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Symmetry and Group Theory
in Chemistry

“Talking of education, people have now a-days’’ (said he) “got a strange opinion
that every thing should be taught by lectures. Now, I cannot see that lectures can
do so much good as reading the books from which the lectures are taken. I know
nothing that can be best taught by lectures. except where experiments are to be
shewn. You may teach chymestry by lectures - You might teach makmg of shoes
by lectures!’’
James Boswell: Life of Samuel Johnson, 1766 (1709-1784)
“Every aspect of the world today - even politics and international relations - is
affected by chemistry”
Linus Pauling, Nobel Prize winner for Chemistry, 1954, and
Nobel Peace Prize, 1962

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ABOUT THE AUTHOR
Mark Ladd hails from Porlock in Somerset, but subsequently, he and his
parents moved to Bridgwater, Somerset, where his initial education was at Dr
John Morgan’s School. He then worked for three years in the analytical
chemistry laboratories of the Royal Ordnance Factory at Bridgwater, and
afterwards served for three years in the Royal Army Ordnance Corps.
He read chemistry at London University, obtaining a BSc (Special) in
1952. He then worked for three years in the ceramic and refractories division
of the research laboratories of the General Electric Company in Wembley,
Middlesex. During that time he obtained an MSc from London University for

work in crystallography.
In 1955 he moved to Battersea Polytechnic as a lecturer, later named
Battersea College of Advanced Technology; and then to the University of
Surrey. He was awarded the degree of PhD from London University for
research in the crystallography of the triterpenoids, with particular reference
to the crystal and molecular structure of euphadienol. In 1979, he was
admitted to the degree of DSc in the Universeity of London for h s research
contributions in the areas of crystallography and solid-state chemistry.
Mark Ladd is the author, or co-author, of many books: Analytical
Chemistry, Radiochemistry, Physical Chemistry, Direct Methods in
Crystallography, Structure Determination by X-ray Crystallography (now
in its third edition), Structure and Bonding in Solid-state Chemistry,
Symmetry in Molecules and Crystals, and Chemical Bonding in Solids and
Fluids, the last three with Ellis Horwood Limited. His Introduction to
Physical Chemistry (Cambridge University Press) is now in its third edition.
He has published over one hundred research papers in crystallography and in
the energetics and solubility of ionic compounds, and he has recently retired
from his position as Reader in the Department of Chemistry at Surrey
University.
His other activities include music: he plays the viola and the double bass in
orchestral and chamber ensembles, and has performed the solo double bass
parts in the Serenata Notturna by Mozart and the Carnival of Animals by
Saint-Saens. He has been an exhibitor, breeder and judge of Dobermanns,
and has trained Dobermanns in obedience. He has written the successful book
Dobermanns: An Owner s Companion, published by the Crowood Press
and, under licence, by Howell Book House, New York. Currently, he is
engaged, in conjunction with the Torch Trust, in the computer transcription
of Bibles into braille in several African languages, and has completed the
whole of the Chichewa (Malawi) Bible.
Mark Ladd is married with two sons, one is a Professor in the Department

of Chemical Engineering at the University of Florida in Gainsville, and the
other is the vicar of St Luke’s Anglican Church in Brickett Wood, St Albans.
He lives in Farnham, Surrey, with his wife and one Dobermann.

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Symmetry and Group Theory
in Chemistry

Mark Ladd, DSc (Lond), FRSC, FInstP
Department of Chemistry
University of Surrey
Guildford
Foreword by
Professor the Lord Lewis, FRS
The Warden
Robinson College
Cambridge

Horwood Publishing
Chichester

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First published in 1998 by
HORWOOD PUBLISHING LIMITED
International Publishers
Coll House, Westergate, Chichester, West Sussex, PO20 6QL

England

COPYRIGHT NOTICE
All Rights Reserved. No part of h s publication may be reproduced, stored in a
retrieval system, or transmitted, in any form or by any means, electronic,
mechanical, photocopying, recording, or otherwise, without the permission of
Horwood Publishing, International Publishers, Coll House, Westergate, Chichester,
West Sussex, England

0M. Ladd, 1998

British Library Cataloguing in Publication Data
A catalogue record of this book is available from the British Library

ISBN 1-898563-39-X

Printed in Great Britain by Martins Printing Group, Bodmin, Cornwall

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Table of contents
Foreword..................................................................................

.................v-VI

Preface ................................................................................................
List of symbols ............................................................................................

... vii

. ~.i.i.-.m. i

1 Symmetry everywhere ......................................................................................
1
1.1 Introduction: Looking for symmetry.............................................................. 1
1.1.1 Symmetry in finite bodies ...................................
.................2
1.1.2 Symmetry in extended patterns .................................................................. 4
1.2 What do we mean by symmetry..................................................................... 5
1.3 Symmetry throughout science .......................................................................
6
1.4 How do we approach symmetry
Problems 1 .................................................................
2 Symmetry operations and symmetry elements ...............................
2.1 Introduction: The tools of symmetry
2.2 Defining symmetry operations, ele

Sign of rotation............................
2.2.4 Reflection symmetry .................
2.2.5 Roto-reflection symmetry..........
2.2.6 Inversion symmetry ................

11
....................

11

......................
.......................................


13
15

.......................................
......................................

18
19

2.2.8 Roto-inversion symmetry ........

2.5.1 Sum,difference and scalar (do
2.5.2 Vector (cross) product of two
2.5.3 Manipulating determinants and matrices .....................................
Matrices and determinants; Cofactors; Addition and subtraction
of matrices; Multiplication of matrices; Inversion of matrices;
Orthogonality ...............................
..............................
2.5.4 Eigenvalues and eigenvectors................................................................... .28
Diagonalization; Similarity transformation; Jacobi diagonalization ..... 30-3 1
....................................... 33
2.5.5 Blockdiagonal and other special matrices
Adjoint and complex conjugate matrices;
matrix; Unitary matrix ......................................................................... 34-35
........................................
.................................. 35
3 Group theory and point groups .............

3.1 Introduction: Groups and group the0
3.2 What is group theory .........................


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..................................
..................................
.....................................

38
38
..38


................................ 38
3.2.1 Group postulates ............................................
Closure; Laws of co
Inverse member ...........
....................................
................. 38-39
.............................
3.2.2 General group definitions............................
................................
3.2.3 Group multiplication tables .
3.2.4 Subgroups and cos
...............................
3.2.5 Symmetry classes and conjugates ........
3.3 Defining, deriving
................................... 46
3.3.1 Deriving point groups.. ............................
Euler's construction
...............................................

............................ 52
3.3.2 Building up the
..........................................
......................... 59
Problems 3................................................................

...............................

67

4 Representations and character tables .........................................
4.1 Introduction: What is a representation ............
........................... 72
4.1.1 Representations on position vectors.......................................
4.1.2 Representations on basis vectors .........................
............................ 75
4.1.3 Representations on atom vectors.. .....
............................................. 77
Unshifted-atom contributions to a re
................................
82
4.1.4 Representations on functions.................
4.1.5 Representations on direct product functions .............................
4.2 A first look at character tables......................
.......................... 86
4.2.1 Orthonormality .............
87
...........................................
4.2.2 Notation for irreducible representations ........
............................. 88

Complex characters ......................................................................
89
4.3 The great orthogonality theorem .....................
............................ 90
4.4 How to reduce a reducible representation .........................................
94
................................ 96
4.5 Constructing a character table.. ..........................
4.6 How we have used the direct product ..............
............................ 103
Problems 4 ............................................................
............................ 104
5 Group theory and wavefunctions...........................

.............................. 108
108
5.1 Introduction: Using the Schrodinger equation ...............................
5.2 Wavefunctions and the Hamiltonian operator............................................ ,109
5.2.1 Properties of wavefunctions ................................
................. 110
5.3 A further excursion into function space..............................................
5.3.1 Defining operators in function space ....................................................... 112
5.4 Using operators with direct products .........................................
115
...............................
117
5.5 When do integrals vanish ................................
5.6 Setting up symmetry-adapted linear combinati
............................... 119
5.6.1 Deriving and using projection operators.......

5.6.2 Deriving symmetry-adapted orbitals for the carbonate ion
Generating a second function for a degenerate representation
5.6.3 Handling complex characters ...........................................
Problems 5...................................................
...................................... 128

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6 Group theory and chemical bonding .............................................................

130

6.1 Introduction: molecular orbitals .................................
Classlfylng molecular orbitals by symmetry ...............
6.2 Setting up LCAO approximations.. ........................................................... .13 1
Function of the Schrijdinger equation ........
....................................... 132
Introducing the variation principle .......
6.2.1 Defining overlap integrals...............................
.............................. 134
6.2.2 Defining Coulomb and resonance inte
.............................. 134
Continuing with the variation principle ..............................
6.2.3 Applying the LCAO method to the oqgen molecule.............................. .137
6.2.4 Bonding and antibonding molecular orbitals and notation....................... 140
Total bond order ................................................
...........142
6.3 P-electron approximations ...... .............................................................. 142
6.3.1 Using the Huckel molecular............................. 143

Benzene.................................................................................................. 144
6.3.2 Further features of the Huckel molecular-orbital theory.......................... .149
........................ 149
ll-Bond order ................................................
Free valence .......................................................................................... .151
Charge distribution ................................................................
152
6.3.3 Altemant and nonaltemant hydrocarbons ..... ..................................... 152
Methylenecyclobutene; methylenecyclopropene ...................................... 153
6 4 4 Huckel's 4n + 2 rule ..........................................
.................156
6.3.4 Working with heteroatoms in the Huckel approximation157
Pyridine...........................................................
6.3.5 More general applications of the LCAO appro
Pentafluoroantimonate(II1) ion ............................
First look at methane ............... ......................................................... 165
......167
6.4 Schemes for hybridization: water
methane .............
....................................... 169
6.4.1 Symmetrical hybrids ............................
Walsh diagrams ...................................
....................................... 173
Further study of methane .....................
6.5 Photoelectron spectroscopy ..........................................................
Sulfur hexafluoride....................................
............................... 178
6.6 Cyclization and correlation ....................
6.7 Group theory and transition-metal compounds.........
..................186

6.7.1 Electronic structure and term symbols........................................
Russell-Saunders coupling.. ..
..................................................... 188
6.7.2 How energy levels are split in a crystal field.......
.................... 192
Weak fields and strong fields ...............
6.7.3 Correlation diagrams in 0, and Tdsymmetry ......................................... .197
........................... 203
'Holes' in d orbitals .....................................
.....205
6.7.4 Ligand-field theory .... ......................................................
Spectral properties............................
............................................... 211
Problems 6 ....................................................................
................... 217

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7 Group theory, molecular vibrations and electron transitions...................... .22 1
7.1 Introduction: How a molecule acquires vibrational energy...........
.............................. 222
7.2 Normal modes of vibration ...................................
7.2.1 Symmetry ofthe normal modes...................................................
7.3 Selection rules in vibrational spectra................
7.3.1 Infrared spectra......................................
Diatomic molecules ..................................................................
................................... 230
7.3.2 Raman spectra .............................................
Polarization of Raman spectra ..

.............................................
7.4 Classlflmg vibrational modes ...................
7.4.1 Combination bands, overtone bands and Fermi resonance............
7.4.2 Using correlation tables with vibrational spectra ..................................... 239
7.4.3 Carbon &oxide as an example of a linear molecule ........................
...................... 241
7.5 Vibrations in gases and solids ....................................
7.6 Electron transitions in chemical species............................................
7.6.1 Electron spin..................................................................................
7.6.2 Electron transitions among degenerate states ......................................... .243
7.6.3 Electron transitions in transition-metal compounds,....................
Problems 7 .......
..................................................................
8 Group theory and crystal symmetry .............................................................

248
8.1 Introduction: two levels of crystal symmetry ...........
.....................
8.2 Crystal systems and crystal classes...................
8.3 Why another symmetry notation ......................
................................ .249
8.4 What is a lattice ........................................................................................ .25 2
8.4.1 Defining and choosing unit cells ............................
................ 253
8.4.2 Why only fourteen Bravais lattices ......................................................... .256
8.4.3 Lattice rotational symmetry degrees are 1, 2, 3, 4 and 6 ....
8.4.4 Translation unit cells ......................................
................................ 261
8.4.5 Wigner-Seitz cells.............................................................
8.5 Translation groups .........

.................
............................... .263
8.6 Space groups........................................................................
8.6.1 Symmorphic space groups...................................................................... ,265
Glide planes and screw axes .........................
................................ 269
8.6.2 And nonsymmorphic space groups.....................................
.............272
Monoclinic nonsymmorphic space groups......... ............................... ,272
Orthorhombic nonsymmorphic space groups .............................
Some useful rules; Tetragonal nonsymmorp
8.7 Applications of space groups..........................
Naphthalene; Biphenyl; Two cubic structures
3 8.8 What is a factor group.................................
8.8.1 Simple factor-group analysis of iron(I1) su
...................... 284
8.8.2 Site-group analysis.......................................
Factor-group method for potassium chro
Problems 8....................................................................................................... 285

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Appendix 1 Stereoviews and models .................................................................
288
A l . l Stereoviews............................................................................................. 288
A1.2 Model with S, symmetry ......................................................................... 289
Appendix 2 Direction cosines and transformation of axes ..................................
A2.1 Direction cosines.....................................................................................
A2.2 Transformation of axes ...........................................................................


291
291
292

Appendix 3 Stereographic projection and spherical trigonometry.......................
A3.1 Stereograms............................................................................................
A3.2 Spherical triangles ..................................................................................
A3.2.1 Formulae for spherical triangles ...........................................................
A3.2.2 Polar spherical triangles .......................................................................
A3.2.3 Example stereograms ...........................................................................
A3.2.4 Stereogram notation .............................................................................

294
294
297
297
298
299
300

Appendix 4 Matrix diagonalization by Jacobi's method ......................................

302

Appendix 5 Spherical polar coordinates.............................................................
A5.1 Coordinates.............................................................................................
A5.2 Volume element......................................................................................
A5.3 Laplacian operator305


305
305
305

Appendix 6 Unitary representations and orthonormal bases ...............................
A6.1 Deriving an unitary representation in C3"................................................
A6.2 Unitary representations from orthonormal bases......................................

307
307
310

Appendix 7 Gamma function ............................................................................. 312
Appendix 8 Overlap integrals ............................................................................

313

Appendix 9 Calculating LCAO coefficients .......................................................

314

Appendix 10 Hybrid orbitals in methane ............................................................

316

Appendix 11 Character tables and correlation tables for point groups ................319
..........................................................
....319
A1 1.1 Character tables ..
oups C and C ; Groups S (n = 4. 6).

Groups C,, (n = 1
Groups C (n = 24); Groups C (n = 2-6); Groups D (n = 26);
Groups D (n = 2-6); Groups D (n = 2-4); Cubic Groups; Groups C and D ..
.................337
A11.2 Correlation tables ..............................................
Groups C (n = 2-4, 6). Groups C (n = 2
.................... 337
Groups D ,T and 0 ..............................
337
A1 1.3 Multiplication properties of irreducible r
General rules; Subscripts on A and B; Doubly-degenerate
representations; Triply-degenerate representations; Linear
.......................................
groups; Direct products of spin multiplicities

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Appendix 12 Study Aids on the Internet ............................................................
A12.1 Computer programs ..............................................................................
Programs .........................................................................................................

338
338
338

Appendix 13 Some useful rotation matrices ....................................................... 342
Twofold symmetry; Threefold symmetry along <111>; .........................................
Threefold symmetry along [OO* 11; Fourfold symmetry;
Sixfold symmetry

Appendix 14 Apologia for a single symmetry notation .......................................
Tutorial solutions .....................................................................................

345

.347-394

References and selected reading ...................................................... ..........395-397
Index

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Foreword
by
Professor the Lord Lewis, FRS
Warden, Robinson College, Cambridge
There is an instant appeal and appreciation of symmetry within a system. The
recognition of symmetry is intuitive but is often difficult to express in any simple
and systematic manner. Group theory is a mathematical device to allow for the
analysis of symmetry in a variety of ways. This book presents a basic mathematical
approach to the expression and understanding of symmetry and its applications to a
variety of problems within the realms of chemistry and physics. The consideration
of the symmetry problems in crystals was one of the first applications in the area of
chemistry and physics, Hwdey observing in the mid-19 century that “the best
example of hexagonal symmetry is furnished by crystals of snow”.
The general occurrence of symmetry is well illustrated in the first chapter of the
book. Its widespread application to a whole variety of human endeavour spreading
from the arts to sciences is a measure of the implicit feeling there is for symmetry
within the human psyche. Taking one speclfic example, let us consider

architecture, which is a discipline that is on the borderline between the arts and the
sciences and has many good examples of the widespread application of symmetry.
In the design and construction of buildings in general there is a basic appeal to
symmetry and this recognition was taken to a logical extreme in the archtecture of
the Egyptians. This applied particularly in the design of temples which were
constructed at one stage with the deliberate intention of introducing a lack of
symmetry; the So called “symmetrophobia”. This itself was a compelling point in
the visual form of the buildings and as such brought these buildings to the attention
of the public and placed them in a unique position compared to other forms of
architecture, consistent with their special function within the community.
The translation of symmetry consideration into mathematical terms and the
application to science has been of considerable use and has allowed for a generality
of approach to wide range of problems. This approach has certainly been of
importance in the study of inorganic chemistry’over the last four decades and is now
considered to be one of the main armaments in dealing with a wide range of
problems in this area; which cover as diverse a series of subjects as basic
spectroscopy, both electronic and vibrational, crystallography, and theoretical
chemistry with particular reference to the bonding properties in molecules. All
these areas are well covered and documented within the present text.
The prime aim of this book is to equip the practising chemist, particularly the
structural chemist, with the knowledge and the confidence to apply symmetry
arguments via the agency of group theory to solving problems in structural
chemistry. The use of symmetry within molecules to determine the structure of
molecules is not new to either the study of inorganic or organic chemistry.
Variation in the charge distribution within molecules was recognised as being
associated with the symmetry of the molecule and the use of techniques such as

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xii

Foreword

dipole moments or polarity within a molecule were readily associated with the
physical properties of compounds. A basic approach used by both in organic and
inorganic chemistry throughout the 19/20 centuries to the solution of a wide range
of problems involving the structure of molecules which depended on the symmetry
of the molecule was the use of isomer counting either as geometrical or optical
isomers. The final proof for the octahedral and planar arrangements of ligands
around a metal centre was the resolution of compounds of metals with these stereochemistries into optically active isomers. The present book develops this approach
giving it the added advantage of a mathematical rigor and applying the arguments
to a range of techniques involving symmetry with particular emphasis on using as
examples molecules that are familiar to the practising chemist.
The text allows the reader to develop the mathematical expertise necessary to apply
this approach. The availability of problem sets at the end of each chapter is
intended to build up the confidence to apply the procedure to examples outside the
text and is a very effective way of testing the mathematical appreciation of the
reader. It is, however, fair to say that the mathematical task set by the text will not
be easy for many students, but it is equally important to emphasise that the effort
that is involved will pay great dividends in the understanding of and application to
many aspects of chemistry. The author is to be congratulated on the clarity and
detail with which he deals with this basic mathematical ground work .
Another interesting feature of the present text is the introduction to computer
techniques for a number of the applications and in particular the use of the internet
for computer programs relevant to certain of the set problems, as well as the use of
stereoviews and models. This allows for a direct application to wide range of data
and is perhaps of particular importance in the area of theoretical chemistry .
In summary, is book provides the “enabling” background to rationalise and
synthesise the use of symmetry to problems in a wide range of chemical

applications, and is a necessary part of any modern course of Chemistry.
J Lewis
Robinson College
Cambridge

June 1998

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PREFACE
This book discusses group theory in the context of molecular and crystal symmetry.
It stems from lecture courses given by the author over a number of years, and covers
both point-group and space-group symmetries, and their applications in chemistry.
Group theory has the power to draw together molecular and crystal symmetry,
which are treated sometimes from slightly Merent viewpoints.
The book is directed towards students meeting symmetry and group theory for the
first time, in the first or second year of a degree course in chemistry, or in a subject
wherein chemistry forms a sigdicant part.
The book presumes a knowledge of the mathematical manipulations appropriate to
an A-level course in this subject: the vector and matrix methods that are used in the
book, that give an elegance and conciseness to the treatment, are introduced with
copious examples. Other mathematical topics are treated in appendices, so as not to
interrupt the flow of the text and to cater for those whose knowledge may already
extend to such material.
Computer power may be said to render some manipulations apparently
unnecessary: but it is very easy to use a sophisticated computer program and obtain
results without necessarily being cognizant of the procedures that are taking place;
the development of such programs, and even better ones, demands this knowledge.
Each chapter contains a set of problems that have been designed to give the reader

practice with the subject matter in various applications; detailed, tutorial solutions to
these problems are provided. In addition, there is a set of programs, outlined in
Appenhx 12, established on the Internet under the web address
www.horwood.net/publish that executes procedures discussed in the text, such as
Huckel molecular-orbital calculations or point-group recognition. A general resume
of the programs is provided under the web address, but otherwise they are selfexplanatory.
Symmetry is discussed in terms of both the Schonflies and the Hermann-Mauguin
symmetry notations. The Hermann-Mauguin notation is not introduced generally
until Chapter 8. By that stage, the concepts of symmetry and its applications will
have been discussed for molecules. Thus, the introduction of the second notation
will be largely on a basis of symmetry that will be, by then, well established and
understood.
A number of molecular and crystal structures is illustrated by stereoscopic
drawings, and instructions for viewing them, including the construction of a
stereoviewer,are provided.
The author has pleasure in expressing his thanks to Professor, The Lord Lewis,
Warden of Robinson College, Cambridge for helpful discussions at the beginning of
the work and for writing the Foreword; to Dr John Burgess, Reader in Inorganic
Chemistry, University of Leicester for encouraging comments and for reading the
manuscript in proof; to various publishers for permission to reproduce those
diagrams that carry appropriate acknowledgements; and finally to Horwood
Publishmg Limited with whom it is a pleasure and privilege to work.
Mark Ladd, 1998
Farnham

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List of symbols
The following list shows most of the symbols that are used herein. It is traditional

that a given symbol, such as k or j, has more than one common usage, but such
duplications have been kept to a minimum within the text.
A, B, . Symmetry operations (operators) in a group
Spectroscopicterm symbol
A
Irreducible representation; matrix; general constant; A-face centred unit
A
cell; member of a group
Matrix, inverse to A
A-'
Matrix, transpose of A
A
A+
Matrix, adjoint to A
A*
Matrix, conjugate to A
R
Cofactor matrix of A
ijth term of cofactor matrix R
A?,
a
Vector along the x axis
Molecular-orbitalenergy level of symmetry type A
a
a
Constant of Morse equation; unit-cell dimension along x axis; a-glide plane
ith component of a vector a
a,
ijth term of matrix A
av

Bohr radius for hydrogen (52.918 pm)
00
B
Irreducible representation; matrix; general constant; magnetic flux density;
B face-centred unit cell; member of a group
Vector along they axis
b
Molecular-orbitalenergy level of symmetry type B
b
Unit-cell dimension along y axis; 6-glide plane
b
C
C-face centred unit cell; member of a group
Rotation symmetry operation (operator) of degree n
crl
Rotation symmetry axis of degree n
cn
Cyclic (point) group of degree n
Cn
C
Vector along the z axis
C
Unit-cell dimension along z axis; c-glide plane
LCAO
coefficients (eigenvectors)
c,I
D
C m); spectroscopicterm symbol
Debye unit (3.3356 x
D-matrix; density

D
D-matrix, conjugate to D
D*
Dissociation energy (theoretical, including zero-point energy)
De
Dissociation energy (experimental)
Do
Dihedral (point) group of order n
Dn
d
dx

d

d Orbital; d wavefunction; differential operator, as in - ;bond length

d2

Second differential operator, as in -

d2

dx2

d
d-Glide plane
det(A) Determinant, IAI, of matrix A

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List of symbols

E
E
E

E
el,

e
eV
F
F
€

f

f
Ax)
G
6

g
H
H
H

ii
&

h

xv

Identity symmetry operation (operator)
Spectroscopicterm symbol
Identity matrix; doubly-degenerate irreducible representation;total energy;
Total electronic energy; electrical field strength
Unit vectors along mutually perpendicular directions i (i = 1,2,3)
Doubly-degeneratemolecular-orbital energy level of symmetry type E
Electronvolt (1.6022 x
J)
Spectroscopicterm symbol
All-face centred unit cell
Free valence
f Orbital; f wavefunction
function; force
function of a variable, x
Spectroscopicterm symbol
Group, as in G{A,B, ..}
Order of subgroup; even (‘gerade’) function; Lande factor
Complete Hamiltonian operator
Nuclear Hamiltonilan operator; spectroscopicterm symbol
Coulomb integral; magnetic field strength
Electronic Hamiltonian operator
Effective electronic Hamiltonian operator
Order of group; hybrid orbital; Hiickel parameter; Planck constant (6.6261
x
J Hz-I); Miller index along x axis
‘Cross-h ’ (= h/27c)

Plane in a crystal or lattice
Form of planes (hkl)
Spectroscopicterm symbol
Ionization energy; body-centred unit cell
Transition moment (integral)
Unit vector along the x axis
Inversion symmetry operation (operator)

Ji

k
k

kB
L
1

Centre of (inversion) symmetry
Infrared
Unit vector along they axis
Combined orbital (I) and spin (s) angular momenta for an electron
Total combined orbital (L)and spin (5‘)angular momenta for multielectron
species
Unit vector along the z axis
Number of symmetry classes in a point group; Hiickel parameter; force
constant; Miller index along y axis
Boltzmann constant (1.3807 x
J K-’)
Total orbital angular momentum
Orbital angular momentum quantum number; Miller index along z axis;

direction cosine along x axis

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List of symbols

xvi
Magnetization

c

“I

Molar mass
Relative molar mass

CS

m,
m

N
NA

N
n
n
-


n

-

n

nP
0
0

P,
p,,

P
P
P
P

P
4

R
R
Rhex

r
r

Minor determinant of q t h term of matrix A
Reflection (mirror) plane symmetry operation (operator)

Reflection (mirror) plane symmetry; direction cosine along y axis
Mass of electron (9.1094 x lo” kg)
Quantum number for resolution of orbital angular momentum about the z
axis (‘magnetic’ quantum number)
Projection of s on the z axis (*%)
Magnetic moment
Normalization constant
Avogadro constant (6.0221 x
mol-’)
Number density
Rotation symmetry operation (operator) of degree n
Dimensionality of a representation; rotation symmetry axis of degree n;
principal quantum number; number of atoms in a species; n-glide plane;
direction cosine along z axis
Roto-inversion symmetry operation (operator) of degree n
Roto-inversion symmetry axis of degree n
Screw (rotation) axis (n = 2, 3,4,6; p < n)
Operator; transformation operator
Octahedral (cubic) (point) group
Projection operator (operating on x)
Projection operator (operating on D(R)$
Position vector
Spectroscopic term symbol
Total bond order; primitive unit cell
p Orbital; p wavefunction
Mobile (p) bond order
Formal charge on an atom
General symmetry operation (operator)
Rhombohedra1 (primitive) unit cell; internuclear distance
Triply-primitive hexagonal unit cell

Vector; unit bond vector
Length of vector r, that is, Irl; spherical polar (radial) coordinate; number
of irreducible representation in a point group; interatomic distance
Equilibrium interatomic distance
s Orbital; s wavefunction; spin quantum number (%) for single electron
Roto-reflection symmetry operation (operator) of degree n
Spectroscopic term symbol
Overlap integral; total spin for multielectron species
Roto-reflection symmetry axis (alternating axis) of degree n
Triply-degenerate irreducible representation

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List of symbols

T

xvii

Tetrahedral (cubic) (point) group
Translation vector
t
Triply-degenerate molecular-orbital energy level of symmetry type T
U
Coordinate of lattice point along x axis
[
Direction in a lattice
<uvw> Form of directions [UVB'I
U

Odd ('ungerade') function
V
Volume of a parallelepipedon; nuclear potential energy function (operator);
coordinate of lattice point along y axis; nuclear potential energy function
(operator)
Molar
volume
vm
\I
Electronic potential energy function (operator)
V
Speed of light (2.9979 x 10' m s-')
V
Vibrational quantum number
W
Coordinate of lattice point along z axis
X
General variable
X
Reference axis; fractional coordinate in unit cell
Anharmonicity constant
Xe
Y
General variable
Reference axis; fractional coordinate in unit cell
Y
Atomic number
z
Reference axis; fractional coordinate in unit cell
Z

a
Interaxial angle y%; general angle; Coulomb integral H for a species with
itself; polarizability; electron spin (+%)
Components
of 3 x 3 polarizability tensor
aij
Interaxial
angle
zAx;general angle; Coulomb integral H between two
P
species; electron spin (-95)
r
Representation; gamma hnction
Interaxial angle xAy; general angle
Y
Triply-degenerate irreducible representation in ,C and D d ; ligand-field
A
energy-splittingparameter
Kronecker's delta
6ij
Complex exponential, as in exp(i2nln); vibrational energy
&
Magnetizability
5
General angle; spherical polar coordinate
e
Volume magnetic susceptibility
K
Eigenvalue; hybrid orbital constant
h

Dipole moment vector; reduced mass; spheroidal coordinate
CL
Components of p (i = x, y, z)
PI
Permeability of a vacuum (4n x 10-7H m-', or J C2m s2 )
PO
V
Frequency; spheroidal coordinate
Wavenumber
u
, and D,h
Doubly-degenerate irreducible representation in C
n
n Bonding molecular orbital
n*
n Antibonding molecular orbital
Electron density; exponent in atomic orbital (= 2Zr/na, )
P
t

n

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xviii

9*
9n


4'
X
Y
\v

v*
0

n
63
V2

List of symbols
Summation; irreducible representation in C, or D,h
General reflexion symmetry operation (operator)
Reflexion symmetry operation (operator) perpendicular to principal C,, axis
Reflexion symmetry operation (operator) containing the principal C,, axis
General reflection symmetry plane; o bonding molecular orbital
(3 Antibonding molecular orbital
Reflexion symmetry plane perpendicular to principal C,, axis
Reflexion symmetry plane containing the principal C, axis
Volume (dz, infinitesimal volume element)
Quadruply-degenerate irreducible representation in C, and Dmh
Molecular orbital or wavefunction; spherical polar coordinate; spheroidal
coordinate
Molecular orbital or wavefunction, conjugate to 0
Molecular orbital or wavefunction
LCAO molecular orbital
Trace, or character, of a matrix; mass magnetic susceptibility
Linear combination of wavefunctionsv, total wavefunction

Atomic orbital or wavefunction
Atomic orbital or wavefunction, conjugate to yl
Angular frequency (= k/p)'
Hybrid orbital
Direct product
Laplacian operator

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1
Symmetry everywhere
Tyger! Tyger! burning bright
In the forests ofthe night,
What immortal hand or eye
Couldframe thy fearful symmetry?
William Blake (1757-1827): The Tyger!

1.1 INTRODUCTION: LOOKING FOR SYMMETRY
Generally, we have little difficulty in recognizing symmetry in two-dimensional
objects such as the outline of a shield, a Maltese cross, a five-petalled Tudor Rose,
or the Star of David. It is a rather different matter when our subject is a threedimensional body. The difficulty stems partly from the fact that we can see
simultaneously all parts of a two-dimensional object, and so appreciate the
relationship of the parts to the whole; it is not quite so easy with a three-dimensional
entity. Secondly, while some three-dimensional objects, such as flowers, pencils and
architectural columns, are simple enough for liS to visualize and to rotate in our
mind's eye, few of us have a natural gift for mentally perceiving and manipulating
more complex three-dimensional objects, like models of the crystal of potassium
hydrogen bistrichloroacetate in Figure 1.1, or of the structure of pentaerythritol


Fig 1.1 Potassium hydrogen bistrichloroacetate (CbC02)2HK

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2

Symmetry Everywhere

[eh.l

Fig 1.2 Stereo view showing the packing of the molecules of pentaerythritol, C(CH20H)4, in
the solid state. Circles in order of increasing size represent H, C and 0 atoms; O-B'O
hydrogen bonds are shown by double lines. The outline of the unit cell (q.v.) is shown, and
the crystal may be regarded as a regular stacking of these unit cells in three dimensions

shown in Figure 1.2. Nevertheless, the art of doing so can be developed with
suitable aids and practice. If, initially, you have problems with three-dimensional
concepts, take heart. You are not alone and, like many before you, you will be
surprised at how swiftly the required facility can be acquired. Engineers, architects
and sculptors may be blessed with a native aptitude for visualization in three
dimensions, but they have learned to develop it, particularly by making and
handling models.
Standard practice reduces a three-dimensional object to one or more twodimensional drawings, such as projections and elevations: it is a cheap method, well
suited for illustrating books and less cumbersome than handling models. This
technique is still important, but to rely on it exclusively tends to delay the
acquisition of a three-dimensional visualization facility. As well as models, we may
make use of stereoscopic image pairs, as with Figure 1.2; notes on the correct
viewing of such illustrations are given in Appendix 1. The power of the stereoscopic
view can be appreciated by covering one half of the figure; the three-dimensional

depth of the image is then unavailable to the eye.
1.1.1 Symmetry in finite bodies
Four quite different objects are illustrated in Figure 1.3. At first, there may not seem
to be any connection between a Dobermann bitch, a Grecian urn, a molecule of 3chlorofluorobenzene and a crystal of potassium tetrathionate. Yet each is an
example of reflection symmetry: a (mirror) symmetry plane, symbol o (Ger. Spiegel
= mirror), can be imagined for each entity, dividing it into halves that are related as
an object is to its mirror image.
If it were possible to perform physically the operation of reflecting the halves of an
object across the symmetryplane dividing them, then the whole object would appear
unchanged after the operation. If we view the Doberma~ from the side its mirror
symmetry would not be evident, although it is still present. If, however, we imagine
a reflecting plane now placed in front of the Dobermann, then the object and her
image together would show c symmetry, across the plane between the object animal
and its mirror image (Figure 1.4). This plane together with that through the

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Sec 1.1]

3

Introduction: Looking for Symmetry

Doberrnann, and her mirror image, combine to give another symmetry element ,
along the line of intersection of the two symmetry planes . We shall consider later
the combinations of symmetry elements.
Often, the apparent symmetry of an object may not be exact, as we see if we
pursue the illustrations in Figure 1.3 a little further . The Doberrnann , beautiful
animal that she is, if scrutinized carefully will be seen not to have perfect c

symmetry; again, only the outline of the urn conforms to mirror symmetry. In a
molecule, the atoms may vibrate anisotropically, that is, with differing amplitudes of
vibration in different directions ; this anisotropy could perturb the exact c symmetry
depicted by the molecular model.
Under a microscope, even the most perfect-looking real crystals can be seen to
have minute flaws that are not in accord with the symmetry of the conceptually

F
001

iT o
110

100

I
I
I

110

110

-

100

Fig.I.3 Examples of reflection symmetry: (a) The Dobennann, Vijentor Seal of Approval at
Valmara (c vertical); (b) Grecian urn (o vertical) ; (c) 3-Chloroflurobenzene molecule (c in
the molecular plane) ; (d) Crystal of potassium tetrathionate; o vertical, relating faces with

Miller indices!l) (hkl) and (h k l) .

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4

[Ch. I

Symmetry Everywhere

Fig 1.4 Vijentor Seal of Approval at Valmara: object and mirror image relationship across a
vertical (J symmetry plane . From a three-dimensional point of view, there are three symmetry
elements here: the (J plane just discussed, the (J plane shown by Figure 1.3a, and an element
arising from their intersection. What is that symmetry element?

perfect crystals shown by drawings such as Figures 1.1 and 1.3d. Then, if we
consider internal symmetry, common alum KAI(S04h.12 H20, for example, which
crystallizes as octahedra, has an internal symmetry that is of a lesser degreee than
that of an octahedron.
1.1.2 Symmetry in extended patterns
If we seek examples of symmetry around us, we soon encounter It III repeating
patterns, as well as in finite bodies. Consider the tiled floor or the brick wall
illustrated by Figure 1.5. Examine such structures at your leisure, but do not be too
critical about the stains on a few of the tiles, or the chip off the occasional brick.
Geometrically perfect tiled floors and brick walls are, like perfect molecules and
crystals, conceptual.
Each of the patterns in Figure 1.5 contains a motif, a tile or a brick, and a
mechanism for repeating it in a regular manner. Ideally, the symmetry of repetition
implies infinite extent , because the indistinguishability of the object before and after

a symmetry operation is the prime requirement of symmetry. The stacking of bricks

(a)

(b)

Fig 1.5 Symmetry in patterns: (a) plan view of a tiled floor; (b) face of a brick wall .

to form a brick wall is limited by the terminations of the building of which the wall
is a part , just as the stacking of the unit cells of a crystal is limited by its faces . In

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Sec 1.2]

What do we mean by Symmetry?

5

both examples, we may utilize satisfactorily the symmetry rules appropriate to
infinite patterns provided that size of the object under examination is very large
compared to the size of the repeating unit itself.
Real molecules and chemical structures, then, rarely have the perfection ascribed
to them by the geometrical illustrations to which we are accustomed. Nevertheless,
we shall find it both important and rewarding to apply symmetry principles to them
as though they were perfect, and so build up a symmetry description of both finite
bodies and infinite patterns in terms of a small number of symmetry concepts.
1.2 WHAT DO WE MEAN BY SYMMETRY?
Symmetry is not an absolute property of a body that exhibits it; the result of a test

for symmetry may depend upon the nature of the examining probe used. For
example, the crystal structure of metallic chromium may be represented by the bodycentred cubic unit cell shown in Figure 1.6a, as derived from an X-ray diffraction
analysis of the the crystal: the atom at the centre of the unit cell is, to X-rays,
identical to those at the corners, and there are two atoms per unit cell. Chromium
has the electronic configuration (lS)2 (2S)2 (Zp)" (3S)2 (3p)6 (3d)5 (4S)I, and the
unpaired electrons in this species are responsible for its paramagnetic property. If a
crystal of chromium is examined by neutron diffraction, the same positions are
found for the atoms. However, the direction of the magnetic moment of the atom at
the centre of the unit cell is opposite to that of the atoms at the corners (Figure
1.6b). X-rays are diffracted by the electronic structure of atoms, but neutron
diffraction arises both by scattering from the atomic nuclei and by magnetic
interactions between the neutrons and the unpaired electrons of the atoms. The
magnetic structure of chromium is based on a primitive (pseudo-body-centred) cubic
unit cell, so it is evident that symmetry under examination by neutrons can differ
from that under examination by X-rays.
In this book, we shall take as a practical definition of symmetry that property ofa
body (or pattern) by which the body (or pattern) can be brought from an initial
spatial position to another, indistinguishable position by means of a certain
operation, known as a symmetry operation. These operations and the results of their
actions on chemical species form the essential subject matter of this book.

Fig. 1.6 Unit cell and environs of the crystal structure of metallic chromium: (a) from X-ray
diffraction, (b) from neutron diffraction. The arrows represent the directions of the magnetic
moments associated with the unpaired electrons in the atoms.

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Symmetry Everywhere


6

[Ch.I

1.3 SYMMETRY THROUGHOUT SCIENCE
The manifestations of symmetry can be observed in many areas of science and,
indeed, throughout nature; they are not confined to the study of molecules and
crystals. In botany, for example, the symmetry inherent in the structures of flowers
and reproductive systems is used as a means of classifying plants, and so plays a
fundamental role in plant taxonomy. In chemistry, symmetry is encountered in
studying individual atoms, molecules and crystals. Curiously, however, although
crystals exhibit only n-fold symmetry (n = 1, 2, 3, 4, 6), molecules (and flowers),
with fivefold or sevenfold symmetry are well known. The reasons for the limitations
on symmetry in crystals will emerge when we study this topic in a later chapter.
Symmetry arises also in mathematics and physics. Consider the equation

r

= 16.

(1.1)

The roots of (1.1) are X = ±2 and X = ±2i, and we can see immediately that these
solutions have a symmetrical distribution about zero. The differential equation

d2Y1dX2 + k'Y = 0

(1.2)

where k is a constant, represents a type encountered in the solution of the

Schrodinger equation for the hydrogen atom, or of the equation for the harmonic
oscillator. The general solution for (1.2) may be written as

Y = A exp(ikX) + B exp( -ikX)

(1.3)

where A and B also are constants. If we consider a reflection symmetry that converts
X into -X, then the solution of (1.2) would become
Y=A exp(-ikX) + B exp(ikX)

(1.4)

Differentiating (1.4) twice with respect to X shows that this equation also is a
solution of (1.2). If, instead of reflection symmetry, we apply to (1.3) a translational
symmetry that converts X into X + t, where t is a constant, we would find that
although the imposed symmetry has translated the function (1.3) along the x axis,
the applicability of the general solution remains.
A single-valued, continuous, one-dimensional, periodic function defined, for
example, between the limits X = ±Y2, can be represented by a series of sine and
cosine terms known as a Fourier series:
h=oo

y

Ao

+ 2

LAh cos (2rrhX)


+

Bh sin(2rrhX),

(1.5)

h=l

where A o is a constant. A typical cosine term, as in Figure 1.7, shows symmetric
behaviour (Y-c = Yc ) with respect to the origin: it is equivalent to a reflection of the
curve across the line X = 0, and is an exemplar of an even function.
In contrast, a typical sine term, illustrated by Figure 1.8, is termed an odd
function, as it is antisymmetric (y_. = - Y.) about the origin; the curve is mapped on
to itself by a rotation of 1800 (twofold rotation) about the pointX= Y= O.

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