Crystal structure
nomenclature [7, 8]
Crystal
Molecules Lattice Molecules per
per unit points [5] Bravais cell,
-" (Z/LP)
per unit points [5] Br;
X-Ray Spectroscopic cell (Z) [7,8] (LP) ZB
SrTiO3
TiO2 (anatase)
ZrO2
a-Al2O3
Cu2O
NH4Ia
P2i/c
/to/.
p
P
1
4
4
2
2 ■
2
1
2
1
1
1
1
a Phase III [6].
a symbol containing P (for primitive) the crystallographic unit
cell and the Bravais unit cell are identical. (An example from
Table 1 is Pm2m for SrTiO3.) Crystal structures designated by
other capital letters (B, C, /, etc.) have crystallographic unit
cells that contain two, three, or four Bravais cells. (An example
from Table 1 is I& /amd for TiO2.) The irreducible
representations obtained from these crystallographic unit cells will
contain two, three, or four times as many vibrations as are needed
to represent the lattice vibrations of the crystal. This problem
of including too many Bravais cells in the crystallographic cell
can be eliminated by dividing the number of molecules per
unit crystallographic cell by a small integer which is identical
to the number of lattice points (LP) in a crystallographic cell
of specific symmetry, as designated by the capital letter in its
symbol. Table 2 gives this number (LP) which reduces the size
of the crystallographic unit cell to the desired Bravais space cell.
This reduction has been included in Table 1.
In summary
number of molecules
in the Bravais space =
cell
number of molecules in
Z crystallographic unit cell
(LP) number of lattice points
(from table 2)

Practical Methods for Selection Rules
TABLE 2
The Number (LP) that Reduces the
Crystallographic Unit Cell to the
Bravais Space Cell
Type of
crystal structure
A
B
C
F
I
P
R
Number (LP)
2
2
2
4
2
1
3 or la
a Here the crystallographic group may have
already been decreased by three; if so, the
crystallographic cell need not be divided.
A simple indication whether to divide by
three is found in the example of a-Al2O3
(Table 1) which contains two molecules per
unit cell. Certainly we should not divide by
three in this case.
3. SITE SYMMETRY OF EACH ATOM
IN THE BRAVAIS CELL
The equilibrium position of each atom lies on a site that has its
own symmetry. This site symmetry, a subgroup of the full
symmetry of the Bravais unit cell, must be ascertained correctly
for each atom. It is easy to do so in some cases, difficult in
others. Let us consider the following examples.
Cu2O
Table 1 states that the symmetry is 0\ and that there are two
Cu2O units in a Bravais cell. There are therefore four equivalent

3. Site Symmetry of Each Atom in the Bravais Cell 5
copper atoms and two equivalent oxygen atoms in the Bravais
unit cell.* Next we turn to the table in Appendix I (p. 179) and
look for the entry 0* in the third column. It is number 224. In
the right-hand column all the possible site symmetries for this
space group are tabulated. They are written as 7^B), 2D3(fD),
ZJdF), . . . , and are given in full in Table 3. They represent
all the possible kinds of site for an 0* crystal, but most of them
will not be occupied in a specific crystal.
TABLE 3
Site Symmetries for the Space Group Designated
O\ or Pn3m or 224 (Cu2O)
Bravais cell
site symmetry
TdB)
3(il /
X/ O/f v ^ /
C3 (8)
X)aA2)
3C2B4)
C.B4)
CiD8)
Number of equivalent
atoms accommodated on
this site of Bravais cell
(number in parentheses)
2
4
6
8
12
12
24
24
48
Number of kinds of
sites of this symmetry
(the coefficient of
column 1)
1
2
1
1
1
1
3
1
a
a No coefficient is needed because there are an infinite number of
Cx sites.
The most useful information is the number contained in
parentheses, for it represents the number of equivalent atoms
which have that particular site symmetry; for example, TdB)
indicates that there are two equivalent atoms occupying sites of
* This rule is always applicable, provided that the equivalent atoms have been
found in a crystallographic cell. This information is provided with the crystallo-
graphic structure in Refs. 7 and 8 (p. 170).

6 Practical Methods for Selection Rules
symmetry Td; similarly, D3dD) indicates the presence of four
equivalent atoms on DZd sites.
Some of the site symmetries have numerical coefficients, such
as 2ZKdD) in Table 3. The coefficient 2 shows the presence of
two different and distinct kinds of DZd site in this unit cell. Each
can accommodate four equivalent atoms. In a given crystal
there may be atoms on one or both sites or on neither. The
second and third columns of Table 3 illustrate these remarks.
For Cu2O the x-ray results show four equivalent copper
atoms and two equivalent oxygen atoms. What will be their site
symmetries? From Table 3 (or from the equivalent entry in
Appendix I, p. 179) we see that only one site symmetry can
accommodate four equivalent atoms—ZKd. Therefore the site
symmetry for copper is DZd. Similarly, only one kind of site can
accommodate just two equivalent atoms—Td. This therefore
is the site symmetry for oxygen. (Note. When selecting the site
symmetry, we must always have the number of equivalent
atoms equal to the accommodational value of the site
symmetry.)
The above example was atypically simple in that there is no
ambiguity in the result. We turn now to another example that
is slightly more difficult.
TiO2 (Anatase)
Table 1 shows us that the space group is designated DH or
/4 /amd9 with two molecules per Bravais unit cell. There are
therefore two equivalent titanium atoms and four equivalent
oxygen atoms in this Bravais cell. From Appendix I (p. 176)
we find that this is spacegroup number 141 which has the site
symmetries 2D2aB), 2C2ftD), Ct,D), 2C2(8), C.(8), and ^A6).
Consider first the two equivalent titanium atoms. Only the
D2d sites accommodate two atoms; therefore it follows directly
that the titanium atoms are on sites of D2d symmetry. There are
two separate kinds of site (coefficient 2), but it is not necessary
for us to know which kind is occupied.

4. Correlation of the Site Group to the Factor Group 7
For the four equivalent oxygen atoms there are two possible
site symmetries—Cih and C2v; both will accommodate four
equivalent atoms. One or the other will be correct, but
additional information is needed to make the choice. For this we
turn again to the crystallographic tables [7, 8], which state that
the oxygen atoms lie on C2v sites. (See Chapter 2 for an
explanation of the use of crystallographic tables.) This additional
information is needed whenever we meet an ambiguity of this
kind.
Table 4 lists the site symmetry of each atom in the various
examples used in this chapter.
TABLE 4
Site Symmetry of Each Atom in Various Examples
TiO2,
Example: anatase SrTiO3, Cu2O a-Al2O3, ZrO23 NH4I
Site of Ti-Z>2d Sr-(\ Cu-D3d Al-C3 Zr-Cx NH4-Z>2d
equivalent O-C2v Ti-O,, O-Td O-C2 O-Q I-C4v
atoms O-D^h
4. CORRELATION OF THE SITE GROUP
TO THE FACTOR GROUP*
The site symmetry for each atom in the lattice has already
been found and the results are summarized in Table 4. The
symmetry species are now identified for each equivalent set of
atom displacements in the site. The displacements we describe
will become the lattice vibrations in the crystal. Knowing the
site species for these displacements, we find that the correlation
tables, which appear in Appendix III (p. 201), relate each
* Note. Different authors vary in their choice of the term factor group, crystal
group, or correlation group to describe crystal symmetry. These terms are
equivalent.

8 Practical Methods for Selection Rules
species of site group to a species of the factor group. This
correlation explicitly identifies the species of the lattice vibration in
the crystal and further allows prediction of infrared or Raman
activity. Using the following molecules as examples, we first
identify the lattice modes in the crystal by obtaining the
irreducible representation that contains the number and species
of the lattice vibrations, and second, we describe the infrared
and Raman activity of each vibration.
TiO2 Crystal
As summarized in Table 4, the two titanium atoms are in
D2d sites and the four oxygen atoms are in C2v sites. In this
example each set of equivalent atoms is treated separately.
Titanium Atoms
First, the vibrational displacements of the titanium atoms in
the lattice can be described as simple motions parallel to the
x, y> or z axis. This simplified description of vibrational mode
allows easy classification into one of the species of the site
symmetry—D2d; for example, the displacements of the titanium
atoms parallel to the z axis will have the same character as the
translation in the z direction. The translation Tz belongs to
the species B2 of the site group. Therefore the atom
displacements parallel to the z axis will also belong to the Bz species.
Similarly, the displacements of titanium atoms along the x axis
will have the same character as Tx and will belong to species E.
It is important to note here that this approach, which classifies
the lattice vibrations as excursions in x,yy and z directions, is no
different from the descriptions used for molecular vibrations
such as bond stretching, bending, and twisting. Of course,
normal vibrations in a crystal or a molecule are far more complex
than this simple displacement picture provides; however, the
importance of this method is found in the simplicity with which
the lattice vibrations can be classified.

4. Correlation of the Site Group to the Factor Group 9
When the species of the site group is identified for each
excursion of an equivalent set of atoms, this information is
integrated via the correlation tables to the species of the crystal
that contain this lattice vibration. To begin this correlation
procedure Table 5 lists a portion of the D2d site group and
identifies the species of the translations TX9 Tv, and Tz; see Appendix
II (p. 183) for character tables. Since the lattice vibrations have
TABLE 5
Species of the Site Group D2d and the Translations
D2d site of
titanium atom
species
Translation
species
Ti atoms
excursion
Motions parallel to
z axis
Motions parallel to
x andjy axes
the same character as the translations, the species that contain
these vibrations can readily be identified and this information
is presented in Table 5. Before applying the correlations of site
to factor group we define some useful terms to help in the
practical application of this method.
1. ty = the number of translations in a site species y. This
number can take the value of zero, one, two, or three,
depending on whether none, one, two, or three translations
are contained in the site species y, respectively. This
information is readily available from the character table in
Appendix II (p. 181). Ry = the number of rotations
included in the site species y. Again this value will be zero,
one, two, or three. The character tables in Appendix II
clearly identify the rotations as Rx, Ry, and Rz.

10 Practical Methods for Selection Rules
2. fy = degrees of vibrational freedom present in each site
species y for an equivalent set of atoms, ions, or molecules.
This can be calculated as follows, where n is the number of
atoms (ions or molecules) in an equivalent set:
fyR = degrees of rotational freedom present in each species
y for an equivalent set of ions or molecules. This can be
calculated by modifying A) to give
fk = Ry ■ n O)
3. ay represents the degrees of freedom contributed by each
site species y to a factor group species £. The value of ay can
be calculated as follows:
B)
The derivation of this equation is not presented here;
however, it is stated in B) that the degrees of freedom in the site are
equal to the degrees of freedom in the factor group for each
equivalent set of atoms, ions, or molecules.
4. Q = the degeneracy of the species £ of the factor group.
An additional superscript y may sometimes be added to show
its correlation to a species of the site group. The usual values
of Cc are summarized in tabular form on the facing page.
There are exceptions to this description of degeneracy for
certain correlations in which separable degeneracy exists.
Without a proof the following modification to the existing

4. Correlation of the Site Group to the Factor Group 11
Speciesa Value of C^
A 1
B 1
E 2
F 3
G 4
H 5
a Usually the species
designation has a superscript, e.g.,
A\ A", or a subscript, e.g.,
Alsp Eg, Flu; however, these
super- and subscripts in no
way describe the degeneracy
of the species and for this
reason are not included here.
correlation tables, Appendix III (p. 201), gives the correct
correlation and ay values:
1. Point group C6, C8ft, C4ft, C5h, CQh, S6, T, and Th do not use
the 2 coefficient which appears in these correlation tables
for the doubly degenerate Et species. These elements, found
in Appendix III, are marked with an asterisk (*) to indicate
the necessity for dropping the 2 coefficient.
2. Only a portion of the T and Th point group correlation
tables is given; however, the table must be modified as
follows:
F A + 2E Fg Ag + 2Eg A + 2E
Fu Au + 2EU A + 2E
Here a coefficient of 2 is added to the E species which must be
doubled because of the separable degeneracy. A double
dagger ($) is used in the table in Appendix III to call

12 Practical Methods for Selection Rules
attention to those entries in which a 2 coefficient must be
added to the Ei entry of the table.
convenient ghegks. It is helpful to check the bookkeeping as
the correlation method progresses. The following equation,
when applied, will help to avoid errors.
3n = (degree of freedom) sifce = 2 fy C)
V
Zn = (degree of freedom)facfcorgroup = £ HCl D)
where a^ = ]Ty S and -W ^s the total number of atoms in the
Bravais cell; i.e., N = 2eq sets n-
The irreducible representation of the crystal gives the number
of lattice vibrations in each species of the factor group. The
total irreducible representation of the crystal, rcrysfc, is the
combined irreducible representation of each equivalent set of
atoms, reqsefc [see G)].
The Peq set is constructed in the following manner:
Teq set = 2 aC ' £ (^)
C
where a^ as previously defined, is the number of lattice
vibrations of the equivalent set of atoms in species f of the factor
group. The total irreducible representation of the crystal,
Fcryst, can be constructed as follows:
pcryst p , -p , /c\
1 — L eq set 1 T~ -1 eq set 2 "T" V°7
This irreducible representation of Pcryst contains the
acoustical vibrations. In the following examples these vibrations are
removed from this representation by simply substracting out the
irreducible representation of the acoustical vibrations:
pcryst _ pcryst pacoust /j\

4. Correlation of the Site Group to the Factor Group 13
Now F^st is the irreducible representation of the lattice
vibrations in the crystal.
This procedure needs only minor modification to include the
intramolecular vibrations and librations for molecular crystals.
Here the irreducible representation of a molecular crystal can
be defined as
rmol cryst -pcryst . t-i i T1 "pacoust /o\
vib — -»■ vib "T 1 mol vib ~r 1 lib ~~ L \fi)
TABLE 6
Titanium Atoms on Site D2d. The Degrees of Vibrational
Freedom for Each Species
(n = 2 atoms/eq set)
Degree of vibrational freedom
D2d species Translation ty fy = n • t7
E
0
0
0
1
2
0
0
0
2
4
The molecular crystal NH4I (p. 25) demonstrates the usefulness
of (8).
Utilizing the above definition, we list in Table 6 the degrees
of vibrational freedom for each species of the site group D2d for
the equivalent set of titanium atoms.
Table 6 indicates the presence of the titanium lattice
vibrations designated as degrees of freedom in species Bz and E.
The next step is to correlate the B% and E species of the site
group D2d to the ZL7l factor group species. The correlation
tables are given in Chapter 2 for D2d to D^ and are also
presented in Refs. 10 and 11.

14
Practical Methods for Selection Rules
By extracting only a portion of these correlation tables in
Chapter 2 (p. 42) we find the following relationship between
the site and factor group species:
site group
species
A*
factor group
species
A complete set of correlation tables needed in this procedure
appears in Appendix III (p. 201). We have chosen to present in
Chapter 2 (p. 42) two specific tables to show primarily how they
are derived and to provide the proper basis for selection of the
correct correlation ta.bles when two or more possibilities exist.
Since only the site species B2 and E contain these translations,
which are like the lattice vibrations in the crystal, the
correlations relating these species to those in this factor group are of
immediate interest. By integrating the site species which contain
the translations into the factor group by use of the correlation
tables it is easy to identify these lattice vibrations in the factor
group species. Table 7 shows this correlation and identifies the
species of the lattice vibration in the crystal.

4. Correlation of the Site Group to the Factor Group
15
TABLE 7
The Correlation for the Lattice Vibrations of the Titanium
Atoms in TiO2 Crystal Between The Site Group D2d and
Factor Group D4A
f
D2d site
species, y
correlation
DAh factor
group ay
species, £ Q a$ — aB + a2
1 1=1
1 1 =
2 1 =
2 }
1
0
0
1
The titanium atom's irreducible representation for the factor
group is obtainable with Equation 5: V = ^ a^ • £, where
#c = 2y av i-e-> ^e number of vibrations in species £. Therefore
the species of the factor group that contains lattice vibration
involving the titanium atom can be written as the following
irreducible representation FTi.
- 1 • Blg + 1 • A2u + 1 - Eg + 1 • Eu.
A check can be made at this point for possible errors by
utilizing Equations 3 and 4.
Equation 3: Degrees of vibrational freedom of equivalent
Ti atoms in site = 3n = 6 = %yfy = 6.
Equation 4: Degrees of vibrational freedom of equivalent
Ti atoms in factor group = 3rc, where
2C aft = 1 + 1 + 2 + 2 - 6 = 3n for
n = 2.

16
Practical Methods for Selection Rules
Oxygen Atoms
By following the same procedure we can obtain the
irreducible representation Foxy for the equivalent set of oxygen
atoms. A summary of the necessary information is given in
Table 8.
TABLE 8
Tabulations of Terms and Correlations Necessary to
Calculate the Lattice Vibrations of the Oxygen Atom in
TiO2 Crystal
C2v site
fy ty species, y
correlation
D4h factor
species.
= aAl + aBl + aB%
4 1 (Tz)Ai
4 1 (Tx,
4 1
1 1
1 0
1 1
1 0
2 2
1 0
1 1
1 0
1 1
2 2
+ 0 +
+0 +
+ 0 +
= 0 + 1 + 1
= 0 + 0 + 0
= 1+0 + 0
= 0 + 0 + 0
1
0
+ o +
+ 1 +
CHECKS
Equation 3:
= Zn =
degrees of freedom for
>the equivalent set of
Equation 4: 2 acQ = ^n = 12 I oxygen atoms.
The number and species of oxygen lattice vibrations can now
be calculated for Foxy =
= 2 H • £ =
0Alu + IA
2U
0B2g
0Blu
2E9
\B
2u
2EU

4. Correlation of the Site Group to the Factor Group 17
COLLECTING TERMS
roxy = Alg + Blg + 2Eg + A2u + B2u + 2EU
The total representation of the crystal, rcryst, can be
calculated by utilizing Equation 6, where rcryst is the sum of the
individual irreducible representation for each set of equivalent
atoms, or
rcryst = rTi + roxy,
+ (Alg + Blg + 2Eg + Aiu + B2u + 2EU)
= Al9 + 2A2u + 2Blg + B2u + 3Eg + 3EU
Applying a check at this point on the vibrational degree of
freedom, we find that Equation 4 gives 3N = ^ eqsets a? • C?,
where N is the number of atoms in the Bravais cell for TiO2
N = 6. Therefore
3 • N = 18 = WAig + 2CMu + 2CBig + \CBu + 3CEa + 3CEa
The acoustical vibrations are included in the irreducible
representation, rTi°2 cryst, given above. Of the 3N degrees of
vibrational freedom, three of these vibrations are acoustical
modes. When we consider only those vibrations at the center of
the Brillouin zone, i.e., k ^ 0, the three acoustical vibrations
have nearly zero frequency. Since vibrations with zero
frequency are of no physical interest here, these acoustical
vibrations can be substracted from the irreducible representation
as suggested in Equation 7:
rcryst -pcryst -pacoust
vib = 1 — 1
The acoustical modes are readily identifiable in factor groups,
since they have the same character as the translation; Table 9
shows this identification.

18 Practical Methods for Selection Rules
TABLE 9
D47i Factor Group, Translation and Acoustical Modes
Translation Acoustical mode
D^h speciesa species species
Eu Tx.y V
a See Appendix II (p. 188) for D±h point group.
Therefore the irreducible representation of the acoustical
vibrations racoust = AZu + Eu.
The results of this factor group analysis, which identifies the
number of lattice vibrations in each species and the spectral
activity, are summarized in Table 10.
Table 10 gives the following selection rules for first-order
Raman and infrared activity in the TiO2 crystal:
raman spectrum : Six fundamental lattice vibrations allowed
one Algy two Blg, and three degenerate
infrared spegtrum: Three fundamental lattice vibrations
allowed one A2u and two degenerate Eu.
One vibration, B2u, will be inactive in both the infrared and
the Raman spectrum.
Also, there will be no coincidences, i.e., the same vibration
mode which is active and observable by both the Raman effect
and in the infrared spectrum.
This completes the original goal of obtaining (a) the number
of lattice vibrations in TiO2 (anatase) and (b) the spectral
activity of these vibrations. Additional information may be had
by studying the polarization properties of Raman scattering.
This procedure, however, is not discussed here.
SrTiO3 Crystal
Now that the step-by-step procedure has been applied to
obtain the molecular vibrations and activity for TiO2 crystal,

TABLE 10
D4h Factor Group Species, Translations, Acoustical Modes, Number of Lattice Vibrations and
Infrared and Raman Activity of TiO2 Crystal, Anatase
Raman
D^h factor Translation Acoustical rTi°2 cryst T™2 Infrared polarization Raman
group species species mode species coefficients coefficientsa activityb tensor species activity0
Air * * i&xx + Kyy)) &ZZ V
Bxl 2 2 (axx - <zyy) V
B a a/
Eg 3 3 **™*,* V
Au Tz 1 2 1 V
B2\ 1 1
Eu TXtV 1 3 2 V
a pTiO2 __ pcryst TiO2 pacoust
V'b = (Alg + 2A2u + 2Blg + B2u + 3Eg + 3EU) - (A2u + EJ
— Alg + A2u + 2Blg + B2u + 3Eg + 2EU.
These coefficients are the number of lattice vibrations present in the species.
b Only those species that contain the translations are infrared active; i.e., A2u and Eu are the only species
that have infrared active vibrations (see Appendix II, p. 188).
c Those species that contain the polarizability tensor can have Raman activity; i.e., Alg> Blg, B2g, and Eg can
have Raman active lattice vibrations. This information is readily available from Appendix II.

20
Practical Methods for Selection Rules
there are several short cuts which, when applied, reduce the
calculation to only a few minutes. SrTiO3 crystal serves as an
example for this simplified procedure.
INFORMATION
crystal: SrTiO3, PmZm-Oxh, ZB = 1 (seeTable 1).
equivalent atom site: Sr-On(n = 1); Ti-Oh (n = 1); oxy-
atoms — Dih {n — 3) (see Table 4).
Irreducible representation of each equivalent set of atoms:
STRONTIUM
3 3(T
Therefore
TITANIUM
r f
Q Q / rj-i >
\ X 1/ Z
0h site
symmetry
species
containing
translation
) Flu
c
0h site
symmetry
species
containing
translation
) F
' r Lu
correlation
nr - Y = 1 • F
correlation
0h factor
group
species
Flu
tu
0h factor
group
species
F
Q
3
3
1
1
Summary:

4. Correlation of the Site Group to the Factor Group 21
OXYGEN
f
D±h site
symmetry
species
containing
translation
correlation
0h factor
group
species
= aA%u + a
Eu
3 1G*
6 2G*
z) ^2w
■*■ oxy ==
A acoust —
77
^_ rlu
F2u
Flu
F2u>
m)
3
3
o
j
1
0
+
+
1
1
Summary :
pSrTiO2
1 cryst vib
_j_ p i p p
Sr "T -1 Ti ~T ■»- oxy -1- acoust
i« + F2u)
The 0A character tables, Appendix II (p. 198), identify Flu
as infrared active and F2u as neither infrared nor Raman active.
Therefore SiTiO3 has three infrared active fundamental
vibrations and no first-order Raman spectrum. In Chapter 3
(p. 64) is an identical irreducible representation by a different,
more laborious method.
Cu2O Crystal
information: 0%-Pn3m9 ZB = 2 (see Table 1).
equivalent atom-site: Cu-DM (n = 4); oxy-Td (n = 2) (see
Table 4).

22 Practical Methods for Selection Rules
Irreducible representation of each equivalent set of atoms:
COPPER
f
D3d site
symmetry
species
containing
translation
correlation
0h factor
group
species Q
= aA
+ a
Eu
4 1(TM)
8 2(TM)
1 1
2 1
3 2
3 1
1 + 0
0 + 1
1 + 1
0 + 1
OXYGEN
2u
f
Td site
symmetry
species
containing
translation
correlation
0h factor
group
species
Q
P
acoust — •* lu
Irreducible representation of the crystal
pCuaO p i p p
-1- vib — x Cu ~r x oxy L acoust
= (A2u +EU+ 2Flu + F2u) + (Flu
= AZu +EU+ 2Flu
2u
F2g.
- (Flu)
Spectral activity: Raman — Fig; infrared — Flu

4. Correlation of the Site Group to the Factor Group 23
Therefore there will be one triply degenerate fundamental
lattice vibration (F2g) active in the Raman effect and two triply
degenerate infrared active lattice vibrations (Flu).
A12O3 Crystal
information : RZc-D\d:> ZB = 2 (see Table 1).
equivalent atom-site: Al-C3 (n = 4); oxy-C2 (n = 6) (see
Table 4 and Chapter 2).
ALUMINUM
Cz site
symmetry
correlation ^ r
species —: >- DZd factor
containing group ay
fy ty translation species Q a^ = aA + aE
\{TZ) A^__ Alg 1 1 = 1 + 0
1 1 = 1+0
8 2G; y) E--- ^<^^c ~E* 2 2-0 + 2
1 1 = 1 + 0
1 1 = 1+0
2 2 = 0 + 2
Here we observe that aA and aE have different values.
Reviewing Equation 3, fy = ay 2/Q gives the values of ay. The
dyS are evaluated in the following manner:
Site species A:
fA = 4 = aA{CAu + CAu + CAlu + CAJ = aAD); .: aA = 1
Site species E:
fE = 8 = aE(CEs + CBu) = aEB + 2); /. aE = 2
Then

24
OXYGEN
Practical Methods for Selection Rules
f
C2 site
symmetry
species
containing
translation
DZd factor
group
species
12
A
A,
1
0
1
1
0
1
0
2
2
0
2
2
r0xy = -di, + 2AZg + 3Eg + Alu + 2A2u + SEU
J- acoust ^ ^2w "f" -^it
Summary: Equations 7 and 8 give
W» = (^l9 + A2g + Alu + A2u + 2Eg + 2Ea)
+ (Alg + 2A2g + 3Eg + Alu + 2A2u + 3EU)
. - (A2u + Eu)
r*}|p. = 2A%* + 3iC + 2Ji°' + 2A£?} + 5E™ +
ZrO2
information: P2 -C\h, ZB = 4 (see Table 1).
equivalent atom site : Zr-Q (re = 4); oxy-Cj (n = 8) (see
Table 3).
* Here we can indicate the activity of each species by a superscript: (R) =
Raman active; (IR) = infrared active; @) = inactive. This information is
available from the character tables, Appendix II (p. 185).

4. Correlation of the Site Group to the Factor Group 25
ZIRCONIUM
f
Cx site
symmetry
species
containing
translation
correlation
C2h factor
> group
symmetry
12
1 3
1 3
1 3
1 3
OXYGEN
Zr = 3Ag + 3Bg + 3AU
3BU
fi
Cx site
symmetry
species
containing
translation
correlation
C2h factor
group
symmetry
24
1 6
1 6
1 6
1 6
Summary:
r0xy = QAg + 6Bg
•*• acoust = A-u ~f" ^u
6BU
92?iR>
NH4I Crystal (phase III)
Durig and co-workers have made extensive investigations of
the molecular crystal NH4I [6]. This crystal not only possesses
lattice vibration but also libration and intramolecular
vibrations of the NH^ group in the crystal. It is worthwhile to

26 Practical Methods for Selection Rules
repeat Durig's calculations, with some modifications, to
demonstrate the usefulness of the correlation method when
applied to a molecular crystal.
A natural division in applying the correlation method to a
molecular crystal can be made as follows:
1. Derive the lattice vibration of the NHj ions and iodine ions.
2. Calculate the libration, i.e., rotation, of the NH+ group in
the crystal.
3. Use the correlation technique to predict the number of
intramolecular vibrations of the NH^ group. *
By combining the irreducible representation obtained from
Parts 1, 2, and 3 and using Equation 8, the total representation
for NH4I is constructed.
Lattice vibrations of NHJ Ion and Iodine Atom
NH+ Ion
information: NHJ, site Du, ZB = 2 (see Table 4).
Du site
symmetry
species D±h factor
containing correlation group
li £ i
fy ty translation c£ species Q ay = aE + a
B%
4 2(T_) E === £„ 2 110
2 1 1 0
2 1G7) #*:=zr £,„ 110 1
110 1
The results are Cf2 -> C'l; the correlation is given in Table 11.
t~\ 7)(R) | yi(IR) I lyxR) I TT'dR)
* The lattice vibrations are sometimes referred to as external vibrations;
the molecular vibrations within a crystal are called internal vibrations.

4. Correlation of the Site Group to the Factor Group 27
This correlation applies to the tables derived in Chapter 2
(p. 40). By repeating a portion of this table we realize that
there are two possible correlations of D2d into D^hy as given in
Table 11.
TABLE 11
The Two Possible Correlations for
D2d into D^
A) B)
Factor group C2 -> C2
B2 Bx
E E
E E
Correlation B) Cfz ~> C'l in Table 11 was correctly used in the
above calculation, but what would the results be if the incorrect
correlation B) Cz -> C'% were used ?
Repeating this calculation but using the other correlation
given in Table 11 which maps D2d——~D4h, we obtain
D2d site
symmetry
species DAh factor
containing correlation group
li f\ i
fy ty translation °f%^c\ species Q a^ = aE + a
2 1 = 1 + 0
2 1 = 1 + 0
11=0+1

28 Practical Methods for Selection Rules
A summary of C'% -> C'z correlation gives
•p z?<R> i jdR) i z?(R) i r^lR)
Compared with the first calculation, C'2 -> C'l correlation gives
-p n(R) , (IR) (R) (IR)
A NHj" = -Big ~T
j
Both irreducible representations predict two infrared and two
Raman-active fundamental vibrations. In this specific instance
TABLE 12a
NH^ Lattice Vibrations
Correlation Correlation Polarizability tensor
Ca -» Cl Cg -* C2 (Appendix II)
7? 77" /v /v
a The result of the use of two different correlations from Table 11.
The polarizability tensors are given for certain species of D±h point
groups.
an improper choice of the correlation table does not alter the
predicted spectral activity. This is not the general rule, as the
reader will find by experience (e.g., see [9]).
When the results are compared by using both correlations in
Table 11, a difference in the presence of the polarizability
tensor is noted (see Table 12).
Summarizing the differences in Table 12, we find that Blg
species contain the polarizability tensor (axx — ayy), whereas
species B2g possesses <xxy. Of course, polarized Raman studies on

4. Correlation of the Site Group to the Factor Group 29
this orientated crystal would detect this difference; however,
the experimental results are extremely difficult to obtain for this
phase of the crystal.
This mistake will not occur if the proper choice, i.e., C2 ~> Cg,
is made in the correlation tables; however, we felt it useful to
include an example of an improper choice to acquaint the
reader with this problem. (Chapter 2 and especially Table 14
contain a description of the correct selection of correlation
tables.)
Lattice Vibrations of Iodine Ions
information : I, site CAv, Z = 2 (Table 4).
IODINE
CAv site
symmetry
species D^h factor
containing correlafcioiV group ^
fy ty translation species C$ a^ = aAi + aE
2 \{TZ) Ax ____^ Alg 11 1 0
"" " -A2u(Tz) 11 10
4 2G*, J E =z Eg 2 10 1
?«(rM) 2101
+
A \ J?
acoust — ^-Zu i -L'w
Summary of the lattice vibration in the NH4I crystal:
7 + A2u + Eg + Eu)
(Alg + A2u + Eg + £J - (^2w + £w)

30 Practical Methods for Selection Rules
Rotations (Librations) of the NH| Ion in the Crystal
The rotations of the NHj ion about the x9y, or z axis have the
same character as the rotations (RX9 Ry, and Rz) contained in
the character table of the D2d site group. Therefore the species
for the rotations parallel to the x, y, or z axis will be easily
identifiable. The correlation method is now applied to relate
the rotations of the site group to the specific species of the factor
group. The following slight modifications are necessary in
treating these librations:
where Ray is the degree of rotational freedom contributed by y
species of the site group. Also, Rac = 2y R(lr
B) rBb = |V£ (ii)
Of course, we note that A0) and A1) are identical to B) and
F), respectively, with only the superscript R added to indicate
rotation.
J Librations
information: NEj-site D2d, ZB == 2 (see Table 4).
D2d site
symmetry
species D±h factor
with co"f*tto% group
fyR Ry rotation 2~* 2 species C? a^ = aAz + aE
11 10
1110
2(RXJ E __ Eg 2 10 1
^2 1 0 1
Eu

4. Correlation of the Site Group to the Factor Group 31
If the incorrect correlation tables C2 -> C2 given in Table 10
were used here to map D2d into DAh, the following irreducible
representation would be obtained:
rrCir°; = A2g + B2u + Eg + Eu.
Comparing these two irreducible representations for the
librations, we note the following difference:
r^|~>C2 indicates a Hbration in species B2u, whereas F^"*0*
does not contain species B2u but instead has species Blu.
Therefore the representation differs in the presence (or absence) of
species B2u and Blu. Neither species Blu nor B2u are infrared or
Raman active; therefore in this specific case the selection rules
are unaffected by the choice of correlation table. The proper
choice of the correct correlation would be important in
predicting the spectral activity of overtone, combinations, and
difference tones for which the symmetry of this Hbration must
be correctly known. Chapter 2 describes a method that relieves
the uncertainty in making this selection by giving the correct
correlation tables for each site at which the ambiguous case
exists.
The reader is referred to Durig's paper [6] for an excellent
application of deuteration studies which distinguished between
the lattice vibration and the Hbration in this crystal.
Intramolecular Vibrations of NHj Ion
The correlation method can be used again to place the
different intramolecular vibrations of the NH^ group into the
proper species of the site group or factor group. First, the number
of intramolecular vibrations of the NHj ion can be obtained by
using Td molecular symmetry of this particular ion. (Details of
this method are described in Chapter 4, p. 79). The
irreducible representation is F = Ax + E + 2F2. These
molecular vibrations are then correlated to the D2d site species and the
site species are integrated into the factor group in the following

32 Practical Methods for Selection Rules
manner. Here, to avoid clutter, each species of Td molecular
symmetry is dealt with separately.
A new column, rvib, which is the degrees of vibration freedom
of the single ion NHj, is introduced. For these molecular
vibrations the following summation is used: vYih = 2y a-fiy =
3n -6 = l'CAi + lCE + 2CV2 - 9 (where Cy is the
degeneracy of the species y of the molecular point group in this
example Td). Also, fy=ZB- vYih, where ZB = number of
NHj molecules in the Bravais cell; therefore, fy becomes the
degree of vibrational freedom in the Bravais cell. This procedure
is summarised on p. 33.
information: NH4 ion Td molecular symmetry, D2d-site
symmetry, and ZLft factor group symmetry. ZB = 2
(see Tables 1 and 4).
The irreducible representation and spectral activity can be
summarized in the following manner:
Crystal Translations Intra- Spectral
symmetry (lattice Acoustical Rotation molecular activity
D4h vibrations) vibration (libration) vibration D^h
Alg 1 2 R
a\
A,., 1 1
1
1
1
2
1
2
1
2
2
2
R
R
R
II
II
final check: Total vibration CN) =9+3+6 + 18= 36,
where N = 12 for the NH4I crystal.

Molecular
symmetry
= * "vib 'vib
correlation
>»
Site
symmetry
™2d
Co —>Co
correlation
^h factor
group
species
2
4
12
E
1
1
1
1
1
1
2
2
1
1
1
1
1
1
1
1
2
2
2
2
£gj vib
B2u) + (B2g + Alu) + 2(Eg + Eu) + 2(Blg + A2u)
= 2Alg + \B2
2Blg + 2B2u + Alu + 2A2u + 2Eg + 2EU

CHAPTER TWO
SITE SYMMETRY
AND CORRELATION TABLES
PART 1 SITE SYMMETRY
The first example that had an ambiguous choice of site
symmetry was the equivalent set of oxygen atoms in TiO2. Recalling
from the x-ray information that four equivalent oxygen atoms
are present in the Bravais cell, we could place this equivalent
set on either a CafcD) or a C2uD) site. Actually the Wyckoff
tables [8] on the published crystallographic data indicate the
site position of each equivalent set of atoms in the following
notation:
Wyckoff's Tables for TiO2
Wyckoff
Atom notation Site position
Ti (a) 0, 0, 0; 0, J, |
Oxygen (e) 0, 0, u; 0, 0, u; 0, J, u + J; 0, |, J - u
35

36
Site Symmetry and Correlation Tables
We could consult the crystallographic tables [7] and identify
the site from the x,y, z coordinates; however, a much simpler
procedure can be followed. Appendix I presents the site
symmetry in alphabetical order. Using TiO2 as an example.
Appendix I gives space group 141, 2D2dB), 2C27iD), C2vD),
2C2(8), CxA6), for Dfh. Noting that DZd, C%ny and C2 appear
twice, we can write the following alphabetical ordering from the
data in this appendix:
Wyckoff notation or
Site in Alphabetical alphabetical ordering Atom
Appendix I order of site position (site)a
9/~) @\ Pi (c)\ o Titiininm
^ 2d\ / 2d\) t«.iiiuxii
(a)
b
c
d
e
2C2(8)
C.(8)
C,(8)
f
g
h
Oxygen
(e)
a Information from WyckofTs table and references cited therein.
The alphabetical letter in parentheses following titanium in
WyckofFs table indicates the site of the atom, i.e., (a) indicates
that the titanium is on site D2d. For titanium this could have
been determined by previous considerations; however, the site
position of the four equivalent oxygen atoms appears to leave us
the choice of either the C2^D) or C2vD) sites, for, as noted before,
both sites will accommodate four equivalent atoms. Wyckoff's
tables give the position of the oxygen atoms on an (e) site.
Examination of the alphabetical tabulation of sites shows that
the (e) site is C2v. Therefore there is no ambiguity in the choice

Site Symmetry and Correlation Tables 37
of site position for atoms, molecules, or ions if all the information
given in crystallographic tables is properly used.
Another example of the proper use of the crystallographic
information is the a-Al2O3 crystal which is Rjc == D\dy Z — 2.
By consulting Appendix I (p. 177) for D\d we find 2>8B), C3iB),
C8D), C<F), C2F), ^A2). Obviously the four equivalent
aluminum atoms can be accommodated only on the C3D) site.
The six equivalent oxygen atoms, however, might be located in
either the QF) or C2F) site. Wyckoff5s table gives the following
information for a-Al2O3:
Wyckoff Site Notation
Aluminum: (c)
Oxygen: (e)
Therefore use of the tables in Appendix I (p. 177) with the
sites arranged in alphabetical order proceeding from left to
right implies the following tabulations:
Site Alphabetical order Atom (site)
D3B) a
C3D) c Aluminum (c)
QF) d
C,F) e Oxygen (e)
f
These two examples illustrate the point that it is a simple matter
to determine the site symmetry of an atom, molecule, or ion in a
crystal lattice from information provided by x-ray experiments.
Again note that all the sites are arranged in alphabetical order
in Appendix I (p. 171). The table in Appendix I is similar to
that provided by Adams [10], except for two major changes:
(a) a reduction in the number of atoms found in the Bravais cell
site from that given for the sites of the crystallographic cell, and

38 Site Symmetry and Correlation Tables
(b) the arrangement of the site in alphabetical order to be
consistent with Wyckoff's ordering and the crystallographic tables.
PART 2 DERIVATION OF CORRELATION TABLES AND THEIR
RELATION TO THE WYGKOFF SITE NOTATION
1. DERIVATION OF CORRELATION TABLE
A selection of correlation tables appears in Appendix III (p.
201); however, we choose to show the derivation of several
tables. By following these examples we can eliminate the
problem of choosing the correct correlation table when two or
more possibilities exist or, in some cases, when no direct
correlation is given in the published tables.
The first example is the simple correlation of the point group
C3v to DZh. First, we write the point group C3v found in
Appendix II (p. 184):
A
A2
E
E
1
1
2
Operations
2C3(z)
1
1
-1
3av
1
-1
0
We note that CZv is a subgroup of DZh. This property was
easily recognized, for we see that Dzn contains the same
symmetry operators as CZv\ i.e., E, 2C3(z), 3^ plus additional
operations &h, 2SZ, and 3C2. To obtain the species of C3v that
correlate with those species of D3h we need only compare the

1. Derivation of Correlation Table
39
character of the operations common to both point groups Dzn
and C3v, which in this case are E, 2C3(z), and 3cfv. To do this we
simply write the partial character table of D3h, including only
the operations common to both C3v and Dzn:
Species of
D3h point
A[
A'[
A',
A'i
E'
E"
the
group
E
1
1
1
1
2
2
Operation
2C3(z)
1
1
1
1
-1
i
—i
-i
i
0
0
- Species of the
C3D point group
A
] a2
Ax
) *
Point group DZh: species A[
Point group C3v: species At
Character of the operation
2C3(z)
3a*
Therefore the correlation is
to DvhA!x.
Character of the operation
E 2C3(z) 3av
Point group D3h: species Ai
Point group CZv: species A2
The correlation is C*VA<> to D*hA".
-1
-1

40 Site Symmetry and Correlation Tables
The other correlations found are summarized below:
c3v
A.\ A^ A2 Jx-y
A'i A2 Ef E
Ao A9. E" E
The correlation between D2d and D±h is a bit more
complicated, since D^h contains two different subgroups that are
identical to D2d. It is best to illustrate this case with the specific
example: first the point group o(D2d can be written as follows:
Point group
A
A2
Bx
B2
E
E
1
1
1
1
2
2S,{z)
1
1
-1
-1
0
St EEE C2
1
1
1
1
—2
2C'Z
1
-1
1
-1
0
1
-1
-1
1
0
Now the operations of ZL7l, which are similar to ZJd, are two
sets or subgroups:
A) E, 2St(z), C2, 2G^ 2aa
B) £,2.S4(z),C8,2C2,2er,
These subgroups differ only in the presence of the C2 and tfd in
subgroup 1 and the replacement of these operations with C2'
and av in subgroup 2. Repeating the procedure already
discussed, we can write the operations common to both point
groups D±h and D2d.

Subgroup 1
ofiV
Ag
Au
Ag
Au
Bi»
B%g
Biu
Eg
K
Subgroup 2
of D,n
Ag
Au
Ag
Au
Blu
B2g
B2u
Eg
Eu
Hi ^*^4V
1 ]
1 __
1
1 —
1 -
1
I
1
2 (
2 (
E 2S4 (
1
1 -
1 ]
1 -
1 -
1
1 -
1
2 (
z) Si = C2
[ 1
1 1
I 1
L 1
I 1
I 1
L 1
I 1
) -2
) -2
I 1
I 1
L 1
I 1
I 1
I 1
I 1
I 1
) -2
2 0-2
2C'2
1
1
-1
-1
1
1
-1
-1
0
0
2CI
1
1
-1
-1
_1
-1
1
1
0
0
1
-1
-1
1
-1
1
1
-1
0
0
2av
1
-1
-1
1
1
I
I
1
0
0
Species
with same
character
in point
group D2d
A1
Bi
A
B2
B\
A
B2
A
E
E
Species
with same
character
in point
group D2d
A
A
B2
B2
A
B1
A1
E
E
41

42 Site Symmetry and Correlation Tables
TABLE 13
The Two Correlations Relating D±h to D2d
AlU Bl Bl
Jj-t „ Jj, /Jo
"iu Ai A2
B2g B2 B±
Eg E E
Eu E E
The correlations that relate Dih to D2d are given in Table 13.
Therefore to choose the correct correlation table we must
consider the symmetry elements in the site group and which of these
symmetry elements coincide within the factor group. This is
exactly the problem we face in selecting the proper correlation
tables in the TiO2 and NH^ ion examples already discussed.
Here we must decide whether the C2 element of the site D2d
coincides with the C2 element of the factor group. Also, in this
correlation the ad plane of the site group must coincide with the
<jd plane of the factor group. If this coincidence occurs, we can
choose the correlation tables marked C2 -^-C'z above or, as in
Appendix III (p. 206), the column marked C2. This, however,
is not the case for TiO2 and NH^ ions, for we find that the C2
element of the site coincides with the C2 operation of the factor
group and that the av plane of the site is the same symmetry
element as the ad plane of the factor group; then we must
correctly choose the correlation tables headed by C% -> C2 as
given above. [See also Appendix III (p. 206), D±h to D2d
column headed by C2.]

2. Relationship to Wyckoff Site Notation 43
2. RELATIONSHIP OF CORRELATION TABLES
TO WYCKOFF SITE NOTATION
There is a need for a simpler method of determining the correct
correlation tables. All this information is contained in the
crystallographic tables, Reference 7, but is not in an
accessible form. Therefore we prepared Table 14, which lists the
Wyckoff sites of interest for some space groups and identifies
the correct correlation tables. The correlation table identification
refers to the tables given in Appendix III.
Crystal TiO25 Space Group D^-141
1. The oxygen is on Wyckoff site (e). Referring to Table 14-Z)^,
we see that site (e) is in column C2, <rv. Here we find that the
relationship between ZLft and C2v given in Appendix III
allows four possible choices:
Dih
Alg
T
C2, av
civ
Al
A,
C2, ad
C2v
Ai
•
C2
c2v
•
a
c2v
Ax
However, Table 14-Z>4^ gives the proper choice as the column
headed by C2, av.
2. Titanium atom is on the Wyckoff site (a). Table 14-D^
for space group 141-D^ has this site (a) in column C2\
Referring to Appendix III (p. 206), we use the correlation
table for D±h to D2d headed by C2.
NHj Ion on Wyckoff Site (a)
In Table 14-D^ for the space group 129-D^ the (a) site is in
column C2. In Appendix III the correlation relating to D^n
to D2d uses the column headed by C2 for this site to factor group
relationship.

TABLE 14
Identification of the Proper Correlation Table to be used in Relating the Site Symmetry
Given by the Wyckoff Description0
Site correlation
Space group
number ^2{z) QO) C%(x) a(xy) o(zx) <?(yz)
16
17
18
20
21
22
23
24
25
26
28
31
35
36
38
D\
Dl
Dl
D\
D\
D\
D\
D\
c\v
civ
dv
Qll
c12
cl4
q, r, s, t
a,b
ij,k
g> A
i,j
c
m, n, o, p
c,d
b
g,h
f> *
ft*
b
a,b
a
e,f
e,j
e,f
a
*>f
g, A
a, b
c
a
e
a
d, e

s s
45

TABLE 14 (continued)
Space group
number
62
63
64
65
66
67
68
69
70
71
72
73
74
7I6
n17
D18
n19
JJ2h
^2h
rJ2
Dll
^2h
D25
n26
rJ7
D28
Space group
number
D{ 89
90
D\
D\
C2(z
e,f, k, I, m
c d e f i
g>l
g> h
e, i,j
g
i,j
c> d, h, i
e
e
i
d
e,fj
c
e
hj
h
f
d, h, h
f
g,h
g
d
c> d,g
\ m, n, o
a,
a,
g>
g
c>
e
c,
e
<?,
f
c
a,
cl
j9k
C2(x)
b,e
b,d
h
d, K i
g> l
f
a{xy)
g
P> q
i
0
n
j
c2
°*
o(zx)
c
0
n
n
m
i
<**
a{yz)
f
f
n
m
m
I
h

s
s-
q Q q q q q
47

TABLE 14 (continued)
Space group
number
C'l
C2
ad
D\n
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
D\d
Did
Z)9
7I0
D11
D\\
D\h
D\h
D\h
D\h
7~\5
n6
^4/1
D]h
Din
Din
D10
7I1
D12
TI3
e,
e,
f,
h
c
h
g
c,
e
f,
f,
h
a.
f
h
g
e
f
k
g
f,g
g>h
e,h
f,g
d
e,f,
m, n, o
c, d, k, I
a> b, c, d9j, ky /, m
b, d9 e9 I, m
a, b, h, i
a, b9 c9 i,j
hk
h
c, d, g, h
d,g
a, b, d, e
a, c9 i,j
g,'h f
e,f, K I
h
f
g,h
g
P> q
m
h
q
n
i
t
P
r
n
k
j
0
71
d,g

TABLE 14 (continued)
Site correlation
136
137
138
139
140
141
142
/)
Dll
Dlt
D17
D18
ZI9
D%2
cy h
f
d
b:
c>
e
tj
dj
a> b>f> g
a, b>f
a, c, d, g, h
dj> K k
d9 e, h, i
^ b9g
b.f
c9d
I
k
I
m
I
Space group
number C9
Space group
number
Space group
number
Jh uv
TQ 111 Dl i j,k c9d9l9m
178 Dl a b
179 D\ a b
180 Di ej g9h ij
181 Dl ej gyh ij
182 Dl a,g b,c,d,h
CL 183 Cl b,e
185
186
a, c
a, b, c
y*h 187
188
189
190
m n
k
h

TABLE 14 (continued)
Site correlation
Space group
number
Ci
191
192
193
194
Dlh gy i
Din
E>th
j.
J
a
a
>k
,g
c, k
b9 d9f9 i
b, c9 d9 h
p, q h,on
I
I k
j K ej
Space group
number
c,
Ol 207
208
209
210
211
212
213
214
O1
O2
O3
O4
O5
O6
O7
O8
h
h,i,j
i
f
g
i
i,j
k, I
g, h
g
h, i
d
d
e,f
d
c,d

ol
221
222
223
224
225
226
227
228
229
230
ol
ol
ol
ol
ol
ol
ol
ol
ol
ol°
g
d
f
f
TABLE 14 (continued)
Site correlation
Space group
number C2 3C2 C2, 2C'Z C2, crA C2, ah C2, <rd crfc crd
iyj h k9 I m
h
b c, d>j f>g>h k
f hj g k
d h,i g j k
c, h e i
h f g
g
L d, h v j k
a a, b, c, . . . , to the space group. Correlation tables are listed in Appendix III.

CHAPTER THREE
THE BHAGAVANTAM AND
VENKATARAYUDU METHOD
Since SrTiO3 has already been treated by the correlation
method, it would be worth while to repeat this calculation, using
the method proposed by Bhagavantam and Venkatarayudu
[3], to determine (a) whether both methods give the same
result and (b) to demonstrate the simplicity of the correlation
method. Only a simple outline of the Bhagavantam and
Venkatarayudu method is given here.
I. The irreducible representation for this crystal can be
obtained as follows:
By definition coR = the number of atoms left invariant under
operation R
%v = the character of the operation i?,
obtained in the following manner
+2 cosS)
53

54 Bhagavantam and Venkatarayudu Method
The angle 0 is defined as follows:
(a) where E is a proper rotation, 0=0°;
360
(b) (+) used for proper rotations, C^ d = —^- ;
360
(c) (—) used for improper rotations, Svi 6 = —- ;
(d) ah is an improper rotation with 0=0°;
(e) i is an improper rotation with d = 180°.
Next, each operation is considered in obtaining ooR and %v.
Crystallographic information for SrTiO3, a perovskite, is
O\-PmZm. First the crystal structure with atoms in the position
of the unit cell shown in Figure 1 must be considered. This
same unit cell is used throughout this discussion.
In the unit cell of this crystal structure (see Figure 1)
• = titanium atom
O = oxygen atom
Sr = strontium
Figure 1. The
crystallographic unit cell of SrTiO3.

Bhagavantam and Venkatarayudu Method 55
Note, (a) Each Ti is shared by 8 Sr; (b) there are 12 oxy
around 1 Sr.
To check:
Atom
Ti-8 per cell, each contributing \ to the unit cell:
8 x i = 1 Ti
oxy-12 atoms, each oxy contributed £ to unit cell;
12 x \ = 3 O
Sr-1 atom in middle of cell
1 Sr atom per unit cell = 1 Sr
Total coR = SrTiO3 for Z = 1.
A. E Operation: Character and number of atoms invariant
under E operations can be found as follows:
All the atoms remained unchanged: .*. coR — 5 (i.e.,
1 Sr + 1 Ti + 3 oxy), ^ = 5( + l + 2 cos 0°) = 5 • 3 =
15.
B. C3 Operation: The illustration that follows shows some of
the C3 elements of symmetry in the unit cell. The list that
accompanies it is a tabulation of the number of atoms left
invariant under all the Cz operations and the irreducible
representation %v.
Comment
There are 7C3's passing through this unit (Figure la); all are
parallel to one another (note that this is not the 8C3 operation

Figure la
Figure Xb
56

Figure lc
Figure Id
57

58
Bhagavantam and Venkatarayudu Method
Figure le
that appears in the character table, but only one of these eight
operations).
Operation Number of atoms invariant, wR
XC3 1 Sr + 2 • | Ti
iTi
iTi
iTi
Total atoms 1 Sr + 1 Ti
However, for all C3 operations 1+2 cos 6 (where d = 120) =
0, where d = 120°.
.'. (oB{\ + 2 cos d) =0

Bhagavantam and Venkatarayudu Method 59
C. Since this example illustrates the procedure followed on
each symmetry operation, only the essentials are presented
in the discussion for obtaining a>R and %v for each operation.
(See Figure \b.)
Operation Number of atoms invariant, coR
D.
2C*
o2
6c2
Total atoms
Operation
1Q
2c4
3C4
4c4
= 1
2
2
2
1
Sr
•*
• i
Sr
+
■ +
i
I
2 •
t
*
2 •
2-ioxy
oxy
oxy
-|Ti
Ti
Ti
Ti
iTi
1 oxy + 1 Ti
1
Ti
Ti
Ti
Sr
+
+
+
(see Figure \c)
ioxy
Joxy
ioxy
2 • i Ti + | oxy
^ ^r — lor + 1 li + lO—o
Operation co^ (see Figure Id)
xCa 1 Sr
2^2 2 • i Ti + i oxy
3Ca 2 • i oxy
4^ 2 ■ i Ti + i oxy
5Cs 2 • i oxy
6Cs 2 • i Ti + i oxy
7Cs 2 • i oxy
8C2 2 • i Ti + i oxy
9Cs 2 • i oxy
.-. 2 ^i? = Sr + Ti +3 oxy =5

60 Bhagavantam and Venkatarayudu Method
F. z-operation
Note. There is a center of symmetry at every atom in the
unit; therefore all the atoms remain invariant under
one of the many z-inversion operations, i.e., 1 Sr +
1 Ti + 3 oxy, for £ coR = 5.
G. Comment
The S& operation yields the same result as the C4 operation,
even though there is the additional reflection. If we note that
Intersection
— of reflection
plane
Figure 1/
there are three reflection planes in the unit cell and all the atoms
lie on one of these reflection planes and the £4 axis, we find
ooR = 3. (See Figure 1*)

H.
Bhagavantam and Venkatarayudu Method 61
Operation a>R (see Figure If)
1 Sr + 2 • i Ti
iTi
iTi
Figure Ig
I. All atoms are invariant under ah\ (See Figure \g)
.". 2 mr = 5
J. Operation coR (see Figure
1ad 2 • i Ti + J oxy
2ad 4 • i Ti + 2 ■ I oxy + Sr
3^ 2-iTi + ioxy

62
Bhagavantam and Venkatarayudu Method
-O
Figure 1/z
The results can be summarized in tabular form:
Class
(Oh factor group)
E
8C3
6C2
6 C4
3C2
i
6^4
8 Se
3 ah
6 ^
5
2
3
3
5
5
3
2
5
3
15
0
-3
3
-5
-15
3
0
5
3
II. Calculation of the number of modes in each of the species.
n(y) = number of modes in each species, y
g = order of the group, g = ^igi
gi = number of elements in each class
Xiy) = the character for the class i and irreducible
representation Ty

Bhagavantam and Venkatarayudu Method 63
= character of the irreducible representation, derived
and tabulated above
Example of its use:
1. A± Species
~ 2
xflg
X$i
Si
E
= i
= 15
= 1
8C3
1
0
8
6C2
1
— 3
6
6C4
1
3
6
1
—5
3
i
1
-15
1
eSi
1
—3
6
8S6
1
0
8
1
5
3
6ad
1
3
6
IXi'Jn'Si = 15 + 0 _ 18 + 18 _ 15 _ 15 _ 18 + 0
8 = 48 + 15 + 18 = 0
Therefore there are no lattice vibrations in Alg species.
2. Flu species
X?1"
Xp
Si
E
o
- 15
= 1
8C3
0
0
8
6C2
-1
-3
6
1
3
6
-1
-5
3
i
-3
-15
1
-1
— 3
6
8*6
1
0
8
3*,
1
5
3
1
3
6
gi = 45 + 0 + 18 + 18 + 15 + 45 + 18 + 0
£=48 +15 + 18=4
Therefore there are 4Flw irreducible representations.
3. F2u species
Xf2u
Xv
Si
E
= 3
- 15
= 1
8C3
0
0
8
6C2
1
— 3
6
6C4
_1
3
6
-1
-5
3
i
-3
-15
1
6St
1
-3
6
8^6
0
0
8
3cr,
1
5
3
6od
-1
3
6
iavt Si =45+0-18-18 + 15+45-18+0
= 48 + 15 - 18 = 1

64 Bhagavantain and Venkatarayudu Method
Therefore there is lFZu irreducible representation.
4. All the other species of 0h give zero irreducible
representations.
5. Summary:
poryst = 4Fiw +i?2w
pcryst = pcryst _ pacoust _ ^ + ^ _ ^
This checks the result in Example 2, p. 21.
It is now easy to compare the two methods to establish that
the correlation method takes only minutes, whereas this
procedure involves much more time.

CHAPTER FOUR
MOLECULAR SELECTION RULES
The molecular selection rules are easily obtained by the
correlation method. Tables which give the number of normal
vibrations in each species for several point groups have already
been published (see Herzberg [12]). We choose to use the
correlation method here to demonstrate its general applicability
to molecules and to provide a completed set of tables (see
Table 24) for all physically possible molecules.
1. APPLICATION OF CORRELATION METHOD
Slight modifications are necessary in the correlation method
applied to crystals before this procedure can be applied to
molecules. The following rules give these modifications for
obtaining the irreducible representative for all normal
vibrations (rmolvib).
65

66 Molecular Selection Rules
1. The molecular symmetry must be known or determined.
Several texts describe these point group classifications for
molecules (for examples see [12]).
2. The site symmetry for all equivalent sets* of atoms in the
molecule must be known or determined. First, if unknown,
the symmetry elements contained in or passing through the
atom must be found. Second, these elements of symmetry
form a complete set of operations belonging to a specific
point group—the site symmetry. Z is the number of atoms
in an equivalent set.
3. Application of the correlation method as already described.
4. The irreducible representation for the molecule (rmo1)
obtained includes both the genuine normal vibrations (rmo1 Vlb)
and nongenuine motions that take the form of pure rotation
and translations of the molecule. These nongenuine motions
can easily be removed by subtraction as follows:
■pmol vib ipm.ol -ptrans -prot
I
5. Identification of the spectral activity of each species of the
molecular point group. (See Appendix II, p. 181.)
2- EXAMPLES
The following examples provide some applications of the
correlation method 'for determining the number of normal
vibrations of a molecule.
Benzene
The symmetry elements present in benzene are shown in
Figure 2. This figure also illustrates those symmetry operations
* Equivalent atoms: definition. A set of identical atoms that can be transferred
into one another by the symmetry operations present in the molecule (see [12],
especially p. 131).

2. Examples
67
CgjCr,
Symmetry Elements:
o;(yz),crh(xy)
V Fig. 2a
Figure 2. Benzene symmetry. The z axis is perpendicular to the plane of the
molecule ah and passes through the inversion point i found at the center of the
hexagon. The z axis contains the elements of symmetry C6, C3, C2, S^ and Ss not
shown in this figure. The molecular point group is D^. The isolated portion
(Figure 2a) at the right, considers only one carbon and one hydrogen atom. This
illustrates the presence of only those symmetry elements E, C2, Gv(yz),
thus the site symmetry of the carbon and hydrogen is C2-y
present in each hydrogen and carbon atom. The symmetry
elements present in each atom immediately allow us to identify \
the site symmetry as C2v; i.e., the operations E, Ca, &vy ah
passing through the hydrogen site describe the point group
CZv.* The correlation of the site symmetry CZv for the hydrogen
atom to the molecular symmetry D6h is given in Table 15. Since
there are six equivalent hydrogen atoms per molecule, Z = 6.
The irreducible representation rH, derived in Table 15 for the
hydrogen atoms, is exactly the same as the irreducible
representation rc for the carbon atoms (this is always true if the site
symmetry and number of equivalent atoms Z are the same for
two nonequivalent atoms).
The total irreducible representation for benzene is the sum
* Note, for example, that the operations, C6, SQ9 i, do not pass through the
hydrogen atoms; therefore they are not included in the description of the site
symmetry of the hydrogen atom.

68
Molecular Selection Rules
TABLE 15
Benzene. The Correlation Between Site and Molecular
Symmetry Species for the Hydrogen Atom (Z = 6)
Site
Molecular
f
symmetry c°7elaA!,on> symmetry
6 1
6 1
6 1
1 = 1
1 =
0 =
1 =
1 =
2 = 1
0 =
1 =
1 = 1
2 = 1
1 =
Ag +
T~
Bla
Note. The site symmetry and Z of carbon are exactly the same as
hydrogen; therefore, FH = Fc.
of the irreducible representation for each equivalent set of
atoms given in Table 15; i.e.,
pmol pH | pC
The irreducible representation for the molecular vibration
pmoi vib can easjjy J3G founc] by subtracting the irreducible
representation of the rotations Frot and the translation Ftrans
from the total representation:
pmol vib __ pmol prot p
-i trans

2. Examples 69
This procedure is outlined in tabular form:
0£, species coefficients (see Table 15)
rH= 1 10 1 12 0 111 21
rc = i 101 120 ill 21
Adding
pnol = 2 202 240 222 42
Subtracting
__prot _ i „ i
ptrans y j
pnol vib:=2 102 140 122 32
or
pmol vib 9^(R) 4_ ^(O) _i_ 9n@) i /rdt) _j_ 4.f(R) _i_ j(IR) i or@)
1 — **^\g T" ^2g ' ^a2g * ^\g ' ^^Zg « ^2W ' 4D1m
Repeating the use of superscripts in Table 9, IR is infrared, R
is Raman, and @) is no spectral activity. This method predicts
four infrared fundamentals, A%u + 3Elu, seven Raman
fundamentals, 2Alg + Elg + 4EZgy and no coincidences in the infrared
and Raman spectra.
1,355-Trichlor obenzene
Figure 3 shows the symmetry elements present in the 1,3,5-
trichlorobenzene and the various site symmetries. The following
summary can be made of the site symmetries.
Ring atom
position Site symmetry Atom in equivalent set
Carbon A, 35 5) C2v, Z=3
Carbon B, 4, 6) C2v, Z = 3
Hydrogen B, 4, 6) C2v, Z - 3
Chlorine A, 3, 5) C2v, Z = 3

Symmetry Elements: E,C2,
^z),crh(xy)
Fig. 3a J
^o;(xy)
Symmetry Elements: E,C2,
crvfo-h(xy)
\^ Fig, 3b ^/
Figure 3. 1,3,5-Trichlorobenzene symmetry. The z axis is perpendicular to the
plane of the molecule ah(xy) and passes through the center of the hexagon formed
by the carbon atoms. Not shown here are the symmetry operations C3 and 6*3
contained in the z axis. The molecular point group is Dzh. The isolated portion
(Figure 3a) shown at the upper right illustrates the presence of symmetry elements
E, C2, <yv{yz)i 0h(xy) for the chlorine (Z = 3) and carbon (Z = 3) atoms in
positions 1, 3, and 5 of the ring. The lower right-hand portion (Figure 3&)
illustrates the presence of symmetry elements E, C2, Gv, Oj^xy) for the carbon (Z = 3)
and hydrogen (Z = 3) atoms in positions 2, 4, and 6 of the benzene ring. Thus the
site symmetry of all these atoms is C2v.
70

2. Examples 71
From the correlation in Table 16a the irreducible
representation for the molecular vibration (rmo1 Vlb) can be
constructed as follows:
Atom's position
a^ species coefficients on benzene ring
A' A' Pf A" A" F"
rcl = 1 i 20 i i i, 3,5
r° = i i 20 i i i, 3,5
Tc - 1 1 2 0 1 1 2,4,6
rH = 1 1 2 0 1 1 2,4,6
^* peq sets _
pmol = 4 4 80 4 4a
prot __ 2 j
ptrans __ j j
pmol vib __- 4 3 7 0 3 3
j-'mol vib a Ar _i Q Af i_ *l fPf _i % A" _|_ 3Wff
where there are 10 infrared fundamentals, IE' + 3 A I, and 14
Raman fundamentals, 4?A[ + IE1 + 3E".
a The same result would have been obtained if the total number of
equivalent atoms with site group C2v had been used. Table 16b
illustrates this by using Ztotal = ZcF) + ZHC) + ZC1C) - 12.
Hence a considerable savings in time would be realized if the total
number of equivalent atoms with specific site symmetries were used
in the correlation procedure.

TABLE 16a
1,3,5-Trichlorobenzene. The Correlation for the Site Group
C2v to the Molecular Symmetry Dzh for the Chlorine Atom
Z-3
r
3
3
3
ty
1
1
1
Site
symmetry
C2v
A
A,
Bit -
B2 - -
rci
Correlation^
= Ai + A2 -
Molecular
symmetry
A'{
\-2E' + Al-
i ■ *•
i
2 = 1
0 =
1 =
+ E"
ay
1
1
\~aB2
1
1
Note. TCI = rc (for carbon in 1, 3, 5 position)
— Fc (for carbon in 2, 4, 6 position)
= rH (for hydrogen in 2, 4, 6 position)
TABLE 16b
1,3,5-Trichlorobenzene. The Correlation Between the Site
Group and Molecular Symmetry Species for All Equivalent
Sites of Atoms with C2v Site Symmetrya
r
1°
12
12
ty
I
1
1
Site
symmetry
c2v
A]^
Bi - ^
B2~^
r =
Correlation^
AA[ + 4A'2 -
Molecular
symmetry
A'l
--E'
f BE' + 4^S
/i.
4
8
0
4
4
+
aA,-
4
= 4
—
=
4E"
ay
4
4
+ ^t
4
4
a Here ZC1 = 3, Zc = 6, ZH = 3 for a total Z of 12 atoms with
C2v site symmetry.
72

2. Examples
73
1,4-Dichlorobenzene
The symmetry operations found in 1,4-dichlorobenzene are
shown in Figure 4. The site symmetries identified in this figure
are summarized below.
Atom on site
Position on Atom in
benzene ring Site symmetry equivalent set
Chlorine
Carbon
Carbon
Hydrogen
1,4
1,4
2, 3, 5, 6
2, 3, 5, 6
C2v,
c2v,
cs,
cs,
Z = 2
Z = 2
Z = 4
Z = 4
The irreducible representations given in Tables 17 and 18 are
summarized below.
pc,ci =
pC,H __
prot
ptrans
Ag
2
4
Blg
0
2
~1
B2g
2
2
-1
BZg
2
4
-1
K
0
2
Bin
2
4
— 1
B2u
2
4
-1
2
2
~1
Position
on ring
1,4
2, 3, 5 and 6
pmoi vib=6
or
pmol vib
52
Thus there are 15 Raman fundamentals, 6Ag + Blg + 3BZg +
5^3^, and 13 infrared fundamentals, 5Blu + 5B2u + 3B3u,
with no expected coincidences.

I C2(z),CTxz
Symmetry Elements: E,C2(z),
Fig. 4a
Symmetry Elements: E,c
Fig.4b
Figure 4. 1,4-Dichlorobenzene symmetry. The x axis is perpendicular to the
plane of the molecule o^yz) and passes through the inversion point i found at the
center of the molecule. Not shown is the presence of the C2{x) operation coincident
with the x axis. The molecular point group is D2h{V^). The isolated portion
(Figure 4a) at the upper right illustrates the presence of the symmetry operations
E, C2, <y{zy), a(xz) contained in the chlorine (Z = 2) and carbon (Z = 2) atoms
found in positions 1 and 4 of the benzene ring. The site symmetries of these atoms
must be C2v. The isolated portion (Figure 4b) at the lower left identifies the
symmetry operations E and cs{zy) contained in the carbon (Z = 4) and hydrogen
(Z = 4) atoms in positions 2, 3, 5, and 6 of the ring. Thus the site symmetry of
these atoms is C-.
74

TABLE 17
1,4-Dichlorobenzene. The Correlation Between the C2v Site
and the Molecular Species D
2h
r
4
0
4
4
1
0
1
1
rc,
Site
symmetry
A
A ^
C1 = 2i4, +
Correlation,
\
-\-
- -\r -
2B2g + 2B3g
Molecular
symmetry
A
A9
Blg
' ~B2g
- - ~BZg
\ K
-:b2u
^3»
ai
2 =
0 =
2 =
2 =
0 =
2
2 -
2 -
+ 25lu + 2B2a
: 2
: 2
ay
V«Bi-
2
2
f flJ?a
2
2
a The number of equivalent atoms Z — 4 includes the equivalent
sets of carbon atoms (Z — 2) and chlorine atoms (Z = 2) in the 1
and 4 positions
TABLE 18
1,4-Dichlorobenzene. The Correlation Between the Site and
Molecular Species for All the Carbon and Hydrogen Atoms
in the 2, 3, 5, and 6 Positions (Z = 8)
f
Site
nm<
symmetry ^g«o£^
Molecular
symmetry
D9
ar =
'2h
16
A!
2
2
= 4
= 4
= 4
75

76 Molecular Selection Rules
Chlorobenzene
Figure 5 illustrates the symmetry operations present in
chlorobenzene. The site symmetries shown in this figure are
summarized:
Site atom
Chlorine
Carbon
Carbon
Hydrogen
Carbon
Hydrogen
Position on ring
2,
2,
1
1
4
4
3,5,
3,5,
6
6
Site symmetry,
c2v,
c2v,
c2v,
cf>
cv
z
Z= 1
Z = 1
Z= 1
Z= 1
Z = 4
Z = 4
Table 19 gives the correlation for the four nonequivalent
atoms on CZv site. The correlation for the remainder of the atoms
on site Cs is given in Table 19.
TABLE 19
Chlorobenzene. The Correlation Between the Site and
Molecular Species of the Chlorine, Hydrogen, and Two Carbons
Atoms (Z = 4)
r
4
4
4
ty
1
1
1
Site
symmetry
pCl,C,H _
Correlation
__ _
Molecular
symmetry
c^
A
+ 4B2
4
4
4

Symmetry Elements: E,C2(z),
V F\g.5a y
6 *
Symmetry Elements: E,
Fig. 5 b
a.
*y
T
-Z
Symmetry Elements:E,C2(z),
V Fig. 5c J
Figure 5. Chlorobenzene symmetry. The x axis is perpendicular to the plane of
the molecule a{zy) and passes through the center of the hexagon formed by the
carbon atoms. The molecular symmetry is C2v. The isolated portion at the upper
right (Figure 5a) illustrates the symmetry elements E, C2(z), a{xz), cr{zy) contained
in the chlorine (Z = 1) and carbon (Z = 1) atoms in position 1 of the benzene
ring; These atoms possess site symmetry C2v. The isolated portion at center right
(Figure 5b) identifies the presence of the symmetry operations E, G(zy) in the
carbon (Z = 4) and hydrogen (Z = 4) atoms in positions 2, 3, 5, and 6 of the
benzene ring. The site symmetry for these atoms is Cs. The isolated portion at the
lower left (Figure 5c) identifies the symmetry operations E, C2(z), o(xz), o{zy)
present in the carbon (Z = 1) and hydrogen (Z = 1) atoms in position 4 of the
benzene ring. The site symmetry of these atoms is C2v-
77

78
Molecular Selection Rules
Chlorobenzene. The Correlation for the Carbon and Hydrogen
Atoms in 25 3, 5, and 6 Position (Z = Z° + ZH = 8)
f
Site
symmetry
Correlation
>
o(xz)~>a{xz)
Molecular
symmetry
aA'
16 2
8 1
8=8 0
4-0 4
8-8 0
4=0 4
4B2
The irreducible representation obtained in Table 19 is
summarized below.
pC,Cl,H =
pC,H __
prot
ptrans
pmol vib
C
4
8
-1
11
&£, species
0
4
— 1
3
coefficient
4
8
-1
I __
10
4
4
-1
-1
6
Result from
Table 19
Table 19
Therefore the irreducible representation of the normal vibration
is
mol vib
There are 30 Raman fundamentals, 11^ + 3^4a +
6525 and 27 infrared fundamentals, IIAX + IOB1 + 6BZ.

2. Examples 79
Ammonium Ion, NHj
The ammonium ion has Td symmetry. Again, the site
symmetry must be determined for each equivalent set of atoms in
the molecule. In the preceding four examples the figure of the
molecule was constructed where the symmetry elements were
identified, which, of course, leads to a description of the site
symmetry for each set of equivalent atoms. There is still another
means of determining the site symmetry.
First, the equivalent sets of atoms are identified. This
describes the number of equivalent atoms (Z) in each set. For
NH+ with Td symmetry there are four hydrogen atoms in one
equivalent set (Z = 4) and one nitrogen atom in the second
equivalent set (Z = 1). Next, in Appendix I (p. 179), the space
group T\ lists the site symmetries and the number of equivalent
atoms that can be accommodated by each site for the molecular
symmetry Td.* This information is tabulated as follows: 7^:
2Td{\), 2D2dC), C,,D), 2Ca,F), C,A2), C.A2), CiB4). Since
only the 7^A) site will accommodate one atom, the nitrogen
atom must be on this site. There are four equivalent hydrogen
atoms which must be on a CZv site. We prefer this simpler,
more rapid means of determining the site symmetries of
equivalent atoms in a molecule. All preceding examples of
benzenes, as well as NH^ ion, are described in Table 20 by this
method. The results are identical to those already obtained for
examples 1 through 5 which consider the symmetry elements
present in equivalent sets of atoms.
Application of the correlation method is given in Tables 21
and 22. The irreducible representation for the normal vibrations
can be constructed as shown on page 83.
General Molecule
The foregoing treatment is applicable to any molecule as the
following example illustrates. First, the molecule of concern is
* The space group to the superscript 1 always describes the site symmetry
present in the molecular group; for example, molecular symmetry Td is described
by Tl D6h by D\h, D3h by D\n, D2h by D\h, C2v by C\v, and so on.

TABLE 20 The Site Symmetry for the Equivalent Sets of Atoms in Various Molecules
Site
symmetry
Molecular Equivalent (from Equivalent atom
Molecule symmetry space group Appendix I) (Z = number of equivalent atoms)a
1. Benzene D6h 191 P^/mmm = ^h 2ZNft(l)
2DWB)
2^C)
2. 1,3,5-Trichloro-
benzene
D*
4C,A2)
3C5F)
Carbon (Z = 6); hydrogen (Z = 6)
Chlorine atom (Z = 3)
in positions 1, 3, and 5
Carbon atom (Z = 3)
in positions 1, 3, and 5
Hydrogen atom (Z = 3)
in positions 2, 4, and 6
Carbon atom (Z = 3)
in positions 2, 4, and 6

3. 1,4-Dichloro-
benzene
D9
6CSD)
4. Chlorobenzene
p
mm2 =
CiD)
Chlorine and carbon atoms (Z = 2)
positions 1 and 4
Carbon and hydrogen atoms (Z = 4)
positions 2, 3, 5, and 6
Chlorine and carbon atom (Z = 1)
position 1
Carbon and hydrogen atom (Z = 1)
position 4
["Carbon and hydrogen atoms (Z = 2)
positions 2 and 6
| Carbon and hydrogen atoms (Z = 2)
positions 3 and 5

TABLE 20 (continued)
Molecule
Molecular
symmetry
Equivalent
space group
Site
symmetry
(from
Appendix I)
(Z =
Equivalent atom
number of equivalent atoms)a
5. NHt
p_ 7
43w ~ L d
2CLF)
Nitrogen atom (Z = 1)
Hydrogen atom (Z = 4)
a The site symmetry for each set of equivalent atoms is described in the column immediately to the left of the
atom. Note the number of equivalent atoms (Z) is always equal to the number in the parentheses of the site
symmetry; for example, in benzene there are six equivalent carbon atoms; hence Z = 6. Therefore the site
must be C2vF) for this is the only site that will accommodate the six equivalent atoms.

2. Examples 83
rH
rN
prot
■ptrans
A
= l
= 0
=
A,
0
0
species
, E
l
0
coefficients
1
0
-1
2
1
I
Result from
(Table 21)
(Table 22)
pmol vib =
for the
NHj ion T = A[
TABLE 21
Ammonium Ion, NH^. The Correlation for the Hydrogen
Atom with Site Symmetry C3v) Z = 4
Site Molecular
symmetry Correlation^ symmetry
J l ^3v l d
1=1+0
0=0+0
0 ^
2 E -I-I— ^^^--^ 1 = 0 + 1
2=1+1
E

84 Molecular Selection Rules
TABLE 22
Nitrogen Atom. The Correlation for the Nitrogen Atom
with Site Symmetry Td9 Z = 1
Molecular
Site symmetry Correlation^ symmetry
f ? Td Td a,
AL Ax 0
A2 A2 0
E E 0
Fx F± 0
F2 1
classifiable as C2v molecular symmetry. Appendix I gives all
possible sites in this molecule, summarized below.
C2v molecular symmetry
Designation of
Equivalent equivalent sets of
Sitea atoms on site atoms on siteb
1 Mo
2 Mxz or Myz
4 Mi
a Space group 25 - Pmm2; C\v: 4C2v(l); 4QB);
^D) (from Appendix I).
b Mx equals the number of equivalent sets of
atoms not on any element of symmetry but
possesses the symmetry element E, i.e., the
identifying operation; Mxz, Myz are the numbers
of equivalent sets of atoms lying on the xz and yz
plane, respectively; Mo is the number of atoms
lying on all symmetry elements.

TABLE 23
Correlation of all the Sites to the C2v Molecular Symmetry
a. C2v sites; MQ is the number of equivalent sets of atoms on all elements of
symmetry
Site
symmetry
Correlation^
Molecular
symmetry
P
MQ
Mo
B*
b. Cs site using axz correlation table; Mxz is the number of equivalent sets of atoms
on the xz plane
r
Site
symmetry
Molecular
symmetry
aA»
A" =■ - I _ _ _. ^^ B
~~~ — — —. D
2MXZ = 2MXZ + 0
= 0 +MXZ
0
Mxz= 0 +MXZ
2MXZ = 2M
c. Cs site using ayz correlation table; Myz is the number of equivalent sets of
atoms on the yz plane
P
Site
symmetry
Molecular
symmetry
aA' + aA"
±Myz 2
2Myz = 2Myz + 0
Myz = 0 + Myz
Myz= 0 +Myz
2Myz = 2Myz + 0
d. General sites; M is the number of equivalent sets of atoms on no elements of
symmetry
Site
symmetry
Molecular
symmetry
12M
3MX
85

86 Molecular Selection Rules
Table 23 lists the correlation of all these sites to the molecular
point group C%v. In site Cs there are two possible correlations.
Here the correlation axz is used when the xz plane passes through
the sets of atoms; the equivalent atoms are designated MXz.
Similarly, Myz is used to designate the equivalent set of atoms
which lies on theyz plane. The resulting irreducible
representations presented in Table 23 can be summarized as follows:
0£, coefficient of species
Irreducible
Site representation A± A2 B± B2 Results from
C2v TM* Mo 0 Mo Mo Table 23a
Cs(axz) T^xz %MXZ Mxz 2MXZ Mxz Table 23b
cslavz) YMv* 2Myz Myz Myz 2Myz Table 23c
Cx TM 3M± 2>MX 3M1 ZMX Table 23d
-rrot -1 -1 -1 C2v character table
_rtrans __ j _{ _j C2v character table
This result is identical to that given by Herzberg [12]. By a
similar procedure the irreducible representation for the normal
modes of vibration can be calculated for all molecular point
groups. Table 24 presents the vibrational contribution of each
set of equivalent atoms Mi to the different normal vibrations
of each species of the point group. The nongenuine vibrations,
i.e., the rotations and translations, are substracted from the
appropriate species and the spectral activity of each species is
identified.
TABLE 24
Number of Normal Vibrations and Selection
Rules for Molecules
Tables for the vibrational contribution of the different sets of
equivalent atoms to the normal vibrations in each species of the
point groups are given here: Mo is the number of equivalent sets
of atoms on all elements of symmetry present in the point group;
Mi is the number of equivalent sets of atoms on a site. The site
identified at the top of each column is described for each M with

2. Examples
87
an appropriate subscript i; N is the number of atoms in the
molecule, ion, or complex. This format is used in all the tables that
follow.
Point Group (space group, if any)
Vibrational contribution of
equivalent sets of atoms on site
Minus
nongenuine
vibrations
Spectral8-
activity
Point
group Site^(rc)
Site?-(m), etc.
Species Mi
Mjy etc.
—6 nonlinear R = Raman
molecules activity
— 5 linear IR = infrared
molecules activity
N = nMi + mMj + • • •, the number of atoms in the molecule, where Mt is
the number of equivalent sets of atoms on site i and each equivalent set contains
n atoms; Mj is the number of equivalent sets of atoms on site,/ and each equivalent
set contains m atoms.
a The selection rules for three- and four-photon Raman interactions are not
present here; however, a paper by J. H. Christie and D. J. Lockwood, J. Ckem.
Phys. 54, 1141 A971) gives some of these selection rules.
C1 (Space Group 1 — PI)
Vibrational contribution of
equivalent sets of atoms on site
Minus
nongenuine Spectral
vibrations activity
A
N =
Mx
Cs (Space Group 6
-Pm)=C
-6
IR, R
Vibrational contribution of
equivalent sets of atoms on site
Minus
nongenuine Spectral
vibrations activity
A'
A"
N
2M0 + SM1
= M0+2M1
-3 IR, R
-3 IR, R

Q (Space Group 2 — C\) ==
Vibrational contribution of
equivalent sets of atoms on site
Minus
nongenuine Spectral
vibrations activity
3M0
Mo + 2M1
R
IR
A
B
N
C2 (Space Group 3 — P2)
Minus
Vibrational contribution of nongenuine Spectral
equivalent sets of atoms on site vibrations activity
2M0 + 3M1
Mo
-2 IR, R
-4 IR, R
C2h (Space Group 10 - P2jm)
Q
N
88
Vibrational contribution of
equivalent sets of atoms on site
Minus
nongenuine
vibrations
C2h(\) C2B) C,B)
Spectral
activity
M2
2M2
Mo + -M2
2M0 + 2M2
2M2
R
R
IR
IR

C2v (Space Group 25 — Pmm2)
Vibrational contribution of
equivalent sets of atoms on site
Minus
nongenuine Spectral
vibrations activity
C2V{\) zx plane yz plane CxD)
2MZ
Myz
2My
-1
1
o
-2
IR, R
R
IR, R
IR, R
N
= MQ
ZJ (Space Group 16 - P222) s F
Vibrational contribution of
equivalent sets of atoms on site
Minus
nongenuine
vibrations
Spectral
activity
C2B)
—— . ii
2JA) -s: axis y axis # axis
A
N
D2d
N
Mo + 2M2* + M2y + 2M2* + 3Afx
= Mo + 2M2z + 2M2y + 2M2X + 4MX
2Jd (Space Group 111 - P42m) =5 1
Vibrational contribution of
equivalent sets of atoms on site
2W1) ^B). Us C.D) Cl(8)
2M2 + Afd + 3Afx
= MQ + 2M2V + 4M2 + 4Md + 8MX
CM CM CM
1 1 1
Minus
nongenuine
vibrations
-1
-1
-2
R
IR, R
IR, R
IR, R
Spectral
activity
R
R
IR, R
IR, R
11 Herzberg identifies this element as ra4.
89

(Space Group 47 — Pmmm) ~ Vh
Minus nongenuine
Vibrational contribution of equivalent sets of atoms on site vibrations Spectral activity
C2VM C8D)
; axis y axis x axis xy plane zx plane yz plane £^(8)
N = MQ -f 2M2z + 2M2l/ + 2M2x + 4Myz + 4MZX + 4M^ + SMX

C3 (Space Group 143 - P3)
Vibrational contribution of
equivalent sets of atoms on site
Minus
nongenuine Spectral
vibrations activity
c8(i)
A
E
N
= Mo
-2
-2
IR, R
IR, R
(Space Group 174 - P6)
Vibrational contribution of
equivalent sets of atoms on site
Minus
nongenuine Spectral
vibrations activity
Csh(\) C,B) C,C) qF)
A'
E'
A"
E"
N
M3 + 2Mh +
M3 + 2Mh +
M3 + Mh +
2M3 + 3Mh +
R
IR, R
IR
R
Czv (Space Group 156 — PSml)
Vibrational contribution of
equivalent sets of atoms on site
Minus
nongenuine Spectral
vibrations activity
Cav C3v(l)
Ai Mo 4
E Mo 4
N - Mo +
- 1MV -f
Mv -\
-3Mv-\
-3AfrH
qF)
- 37kf,
- 3AT,
- 6MX
- 6MJ
-1
-1
-2
IR, R
IR, R
91

D3 (Space Group 149 - P312)
Dz
E
N
Vibrational contribution of
equivalent sets of atoms on site
JJz\i) C3vw °2Wy uivD;
Mo + M9 + ZMy
^/f _i_ j^/£ _i_ 2-/V^o H~ 3Ai"i
Mn + 2AfQ + 3Afo + 6M-,
= M0+2M3+3M.+ 6M,
Minus
nongenuine
vibrations
-2
—2
Spectral
activity
R
IR
IR, R
(Space Group 162 — P3\m) = S6v
Minus
Vibrational contribution of nongenuine Spectral
equivalent sets of atoms on site vibrations activity
,A) CzvB)* C2F) CsF) qA2)
Ms,, + M2 + 2Md + 3M! R
2M2+ Md+ 3M1 -1
M2-
R
Mo + 71^3^ + 2^2 + 2^^^+ 3MX —1 IR
Mo + Af3v + 3M2 + 3Md + 6MX -1 IR
iV = Mo + 2M3^ + 6M2 + 6Md + 12MX
' Herzberg identifies this element as m6 [12].
92

Minus nongenuine
Vibrational contribution of equivalent sets of atoms on site vibrations Spectral activity
C3^B) C2VC) h plane v plane ^A2)
A[ Mzv+ M2v + 2Mh + 2MV + 3 Mi R
4 M2V+ 2Mh + Mv + 3M± -I
E' Mo + M3V+2M2V+ 4Mh + 3MV + 6Afx -1 IR, R
A'[ Mh + Mv + 3M-L
4 Mo + M8t,+ M2V+ M^ + 2MV + 3MX -1 IR
^ M3V+ M2V+ 2Mh + 3ATV + 6M1 -1 R
AT = Mo + 2M3V + 3M2V + 6Mh + 6MV
s

C4 (Space Group 75 — P4)
A
B
E
N
Qa
1
N
n
4i>
A
Bx
B*
E
N
Vibrational contribution of
equivalent sets of atoms on site
Q(l) CxD)
MQ + 3MX
Mo + 3M2
= Mo + 4M2
Qa (Space Group 83 —
Vibrational contribution of
equivalent sets of atoms on site
C47i(l) QB) CsD) q(8)
2Mh + 3Afx
Mo + M4 + MA + 3^i
0 • 4 ft ' 1
= Mq + 2Af4 + 4Af^ + 8M^
CAv (Space Group 99 —
Vibrational contribution of
equivalent sets of atoms on site
C.D)
C4v(l) y plane t/plane ^(8)
Mo + 2MV + 2Md + 3MX
M.v + 2A/^ -f- 3AjT^
Mo + 3MV + 3Md + 6M1
= Mo + 4MV + 4Md + 8Mt
Minus
nongenuine
vibrations
-2
-2
P4/m)
Minus
nongenuine
vibrations
-1
-1
-1
P4mm)
Minus
nongenuine
vibrations
-1
-1
-2
Spectral
activity
IR, R
R
IR, R
Spectral
activity
R
R
R
IR
IR
Spectral
activity
IR, R
R
R
IR, R
94

Z>4 (Space Group 89 — P422)
Vibrational contribution of
equivalent sets of atoms on site
Minus
nongenuine Spectral
vibrations activity
Z>4A) C4B) Cgaxi
E
N
Af£
Mo + M4
Mo + 2M4
j + 3Afx
-2
-2
R
IR
R
R
IR, R
Vibrational contribution of
equivalent sets of atoms on site
Minus
nongenuine
vibrations
C2(S)
C^axis Cs(8)
Spectral
activity
2Md
3M1
6M1
-1
R
-1
-1
-1
IR
IR
R
R
N
Mo
Herzberg identifies this element as
95

D4h (Space Group 123 — PAjmmm)
Minus nongenuine
Vibrational contribution of equivalent sets of atoms on site vibrations Spectral activity
C2VD) C,(8)
Cj axis C2 axis h plane v plane d plane CjA6)
R
-1
R
R
-1 R
+ M4v+ Mzv+ M2v + Mh + 2MV + 2Md + 3MX ~1 IR
M2v + Mh + Mv + 2Md 4- 3MX
M'2v + Mh + 2MV + Md + 3MX
+ M4v + 2MgV -f 2M^ + 4Mft + 3MV + 3Md + 6Mt — 1 IR
N = Af0 + 2M41, + 4M2-y + 4M^ + 8ATA -f 8M^ -f SMd
M2v +
Mgv +
Mfzv -f
Afgv
2M^4
^2-y "^
" M'2'v -+
■ ^ 4
Mlv 4
4
■ 2M2v 4
■ 2Af -4
• 2AjT -4
■ 2AfA H
^ H
- a4 -
- Mh -
■ ^Mn ~
- 2MV 4
- Mv 4
h 3MV 4
h Mv 4
h Mv 4
h 2MV 4
h 3MV 4
~ Jma 4~
- 2Md 4-
- 3Mrf +
- M^ 4-
- 2Md 4-
h 2Md 4-
h Afd 4-
- 3Md 4-
3M1
3MX
6MX
3Afx
3MX
3Mi
3MX

(Space Group 81 - Pi)
Vibrational contribution of
equivalent sets of atoms on site
Minus
nongenuine
vibrations
C2B) qD)
Spectral
activity
A
B
E
N
h M2 + 3Afx
h 2M2 + 3Af2
h 2M2 + 4MX
-1
-1
-2
R
IR, R
IR, R
Vibrational contribution of
equivalent sets of atoms on site
C5AA) C5B) CsE)
Minus
nongenuine
vibrations
Spectral
activity
A"
N
Mo
Mo
= MQ
ZMh+ 3M1
Mh+ 3M1
Mh+ 3M1
Mh+ 3M1
R
IR
R
IR
R
2M5 + 5Mh
\0M1
97

A,
Vibrational contribution of
equivalent sets of atoms on site
Minus
nongenuine Specjral
vibrations activity
csE) q(io)
N
MQ + 2MV
Mo
= Mo
3MX
1
6M1
-1 IR, R
-1
-2 IR, R
R
Vibrational contribution of
equivalent sets of atoms on site
Minus
nongenuine Spectral
vibrations activity
CBB) C2E)
Mo
M5
M5
6M1
R
IR
IR, R
R
= Mo + 2M5 + 5M2
98

Minus nongenuine
Vibrational contribution of equivalent sets of atoms on site vibrations Spectral activity
Dbh{\) C5vB) C2vE) h plane v plane CxB0)
A\ Mxv + M9/» -f %Mh + 2Mn -f 3M, R
"PI j\/f _i_ iVjTc ~\~ 2^Wo ~ ^fyfjt ~\~ 3^A. —I— GiV^fi 1 IR
E'2 2M2V + 4Mh + 3M^ + 6M2 R
^4'i AfA + Mv + 3M±
A'2 Mo + M5v+ M2V+ Mh+ ZMV+ 3M1 -1 IR
E'[ M5v + M2V + 2Mh + 3MV + 6Ma — 1 R
j\T = Mo + 2Af5v + 5M2v + 10MA + 10Mv -

CQ (Space Group 168 - P6)
Vibrational contribution of
equivalent sets of atoms on site
Minus
nongenuine Spectral
vibrations activity
c6(i)
A
B
Mo
Mo
-2
IR, R
-2 IR, R
R
N
= Mo
(Space Group 175 — P6/m)
C6;
Vibrational contribution of
equivalent sets of atoms on site
Minus
nongenuine Spectral
vibrations activity
C6B) CsF)
Mo
2Mh +
6Mh +
-1
-1
-1
-1
R
R
R
IR
IR
N
100

C6v (Space Group 183 — P6mm)
Minus
Vibrational contribution of nongenuine Spectral
equivalent sets of atoms on site vibrations activity
v plane d plane
1MV
3M1
6M1
-1
2
IR, R
IR, R
E2 3MV
N = Mo + 6MV
£>*
+ 3A
+ 6A
(Space
fa + 6M1
Id + 12M1
Group 177 —
R
P622)
Minus
Vibrational contribution of nongenuine Spectral
equivalent sets of atoms on site vibrations activity
C2F)
'2axis Cjaxis qA2)
-Y
Mo
= Mo + 2M6
fg + 2Mj + 3Afj_ -2
^2 + M2 + 3M1
f' + 6M" + 12M.
R
IR
IR, R
R
101

(Space Group 191 — PS/mmm)
Minus nongenuine
Vibrational contribution of equivalent sets of atoms on site vibrations Spectral activity
C2vF) C,A2)
D6h(\) C6<yB) Cgaxis Cg'axis h plane v plane d plane CxB4)
^v h V d t
Kv + Kv + 2MA + 3MV + 3Md + 6Af! -1 R
Alg MQ + M'2V + M2'v + 2Mh + 2MV + 2Md + 3Afx
4>, MJ, 4- Mj, + 2MA + Mv + Md + 3MX
J?lg ^ M^ + M^ + Mv + 2Md + 3MX
3MV + 3Md + 6MX R
AZ "" "^ ^ -f- Mv + Md + 3Mt
J? A/' -f- Af" -4- 2Affc -4- 2 A/ -4- iW^» -4- 3JM
Bo«, ML, + Afo« 4- 2Af», 4- Mn + 2Ma 4- 3Af,
jEi«, A/n 4* Me 4* 2Afo« 4" 2A^O« ~{~ 4Af», 4* 3Af4, 4~ 3A/,» + 6Aft — 1 IR
JV = Af0 + 2M6 4- 6M'2V

= Czi (Space Group 147 — P3)
A,
M3
+ M3
Minus
Vibrational contribution of nongenuine Spectral
equivalent sets of atoms on site vibrations activity
__ — _ _
Mx -1 R
Mx -1 IR
Eu* MQ + M3 + 3MX — 1 IR
iV = Mq + 2Af3 + 6Afj
* Herzberg identifies ^4M as Bui Eg as £y, and Eu as £'lM.
Minus
Vibrational contribution of nongenuine Spectral
equivalent sets of atoms on site vibrations activity
j j^
-1 IR
-2 IR
R
Ez M4 + 3MX R
N = Mo + 2M4 + 8MX
Cn: general case in which n is an odd number > 5 (^ 00)
Minus
Vibrational contribution of nongenuine Spectral
equivalent sets of atoms on site vibrations activity
Cn C»0) Ci(»)
A MQ + 3M1 -2 IR, R
£x Mo + 3MX -2 IR, R
£9 ZM, R
*1
*8A)
Mo-
Mo-
C4B)
M4-
f M4 -
f M4 -
f- 3MX
r* 3MX
f- 3MX
3M1
iV = Mo + nMx
103

Cnll: general case in which n is odd > 7 (^ oo)
Minus
Vibrational contribution of nongenuine Spectral
equivalent sets of atoms on site vibrations activity
Cnn{\) Cn{2) Cs(n)
Af
A"
Mo
Mn + 2Mh
Mn+ Mh
Mn+ Mh+
3M1
3M1
h+ x
2Mh+ 3M,
Mh+ 3M±
2Mh+ 3M1
Mh+ ZMX
1
1
1
1
R
IR
IR
R
R
E(n-l)/2
N
= Mn
2Mh
3M,
Mh+ 3M1
2Mn + nMh
2nM±
Cnv: general case for odd w's in which n > 7 (^ oo)
Minus
Vibrational contribution of nongenuine Spectral
equivalent sets of atoms on site vibrations activity
Mv+ 3M1
3MV+ 6MX
3Af,, + 6Af,
-1 IR, R
-1
-2 IR, R
R
2nMx
104

Dn: general case in which n is an odd number > 7 (^ oo)
Minus
Vibrational contribution of nongenuine Spectral
equivalent sets of atoms on site vibrations activity
Dn Dn{\) CnB) C2(n) C±[2n)
Ax Mn+ M2 + %M1 R
A2 Mo + Mn + 2M2 + 3MX -2 IR
Ex Mo + Mw + 3M2 + 6Afx —2 IR, R
2M5
c?: general case in which n is an odd number > 5 (# 00)
Minus
Vibrational contribution of nongenuine Spectral
equivalent sets of atoms on site vibrations activity
Elg
Elu
E2g
E2u
Mo +
CnvB)
Mnv +
Mnv +
Mnv +
Mnv +
C2B«)
^2 +
M2X
2M2 +
3M2 +
3M2 +
3M2 +
3M2 +
CsBn)
2Md +
MI +
2Md +
3Md +
3Md +
3Md +
3M,
3m\
3M1 —1
o.Ax-f — 1
6^X-| — 1
R
IR
R
IR
R
Mo + 2Mnv + 2«M2 + 2nMd + 4nM1
105

§5
Dnh: general case in which n — odd number > 7 (^ oo)
Vibrational contribution of equivalent sets of atoms on site
Minus nongenuine
vibrations Spectral activity
Dnh(l) Cnv{2) C2v(n) h plane v plane
Mnv + M2V + 2Mh + 2MV
M2V + 2Mh + Mv
-1
R
Mo + Mnv+ M2V+
Mo + Mnv + 2M2V+
2Mh
M2
6M1
IR
IR
R
R
K
K
N
3MV + 6M1
3Mm + 6M,
"I 2
2nMh + 2nMv

Cn: general case in which n is an even number > 8 (^ oo)
Minus
Vibrational contribution of nongenuine Spectral
equivalent sets of atoms on site vibrations activity
A
B
Ei
Mo-
MQ -
MA*!
3M2
hSMx
IR, R
IR, R
R
Enf2-
N
= Mo
3^f1
E(nfz-l)g
E(nlz~l)u
Cnft: general case in which n is even > 8 (^ oo)
Ag
Au
Bg
K
E2
Vibrational contribution of
equivalent sets of atoms on site
C«kV) CnB) C,(«) C^n)
Mn+ 2Mh+ ZMX
Mo + Mn+ Mh+ 3M1
Mh+ ZMX
2Mh+ 3MX
Mn + Mh + 3M1
Mo + Mn+ 2Mh+ 3M1
2Mh+ 3M1
Mh+ 3M,
Minus
nongenuine
vibrations
j
-1
-1
-1
Spectral
activity
R
IR
...
R
IR
R
h
2Mh+
X
3MX
Mo + 2Mn + nMh + 2nMx
a When (n/2 — 1) is odd, then a — 1, 6 = 2; however, if nj2 — 1 is even, then
« = 2, *= 1.
107

Cnv: general case for even w's in which n > 8 (but ^ oo)
Minus
Vibrational contribution of nongenuine Spectral
equivalent sets of atoms on site vibrations activity
Cnv Cnv(l) v plane d plane C1Bn)
A± Mo + 2MV + 2Md + 3^ — 1 IR, R
4a Mv + Md + 3MX -1
£2 Mv + 2Md + 3Af2
E± Mo + 3MV + 3Md + 6Mj_ -2 IR, R
£o 3M« + 3M^ + 6M, R
3MV
iV = Mn + »Af«
^: general case in which n is an even number > 8 (^ oo)
Minus
Vibrational contribution of nongenuine Spectral
equivalent sets of atoms on site vibrations activity
C2(n)
Dn(l) Cn{2) C^axis C^axis Cx{2n)
Bx
B*
Ex
^2
*3
Mo -
Mo -
Mn +
\- Mn+ 2
2
t- 2MK + 2
M2 + M2 -f-
!Mg + 2Mg +
Mg + 2Mg +
IMg + Mg +
\M'% + 3Afg +
;m^ + 3M'l +
fM^ + 3Ml +
R
IR
IR, R
R
108

general case in which n is an even number > 6 (=fi oo)
Dna
A
Bi
Ei
E3
E5
En-1
N
Vibrational contribution of
equivalent sets of atoms on site
Dnd{\) CnvB)
Mnv-
Mo + Mnv ■
Mo + Mnv
MnV
= Mo + 2Mnv
C2Bn)
C2 axis CsBn) C^
i- M^+ 2Md +
2Mf2 + Md +
M'2 + Md +
+ 2M'2 + 2Md +
3^2 + 3Md +
4- 3Mg + 3Md +
3Mf2 + 3Md +
3Mg + 3Md +
3^2+ 3^ +
-f 2wMg + 2nMd + -
Minus
nongenuine
iD»)
3Mt
3^
3M7
3MX
6MX
6M1
6M±
^nMx
vibrations
-1
-1
-1
-1
Spectral
activity
R
IR
IR
R
R
...a
a For n > 6 the character of the polarizability tensor should be determined for
each point group. See [11] for details.
109

Dnh\ general case in which n is even and > 8 (^ oo)
A*
A,
eZ
Mo
Mo
Vibrational contribution
CwB)
^n +
C2 axis C2 axis
• AfL + ^2tf
M2v + Af2t,
of equivalent sets of atoms on site
h plane
+ 2A4-
C.B»)
z; plane
f 2M^
f Mv
(/plane CxD«) ]
+ ^+ 3M^
+ 2Md + 3M,
+ Md + 3^rx
+ Md + 3MX
+ 3Md + 6M!
Minus nongenuine
vibrations
-
Spectral activity
R
IR
R
IR

E2g 2M'%V 4- 2M%V + 4Mh + 3MV + 3Md + 6M2
E2u M'%v + Mlv 4- 2Mh + 3MV 4- 3Md + 6M1
E3g M^v + Mly 4- 2Mh + 3MV + 3Md + 6MX
^v + aMlo 4- ^MA + 3MV + 3Md +
^v 4- /AfJ, + cMh + 3MV + 3Md 4-
N = AfQ 4. 2Mn 4- nM^ 4- wAf^ 4- 2nMh 4- 2«Afv 4- 2nMd 4-
■■ =!-■
When i is an odd number, a = I, b — 2, c — 4,/= 2; however, if 1 is an even number, then a = 2, £ = 4,/= 1, <r = 2

Vibrational contribution of
equivalent sets of atoms on site
Minus
nongenuine Spectral
vibrations activity
-1
IR, R
= n
-2 IR, R
R
N
D,
coh
Vibrational contribution of
equivalent sets of atoms on site
Minus
nongenuine
vibrations
Spectral
activity
Mo
0°°
R
IR
R
IR
R
N
= Mo +2Moo
T (Space Group 195 - P23)
112
Vibrational contribution of
equivalent sets of atoms on site
Minus
nongenuine
vibrations
T{\) C3D) CaF) qA2)
Spectral
activity
A
E
F
N
= Mo~
M34
h 3Af8M
h4M3H
- M2 4-
- M2 -f
- 5M2-f
- 6M2 +
3A^i
9M1
■\2M1
-2
R
R
IR, R

Td (Space Group 215 — P43m)
*i
E
Vibrational contribution of
equivalent sets of atoms on site
Td{\) C3vD) C2vF) C2A2) C
Ms + M2 + 2Md +
Md +
M3 + 2M2 + 4Afd +
Mo + 2M3 + 3Af2 + 5Md +
?xB4)
3Af,
3^
6Mt
9MX
9Mi
Minus
nongenuine
vibrations
— 1
Spectral
activity
R
R
...
IR,R
¥
== Mo
4M3 + 6M2
UMd
rft (Space Group 200 - Pm3)
N
Minus
Eg
Au
Eu
Vibrational contribution of
equivalent sets of atoms on site
M2v +
M2v +
2M2v +
MQ + 3M2v +
C3(8)
M3-f
M3 +
3Af34
M3 +
M3 +
3M3 +
C,A2) C
■ 2Mh +
- ™h +
■ Mh +
■ Mh +
■ 5Mh +
nongenuine
vibrations
iB4)
3MX
3MX
9MX -1
3MX
3MX
9MX -1
Spectral
activity
R
R
R
IR
\2Mh
0 (Space Group 207 - P432)
Minus
Vibrational contribution of nongenuine Spectral
equivalent sets of atoms on site vibrations activity
0A) C4F) C3(8) C2A2) CxB4)
^4+ M3+ M2+ 3MX
M3 + Afa + 3MX
M4 + 2M3 + 2M2 + 6MX
Mo + 3M4 + 3M3 + 5M2 + 9Mt
2M4 + 3M3 + 5M2 + 9Afx
-2
R
R
IR
R
N
= Mo + 6M4 + 8M3 + 12M2
113

Oh (Space Group 221 — Pm3m)
Minus nongenuine
Vibrational contribution of equivalent sets of atoms on site vibrations Spectral activity
(C'2->C2
kd~>^ C,B4)
V ^* i -*~^- .^
Oh Oh(\) C4vF) C8,(8) C2V{12) d plane A plane CxD8)
Alg MAv + Mzv + M2V + 2Md + 2Mh + 3MX R
^iw ^ + -^a + 3Afx
^2« M3v+ M21; + 2Md + AfA + 3MX
/? A^* 4— A'/' 4- *? A/F 4— A^A/f 4- c>A'/t. 4— 9A^f<
W = Mo

/
A
F,
F2
G
H
N
\
Ag
Au
Fig
Flu
F2g
F2«
Glg
Glu
Hg
H
N
/
1
Ih<D
M +
= M +
1
Vibrational contribution of
equivalent
"A) <
M0 +
V:
equi<
C5vA2
M5v
M5v
2M5v
M5v
M5v
M5v
2M5v
M5v
12M5v
-sA2)
M6 4
3M5 -f
M5 -f
2M5 -|
3M5-j
12M54
tbrational
valent sets
sets of atoms on site
C3B0)
- ^3H
- 3M3H
- 3M3 H
- 4M3 H
- 5M3H
- 20Mg H
contribution
of atoms on
) C3vB0) C2vC(
+ M3v
+ M3v
+ 2M3v
M3v
+ 2M3v
+ ^v
+ 2M3v
+ 3M3v
+ 2M3v
+ V
+ 2M2v
+ 3M2v
+ 2M2v
+ 3M2v
+ 3M2v
+ 3M2v
+ 4M2v
+ 3M2v
~»
C2C0) ^F0)
h M2+ 3MX
h 5M2+ 9MX
h 5M2+ 9MX
h 6MZ + 12MX
h 7M2 + 15^^*3^
h 30M2 + 60M1
of
site
)) CsF0) C1A20;
+ 2M, + 3MX
\ + 3M1
+ 4^ + 9MX
+ 5Mh + 9MX
+ 4^ + 9M1
+ 5Mh + 9Mi
+ 6Mh + 12M1
+ 6Mh + 12KX
+ 8^ + 15MX
+ 7^ + 15M1
+ 60M, + 120M.
n 1
Minus
nongenuine
vibrations
-2
Minus
nongenuine
vibrations
)
"I
"I
Spectral
activity
R
IR
R
Spectral
activity
R
-
IR
-
-
R
115

CHAPTER FIVE
APPLICATION AND SPECIAL CASES
Example 1. Ionic crystals containing linear molecular groups.
CASE A. The NH4N3 crystal (Figure 6). X-ray information*:
Pmna'Du (Space Group 53) Z - 4 and ZB = 4.
Site symmetries ~ . . /rr .. ,.N
7 Correlation (Table 14)
Wyckoff Schonflies elating site to factor group
NH+ g C2
C2h Ca(x)
N3
* L. K. Frevel, Z. Krist., A94, 197 A936).
117

ooo-—®—ooo
ooo-
Figure 6. The NH4N3 crystal. This projection shows only the ab plane. Two of
the Nj" ions are parallel to the a(x) axis, whereas the second set of Njj" ions is in
the bc(yz) plane. Here the tilted N^" ions in the be plane are represented by the
following: ^^^
( 0 V-nitrogen atom on the ab plane
itrogen atom above the ab plane shown
■nitrogen atom below the ab plane shown
The position of the NH£ ions is not described. The NH£ ions are not found in the
ab plane shown but are present in the crystallographic unit cell.
118

Irreducible Representations of the Ammonium Ion
119
Irreducible Representations of the Ammonium Ion
The irreducible representations for the translation, libration,
and intramolecular vibrations of NH^" are obtained in the
standard way and are listed here:
NHJ Ion Translation
with ZB = 4
Site C,
Correlation £) factor
O2(y) group
4
8
Tv) £?c\>>
1 +
0 +
= 1 + 0
= 0 + 2
= 1+0
= 0 + 2
= 1+0
= 0 + 2
trans
NHJ
M
= ^
ion
Ry
! 1 O D { T> _i Q D
Libration, FnH+; Z =
Correlation
+
4
A + 2S..H
ZJ^ factor
group
4 l(Rz)
8
1
2
1
2
1
2
1
2
1 + 0
0 + 2
1 + 0
0 + 2
1 + 0
0 + 2
1 + 0
0 + 2

120 Application and Special Gases
Summary:
= Ag + 2Blg + B2g + 2B3g + AU + 2Blu + B2u + 2B3u
ZNH+m° m , ZB = 4; Molecular Symmetry ofNHt ion is Td
viba
Molecular
symmetry Correlation ^
ofNH+ ^
Site
symmetry
ion T,
Correlation
C2(V)
d
Factor
group
D2h
intramol
+
5B
2g
5AU
4BS
a We have already obtained the irreducible representations for the NH^~ ion,
i.e., rHN+ = A1 + E + 2F2. Here we have four NH^" ions; therefore this
irreducible representation is multiplied by four, or T^g-H = 4A
4E + 8F2;
v-± is the N—H symmetrical stretches, v2 is the degenerate bendings of the
HNH bond, i>3 is the asymmetrical stretching of the NH bonds, and v± is the
triply degenerate HNH angle bendings. Note the intramolecular vibration in the
crystal is far more complex than this simple description for the isolated ion.
to To conserve space in some of the following examples, the individual a 's are
written atop each ray relating the different species.
Correlations Relating D^ to C2h for the Azide Ions
Two N^~ ions lie along the a axis in Site a; the remaining two
NjJ" ions are found in the be plane in Site b. In Site a the molecular
C^ axis is coincident with the C2 axis of the C2h site group and
in Site b a C2 axis of the molecule lies along the C2 axis of the

Irreducible Representations for N3 in Site a 121
C^n s^e group- The correlation relating Dooh to CZh can be
constructed according to the method given in Chapter 2:
Site a: Site b:
n,
A
2Ag
2~^~ A Z?
S~~ A A
u -**« ""«
£*-> nt *1<,I "t"
Irreducible Representations for N3 in Site a
Translations of the N3 /<?#, FN-(^««a); ^^ ZB = 2
Molecular Correlation Correlation ^2A
symmetry >• Site C2ft > factor
P t* Dmh a*»°* °*» group
2 1G;)
4 2(^7;) ntt-
trans _
L NJ (Site a) — ^w " ^-°1m i" 4-°2w "r ■
Libration of the N3 /ow, rN-(^«ea)/ «;zVA Z"8 = 2
Since the Ng" ion is linear, there can be no molecular rotation
of this ion about the axis containing all three nitrogen atoms,

122 Application and Special Gases
i.e., the degree of freedom involving Rz is zero.
Molecular Correlation Correlation D*
symmetry > Site C*h >- factor
II,
Intramolecular {Internal) Vibrations, V^-{Sitea); ZB = 2
To determine the number of molecular modes for the N7 ion,
we refer to Table 24 and point group D^h, In Table 24 we
substitute MQ = 1 and M^ — 1.
The irreducible representation of the intramolecular
vibrations for one N^ group is
rN- = ^ + sj + nuforZ = i.
However, ZB = 2, and therefore we multiply the above by two
to obtain the correct irreducible representations for the
intramolecular vibrations of the N^ ions in Site a in this crystal. Thus
P
Molecular
symmetry Correlation Correlation *actor
.. ,T_ >■ symmetry >• group
ofionNg, Coo,"* r C^x) 6n v
1
0
V _- -LV
2 2(»1)
2 2(v3)
4 2{v2)
a Vibrations are described as different combinations of the v^ symmetrical
stretching and contraction of the ions, i>2> as modes involving combinations of the
degenerate bending of the ions, and v3, as combinations of the asymmetrical stretch.

Irreducible Representations for N^~ in Site b
123
The irreducible representation for the intramolecular
vibrations of the N7 in Site a is
B3g +AU+ 2B1U + 2B2u + B3u
Irreducible Representations for N3 in Site b
.trans
Translations of the N3 Ion, r^iSiteb); with ZB = 2
fy
Molecular Correlation Correlation
symmetry >- Site C9Jl >
D G2,av 2h C2(x)
factor
group
\{TZ)
, ry)
_trans
==Au
BZ
lib
Libration of the N3 Ion, T^-(Sit6b); with Z = 2
Molecular Correlation Correlation °»
symmetry y- Site C2ft —— > factor
fh
group a?
4 2(TX, Ty)
^

124 Application and Special Cases
Intramolecular (Internal) Vibrations, T™™^eh); ZB = 2
As already discussed for N^ in Site a, the intramolecular
vibrations for N7 in Site b are also given by F2N- =22^" +
r
Molecular c. _
^ . .. Site ~ , ,. r actor
symmetry Correlation Correlation
1. ,, >• symmetry >■ group
ofionN3, c2,<rv n C*W n
2 2(Vl) 2+
2 2(v3) S+
4 2(,2) ^u
ite 6) = Ag + ^39 + ^« + 2-SlM + 2S2u + B3u
Summary of the Irreducible Representation for
NH4N3
trans
1 n~
trans
NH4
rNH+
pintramol
pinteamol
pacoust
Fcr°ytt = K
Ag
0
1
1
1
2
5
0
Ug +
*i.
0
2
3
2
0
4
0
"Si, H
0
1
3
1
0
5
0
- 102?2, ■
r for species
0
2
1
2
2
4
0
2
1
0
1
2
5
0
+ 11*8,+
+ 15
:
*lu
4
2
0
2
4
4
-1
*1« +1
B*u
4
1
0
1
4
5
-1
4B -f
2
2
0
2
2
4
-1
■H*8«

Summary for NH4N3 Molecule
K F H
oo o
125
Figure 7. The KHF2 crystal. Shown here is the ab(xy) plane. The FHF~ ions
are situated in the ab plane; however, the C^ axis of the ions is not parallel to
either the a or b axis. The numbers within the circled atom designate the position
of the atom in the c plane.
case B. The KHF2 crystal. (Figure 7). X-ray information:
Dt-h,mcm (Space Group 140). Z = 4, ZB = 2.
Atom
Site symmetry
WyckofF Schonflies
Correlation (Table 14)
relating site to factor group
K
H
F
a
d
h
D
C..
2h
First we obtain the irreducible representation for the crystal,
disregarding the presence of the covalently bonded linear ion
(FHF)~; second we treat the molecular crystal identifying
libration and internal vibration of the molecular ion (FHF) ~.

126 Application and Special Cases
The total irreducible representation of the crystals must be the
same in each of the proposed treatments.
Irreducible Representation of the Crystal
(disregarding the FHF~ molecular ion)
Irreducible Representation of the Translation Pr*118, the
Irreducible Representation of the Potassium Ions, ZB = 2
Site Factor
Correlation
symmetry >- group
fy Ty D D
4 2G;, Ty) E = .- .- _- : _- -
Summary: IT** = Au + A2u + Eg + Eu
Fh?8, the Irreducible Representation of the Hydrogen Atoms,
ZB =2
r
Site
Correlation ractor STOUD
symmetry ^ °
°; u
2
2
2
1G-,)
1G",)
1G-.)
^lw —.
5 -
A
EU
1
2

Irreducible Representation for the KHF2 Crystal 127
Summary: Y%*?* = A2u + Blu + 2EU
Tf-71*, the Irreducible Representation of the Fluorine Atoms,
ZB -4
f T?
Site
Correlation Factor fiTOUD
symmetry ^ ° r
° D
4 HTZ) At ^— Au 1
4 l(rx) 5X =\r^^~- A2g 1
4 1(T.) 52.--X ~^^^^---5,. 1
1
1
0
1
1
0
2
■ptrans j ,j tz? iZ? iJFiJ ?z> iOZ?
Irreducible Representation for the Molecular Crystal
KHF2 Considering the Presence of the Molecular Ion
(FHF)-
Note that the T^+ns is identical to that derived in the first part.
The translation of the (FHF)~ molecular ion is the same as the
irreducible representation for the translation of the hydrogen
o+,-n™ 4 ~ -ptrans T-itrans
atom, i.e., i (FHF)- = 1 H .
For the librations and intramolecular vibrations of the (FHF) ~
ion we require the correlation of the molecular point group
Dooh to the site group D2h. The C2(z) axis of the D2h site is
coincident with a Cz axis of the D^ group. For one-half the ions
the C^ axis of the ion lies along the C2(x) axis of the DZh site,
whereas for the other half it is aligned along the C2{y) axis.

128 Application and Special Cases
Using the method outlined in Chapter 2, the following
correlations can be constructed:
2+
27
Ug
Ag
C00A
Ag
Ag +
g
%Zg
■ ^2g
A,
B3g
Ag + B3g
TT-1 A A
u -^lu "T" ^3w -"lw I ^2
Since the difference between the two correlations involves
only an interchange of the 2 and 3 subscripts of the B species
(i.e.3 designation of the x andjy axes is arbitrary), both will give
the same final irreducible representations for the librations
and intramolecular vibrations of the (FHF)~ ion in the D%h
site group and Dih factor group, and either can be used to
obtain the final result.
Irreducible Representation for the Libration of the (FHF)~ Ion
This linear ion has only two degrees of rotational freedom,
Rx and Ry.
Molecular Correlation Site Correlation Factor
symmetry >■ symmetry y group
2 (*«,*, . _ ^
""^ ~~ 1
1

Irreducible Representation for the KHF2 Crystal 129
The Intramolecular or Internal Vibrations of the (FHF)~ Molecular
Ion (FHF)~ Possesses D(X)h Molecular Symmetry
Using Table 24 for this ion and substituting Mo = 1 and
Mao ^ 1> tne irreducible representation for this ion molecule is
r = s+ + s+ + uu.
However, there are two molecular ions per Bravais cell;
therefore the irreducible representation is multiplied by 2:
r = 2SJ + 2Si + 2nw.
The following correlation relates the molecular symmetry to
the factor group symmetry:
Molecular Site Factor
Correlation Correlation
symmetry >- symmetry 7, >- group
D D D
n.<S>
A2u + Blu + 2EU
Summarizing these results, we have the following:
1. The irreducible representation for the ionic crystal KHF2
J.
p trans
p trans
p trans
pacoust
pcryst
0
0
1
0
Ag
Ag
1
0
1
0
Blg
0
0
1
0
Ag-i
B2g
0
0
1
0
-Bl9
a^ for
Eg -
1
0
1
0
4- B A
species
0
0
0
0
-22?,
An
1
1
1
-1
+ 2,
Blu
0
1
1
0
B*u
0
0
0
0
2£lw4
1
2
2
— 1
-4£u

130 Application and Special Gases
2. The irreducible representation for the molecular crystal
KHF2 in which (FHF)~ molecular ions are identified
ptrans
p trans
1 (FHF)~
plib
1 (FHF)~
pintramol
■ (FHF)~
pacoust
pcryst
1 total —
A,
0
0
0
l
0
Ag
r Ag
1
0
1
0
0
+ 2A
Blg
0
0
1
0
0
2, +
B2g
0
0
0
1
0
Bl9-\
H
Eg
1
0
1
0
0
for species
An
0
0
0
0
0
, + 22?,
An
1
1
0
1
-1
+ 2j
Bin
0
1
0
1
0
*2u + '
B2u
0
0
0
0
0
Eu
1
2
0
2
-1
M2?«
The total irreducible representations in parts 1 and 2 derived
above are identical. This must always be the case, regardless of
whether covalent or ionic bonding is present (we have more to
say about this in Example 4 of this chapter). Indeed, only one
correlation which relates the molecular ion symmetry to the site
group symmetry is correct; however, there are possibilities of
other incorrect choice. Should an incorrect choice be made, the
total irreducible representation for the molecular crystal will
not be identical to the correct total irreducible representation
derived from the ionic crystal. This method provides an excellent
cross check to avoid errors.
Example 2. Identical atoms with different sites: B^C
Information: D\d-R^m (Space Group No. 166), Z = 9.
site symmetries:
Atom
1C
2C
6B
6B
Wyckoff
notation
b
c
h
h
Schonflies
description
c3v
cs
cs

Irreducible Representation for the KHF2 Crystal 131
It was noted in Table 2 that R type crystal structures may be
divided by 3 or 1. We found in the case of a-Al2O3 that Z = 2;
therefore, no reduction was necessary. In the case of B^C,
however, where Z = 9, we must divide by 3 to reduce the
crystallographic unit cell to the Bravais space cell; i.e.,
Care must always be exercised in the reduction of /?-space
groups to the Bravais cell.
Since the carbon atoms have different site symmetries, the
irreducible representation for the carbon atoms is
SiteZ>3d Site C3v
and the total irreducible representation for the Bfi crystal
The irreducible representation for each site and specific atom
in the above equation can be obtained as follows:
Site D2d
Correlation factor
species y group
ty species £ C^
1 1G-,) Ai% A2u 11=1+0
2 2{Tx,Ty) En Eu 2 1=0 + 1
Eu.

132
Application and Special Cases
Site CS
r
Site CZv
species y
Correlation
1J3d
factor
group
species I
ar =
UE
2
4
1 i
2 1
1 1
2 1
1 + 0
0 + i
1 + 0
0 + 1
SiteO3v
■En
fy tv
24 2(Tx,Ty)
12 1G-,)
Site Cs correlation
species y >
A'
j.k
factor
group
species £
A
- A2g
^^ F
\ A2u
c,
1
1
2
1
1
2
2 =
6 =
2 =
4 ==
6 =
A!
4
0
4
0
4
4
+
+
+
+
+
+
+
0
2
2
2
0
2
= 4A
lg
2A2
6Eg
4A
2u
Summarizing the above results to obtain the irreducible
representation for the crystal, we have
rB.Ccryst
SiteD3d SiteC3v
lEg
6A
2u
- Eu

Irreducible Representation for the KHF2 Crystal 133
where racoust - A%u + Eu. Therefore the spectral activity of
the jB4C crystal is
= 54? + 7£<R) + 545° + 7£<m> + 2A\£> + 242'
Example 3. Use of correlation tables for species with
separable degeneracy. R2O3-bixbyite structures such as MnaO3.
Information: Tl-IaZ (Space Group 206); Z = 16, therefore
SITE
o
; SYMMETRIES
Atom
4R
12 R
24oxy
Wyckoff
notation
b
d
e
Schonflies
designation
^6 = CZi
Note: As in Case 3, there are two different and distinct sites for
the R atoms. Therefore the irreducible representations of the
Site SQ Site C2
crystal is T^03 cryst = TR + TR + roxy. The irreducible
representation for the different sites and atoms follow:
Site
Correlation ?
j-9 v species y Y _ o-t — ;
/7 r ' species 4 C? a^ + a
1 1=1+0
8 2G;, Tv) Eu - -^^^- -Eu 2 1=0+1
3 3 = i + 2a
6
.-. rB = Au + Eu + 3FU
a It is to be noted here that the correlation tables given in Appendix
III must be modified because of the separable degeneracies of the

134
Application and Special Cases
group. The modification involves the addition of a coefficient 2 to
the Et species of point group S6 relating Fg and Fu of the point
group Tk as follows:
Aa+2Eg
Note that a forked ( c) ray is used in the diagram
above from the Eu to the Fu species to indicate the presence of the
coefficient 2. The coefficient 2 appears in the calculation of ay as
follows:
= 8 = a
rE.
K.
Bm
Tabulation of the a%
K,
= aEu [*-«CEu + (»-Cr% + *»CFa)\,
collecting terms
av = 1.
Th factor group
species I
x coefficient
1x1=1
1x2-2
Site
r
SiteC2 correlation
species y
Th factor
group
species £
aB
12
24 2{TxiTy)
Site C2
5Fa
1
2
3
1
2
3
1 + 0
la + 0
1 + 4
1 + 0
la + 0
1 + 4
Eu.+ 5FU

Irreducible Representation for the KHF2 Crystal 135
a Note that the coefficient 2 is removed for the E species because of
the separable degeneracy; therefore the modified correlation table
is
A
A
In Appendix III those species that do not use the coefficient 2 are
designated by an asterisk (*).
roxy, see p. 136-7 for correlation diagram.
Therefore to calculate aA
f =12= aA{CAg + CEs + CFg + CFg + CFa + CAu + CEu
+ CFu + CFti + CFJ, collecting terms
= aA{\ + 2 + 9 + 1 + 2 + 9}
= aA{24}
aA =3
Therefore
roxy = 3Ag + 3Eg + 9Fg + SAU + 3EU + 9FU
SiteS6 SiteC2
rR2O3 crystal = pR _|_ pR + p^ _ pacoust
~ Fu
where racoust = Fu. Summarizing the selection rule for R2O3,
we have
5£<
<0)

* oxy
r
Site symmetry
Ci species y
Correlation8 Th factor
** group species £
~ aA
X (coefficient)
72 3G;, Ty, T,)
Ag
Eg
F,
K
Eu
F,,
1
2
3
1
2
3
3 =
3 =
9 =
3 =
3 =
9 =
3
3
3
3
3
3
X 1
X 1
X 3
X 1
X 1
X 3
a The correlation for this is

x (comment)
A
En A [the coefficient 2 marked with an asterisk (*) is disregarded]
3^4 [note the ray for A to Fg and Fu has three prongs ( —
three coefficient]
A
Eu A [the coefficient 2 marked with an asterisk (*) is disregarded]
Fn, ZA (same as Fg species)
-e-) to indicate the presence of the

138 Application and Special Gases
We realize it is possible to write out formally an equation that
generates all the ayys regardless of the presence of separable
degeneracy; however, the additional coefficients, often 1, of the
equation distracts from the simplicity and ease of applying the
correlation method. Therefore we have chosen to discuss its
application when separable degeneracy occurs as a special case
and to provide these examples to illustrate its alternate
application.
Example 4. M3A15O12-Garnet Structure. Information: 0™-
7O3d(Space Group No. 230), Z - 8, ZB = 8/2 = 4.
SITE SYMMETRIES
Correlation
Wyckoff Schonflies table to be used
Atom notation designation (see Table 14)
Direct
Direct
Direct
case 1. M3A15O12 as an ionic crystal. In this treatment no
consideration is given to the possible presence of (A1O4) or
(A1O6) molecular groups. Without some chemical knowledge or
experimental information regarding the presence of covalent
bonding it is meaningless to discuss group librations and
translations. The following irreducible representations are derived
for the ionic crystal, disregarding any possible covalent bonding:
8A1
12 M
12 Al
48 O
a
c
d
h

Irreducible Representation for the KHF2 Crystal
139
Site S6
TM, 8 Al Atoms in Site S6
Site
symmetry correlation.
species y
factor
group
species £
ar =
8 1G-.)
16 2(TX, Tv)
1 =
1 + 0
1 + 0
0 + 2
1 + 2
1 +2
Site S6
Tai = Alu + A2
TM, 12 M Atoms in Site
2EU + 3F1U + 3F2u
Site
Symmetry Correlation^
;
species y
factor
group
species £ Cc
aBi
+ a
aB>
12
12 1G-,)
12 1G;)
0 + 0 + 0
1+0 + 0
i+o + o
1 + 1 + 1
0 + i + i
0 + 0 + 0
1+0 + 0
1+0 + 0
1 + 1 + 1
o + i + i
= A2g +Eg+ 3F
lg
2F
ig
3F
lu
2F
2u

140
Site 6*4
TA1? 12 Al Atoms in Site S4
Application and Special Cases
Site
symmetry Correiationv
P ty species y
If) 1 / rj-i \ 7")
24 2G;, 7;) ^.= ^§^sT"
\
oh
factor
group
species £
A.
— A2q
^E9
~~ Flg
"^ F29
^ A\u
\ A2u
V\ U
\
s lw
Ft*
1
1
2
3
3
1
1
2
3
3
0 = 0 +
1 = 1 +
1 = 1 +
2 = 0 +
3=1 +
1 = 1 +
0 = 0 +
1 = 1 +
3=1 +
2 = 0 +
0
0
0
2
2
0
0
0
2
2
TA1 = A2g +Eg+ 2Flg + 3F2g + Alu + Eu + 3Flu
Summarizing a^ for species, we have the following:
Site £6
PA1 =
PM =
Site ^4
PA1 =
pa _
x oxy
pacoust
rcry vib ==
0
0
0
3
0
34
•^2ff
0
1
1
3
0
? +
E9
0
1
1
6
0
5^2°
0
3
2
9
0
+ I
F*9
0
2
3
9
0
8E™
1
0
1
3
0
+ 14J
^2M
1
1
0
3
0
10£lo) +
2
1
1
6
0
■ 17^R
3
3
3
9
-1
r
) _|_
3
2
2
9
0
1 BJ?@)
ior2u
°Site table summarized on p. 141.

roxy All Oxygen Atoms of General Sites Cx
Site symmetry
Q species y
Correlation
Oh factor
group species £
U x
(coefficient)
144 3G;, Ty, Tz)
1
1
2
3
3
1
1
2
3
3
3-3
3-3
6-3
9 = 3
9-3
3-3
3-3
6 = 3
9-3
9-3
X
X
X
X
X
X
X
X
X
X
1
1
2
3
3
1
1
2
3
3
Foxy — oAx
3A
2g
6Eg + 9F
U
9F
2g
3Alu + SA2u + 6EU + 9Flu + 9F
iu

142
Application and Special Cases
case 2. M3A12(A1O4) 3. Here we have the same crystal structure
as in Case 1 of this example, except that covalently bonded
groups AIO4 of Td symmetry are identified. How do the selection
rules change ? Indeed, there are now the external librations and
translations of the A1O4 group as well as its internal vibrations
or intramolecular modes. The following example will answer
this question:
1. FM is the same as in Case 1, Example 4.
2. F^*6 6*6 is the same as in Case 1, Example 4.
3. Therefore the irreducible representation for the external
vibration (libration and translation) and internal vibrations
must be calculated for the A1O4 group.
1 A104
fk
19
1Z
24 :
1 \-K-z}
Site
symmetry
species y
A
■ *N
oh
Correlation factor
group
species £
A
"^-^ A
V^V ^^^\ 9
<\F±u
1
1
1
2
3
3
1
1
2
3
3
H "a
i i
i — i
0 = 0
1 = 1
3 = 1
2 = 0
0 = 0
1 = 1
1 = 1
2 = 0
3 = 1
a7
1 A
\ "
+ o
+ o
+ 2
+ 2
+ o
+ o
+ o
+ 2
+ 2
= Alg +Eg+ SFlg + 2F
2g
2u
Eu + 2F
lu
3F
2u

Irreducible Representation for the KHF2 Crystal
143
pjntern vib
1 A104
Moleculara
Correlation
(f7) Td
/1O\ A 12
B4) E - - - - - -1 - -
G2) F2 — 36—
Site
Correlation
A
— E ^^^\^\N^-
N
oh
factor
group
A9
Z~'F2g
\^Alu
\ A2u
^ Eu
^Flu
1
i
i
2
3
3
1
1
2
3
3
H
2
Q
o
5
6
7
3
2
5
7
6
a The irreducible representation given in Chapter 4 for NHj
(p. 83) is identical to that for (A1O4).
rmtern vib = j^ + 3^ + 5^ + ^ + lF2g + 3Alu + 2A2u
+ 5EU + 7Flu + 6F2U
Site S^
rAiao4S is e(lual to rAi derived in Case 2, (p. 139).
Summarizing the irreducible representation derived atoms,
we have the following:
rM
Site S6
Tai —
■plib..
AIO4
pintern vib
AIO4
■p trans
1 AIO4 —
pacoust
p «
total vib *
Alg
0
0
1
2
0
0
lAffi
A29
1
0
0
3
1
0
+ 54
1
0
1
5
1
0
g> +
3
0
3
6
2
0
8£CR
\JX-J g
5 for
2
0
2
7
3
0
'> +
species:
0
1
0
3
1
0
14F{0>
10i4°
-^2u
1
1
1
2
0
0
>+ J
K
1
2
1
5
1
0
:if^
Fiu
3
3
2
7
3
-1
">+ 1
F2u
2
3
3
6
2
0

144 Application and Special Gases
Here we see the same total irreducible representation
obtained in Case 1, summarized on p. 140; because of the co-
valent bonding, however, some motions are now identified as
librations, translations, and intramolecular vibrations of the
A1O4. This example illustrates that the total irreducible
representation must be the same regardless of the presence or
absence of covalent bonding in certain groups.
case 3. Case 3 considers the possible structure M3A13(A1O6J
with A1O6 covalently bonded groups of Oh symmetry. A
summary of the irreducible representations follows:
■ptrans
1 (A106)
(■A-lOg)
T^int6rn vib
pacoust vib
Aotal vib
Al9
= 0
- 0
= 0
= 1
— 2
= 0
A2g
1
1
0
1
2
0
+ 5A
+ ^
1
1
0
2
4
0
Vs
3
2
0
3
6
0
OJZlg
for species
2
3
0
3
6
0
0
1
1
0
3
0
4/r(o:
^2M
1
0
1
0
3
0
) + 1
) _j_ j
K
l
l
2
0
6
0
3
3
3
0
9
-1
^2W
2
2
3
0
9
0
Again this total irreducible representation is the same as that
found in Cases 2 and 3 of Example 4 on pp. 140 and 143. We
are now describing different librations and translations in
intramolecular vibration of the (A1O6) group.
These examples demonstrate the importance of knowing the
nature of the bonding in a crystal for describing the vibrational
modes.

Irreducible Representation for the KHF2 Crystal 145
Example 5. Polymers: In the preceding applications of the
Winston-Halford* site group approach which utilizes the
correlation method it has been assumed that the center of gravity
of a molecule is located on a crystallographic special position
of the space group. The properties of this special position are
unique in that the point group operations which leave this
position invariant form a group that is a subgroup of both the
space and molecular groups. For an infinite chain polymer,
however, it is evident that the center of gravity position is
meaningless; therefore an "axis of gravity" point must be
selected in the unit cell. The axis of gravity, which must be used
instead of the center of gravity in polymers, can contain several
operations that are not contained in the various molecular
point groups, e.g., translation, glide reflections, and screw
rotation. Each of these operations will leave the axis of gravity
unchanged. The complete set of all the operations that leave
the axis of gravity invariant constitutes the site group of the line
group which may or may not be isomorphous to any subgroup
of the space group. It will, however, be isomorphic to some
subgroup of the factor group of the crystallographic space
group.
To begin the derivation of the selection rules for polymers we
must first consider the polymer as an isolated infinite chain and
classify its symmetry according to a line group analysis. To begin
this procedure the line group is first correlated to the polymer
site group and immediately followed by the correlation of the
polymer site group to the factor group of the crystallographic
space group. Both Tobinf and ZbindenJ give a very good
discussion on the theory of line groups and therefore it is not
repeated herein.
* H. Winston and R. S. Halford, J. Chem. Phys. 17, 607 A949).
t M. Tobin, J. Chem. Phys. 23, 891 A955); J. Mol. Spectry. 4, 349 A960).
% R. Zbinden, Infrared Spectroscopy of High Polymers, Academic, New York, N.Y.,
1964.

146
Application and Special Cases
data: D£-Pn
case 1. Crystalline polyethylene. X-ray
(Space group 62).* There are two chains each consisting of
—(CHa—CH2)— units passing through the crystallographic
unit cell. The symmetry elements of the planar zig-zag chain
of polyethylene are shown in Figure 8a. The line group consists
of the following operations: translations along the chain axis, a
(a) Polyethylene
V
x(a) ,
<rfl(xz)
f\
a(xy)
C2(x)
cr(xy)
C2(x|
z(c)
y(b)
H
---c|(z)
(b) Polyvinyl Chloride
cr
C2(z)
\\Jt
cr(xz)
4
H
Cl
C2(z) cr(xz)
Figure 8. {a) A line group representation for crystalline polyethylene. The line
group described within the brackets [ ] possesses the following elements of
symmetry: a C2(y) rotation axis parallel to they axis which passes through the center
of the G—G bond; a center of inversion, i9 at the midpoint of the G—G bond;
C2(x) rotation axis parallel to the x axis and passing through the carbon atoms;
a(xy) reflection plane parallel to the xy plane and passing through the carbon and
hydrogen atoms; and the identity element E. (b) A line group representation for
crystalline polyvinyl chloride. The line group described within the brackets [ ]
has the following elements of symmetry: E, the identity; C2(z) rotation axis
parallel to the z axis and passing through the carbon atoms of the GH2 group;
a reflection plane a{xz) parallel to the xz plane and passing through the hydrogen,
chlorine and carbon atoms of the GHC1 group.
* G. W. Bunn, Trans. Faraday Soc. 35, 483 A939).

Irreducible Representation for the KHF2 Crystal 147
center of inversion, i; a glide reflection, ag{ yz); a twofold screw-
rotation, Csz{z); two reflection planes, ct{xy) and a{xz); and two
twofold rotation axes, Cz(x) and C2{y). These eight operations
form a group of order 8 and this group is isomorphic to the DZh
point group.
Referring to Figure 8a, we can describe the local symmetry of
the carbon and hydrogen atoms of the {CH2—CHa} unit as
follows:
1. Two carbon atoms contain the operations E, C2(x),
a{xy), g(xz). These are the operations of the CZv point group;
therefore the site symmetry of the carbon atoms is isomorphic
to CZv.
2. Four hydrogen atoms contain the operations E, a{xy).
They are isomorphic to the point group Cs.
3. Correlations to be used are found in the following manner:
C2v -> DZh. The C2 axis of the CZv group is coincident with the
Cz(x) o(DZh; therefore CZv —-> D
C2(x)
Zh-
Cs —* D2h. The cth plane of Cs is the same as the <y{xy) plane of
D2h; therefore Cs > D2n.
For carbon atoms Z = 2 in the line group the irreducible
representation is derived as follows:
2 ur.) ^^ ^o 1 = 1 + 0 + 0
f i + o
2 urj 5, c - \ ^ -52O i = o+o+ l
f o + o
2 ur.) *,d "-\ ^ o = o+o + o
f o + l
f 1 + 0
f 0 + 0
^carbon (line group) = ^g + -Si9 + 52i7 + £lu + £2m + B3u

148 Application and Special Gases
For the hydrogen atom Z = 4 in the line group the
irreducible representation is
r
«A'
2=2 + 0
2=2 + 0
1 = 0+1
1 = 0+1
1 = 0+1
1 = 0+1
2=2 + 0
2=2 + 0
■^hydrogen (line group) ~ ^4g + 2Blg + B%g + BZg + Au
+ Blu + 2B2u + 2BZu
The total irreducible representation derived for the carbon and
hydrogen atoms of the line group is
line group carbon (line group) ~r~ x hydrogen (line group)
Regroup = 3 A* + 35X, + 2B2g + Bzg + Au + 2Blu
+ 3B2u + 3B3u
The irreducible representation of the translations for the line
group in this species of DZh is
■ptrans r> i p i_ r>
1 line group ~ -°lu "t" ^2W "T ^>3u
The only rotation possible is about the chain axis, i.e., the
z axis. Since Rz is contained in the Blg species of D2h, the
libration of this polymer will be
plib _ D
line group ^9

Irreducible Representation for the KHF2 Crystal 149
Therefore the irreducible representation for the intramolecular
vibration can be obtained for the line group:
pintramol vib ptot ptrans plib
A line group line group l line group line group
= 3 Ag + 2Blg + 2Big + B3g + Au + Blu + 2B2u
Next we must determine which elements of the factor group
D2h of the space group D^h leave the axis of gravity invariant.
The factor group operations include E, Cl(x), C2{y), Cs2(z),
0g(xz), <yg(yz), <y(xy), and i. Referring to Figure 8a, we recognize
the invariant operation as E, Cs2(z)> tf(ry), and i. These
operations, E, C2y a{xy), and i9 are isomorphic to the C2h point group;
therefore the site group of the polymer is CZh. The correlation
that relates the line group D2h to the polymer site group CZh to
the D2h space group is C2(z), since the C2(z) axis is the coincident
axis in all these groups. The correlation is D%h line group
——> C2h site > D2h factor group. Therefore the irreducible
representations are derived as shown on page 150.
Tfcot - 6Ag + 6Blg + 3B2g + 3BBg + 3AU
+ 3Blu + 6B2u + 6B3u
There are no pure rotational modes of the polymer chain in the
three-dimensional case:
coust
pintramol vib __ ptot pa
= 6Af + 6Bfg + 3Bf9 + 3Bfg + 3^2
+ 2Blu + 5B2u + 5BZu
case 2. Syndiotatic crystalline polyvinylchloride. X-ray
data*: DH-Pbcm (Space group No. 57) with two chains each
* G. Natta and P. Gorradini, J. Polymer Sci. 20, 251 A956).

ptot a
line group
3
3
2
1
1
2
3
3
X
X
X
X
X
X
X
X
X
Number of
chains
per unit
cell
o
2 =
2 =
2 =
2 =
2 =
2
2 =
6
6
4
2
2
4
6
6
Line group
Polymer
site group
C9h
Factor
group of
space group
6
6
3
3
3
3
6
6
Previously derived on page 148. Note the degrees of freedom/7 for each species is the
rptot i v (Number of chains)
1 line group/ X |per unit ceU |-

Irreducible Representation for the KHF2 Crystal
151
—(CH2CHC1)— in unit cell. The symmetry elements shown in
Figure 8b include the identity £, glide plane ag{yz), mirror
plane <y{xz) through the CHGl group and twofold axis Ca(z)
bisecting the CH2 group. The operations form a group iso-
morphic to C2v.
Local symmetries for the atoms of the line group are the
following:
Atoms
Carbon (CH2 group)
Hydrogen (CH2 group)
Carbon (CHC1 group)
Hydrogen (CHC1 group)
Chlorine (CHC1 group)
Operation
leaving
atom-invariant
E,C,
E
E, a
E,a
E,a
Symmetry
of atom in
equivalent
set
c,
cs
cs
cs
Correlation
needed
direct
direct
a(xz)
a(xz)
a(xz)
Irreducible representations of line groups:
For carbon atoms of CH2 group Z = 2
r
Line group
Co,, ar
2G;, Ty)
B =_ . . . _
~ ~ B9
■ C(CHa)
= A1
2B1

152 Application and Special Cases
For hydrogen atoms of CH2 group Z = 4
Line group
fy ty C"i C2v #£
12 3G;, Ty, T,)
A
B1
B,
3
3
3
3
= 3^! + 3A2 + 3B± + 3B2
In the irreducible representations for carbon, hydrogen, and
chlorine of the (CHCl) group the symmetry of all six atoms is
Cs; therefore the derivation can be combined as follows:
Line group
a{xz) ' f-i
12 2(Tx,Ty) Ar ^__ A± 6
-i4a 3
6 1C",) A"=--W~' ^^ B± 6
*~ - x>2 ^
GHC1 ~~* °^1 I ^^2 ~T OjD! -j- Ox>2
Combining the above to obtain the total irreducible
representation,
pt0t P ! p _1 P
we have
rtot 1A/4 I 1 A i 11Z? I QD
Writing the irreducible representation for the following
translations and librations,
ptrans = At + B± + B2
Flib = Bx (rotation about the polymer chain or y axis)

Irreducible Representation for the KHF2 Crystal 153
we obtain the irreducible representation for the intramolecular
vibrations of the line group as before:
pintramol vib __ ptot ptrans plib
pintramol vib =
for the line group of syndiotatic crystalline polyvinylchloride.
The D2h factor group operations of the space group which
leaves the axis of gravity of the polymer chain invariant are the
identity element E, a. mirror plane ct(xz)> a glide plane ag{yz)>
and a twofold axis Ca(z). These symmetry elements are from
the C2v point group. Here we find a unique situation in which the
symmetry of the line group C%v is the same, i.e., homomorphic,
to the polymer site symmetry. Note from Figure 8b that the
C2(z) axis of the line group is coincident with the Cz(y) axis
of DZh\ therefore the correlation is
polymer site
The irreducible representation for the intramolecular vibrations
in crystalline syndiotatic polyvinylchloride are derived as
shown on page 154.
As derived before,
racoust n I p i n
and
= ptot _ pacoust (there are no Hbrations for the
infinite polymer chains)
Therefore the irreducible representation for syndiotatic
crystalline polyvinylchloride is
intramol vib = 10^R + ggR + y^R
+ 1A°U

iine group
Number of
chains
per unit
r
Line group
Polymer
site group
CJv)
Factor
group of
space group
D
2h
10 x 2
7 x 2
11 x 2
8 x 2
20
14
22
16
B,
10
8
7
11
7
11
10
8
-not
7B
2g
UBSg + 7AU
UBlu + 10B2u + 8BSu

Irreducible Representation for the KHF2 Crystal
155
case 3. Helical molecules. The IR and Raman selection rules
for polymers possessing helical symmetry have been discussed
by Higgs,* by Liang and Krimm,f and by Tadokoro.J The
symmetry of a helical chain may be treated by using the cyclic
factor group CB7rmjn) or the dihedral factor group DB7rm/rcK
where n is the number of monomer units per m turns. These
factor groups are isomorphous to the point groups Cn and Dn,
respectively.
Type 1. Crystalline isotatic polypropylene. X-ray data§:
3(—CH2—CH—) units per one turn of helix. Factor group:
I
CH3
CBtt/3) which is isomorphous to C3.
The local site symmetries of all the atoms are Cx (i.e., no atoms
lie on the C3 axis).
r
Atom
symmetry
Factor
group
symmetry
c.
81 3G;, Ty, T.)
27
27
x, Ty)
rtot =
ptrans =
Rotation can occur about the helical axis only:
.-. rrot = a
* P. W. Higgs, Proc. Roy. Soc. {London) 220A, 472 A953).
t C. Y. Liang and S. Krimm, J. Chem. Phys. 25, 563 A956).
% H. Tadokoro, J. Chem. Phys. 33, 1558 A960).
§ G. Natta and P. Corradini, Nuovo Cimento 15, 40 A960).

156
Application and Special Gases
The intramolecular vibrations are found by subtracting the
rtransand rrotfrom Ttot:
Tintramol vib
= 25A + 26E
Type 2. Crystalline polyethylene oxide. X-ray data*:
7(—CH2CH2—O—) units per two or five turns of the helix.
Factor group: Z)Dtt/7) or D{\QttJ1) for two and five turns of the
helix, respectively. Both factor groups are isomorphic to ZO.
The local site symmetries are
Atom Site symmetry
C's
H's
O's
For oxygen atoms of local symmetry C2 the correlation is the
following:
r
7
14
f
HT.)
2(TX, Tt
Oxygen
symmetry
4
Factor
group
symmetry
D7
4
1
2
3
3
3
= 0 -
1
- 1 -
= 1 -
f o
f 2
f 2
f 2
f 2
roxy =AX+ 2A2 + 3E± + 3E2 + 3E3
The correlation for the carbon and hydrogen atoms on
general sites C1 is shown on the facing page.
* H. Tadokoro, Y. Ghatani, T. Yoshihara, and S. Murahashi, Makromol. Chem.
73, 109 A964).

Symmetry of
carbon atoms
r
Factor group
symmetry
C ~ aA X (coefficient)
126 3(!TX) Tv, T,)
1
1
2
2
2
9
9
18
18
18
9
9
9
9
9
1
1
2
2
2
lfy = 126 = aJCAl + CAi + 2CEl + 2CEt + 2CEa}; therefore aA = V¥ = 9

158
Application and Special Gases
The irreducible representation for these carbons and
hydrogens is
rCiH - 9A1 + 9A2 + 1SE± + 18E2 + 18£3
The total irreducible representation rtot = Toxy + Tc H:
nAm
21Ef
rtot =
ptrans = A% + Et
prot-ckain axis A
The irreducible representation for the intramolecular vibration
is
pintramol vib ptot ptrans prot
pintamolvib
Type 3. General equations for helical molecules
1. For helices with symmetry isomorphic to Cn: n monomer
units (containing p atoms) and m turns in helix. Here the factor
groups are isomorphous to Cn—no atoms lie on helical axis,
i.e., all local symmetry Cx; therefore/7 = 3n x p.
For n odd the general irreducible representation can be
written as follows:
r
Symmetry
of
monomer
atoms C1
Factor
group
symmetry
3np 3G;, Tv, Tz) A
2 3p

Irreducible Representation for the KHF2 Crystal 159
Summarizing the total irreducible representation, we have
rtot = 3pAn,m + 3^ir.k + 3pgR + 3pEo + ...
and we write the irreducible representation for translation and
rotation:
ptrans = A + ^
ptot = A
These are subtracted from rtot to give
pintramolvib = ^ _ 2)A + Bp -
3pEz
3pEin_1)/2
For n odd
r
Symmetry
of
monomer
atoms Ci
Factor
group
symmetry
3np
T T )
1 V> 2 z)
(n/2-1)
1 3p
1 3p
2 3p
2 3p
2 3d
2 3p
Summarizing, we find
ptot __
ptrans = A + Ei
profc = A

160
Application and Special Gases
Therefore the irreducible representation for the intramolecular
vibration can be written in the general form
intramolvib
+ ZpB + C/>
+ 3pE2
2. For helices with symmetry isomorphic to Dn with n
monomer units, each containing p atoms and m turns in the
helix; the factor groups are isomorphic to Dn.
To derive the irreducible representation we assume that all
atoms have Cx symmetry except those lying on C2 axes
perpendicular to the helical axis. Let q = number of atoms lying
on Ca axis per monomer unit; then p — q — number of atoms
with Cx local symmetry.
For n odd
r
Symmetry
of
monomer
atoms Co
Factor
group
symmetry
= aA
2nq 2(Tx,Ty)
B *=
1 q
1 2q
2 3q
2 3q
2 3q
q
0
q
q
0
2q
2q
2q
n-D/2 2 Sq = q + 2q
Therefore for q atoms per monomer unit lying on Ca axis
Fo, = qA1 + 2qA2 + 3qE1 + 3?£2 + 3qE3

r
Symmetry of
monomer atoms
ty Cx
Factor group
symmetry
a^ = aA X coefficient
S(Tx,Tv,Te)
1
2
2
2
HP-q) HP-q)
2
HP-
2
HP-
HP-
q)
q) =
q)-
2
3(p-
2
3(p-
2
3(p-
2
3(p-
■q)
q)
■q)
q)
x 1
X 1
X 2
x 2
X 2

162 Application and Special Gases
Next, for atoms on general sites Cl9
f = Sn(p -})=4i[lxl + lxl + (S^)B X
HP-q)
a a —
2
For atoms in Cx sites the irreducible representation is
(A, + A2) + 3(p - q)[Ex + E2+E3
2
Therefore total irreducible representation can be written
rtot _PP-9\A , CP + 9\A
+ 3pE3 + • • * + 3pE(jl_1)/2
■ptrans a \ f?
prot helix axis a
•pintramol vib
Cp - l)Eln'R + 3pEf
+--' +3pE°n_1)/2
For n even there is no general formulation that will apply to
all the Dn groups, and each case must be worked out as in
Type 2 already described.
Type 4. Layer structures. Here we use graphite as our example.
X-ray data: Clv(P§zmc), Z = 4. The carbon atoms are on

Irreducible Representation for the KHF2 Crystal 163
Sites a and b
1. First the isolated sheet or layer is considered. (This is
analogous to the line group considerations in linear chains.)
Sheet symmetry: D6h
Symmetry of carbon atoms in sheet: DZh
Correlation: DZh —y-> D6h
C2
There are two carbon atoms in
P f
4 2G;,
2 \(TZ)
ptot
1 sheet
p trans
x sheet
pintramol
sheet
Carbon
atoms site
symmetries
r-n \ rpf _
Tv) E ^- ^-
All
= B2g + A2u + E2g
ptot ptrans
sheet sheet
the repeat unit
Symmetry
of layer
r °*
+ Elu
K + K
per
1 =
1 =
1 =
1 =
sheet:
' aE' +
■- 1 +
= 1 +
= o +
= 0 +
It
0
0
1
1
There are no librations of an infinite sheet.
Tuinstra and Koenig* observed one Raman scattering
frequency and no infrared absorption. These observations are
consistent with selection rules predicted for a layer.
2. If there is interaction between layers in graphite, the
selection rules would be derived as follows: the sheet symmetry
DQJl is correlated to the site symmetry of the layer CZv which in
turn is correlated to the factor group symmetry C6v. The degrees
* F. Tuinstra and J. L. Koenig, J. Chem. Phys. 53, 1126 A970).

164 Application and Special Cases
of freedom/7 = (number of sheets = 2) X (Cy, the degeneracy
of each species of the sheet symmetry Deh).
r
o
4
2
4
Sheet
symmetry
2
- ^ <j
At1
Layer site
symmetry —
A
^^-^
Factor
group
->• symmetry
E1
a(
o
2
2
2
Therefore the total irreducible representation is
■ptrans a \ T?
space group 1 ' 1
There are no rotations. The irreducible representations of this
layered structure is
■pintramol "ntot "ptrans ATR,~R \ odO _i_ i^IR.R i orR
space group 1 "• "^ 1 ' 1 ' ^^2
The above representation for intramolecular vibrations predicts
two infrared active vibrations and four Raman active
frequencies. As previously noted from Koenig's work, there was
only one Raman frequency observed; therefore it must be
concluded that there is weak interaction between the layers
and no coupling of vibrational modes.
Example 6. Selection rules for some common inorganic
structures. The following information is readily available in the
literature. The irreducible representation and selection rules
are the same for all inorganic crystals that contain the same
crystallographic structure. These examples are presented to

Irreducible Representation for the KHF2 Crystal
165
allow the reader to test his ability to apply the practical rules
presented here and/or to save him some time if the selection
rules are needed for certain types of inorganic structure.
aragonite strugture. Representative example CaCO3. X-
ray data D\l-Pnma (Space Group 62), ZB = 4.
Number of
atoms in
eq. site
4
4
4
8
T
Atom
Ca
C
Oxy
Oxy
Wyckoff
site
c
c
c
d
Af + 6< +
O DR I Q pH
- OX>30 -f- OJJlu
Schonflies
notation
for site
c,
Cs
cs
9Bfa
1 + 5Bl* + 8
Correlation
azx
Gzx
inactive
p. A
rutile strugture. Representative example TiO2. X-ray data
K-Pi2/mnm (Space Group 136), ZB = 2.
Number of
atoms in
eq. site
2
4
active
F
inactive
Atom
Ti
Oxy
= Afg -
= A2g -
WyckofF
site
a
f
r -Dig ~r -E>2g "T
i O Z?
Schonflies
notation
for site
c2v
-Ef+Al«-
Correlation
r"
acoust
— -^2m i ■"«

166
Application and Special Cases
anatase structure. Representative example TiO2. X-ray
data D^I,1/amd (Space Group 141), ZB = 2.
Number of
atoms in
eq. site
Atom
Schonflies
Wyckoff notation
site for site Correlation
2
4
Ti
Ox
a
e
c'i
active
3E
■R
2Eln
inactive
J2u
p A
x acoust ~ ^2
ilmenite structure. Representative example FeTiO3. X-ray-
data Cti-Rs (Space Group 148), ZB = 2.
Number of
atoms in
eq. site
2
2
6
Atom
Fe
Ti
Ox
Wyckoff
site
c
c
f
Schonflies
notation
for site
ca
c3
= 5A* + 5Ef + AA™ +
acoust

Irreducible Representation for the KHF2 Crystal 167
calcite structure. Representative example CaCO3. X-ray
data Dla-Rzc (Space Group 167), ZB - 2.
Number of
atoms in
eq. site
2
2
6
Atom
Ca
C
Oxy
Wyckoff
site
b
a
e
Schonfiies
notation
for site
s*
c.
rinactive
"P A i J?
acoust ""-Zu T* J-Ju
wurtzite structure. Representative example ZnS. X-ray
data Civ-PQMeIm (Space Group 186), ZB = 2.
Number of Schonfiies
atoms in WyckofF notation
eq. site Atom site for site Correlation
2 Zn b C3v ov
2 S b C, a,,
-p jR.IR , t?R,1
1 active = Al + ^1
" inactive ^13
acoust 1 "•" 1

168 Application and Special Gases
pyrite structure. Representative example FeS2. X-ray data
T%-Pa3 (Space Group 205), ZB = 4.
Number of Schonflies
atoms in Wyckoff notation
eq. site Atom site for site
4 Fe b S%
8 S c C,
^active = Ag + Eg + 3Fg
inactive ^-^-u "i ^-^u
F = F
acoust u
zing blende structure. Representative example ZnS. X-ray
data 7t-Frsm (Space Group 216), ZB = 1.
Number of Schonflies
atoms in Wyckoff notation
eq. site Atom site for site
1 Zn a Td
1 S b Ta
p
active

Irreducible Representation for the KHF2 Crystal 169
fluorite structure. Representative example CaF2. X-ray
data Ol-FmZm (Space Group 225), ZB = 1.
Number of Schonflies
atoms in Wyckoff notation
eq. site Atom site for site
1 Ca b 0h
2 F c Ta
r
active
t-TR
r = f,
acoust lw*
spinel structure. Representative example Al2MgO4. X-ray
data Ol-FdZm (Space Group 227), ZB - 2.
Number of Schonflies
atoms in Wyckoff notation
eq. site Atom site for site
4 Al d DM
2 Mg a Td
8 Oxy e CZv
^active = Fl9 + 2^2M + 2£u + 2F2u,
r — f
acoust lw

REFERENCES
1. D. F. Hornig, J. Chem. Phys. 16, 1063 A948).
2. H. Winston and R. S. Halford, J. Chem. Phys. 17, 607 A949).
3. S. Bhagavantam and T. Venkatarayudu, Theory of Groups and Its
Application to Physical Problems, Bangalore Press, Bangalore City, India,
2nded., 1951.
4. Charles Kittel, Introduction to Solid State Physics, Wiley, New York,
4th ed.
5. John C. Slater, Quantum Theory of Molecules and Solids, McGraw-Hill,
New York, Vols. I, II, and III.
6. J. R. Durig and D. J. Antion, J. Chem. Phys. 51, 3639 A969).
7. International Tables for X-Ray Crystallography, Vol. 1, N. F. M. Henry
and K. Lonsdale, Eds., 1965, Kynoch, Birmingham, England, 2nd ed.
8. R. W. C. Wyckoff, Crystal Structures, Wiley-Interscience, New York,
Vols. I and II. 1964.
9. R. K. Khanna and C. W. Reimann, A Simplified Method for Symmetry
Classification of Vibrational Modes of Molecules {Correlation Method), Spectra-
Physics Raman Tech. Bull., No. 3, 1970, Spectra-Physics, 1250 West
Middlefield Road, Mountain View, Calif. 94040.
10. David M. Adams, Metal-Ligand and Related Vibrations, St. Martin's,
New York, 1968.
11. E. Bright Wilson, Jr., J. C. Decius, and Paul C. Cross, Molecular
Vibrations, McGraw-Hill, New York, 1955.
12. G. Herzberg, Molecular Spectra and Molecular Structure. Vol. II. Infrared
and Raman Spectra of Polyatomic Molecules, Van Nostrand, Princeton,
N.J., 1945.
170

APPENDIX I
SITE SYMMETRY TABLE
FOR THE BRAVAIS SPACE CELL
The following table was compiled from the site symmetry tables
for crystallographic groups found in the International Tables for
X-ray Crystallography, Vol. I. Please note that the table is modified
and should be used only for the Bravais space cell and the Halford
site symmetry correlation method suggested in this book. The
site symmetries are arranged in alphabetical order, reading from
left to right. We refer the reader to Chapter 2 (p. 36) for an
explanation of this ordering. Note also that no coefficient
precedes the general site Cx because an infinite number of these
sites are present.
171

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
Space group
PI
PI
P2
P2,
B2 or C2
Pm
Pb or Pc
Bm or Cm
Bb or Cc
P2\m
P2,\m
B2\m or C2jm
P2\b or P2\c
P2,jb or P2Jc
B2jb or C2\c
P222
P222X
P21212
P212121
C222X
C222
F222
1222
12,2,2,
Pmm2
Pmc2,
Pcc2
Pma2
Pca2,
Pnc2
Pmn2,
Pba2
Pna2,
Pnn2
Cmm2
C1
C1
cl
to to
cl
c1
cl
cl
c4
L2h
c\n
ck
CL
CL
CL
D\
D\
Dl
D2
D\
D\
D\
D\
Dl
clv
c\v
clv
c\v
ctv
c1
clv
Clv
Clv
ell
Site symmetries
Ci(l)
80,AM^B)
4C2A);C1B)
CiB)
20,A); 0^2)
2CSA);C1B)
CiB)
0,AM^B)
CiB)
8O2ft(l);4C2B); 20,BM 0^4)
40,B); C,B); CxD)
4Cu(l); 20,B); 2C2B); 0,B); CxD)
40,B); 20,BM 0^4)
40,BM 0,D)
40,BM 0,BM 0,D)
*-)jt-^2\ / ' ^ 2\ ) 9 1\ /
40,BM 0,D)
20,BM 0,D)
0,D)
20,B); 0,D)
4ZJA);7C2B);C,D)
4Z),A);6C2B);O1D)
4D,AM 60,B); 0,D)
30,B); 0,D)
4Cgv(l); 4OsB); 0,D)
20,BM 0^4)
40,B); 0,D)
2C2B);CSB);C,D)
C,D)
2C2B);O,D)
0,BM 0^4)
20,B); 0,D)
C,D)
2C2B);C,D)
20^AM 0,B); 2CSB); 0,D)
172

Space group
36 Cmc21
37 Ccc2
38 Amm2
39 Abm2
40 Ama2
41 Aba!
42 Fmm2
43 7ta/2
44 7mm2
45 Iba2
46 7ma2
47 Pmmm
48 Pww
49 Pcctyi
50 Pte
51 Pmma
52 Ptttttf
53 Pmwfl
54 p££#
55 Pbam
56 Paw
57 Pbcm
58 Pmzm
59 Pmmw
60 Pbcn
61 Pfoa
62 Pnma
Do dmcm
64 Cmc<z
65 Cmmm
66 Cccm
67 Cmma
68 Ccftz
C12
C2tf
C14
cj;
c17
c^
cJJ
c20
c21
c|2
dJa
Dth
D2h
D\h
7M
U2h
Dth
Dlh
Dlh
Dlh
DH
D11
2I2
2fi
7~\14
7I6
JJ2h
2I7
7I8
D2h
7J0
1J2h
DH
2J2
Site symmetries
0,BM 0,D)
30,BM 0,D)
2CMA);3C,B);C1D)
202BM 0,BM 0,D)
0,BM 0,BM 0,D)
C2B);C,D)
C2,A);C2B); 20,BM 0,D)
C2B);C,D)
2C2,A); 20,B); 0,D)
2O2B);O,D)
C2B); 0,B); 0,D)
8D2h(l); 12C21,B); 60,D); 0,(8)
4ZJB);2C,D);6C2D);C,(8)
4C2ftB); 4Z>2B); 8C2D); 0,D); 0,(8)
4^BM 20,DM 60,DM 0,(8)
4C2AB); 2C2,B); 2C2D); 30,D); 0,(8)
20,D); 20,D); 0,(8)
40^BM 30,DM 0,DM 0,(8)
20,D); 30,D); 0,(8)
4O,ftB);2C2D);2C,D);C,(8)
2C,D);2C2D);C,(8)
20,D); C2D); 0,D); 0,(8)
4C2ftB);2O2D);C,D);C,(8)
2O2sB); 20,D); 2CSD); 0,(8)
20,D); C2D); 0,(8)
20,D); 0,(8)
20,D); 0,D); 0,(8)
2C2ftB); C2VB); 0,D); C2D); 20,D); 0,(8)
2C2AB); 0,D); 2C2D); 0,D); 0,(8)
4D2h(l); 2C2ftB); 6C,,B); 0,D); 4C5D);
20,B); 4O2,B); 5C2D); 0,D); 0,(8)
20,B); 4C2ftB); C,,B); 5C2D); 20,D);
20,B); 20,D); 4C2D); 0,(8)
173

Space group
69 Fmmm
70 Fddd
71 Immm
72 Ibam
73 Ibca
74 Imma
75 P4
76 P4X
77P42
78P43
79 74
80 74X
81 P4
82 74
83 P4/m
84 P42/m
85 P4/n
86 P4Jn
87 74/m
88 74^
89 P422
90 P42X2
91 P4X22
92 P41212
93 P4222
94 P422X2
95 P4322
96 P432X2
97 7422
98 74X22
99 P4mm
100 PUm
101 P42cm
D23
7J4
Dll
2J6
D21
D28
C\
ct
cl
ct
cl
s\
s2
c1
c2
c3
cth
cl
D\
D\
D\
d\
Dl
D\
Dl
Dl
Dl
D™
Clv
C2
C3
Site symmetries
2D2AA); 3C,,B); D2B); 3C,,B); 3C2D);
2^BM 20,DM 30,DM 0,(8)
4ZJft(l); 6^,B); 0,D); 3CSD); 0,(8)
2Z»2B); 2CaB); 0,D); 40,D); CsD); 0,(8)
20,DM 30,DM 0,(8)
2C4A); 0,B); 0,D)
0,D)
30,B); 0,D)
0,D)
O4A);C2B);C,D)
O2B);C,D)
«4A); 30,B); 0,D)
4S4A);2C2B);C,D)
4C4ft(l); 2C,AB); 2C4B); 0,D); 2CSD); 0,(8)
4O»B) 5 254B); 30,D); CSD); 0,(8)
2S4B) ;C4B); 20,D); C2D); 0,(8).
2O*(l); C2ftB); 54B); C4B); 0,D); 0,D);
254B);2OD);OD);O(8)
4D4A); 2D,B); 2O4B); 70,D); 0,(8)
2D,B);C4B);3OD);O(8)
30,D); 0,(8)
0,D); 0,(8)
6Z>,B); 90D); 0(8)
2Z),B); 40,D); 0,(8)
IC (A.\ • C* SR\
OL/2^ttJ , Uj^Oj
0D); 0(8)
2Z>4A); 2D2B); C4B); 5C2D); 0,(8)
2£),B);4O,D);C,(8)
2C4r(l); C2v{2); 3OsD); 0,(8)
OB); 0.B) ;OD); 0(8)
2C,,B);C,D);CSD);C,(8)
174

Space group
102 P42nm
103 P4cc
104 P4nc
105 P42mc
106 P4zbc
107 /4mm
108 I4cm
109 I\md
110 I\cd
111 P42m
112 P42*
113 P42xm
114 P42xc
115 P4m2
116 P4*2
117 PU2
118 P4n2
119 74m2
120 I4c2
121 /42m
122 Il2d
123 P4\mmm
124 P4\mcc
125 P4\nbm
126 P4\nnc
127 P4/m6m
1 O Q DA 1 ~,~
I/O LiT\V(lYlC
129 P4/nmm
Ctv
Clv
Ctv
Clv
clv
Clv
4v
Cll
Clv
D\d
Dtd
Dld
D\d
D\d
Dtd
Dld
Dld
K
D\\
D\l
D\h
D\h
D\h
DU
DL
Dk
Din
Site symmetries
C2eB);C2D);CsD);C1(8)
2C4B);C2D);C1(8)
C4B);QD);C1(8)
3(^B); 2CtD);C1(8)
2C2D); ^(8)
CivW;C2vB); 2CSD);C1(8)
C4B);C2KB);CSD);C1(8)
^BM^DM^(8)
C2D);q(8)
4D2d{\); 2Z>2B); 2C2BB); 5C2D); CsD); q(8)
423,B); 254B);7C,D);C1(8)
2J4B);CtoB);C,D);C,D);Cl(8)
254B); 2C2D); q(8)
4Z)MA); 3C2sB); 2C2D); 2CSD); ^(8)
223,B);2J4B);5C,D);C1(8)
254B); 2ZJB); 4^DM^(8)
2J4B); 223,B); 4C,D); ^(8)
423MA); 2CtoB); 2C2D); C,D); q(8)
n /'9^ • 9 c /'9^ • d fo\ • 4./° (&\ • r* ^^
X/2\^'// j ^^4\^/ 5 -^2\ / > ^^2v^/ s ^iv. /
223^A); 23,B); J4B); C2sB); 3C2D); C,D);
Cx(8)
2S4B);2C,D);Cl(8)
423tt(l); 2D2AB); 2C4rB); 7C2,,D); 5CS(8);
234B); C4AB); 234B); C4ftB); C2ftD); 23,D);
2C4D);4C,(8);C,(8);C1A6)
2£>4B); 2D2dB); 2C2ftD); C4D); Civ{4);
4C2(8);CS(8);C1A6)
2Z>4B); 23,D); 54D); C4D); Q(8); 4C2(8);
2C4ftB); 2Z32ftB); C4D); 3C2l)D); 3CS(8);
2C4ftB); C2hD); 23,D); C4D); 2C2(8); Cf(8);
2232dB); C4,B); 2C2AD); C2,D); 2C2(8);
2C,(8);C1A6)
175

Space group
130 P4\ncc
loi fr2f nitric
132 P42jmcm
133 P42\nbc
134 P42\nnm
135 P42/mfo
136 P42jmnm
137 P42jnmc
138 P42\ncm
139 /4/wmm
140 /4/mcm
141 ^/^m^
142 J^/aci
143 P3
144 P3X
145 P32
146 R3
147 P3
148 R3
149 P312
150 P321
151 P3X12
152 P3X21
153 P3212
154 P3221
155 £32
156 P3m/
157 P31m
Din
D\h
D\l
D\l
D\l
D\l
Djk
D\l
D\l
D\l
D\l
C1
r2
°3
cl
ch
ch
#*.
D\
D\
D%
D\
D%
Dl
ch
cl
Site symmetries
0,D); J4D); C4D); C,(8); 2C2(8); CxA6)
402ftB); 202dB); 7C21)D); 0,(8); 3CS(8);
^2/,B); 02<*B); 02ftB); 0MB); 0,D); C2ftD):
40^D); 30,(8); 2C,(8); ^A6)
302D); 54D); Q(8); 5C2(8); G,A6)
2DMB); 202D); 2C27lD); CtoD); 5C2(8);
C»D); 54D); C2ftD); 0,D); 3C2(8); 0,(8);
20,»B); C,hD); J4D); 3C2,D); 0,(8); 2CS(8);
20MB); 2C2,D); 0,(8); 0,(8); 0,(8); C,A6)
02D); 54D); 2C2ftD); C29D); 3C2(8); 0,(8);
C,A6)
204ft(l); 02ftB); 02dB); CivB); C2ftD);
4C2,,D);C2(8);3CS(8);C,A6)
04B); 0MB); C4AB); 0ttB); 0^D); C4D);
2C2eD);2C2(8);2Cs(8);C,A6)
202dB); 2C2ftD); C*D); 2C2(8); 0,(8); C,A6)
>S4D); 02D); 0,(8); 3C2(8); 0,A6)
30,A); 0^3)
0,C)
0,C)
0,A); 0^3)
2C,,A); 20,B); 20,C); qF)
2C,,A); 0,B); 20,C); CiF)
603A);3C3B);2C2C);C,F)
203A);2C3B);2C2C);C,F)
2C2C);C,F)
20,CM 0,F)
20,C); 0,F)
203A);C3B);2C2C);C,F)
3C3,A);CSC);C,F)
Cw(l); 0,B); 0,C); ^F)
176

Space group
158 P3cl
159 P3\c
160 R3m
161 R3c
162 P31m
163 P3\c
164 P3ml
165 P3d
166 R3m
167 R3c
168 P6
169 P6X
170 P65
171 P62
172 P64
173 P63
174 P6
175 P6/?n
176 P63/m
177 P622
178 P6X22
179 P6522
180 P6222
181 P6422
182 P6322
183 P6mm
184 P6<*
185 P6scm
186 P6zmc
187 P6m2
cl
civ
Czv
cl
D\d
Bid
DL
DU
Did
ci
ci
ci
ci
ci
ci
C\h
ch
Dl
Dl
Dl
Di
Dl
Dt
Civ
C2
c3
civ
D\n
Site symmetries
3C,B);C1F)
20,B); CxF)
CSv(l); CsC); (^F)
03BM^F)
2D3d(l); 2D3B); C3v{2); 2C2ftC); C,D);
2C2F);CSF) 5^A2)
D3{2); C3{B); 2D3B); 2C3D); QF); C2F);
2I>MA); 2C3SB); 2C2ftC); 2C2F); C,F);
D3B); C3iB); 2C3D); C<F); C2F); CxA2)
2ZKd(l); C3v(z); /C2^(j); 2C2F); Cs(o); ^(Iz)
JJo[Z): Oq,-(z); Cq(^); t/,-(o): C9(b); C-,(lz)
Ci(l);C8B);C8C);C1F)
CxF)
C,F)
2C2C); CxF)
2C,C);C1F)
20,BM^F)
6C3AA);3C3B);2CSC);C1F)
2C6ft(l); 2C3AB); C6B); 2C2ftC); 0,D);
C3ftB); CMB); 2C3ftB); 2C3D); 0,F); 0.F);
22>,A); 2D3B); 0,B); 2Z),C); 0,D); 5C2F);
20,FM ^A2)
20,F); 0^12)
42JC);6C2F);C1A2)
4i),C);6C,F); 0^12)
4ZKB);2C3D);2C2F);C1A2)
0e«,(l); C3^B); C2^C); 2CSF); CXA2)
C6B);C8D);C2F) 5 0^12)
C«.B); 0,D); 0,F); ^A2)
208,BM 0,F); ^A2)
65^A); 3C,,B); 2C,,C); 30,F); 0^12)
177

Space group
188 P6c2
189 P62m
190 P62<:
191 P^jmrnm
192 P6/W
193 P6Jmcm
194 P6Jmmc
195 P23
196 F23
197 723
198 P2X3
199 72X3
200 Pm3
201 Pn3
202 Fm3
203 Fd3
204 7m3
205 Pa3
206 7*3
207 P432
208 P4232
209 F432
210 F\32
211 7432
212 P4332
213 P4X32
Din
Din
Din
Din
Din
Din
Din
T1
T2
T3
r4
T5
rpl
rp2
n
n
n
rpl
o1
o2
o3
o4
o5
o6
o7
Site symmetries
0,B); C3AB); 0,B); C3AB); 0,B); C3AB);
3C3D);C2F);CSF); C,A2)
22>3AA); 2C3AB); O3J)B); 2O2t)C); C,D);
3CSF);C,A2)
0,B); 3C3AB); 2C3D); O2F); C,F); C,A2)
20tt(l); 22KAB); C6t>B); 22JAC); C3l)D);
5C2eF);4CsA2);C,B4)
0,B); C6AB); 0,D); C3AD); C6D); 0,F);
C2ft(o); C3(o); JC2(lz); Cs(l/); Lx\l<±)
2KAB); 0MB); C»D); 0,D); C6D); C2ftF);
C2,F); C,(8); C,A2); 2CtA2); C,B4)
0MB); 32)^B); 2C3rD); C2AF); CivF);
C2A2); 2C,A2) 5^B4)
2 T(l); 202C); C3D); 4C2F); C,A2)
47-(l); C3D);2C2F); 0,A2)
C3D); 0,A2)
O3D);C2F);O,A2)
2rA(l); 20»C); 4(^,F); C,(8); 2CtA2);
TB); 2C8iD); 0,F); 0,(8); 20,A2); 0,B4)
27\A); T{2); C2AF); C2eF); 0,(8); 0,A2);
2 T{2); 2C8ID); C8(8); 0,A2); 0,B4)
0,B4)
2C3l.D);C3(8);C,B4)
2C3iD);C3(8);C2A2); 0,B4)
20A); 204C); 2C4F); C3(8); 30,A2); 0,B4)
. T{2); 208D); 302F); 0,(8); 5C2A2); 0,B4)
20A); rB);2JF);C4F);C3(8);3C2A2);
2 T{2); 20,D); 0,(8); 20,A2); 0,B4)
0A); 04C); 0,D); 0,F); C4F); 0,(8);
20,D); 03(8M 02A2M 0,B4)
22KD);C3(8);C2( 12); 0,B4)
178

Space group Site symmetries
08 20,D);20,F); 0,(8) 5 30,A2); CxB4)
7* 27^A); 20MC); CwD); 2C2,,F); 0,A2M
0.A2) 5 0^24)
7* 4rd(l); C3))D); 2C,,F); CsA2); CxB4)
T* 7^A); 0MC); CwD); J4F); C*F); C2A2);
CSA2);CXB4)
T* T{2); 0,F); 2S4F); C,(8); 3C2A2); qB4)
7* 2 TB); 2^4F); C,(8); 2C2A2); qB4)
7* 25,F); C,(8); C2A2); CxB4)
Oj 20A(l); 2D4ftC); 2C4,F); C3r(8); 3^,A2);
3^B4M^D8)
O\ 0B); D4F); Cw(8)j J4A2); C4A2); C,A6);
2^B4M^D8)
0* 7^B); D2hF); 2D2dF); D3(8); 3C2,A2);
C3A6);C2B4) 5 0,B4M^D8)
0* Td{2); 2Z>3dD); D2dF); C3,(8); 0,A2) 5
C2sA2);3C2B4);CsB4);C1D8)
0\ 20»(l); TdB); DuF); C4l)F); C3v(8);
30^,A2M 20,B4M 0^48)
0« 0B); 7\B); D2dF); C4ftF); CtoA2); C4A2);
0,A6M 0,B4M C,B4); ^D8)
0\ 2 TdB); 2£>3dD) 5 0^(8); C2v{\2); CsB4);
0,B4M 0^48)
Oj TD); 0,(8); CM(8); J4A2); 0,A6); 2C2B4);
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
/4X32
P43m
F43m
/43m
P43n
F43c
I43d
Pm3m
Pn3n
Pm3n
Pn3m
Fm3m
Fm3c
Fd3m
Fd3c
Im3m
Ia3d
0\ 0h{\); 0ttC); D3dD); 0MF); 0^F); C3v(8);
2C2^A2); C2B4); 2CSB4); 0^D8)
Of C8i(8M 0,(8); 0,A2M 54A2); 0,A6M
20,B4M 0^48)
Note the following equivalent nomenclatures:
dI - vd
179

APPENDIX II
CHARACTER TABLES
The symmetry operations present in each point group are
described in References 3, 11, and 12. The species of the
translations Txy Tyy and Tz in the x,j, and z directions, respectively,
and the rotations Rx, Ry, and Rz in a right-handed system
parallel to the x, y and z axes, respectively, are given for each point
group. Also, the species of the polarization tensor elements,
a#, is identified for each point group. This allows us to determine
immediately the spectral activity of each species in a point
group; for example, all those species that contain a translation
will be infrared active, whereas those species containing an
element of the polarization tensor oc# will have Raman activity.
The combination levels v{ + vh difference tones, vt — vjy and
overtone levels Bvi9 2vjy etc.) symmetry can be determined by
using these tables. A description of this procedure is given in
Wilson, Decius and Cross, [11], especially Chapter 7.
181

A'
A"
1 1
1 -1
Rx, Ry
A
A
E
1
1
i
1
-1
Ry,
T }
Rz
axx, a,w, a2Z, *„, «.xz, xyz
c2
A
B
E
1
1
Cm
1
I
35
^055 "* y I
R, ■
■Kx> ^y
*xx> *yy> *zz> *xy
&*XZ) &yz
c*h
Ag
Bg
Au
Bu
E
1
1
1
1
c2
1
-1
1
-1
i
1
1
-1
-1
°*
1
-1
-1
1
Rz
Tz
Tx,Ty
a«*» *yy> «-zz> *xy
<*-xz> Kyz
c2v
A
A
B2
E
1
1
1
1
c2
1
1
-1
-1
av(zx)
1
— 1
1
-1
1
-1
-1
1
Tz
Tx; Ry
Ty;Rx
a**5 «w> *zz
a*2
OLyz
A
Bx
B2
Bz
E
1
1
1
1
CmW
1
1
-1
-1
Cm&)
1
-1
1
—1
1
-1
— 1
1
1 z>Kz
Ty, Ry
TX,RX
OLxy
<xxz
*yz
182

to*
<M
^H -H ^H ^ O
_H »-H r-X ^ CN
I
— -H rt -. O
r—« r-H f—i r—J O*I
If
n a 8
ftj a^ ct;
1 1 1
I 1 1
1 1 1
1
1 1
1 1
1 1
E-rE-Tf-,8
1
1
1
! 1 1
1 1
1 1
1 1
8
■ft-
o. a a
183

184

E-H £
ft!
CO
r-< -< O -H ^H O
^ r-4 O —I ^ O
I I I
^ ^ CM i-< ^ CM
I I I
^ *-< O ^H r-H O
I
CM
I
CM
^h _., CM *-* —i CM
-H -H O -^ —*
r^ r-H CM —< —• CM
«—t 'to ««^
185

8 « „
8 8
OS h
a 8
III I
T 7 7 7
i - i i | -
rt " " 7 7 7 7
I I '- II-
CM
I I
^-H 1 1 O
I I
cf
3
r-l <M
cq cq
186

CM
CM
CJ
II
CM
H 8
^77
_ _ ,-H
1
r_H i 1 r—J
^H ^H ^
— -H -*
W (N r-i
ftf
^ o
1
-h CM
1
^ o
1
r-< CM
CO 00
CM
CM
ftf
r-H ^H r-H ^H O O O
I I
r—t i—I r—I r—( O O O
_< _h ^ CM CM CM
I I
*-h —« —• I CM O I CM
-H -H —< r-l O CM O
—I -H r-i | CM O I CM
*-H r-H CM CM CM
187

b
(M
8 1
8 **
I I
CM r-H r-H ^H r-^ CM
I I I I
ll M
H H CM
r 7 7 71
■f- I
8 8
51
7T
o
CM
I I I
f—< r—\ ~-< CM i—• »-h
I I •
I I
I I
fel
rH <M t-I
t-(<Mt-H<M
188

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200

APPENDIX III
CORRELATION TABLES
We have not attempted to list all possible correlations between
the different point groups; for example, if the correlation
between D^ and 0h is desired, it can be carried out in two or,
if necessary, more steps. First we would write the correlation
from Dooh to ZLA; then the correlations that relates from
ZL7t to 0h would give the desired relationship between D aoh
and 0h, The reader can also construct the correlation tables
(see Chapter 2 for details) from the character tables given in
Appendix II or from the group theory of the individual point
groups. Since the correlations depend on certain choices in
axes, planes, or both, this choice is identified above each
subgroup of a point group. This information is of great importance
in applying Table 14; for example, in the correlation table for
201

202
Appendix III
C2v there are two columns relating the Cs subgroup. These are
given as follows.
(This portion identifies the plane in
C2v which contains the equivalent set
of atoms of site symmetry Cs. (See
Chapter 2, p. 38, for an example.)
A
A,
Bt
B2
a(zx)
cs
A'
A"
A'
A"
o(yz
cs
A'
A"
A"
A'
Those species of the point groups C9h9 C^h, C5h, CQh, C6, £4, S6,
S8, T, and Th marked with an asterisk (*) will not use the
coefficient 2 of the Ei species in this correlation procedure only.
Also, for those species of the point group 7* and Th marked with
a double dagger (J) a coefficient 2 will be added to the E4
term related to the Ft species of the point group. Several
examples, in particular 3 and 4, which illustrate this change,
are given in Chapter 5.
^2h
A,
Bg
Au
Bu
c,
A
B
A
B
cs
A'
A"
A"
A'
Ag
A
Au
Au
^2v
Ax
A,
B1
B2
c%
A
A
B
B
a{zx)
cs
A'
A"
A'
A"
a(yz)
cs
A'
A"
A"
A'
A
B1
B,
Bz
ci
A
A
B
B
pV
^2
A
B
A
B
px
^2
A
B
B
A
A
A2
B2
E
c
A
A
B
B
E .
2-+C2(z
D2
A
Bx
A
B\
^2+B3
) c2
p s~t
^2v ^2
A± A
-A-2 -^i
A A
/In /I
A A
-ill jCX
Did O Z?
Jj-i-f-JDo ^13
C2
c2
A
B
A
B
A+B
Cs
A'
A"
A"
A'
A'+A"

a(zx)
b
o°
o*
cl
ci
<M
•< 5; 5:
« 5: «k
^«
***
r-l <M <M
iH Ol
O> tH ©4
X!
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xT
Cd
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ft;
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xj
3
3
xj
«r
3
(M
3
3
XJ
X!
«,
xj
ft;
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(N
ft;
3
Of
xj
x|
3
«r
CO
ft;
CO
ft;
203

A'
Ef
A"
E"
C*
A
E
A
E
cs
Ar
2A'*
A"
2A"*
Ci
A
2 A*
A
2A*
Ax
A2
E
c3
A
A
E
A'
cs
A'
A"
+ A"
^3
A
E
Cs
A
A
E
A
c2
A
B
+
B
D3d
Alg
A2g
Eg
Aiu
eI
Ax
E
Ax
E
c3v
Ax
A,
E
A2
Ax
E
Ag
A
Eg
Au
Au
Eu
■ cz
A
A
E
A
A
E
Cu
Ag
Bg
Ag + Bg
Au
Au + B,
A
« A
c2
A
B
+
A
B
+
B
B
A'
A'
cs
A'
A"
+ A"
A"
A'
+ A"
ct
AB
Ag
2A,
Au
2AU
D3h
A'x
A'z
E'
A'i
Al
E"
cSh
A'
A'
E'
A"
A"
E"
D3
Ax
A2
E
Ax
A,
E
c3v
Ax
A,
E
A2
Ax
E
Ax
At
> av{zy)
c2v
Ax
B%
+ B%
A,
+ Bx
C3
A
A
E
A
A
E
A
A
c2
A
B
+
A
B
+
B
B
cs
Ax
A'
2A'
A"
A"
2A"
A'
A'
ct
A'
A"
+
A"
A'
+
A"
A"
A
B
E
C,
A
A
2B
C*n
A*
Bg
Au
Bu
K
A
B
E
A
B
E
s,
A
B
E
B
A
E
C>2h
A9
Ag
2B*
Au
Au
2B*
c2
A
A
2B*
A
A
2B*
cs
A'
A'
2A"*
A"
A"
2A'*
Q
Ag
Ag
2A*
Au
Au
2At
Cx
A
A
2 A*
A
A
2A*
* The coefficient 2 is not to be used in this correlation method; see p. 202 and
Chapter 5, examples 3 and 4.
204

c
A
A2
Bx
B2
E
c*
A
A
B
B
E
C2v
A
A2
A
Bx+Bz
cl
Ax
A2
A
B,+B2
c2
A
A
A
A
2B
C,
A'
A"
A'
A"
A' + A"
c.
A'
A"
A"
A'
A' + A"
At
A,
Bx
Bz
E
c2
D2
A
B1
A
Bx
B2 + B3 J
c2
A
Bi
B1
A
B2 + B
c,
A
A
B
B
c2
A
A
A
A
2B
c;
c2
A
B
A
B
A + B
c2
c2
A
B
B
A
A+B
c2
Ax
A2
B2
Ex
E2
D,
Ax
A2
Ax
A2
E
■#i +B2
E
A
A2
A2
E
D 1 Z?
151 + ^2
E
A
A
B
B
Ei
E2
Ez
c,
A
A
A
A
E
2B
E
&2V
Ax
A2
A
Bx + B2
Ax + A%
Bx+B2
c2
A
A
A
A
2B
2A
2B
A
A
A
c2
A
B
A
B
+ B
+ B
+ B
A'
A'
A'
C$
A'
A"
A"
A'
+ A"
+ A"
+ A"
205

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206

A
B
E
A
A
2B*
Ct
A
A
2 A*
A'
K
E'2
A"
E'i
El
ct
A
Ex
E2
A
Ex
E2
cs
A'
2A'*
2A'*
A"
2A"*
2A"*
Cx
A
2 A*
2 A*
A
2 A*
2A*
Ax
A2
Ex
E,
cs
A
A
Ex
E,
A'
A'
cs
A'
A"
+
+
A"
A"
Ax
As
Ex
E2
Q
A
A
Ex
E2
A
A
ct
A
B
+
+
B
B
Ax,
Aig
Ex,
Eig
^2w
Exu
Eiu
D5
Ax
A2
Ex
E2
Ax
A2
Ex
E2
Ax
A,
Ex
E2
A*
Ax
Ex
E,
c5
A
A
Ex
E,
A
A
Ex
E2
A
A
A
A
c2
A
B
+
+
A
B
+
+
B
B
B
B
A'
A'
A'
A'
cs
A'
A"
+
+
A"
A'
+
+
A"
A"
A"
A"
Ct
Ag
AB
2Ag
ZAg
Au
Au
2AU
2AU
A'x
A'z
El
E'2
A'i
Al
E'i
El
D,
Ax
A2
Ex
E2
Ax
e\
E,
c»
Ax
A2
Ex
E2
A2
Ax
Ex
E2
Af
A
E{
E2
A"
A"
E'i
El
A
A
E1 i
E2 j
A
A
E1 j
E2 2
->■ a(zx)
c2v
A
Bx
k + B2
h + &2
B2
12 + B2
A
A
A
A
c2
A
B
+
+
A
B
+
B
B
B
B
Cs
A'
A!
2A!
2A'
A"
A"
2A"
2A"
A'
A!
A
A!
Cs
A1
A"
+
A"
A9
+
+
A"
A"
A"
A"
* The coefficient 2 is not to be used in this correlation method; see p. 202
and Chapter 5, examples 3 and 4.
207

c.
A
B
Ei
c,
A
A
E
E
c2
A
B
2B*
2 A*
Ci
A
A
2 A*
2 A*
c*h
Ag
Elg
E2g
Au
Bu
Exu
E2u
C6
A
B
Ex
A
B
Ex
E2
c3h
A'
A"
E"
E'
A"
A'
E'
E"
$6 = C3i
Ag
e\
Eg
Au
Eu
E
C™
A9
Bg
2B*
2A*
Au
Bu
2BZ
2AZ
c3
A
A
E
E
A
A
E
E
c,
A
B
2B*
2 A*
A
B
2B*
2 A*
Cs
A'
A"
2A"*
2A'*
A"
A'
2A'*
2A"*
Ci
Ag
Ag
2A*
2A*
Au
Au
2At
2AZ
Cx
A
A
2 A*
2 A*
A
A
2 A*
2 A*
c«v
Ax
A2
Bx
B2
Ex
E2
c*
A
A
B
B
Ex
E2
c3v
Ax
A2
Ax
A2
E
E
c»
Ax
A%
Ax
E
E
av ->■ a(zx)
c2v
Ax
A2
Bx
B2
Bt +B2
Ax + Az
c3
A
A
A
A
E
E
c2
A
A
B
B
IB
2A
A'
A'
<*»
Cs
A'
A"
A'
A"
+
+
A"
A"
A'
A'
c.
A'
A"
A"
A'
+
+
A"
A"
A
A
B
B
E1
E2
C'z
Dz
A
A2
A
E
E
ci
£3
A
E
E
A
Bi
B3
B2 + B
A + Bx
C3
A
A
A
A
3 E
E
c2
c2
A
A
B
B
2B
2A
A
A
c2
c2
A
B
A
B
+
+
B
B
A
A
^2
c2
A
B
B
A
+
+
B
B
Ax
Ex
* The coefficient 2 is not to be used in this correlation method; see p. 202 and
Chapter 5, examples 3 and 4.
208

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cq cq cq
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rH:
CM
q
CO rH
j , r-f
«S 53) 3 3
^ f^ TjH TjH T
^ (^ TjH "^ T
r-f CM
^] fjj TJH Trt T
i-t eM
co o>
I S rH
3 3
jq tq ^j ^ t
«^ to, ^ "^ c
rH CM N rH
.^q ^q ^ tjh c
: ^. 5: rHt CM*.
■^ ^ "^ "^ T
: v 5: tH5: <M«^
rH CM rH CM
to & 3 9
-t} tt) ^ "^ C
00 (vT1
H ^J M [^
H T^ fj] Ct^
H CM r ,.
^ "^q to ^
CM rH r ,.
c "^ to ^
tH_ <M
CO rH
cm eo . i
CM 3 Sr »
jh r^< tq to,
S S 3 3
^ bq bq
q £q faT^
<Nv.tHv. %
tH»« CM»» %
d "^! to. to.
rH <N rH CM
q Dq tq wq
3 3 3 3
q cq tq to.
210

to O
to AJ
to^O"
bV
CM CM
<M CM
CM CM
ear r-t
GJ t-4
fiq "^
Dq "^
i-ic<itH<nj ( ear
r-l r-r
cq ^
a a a ^ ^s a 3^
» S 3 3
CM CM
CM CM
QQ oq
CM ^
oq^Dq3
CM CM
d t-I rH _i I
r-i e5f
Dq ^
1-4 (M
rH »H
n ea h j 1
Dq Dq *i \
Dq1^1
211

c3i = s6
Ag
Eg
Au
cs
A
E
A
E
Q
Ag
2A*
Au
2AU*
Cx
A
2 A*
A
2 A*
A
B
c,
A
A
E
2Z?*
E
c2
A
A
25*
2 A*
2B*
Ci
A
A
2 A*
2 A*
2 A*
AX = V+
A2 = 1r
Ex = II
E2 = A
E3 = 0
£4 = r
^2
-E2
B2 + B:
E2
clv
Ax
A2
E
^1 + ^2
L ^
A + A2
CSv
A
A2
E
E
A + A
E
c2v
A
Z? 1 Z?
■oi -r ^2
A + A
Bx + B2
s+
\
s~
n.
.A".
^2g A
Elg Ex
E2g E2
A2u A^
Au A
77" 771
^2u £2
^2
Ax
A,
E
E
Alg
A2g
K
Blg + B2
Azu
Alu
Eu
civ
Ax
A2
E
o Bx + B2
Ax
A2
E
„ Bx-\- Be,
A 2+ = ax
A2 S~ = ^2
^i + B2 U^E1
Ax + A2 A == E2
^ s+ = Ax
A V— — J
^ + .b2 n = j?!
T
A
E
F
D2
A
2 A*
Bx -f B2 + B3
Cs
A
E
A + Et
A
2 A*
A + 2B
Ci
A
2 A*
3A
* The coefficient 2 is not to be used in this correlation method; see p. and
Chapter 1, examples 3 and 4.
f The z axis of C^v and D^n groups must coincide with z axis of point group.
t p. 202, add the coefficient 2 to E{ species.
212

A
A2
E
Fx
F2
T
A
A
E
F
F
A
A
B2
D2c
A
+
+
+
i
Bx
E
E
c3v
A,
E
A2 +
A +
E
E
A
A
B
A
B
+
+
B
E Bx-
^2
A
A
2A
hB2 + l
\-B2 + i
53 A-
c2v
A
k + A
- &i + B2
f Bx + B2
A
A
c3
A
A
E
+
+
E
E
A
A
c2
A
A
2A
+ 2B
+ 2B
A'
A'
2A'
Cs
A!
A!1
+
+
+
A"
2A"
A"
Dt
2h
A
E
F
A
E
F
2Ag*
Z
2AU*
A
2A*
A2
2A*
2Ag*
A**
Au + 2BU
A
E
A
2,4*
A + 2B
A
2A*
A + 2B
A'
2Af*
A' + 2A"
A"
2A"*
2A! + A!'
* The coefficient 2 is not to be used in this correlation method; see p. j 53and Chapter 5, examples 3 and 4.
J p. 202, add the coefficient 2 to E{ species.
2A*
SAn
3AU
A
2A*
3A
A
2A*
SA

0
A
A
E
E2
T
A
A
E
F
F
A
A
B*
A
Bx
+ Bt
+ E
+ E
A
A
A
A
E
s +
L +
E
E
A
A
B
Q
A
B
+
+
B
JCj Jj
D2
A
A
2A
±+B2 + l
X+B2 + 1
C2, 2C2
A
Bx
A + Bx
53 B1 + B2 + i
3Z A + B2 + B
33 A
3 A
c*
A
A
E
+
+
E
E
A
A
c2
A
A
2A
+ 2B
+ 2B
A
A
2A
c,
A
B
+
+
+
B
2B
B
On
Alg
E
F1
FZg
AL
A2u
E
Fxu
CORRELATION TABLE FOR Oh AND
0
At
A
E
Ft
F2
Ax
A
E
Td
A
A
E
Fx
A
A
E
F2
Ft
A
A
Fg
Au
A
Fu
T
A
A
E
F
F
A
A
E
F
F
J>.d D* C3V
A A A
rp a i^ n rp
^9 ^19 "t" nlg ■&
2g i" &g A2g -f- tLg A2 -f-
Ag H" Eg B2g + Eg A1Jr
Au Biu A
Eu Alu -f- Blu E
Au + Eu A2u + Eu Ax +
Alu + Eu B2u + Eu A2 +
ITS
E A
E A
E A
E A
SUBGROUPS
^3
A
E
2 +
1 +
Ax
A*
E
2 +
1 +
E
E
E
E
Ag
Ag
Au
Aa
-s.
Ag
Ag
+ Eg
+ E9
Au
Au
Eu
+ eI

I?
1? *
oaq
-^ fiq
^ cq
rH r-i
^ cq
rH r-i
^ cq
cq
cq
+
«f
^*
CqH
rH
rH
cq
of
T-f
rH
cq
to.
to, to,
+ +^
^ cq
to. to.
(M 93
+ +0
tq bq
©a e<i
to, to.
cq
oq
cqS
3
cq"
r-t rH i
^ cq i
rH
q cq +
93
rH
cq
rH ea |
q ^ +
03
ccT
rH
cj "^d to
cq
cq
+
+
+
oq
3
+
to.
©3
ft)
IT
CO ^
Jo"
IT
CM *3
CO ^H
CM
c?
rH rH .
H
(M
^of +
to
CM
eo
cq
+
+
CO
cq
+
tH
cq
«T
•+
cq*
i
of
cq
^r
cq
cq^
+
G)
cq^
cq
of
"t
of
'ft?
CO
cq
+
cq"
CO
cq
+
cq"
+
C3
rH
rH
cq
+
cq
t
to
cq
+
cq^
Co
cqW
~t
of
cq"
co eo
cq cq
cqM+ +
t^ ftj*_[_of cq1
^ + +
CO CO
cq cq
£\| ^M **H
r-i rH
cq cq
of of
«f++
^■of+of^
cq cq
^•V + «fof
N N « «j _i
CM
co 3
o^H- +
^ cq "t Qq Q2*
3 ^
cq ^q
3 3
cq cq
3 3 3 3 3
^ ^ 5 of of
3 3
of of
215

' CM CO CO CM CO CO
cq cm ^Q ^q cm ^Q
CM
CM CM
^ CM "^
CM CM
i ^ 4.°* * * fe
' CM v. 4" CM .
[CM 3 » 3CM CM
o
CMCMCMCMCMCMCMCM
CM CM CO CO
CM CM
CM CM
^
216

AUTHOR INDEX
Adama, D. M., 37, 171
Antion,D. J., 170
Bentley, F. F., 1
Bhagavantam, S., 1, 53
Bunn, C. W., 146
Corradini, P., 149,155
Cross, P. C, 170, 181
Decius,J. C, 170, 181
Durig,J.R., 25,31,171
Fateley, W. G., 1
Frevel, L. K., 117
Halford, R. S., 1,145, 171
Henry, N.F.M., 170
Herzberg, G., 65, 86, 171
Higgs,P.W., 155
Hornig,D.F., 1,171
Khanna, R. K., 170
Kittel,C, 170
Koenig,J. L., 163
Krimm,S., 155
Liang, C.Y., 155
Lonsdale,K., 170
McDevitt, N. T., 1
Natta,G., 149,155
Reimann, C. W., 170
Slater, J. S., 170
Tadokoro, H., 155
217

218 Author Index
Tobin, M., 145 Wilson, E. B., 170, 181
Tuinstra, F., 163 Winston, H., 1,145, 171
Wyckoff, R.W.C., 170
Venkatarayudu, T., 1, 53
Zbinden,R., 145

SUBJECT INDEX
Aluminum oxide, alpha, 3,4, 7, 23,
37
aluminum atom, 23
lattice vibrations, 23
equivalent atom-site, 23
infrared activity, 24
oxygen atom, 24
lattice vibration, 24
raman activity, 24
Ammonium azide, 117
ammonium ion, 119
intramolecular vibrations, 120
lattice vibration, 119
libration, 119
irreducible representation for, 124
Ammonium iodide, 3, 7, 25
acoustical vibrations, 32
intramolecular vibrations, 32
iodine ion, 29
infrared activity, 29
lattice vibrations, 29
raman activity, 29
lattice vibrations, 29
NHj ion, 26,42,43
infrared activity, 28
intramolecular vibrations, 31
lattice vibrations, 26
raman activity, 28
rotations, 30
rotational vibrations, 32
Ammonium ion, 79
hydrogen atom, 83
activity, 83
intramolecular activity, 83
nitrogen atom, 84
activity, 84
Aragonite structure, 165
Axis of gravity, 145
219

220
Subject Index
azide ions, 120
intramolecular vibrations, 122
ions in site a, 121
ions in sitefe, 122
Benzene, 66
hydrogen atoms, 68
activity, 68
infrared activity, 69
raman activity, 69
site symmetry, 61
Bixybyite structure, 133
lattice vibrations, 135
separable degeneracy, 133
site symmetries, 133
Boron carbide, 130
irreducible representation, 132
site symmetries, 130
Bravais, space cell, 2, 3, 4,181
cell, 4,12,37
Brillouin zone, 17
Calcite structure, 167
Calcium carbonate, 165, 167
Calcium fluoride, 169
Center of gravity, 145
Character tables, 181
Chlorobenzene, 76
infrared activity, 78
raman activity, 78
site symmetries, 76
Correlation tables, 14
derivative, 37
relationship to site notation, 43
Crystal structure, 2
Cuprous oxide, 3, 4, 6, 7, 21
copper atom, 22
lattice vibrations, 22
equivalent atom-site, 21
infrared activity, 22
oxygen atom, 22
lattice vibrations, 22
raman activity, 22
1,4-Dichlorobenzene, 73
hydrogen atoms, 75
activity, 75
infrared activity, 73
raman activity, 73
site symmetries, 73
Equivalent atoms, 5
External vibrations, 26
Fluorite structure, 169
Garnet structure, 138
AlO4ion, 142
internal vibrations, 143
lattice, 143
libration, 142
A1O6 ion, 144
lattice vibrations, 140
site symmetries, 138
Graphite, 162
interaction between layers, 163
activity, 164
isolated layer, 162
intramolecular vibrations, 162
Helical axis, 155
Helical molecules, 155
general, 158
Ilmenite structure, 166
Internal vibrations, 26
Intramolecular vibrations, 25

Subject Index
221
irreducible representations, 65
nongenuine motions, 66
Ironsulfide, 168
Iron titanate, 166
Irreducible representation, 12,13
Lattice points, 3
Lattice vibrations, 2, 9,13,14
Libration, 25
Molecular crystal, 13
Molecular selection rules, 65
Perovskite, 54
Polarizability tensor, 28
Polyethylene, 146
irreducible representation, 149
line group, 146
activity, 148
Polyethylene oxide, 156
irreducible representation, 158
Polyvinylchloride, syndiotatic,
149
intramolecular vibrations, 153
irreducible representation, 154
Potassium hydrogen bifluoride,
125
ionic crystal, 129
with molecular ion, 127
intramolecular vibration, 129
irreducible representation, 130
libration of, 128
Pyrite structure, 168
Raman interactions, 87
References, 170
Rotational freedom, 10
Rotations, 9
improper, 54
polymer chain, 148, 152
proper, 54
Rutile structure, 165
Separable degeneracy, 10,11
example, 133
Site symmetry, 4, 5, 35
equivalent sets of atoms, 66
tables, 181
Spinel structure, 169
Strontium titanate, 3, 7,18, 54
equivalent atom-site, 20
infrared activity, 21
irreducible representation, 53, 64
oxygen atom, 21
lattice vibrations, 21
raman activity, 21, 64
strontium atom, 20
lattice vibrations, 20
titanium atom, 20
lattice vibrations, 20
Titanium dioxide, 165
acoustical vibrations, 17
anatase, 3, 6, 7, 15,16,18, 19,35,
36,42,43,166
equivalent atom-site, 7
infrared activity, 18
lattice vibrations, 16
oxygen atom, 16,36,43
raman activity, 18
titanium atom, 8, 13,15, 36
vibrational displacements, 8
lattice vibrations, 15
Translations, 9
polymer chain, 148,152
1,3,5-Trichlorobenzene, 69
activity, 72
chlorine atoms, 72

222
Subject Index
infrared activity, 71
raman activity, 71
site symmetries, 69
Unit cell, crystallographic, 2,4, 37
Vibrational freedom, 10
Wurtzite structure, 167
Wyckoff, tables, 35, 37
notation, 36, 38,43
Zinc blende structure, 168
Zincsulfide, 167,168
Zirconium dioxide, 3, 24
equivalent atom-site, 24
infrared activity, 25
oxygen atom, 25
lattice vibrations, 25
raman activity, 25
zirconium atom, 25
lattice vibrations, 25