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

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 Morgans 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 Lukes Anglican Church in Brickett Wood, St Albans. He lives in Farnham, Surrey, with his wife and one Dobermann.

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

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

0 M. 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

Table of contents

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

Preface ................................................................................................ ... vii

List of symbols ............................................................................................ . ~ i i - m i . . . . .

1 Symmetry everywhere ...................................................................................... 1 1.1 Introduction: Looking for symmetry .............................................................. 1

................. 2

1.2 What do we mean by symmetry ..................................................................... 5 1.3 Symmetry throughout science ....................................................................... 6 1.4 How do we approach symmetry Problems 1 .................................................................

1.1.1 Symmetry in finite bodies ................................... 1.1.2 Symmetry in extended patterns .................................................................. 4

2 Symmetry operations and symmetry elements ............................... 11 2.1 Introduction: The tools of symmetry 2.2 Defining symmetry operations, ele

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

...................... 13 ....................................... 15 Sign of rotation ............................

2.2.4 Reflection symmetry ................. 2.2.5 Roto-reflection symmetry.. ........ 2.2.6 Inversion symmetry ................

2.2.8 Roto-inversion symmetry ........ ....................................... 18 ...................................... 19

2.5.1 Sum, difference and scalar (do 2.5.2 Vector (cross) product of two 2.5.3 Manipulating determinants and matrices .....................................

Orthogonality ............................... .............................. 2.5.4 Eigenvalues and eigenvectors.. ................................................................. .28

Diagonalization; Similarity transformation; Jacobi diagonalization ..... 30-3 1

Matrices and determinants; Cofactors; Addition and subtraction of matrices; Multiplication of matrices; Inversion of matrices;

2.5.5 Blockdiagonal and other special matrices ....................................... 33 Adjoint and complex conjugate matrices; matrix; Unitary matrix ......................................................................... 34-35

........................................ 35

.................................. 38

.................................. 38 ..................................... ..38

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

3 Group theory and point groups ............. 3.1 Introduction: Groups and group the0 3.2 What is group theory .........................

3.2.1 Group postulates ............................................

3.2.2 General group definitions.. .......................... 3.2.3 Group multiplication tables .

3.2.5 Symmetry classes and conjugates ........

3.3.1 Deriving point groups.. ............................

3.3.2 Building up the

................................ 38

Inverse member ........... .................................... ................. 38-39 Closure; Laws of co

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

3.2.4 Subgroups and cos

3.3 Defining, deriving ...............................

................................... 46

............................ 52

......................... 59

Problems 3 ................................................................ ............................... 67

Euler's construction ...............................................

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

4 Representations and character tables .........................................

4.1.1 Representations on position vectors ....................................... 4.1 Introduction: What is a representation ............ ........................... 72

4.1.2 Representations on basis vectors ......................... ............................ 75 4.1.3 Representations on atom vectors.. ..... ............................................. 77

4.1.4 Representations on functions.. ............... ................................ 82 4.1.5 Representations on direct product functions ............................. 4.2 A first look at character tables.. .................... .......................... 86

87 4.2.1 Orthonormality ............. ........................................... 4.2.2 Notation for irreducible representations ........ ............................. 88

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

Unshifted-atom contributions to a re

5 Group theory and wavefunctions ........................... .............................. 108 108

................. 110

5.1 Introduction: Using the Schrodinger equation ............................... 5.2 Wavefunctions and the Hamiltonian operator.. .......................................... ,109 5.2.1 Properties of wavefunctions ................................

5.3.1 Defining operators in function space ....................................................... 112

5.5 When do integrals vanish ................................

5.3 A further excursion into function space.. ............................................

5.4 Using operators with direct products .........................................

5.6 Setting up symmetry-adapted linear combinati 5.6.1 Deriving and using projection operators.. ..... 5.6.2 Deriving symmetry-adapted orbitals for the carbonate ion

5.6.3 Handling complex characters ........................................... Problems 5 ................................................... ...................................... 128

115 ............................... 117

............................... 119

Generating a second function for a degenerate representation

6 Group theory and chemical bonding ............................................................. 130

6.2 Setting up LCAO approximations.. ........................................................... .13 1 Function of the Schrijdinger equation ........

6.2.1 Defining overlap integrals.. .............................

6.1 Introduction: molecular orbitals ................................. Classlfylng molecular orbitals by symmetry ...............

....................................... 132 Introducing the variation principle .......

.............................. 134 6.2.2 Defining Coulomb and resonance inte .............................. 134

6.2.3 Applying the LCAO method to the oqgen molecule.. ............................ .137 6.2.4 Bonding and antibonding molecular orbitals and notation ....................... 140

........... 142 6.3 P-electron approximations ...... .............................................................. 142

............................. 143 Benzene .................................................................................................. 144

6.3.2 Further features of the Huckel molecular-orbital theory.. ........................ .149 ll-Bond order ................................................ ........................ 149 Free valence .......................................................................................... .15 1 Charge distribution ................................................................ 152

..................................... 152 Methylenecyclobutene; methylenecyclopropene ...................................... 1 5 3

................. 156

Continuing with the variation principle ..............................

Total bond order ................................................

6.3.1 Using the Huckel molecular-

6.3.3 Altemant and nonaltemant hydrocarbons .....

6 4 4 Huckel's 4n + 2 rule .......................................... 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

....................................... 169

....................................... 173

Sulfur hexafluoride. ................................... ............................... 178

.................. 186

6.4 Schemes for hybridization: water methane ............. ...... 167 6.4.1 Symmetrical hybrids ............................

Walsh diagrams ................................... Further study of methane .....................

6.5 Photoelectron spectroscopy ..........................................................

6.6 Cyclization and correlation ....................

6.7.1 Electronic structure and term symbols ........................................ 6.7 Group theory and transition-metal compounds.. .......

Russell-Saunders coupling.. .. ..................................................... 188 6.7.2 How energy levels are split in a crystal field.. .....

6.7.3 Correlation diagrams in 0, and Td symmetry ......................................... .197

6.7.4 Ligand-field theory .... ...................................................... ..... 205 Spectral properties ............................ ............................................... 211

Problems 6 .................................................................... ................... 217

.................... 192 Weak fields and strong fields ...............

'Holes' in d orbitals ..................................... ........................... 203

7 Group theory, molecular vibrations and electron transitions.. .................... .22 1 7.1 Introduction: How a molecule acquires vibrational energy.. ......... 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 .................................................................. 7.3.2 Raman spectra .............................................

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

.............................. 222

................................... 230 Polarization of Raman spectra . . .............................................

7.4 Classlflmg vibrational modes ...................

...................... 241

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 ........................................................................................ .2 5 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 ....

8.5 Translation groups ......... ................. ............................... .263

8.6.1 Symmorphic space groups.. .................................................................... ,265

8.6.2 And nonsymmorphic space groups.. ...................................

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

8.4.4 Translation unit cells ...................................... 8.4.5 Wigner-Seitz cells .............................................................

8.6 Space groups ........................................................................

................................ 261

Glide planes and screw axes .........................

Monoclinic nonsymmorphic space groups.. ....... 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

................................ 269 ............ .272

............................... ,272

8.8.2 Site-group analysis.. ..................................... ...................... 284

Problems 8 ....................................................................................................... 285 Factor-group method for potassium chro

Appendix 1 Stereoviews and models ................................................................. 288 Al . l Stereoviews ............................................................................................. 288 A1.2 Model with S, symmetry ......................................................................... 289

Appendix 2 Direction cosines and transformation of axes .................................. 291 A2.1 Direction cosines ..................................................................................... 291 A2.2 Transformation of axes ........................................................................... 292

Appendix 3 Stereographic projection and spherical trigonometry ....................... 294 A3.1 Stereograms ............................................................................................ 294 A3.2 Spherical triangles .................................................................................. 297 A3.2.1 Formulae for spherical triangles ........................................................... 297 A3.2.2 Polar spherical triangles ....................................................................... 298 A3.2.3 Example stereograms ........................................................................... 299 A3.2.4 Stereogram notation ............................................................................. 300

Appendix 4 Matrix diagonalization by Jacobi's method ...................................... 302

Appendix 5 Spherical polar coordinates ............................................................. 305 A5.1 Coordinates ............................................................................................. 305 A5.2 Volume element ...................................................................................... 305 A5.3 Laplacian operator305

Appendix 6 Unitary representations and orthonormal bases ............................... 307 A6.1 Deriving an unitary representation in C3" ................................................ 307 A6.2 Unitary representations from orthonormal bases ...................................... 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 A1 1.1 Character tables . . .......................................................... .... 319

Groups C,, (n = 1 oups C and C ; Groups S (n = 4. 6). 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 Groups C (n = 2-4, 6). Groups C (n = 2

.................... 337 337

A11.2 Correlation tables ..............................................

Groups D , T and 0 .............................. 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 .......................................

Appendix 12 Study Aids on the Internet ............................................................ 338 A12.1 Computer programs .............................................................................. 338 Programs 338 .........................................................................................................

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

Appendix 14 Apologia for a single symmetry notation ....................................... 345

Tutorial solutions ..................................................................................... .34 7-394

References and selected reading ...................................................... .......... 3 95-3 97

Index

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 chemistryover 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

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 stereo- chemistries 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

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 self- explanatory.

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

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, . A A

A-'

A+

- A

A* R A?, a a a a, av 0 0 B

b b b C crl cn C n C

C

c,I D D D* De Do D n

d

d2

d det(A)

Symmetry operations (operators) in a group Spectroscopic term symbol Irreducible representation; matrix; general constant; A-face centred unit

cell; member of a group Matrix, inverse to A Matrix, transpose of A Matrix, adjoint to A Matrix, conjugate to A Cofactor matrix of A ijth term of cofactor matrix R Vector along the x axis Molecular-orbital energy level of symmetry type A Constant of Morse equation; unit-cell dimension along x axis; a-glide plane ith component of a vector a ijth term of matrix A Bohr radius for hydrogen (52.918 pm) Irreducible representation; matrix; general constant; magnetic flux density; B face-centred unit cell; member of a group Vector along they axis Molecular-orbital energy level of symmetry type B Unit-cell dimension along y axis; 6-glide plane C-face centred unit cell; member of a group Rotation symmetry operation (operator) of degree n Rotation symmetry axis of degree n Cyclic (point) group of degree n Vector along the z axis Unit-cell dimension along z axis; c-glide plane LCAO coefficients (eigenvectors) Debye unit (3.3356 x D-matrix; density D-matrix, conjugate to D Dissociation energy (theoretical, including zero-point energy) Dissociation energy (experimental) Dihedral (point) group of order n

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

d 2 Second differential operator, as in - dx2

d-Glide plane Determinant, IAI, of matrix A

C m); spectroscopic term symbol

d dx

List of symbols xv

E E E E el, e eV F F f f A x ) G 6 g H H H ii & h

k k

kB L 1

Identity symmetry operation (operator) Spectroscopic term 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-degenerate molecular-orbital energy level of symmetry type E Electronvolt (1.6022 x J) Spectroscopic term symbol All-face centred unit cell Free valence f Orbital; f wavefunction function; force function of a variable, x Spectroscopic term symbol Group, as in G{A, B, ..} Order of subgroup; even (gerade) function; Lande factor Complete Hamiltonian operator Nuclear Hamiltonilan operator; spectroscopic term 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

Cross-h (= h/27c) Plane in a crystal or lattice Form of planes (hkl) Spectroscopic term symbol Ionization energy; body-centred unit cell Transition moment (integral) Unit vector along the x axis Inversion symmetry operation (operator)

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

Unit vector along the z axis Number of symmetry classes in a point group; Hiickel parameter; force

Boltzmann constant (1.3807 x Total orbital angular momentum Orbital angular momentum quantum number; Miller index along z axis;

direction cosine along x axis

J Hz-I); Miller index along x axis

Ji

species

constant; Miller index along y axis J K-)

xvi List of symbols

m, m N N A

N n n

- n n

nP 0 0 P, p,, P P P P P 4 R R

r r

-

Rhex

Magnetization

Molar mass Relative molar mass CS 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;

c I

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

List of symbols xvii

T Tetrahedral (cubic) (point) group t Translation vector t U [ Direction in a lattice Form of directions [UVB'I

Triply-degenerate molecular-orbital energy level of symmetry type T Coordinate of lattice point along x axis

U

V

v m

\I V

V

W X

Xe

Y Y z

X

Z a

aij

P

r Y A

6ij

5 e

&

K

h CL PI PO V

u - n n*

P

n

Odd ('ungerade') function Volume of a parallelepipedon; nuclear potential energy function (operator);

coordinate of lattice point along y axis; nuclear potential energy function (operator)

Molar volume Electronic potential energy function (operator) Speed of light (2.9979 x 10' m s-') Vibrational quantum number Coordinate of lattice point along z axis General variable Reference axis; fractional coordinate in unit cell Anharmonicity constant General variable Reference axis; fractional coordinate in unit cell Atomic number Reference axis; fractional coordinate in unit cell Interaxial angle y%; general angle; Coulomb integral H for a species with

itself; polarizability; electron spin (+%) Components of 3 x 3 polarizability tensor Interaxial angle zAx; general angle; Coulomb integral H between two

Representation; gamma hnction Interaxial angle xAy; general angle Triply-degenerate irreducible representation in C, and D d ; ligand-field

Kronecker's delta Complex exponential, as in exp(i2nln); vibrational energy Magnetizability General angle; spherical polar coordinate Volume magnetic susceptibility Eigenvalue; hybrid orbital constant Dipole moment vector; reduced mass; spheroidal coordinate Components of p (i = x, y, z) Permeability of a vacuum (4n x 1 0-7 H m-', or J C2 m s2 ) Frequency; spheroidal coordinate Wavenumber Doubly-degenerate irreducible representation in C,, and D,h n Bonding molecular orbital n Antibonding molecular orbital Electron density; exponent in atomic orbital (= 2Zr/na, )

species; electron spin (-95)

energy-splitting parameter

xviii List of symbols

9* 9n

4' X Y \v v* 0

n 63 V2

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

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

coordinate

1Symmetry everywhere

Tyger! Tyger! burning brightIn the forests ofthe night,What immortal hand or eyeCouldframe thy fearful symmetry?William Blake (1757-1827): The Tyger!

1.1 INTRODUCTION: LOOKING FOR SYMMETRYGenerally, we have little difficulty in recognizing symmetry in two-dimensionalobjects 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 three-dimensional body. The difficulty stems partly from the fact that we can seesimultaneously all parts of a two-dimensional object, and so appreciate therelationship of the parts to the whole; it is not quite so easy with a three-dimensionalentity. Secondly, while some three-dimensional objects, such as flowers, pencils andarchitectural columns, are simple enough for liS to visualize and to rotate in ourmind's eye, few of us have a natural gift for mentally perceiving and manipulatingmore complex three-dimensional objects, like models of the crystal of potassiumhydrogen bistrichloroacetate in Figure 1.1, or of the structure of pentaerythritol

Fig 1.1 Potassiumhydrogen bistrichloroacetate (CbC02)2HK

2 Symmetry Everywhere [eh.l

Fig 1.2 Stereo view showing the packing of the molecules of pentaerythritol, C(CH20H)4, inthe solid state. Circles in order of increasing size represent H, C and 0 atoms; O-B'Ohydrogen bonds are shown by double lines. The outline of the unit cell (q.v.) is shown, andthe 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 withsuitable aids and practice. If, initially, you have problems with three-dimensionalconcepts, take heart. You are not alone and, like many before you, you will besurprised at how swiftly the required facility can be acquired. Engineers, architectsand sculptors may be blessed with a native aptitude for visualization in threedimensions, but they have learned to develop it, particularly by making andhandling models.

Standard practice reduces a three-dimensional object to one or more two-dimensional drawings, such as projections and elevations: it is a cheap method, wellsuited for illustrating books and less cumbersome than handling models. Thistechnique is still important, but to rely on it exclusively tends to delay theacquisition of a three-dimensional visualization facility. As well as models, we maymake use of stereoscopic image pairs, as with Figure 1.2; notes on the correctviewing of such illustrations are given in Appendix 1. The power of the stereoscopicview can be appreciated by covering one half of the figure; the three-dimensionaldepth of the image is then unavailable to the eye.

1.1.1 Symmetry in finite bodiesFour quite different objectsare illustrated in Figure 1.3. At first, there may not seemto be any connection between a Dobermann bitch, a Grecian urn, a molecule of 3-chlorofluorobenzene and a crystal of potassium tetrathionate. Yet each is anexample of reflection symmetry: a (mirror) symmetryplane, symbol o (Ger. Spiegel= mirror), can be imagined for each entity, dividing it into halves that are related asan object is to its mirror image.

If it were possible to perform physically the operation of reflecting the halves of anobject across the symmetryplane dividing them, then the whole object would appearunchanged after the operation. If we view the Doberma~ from the side its mirrorsymmetry would not be evident, although it is still present. If, however, we imaginea reflecting plane now placed in front of the Dobermann, then the object and herimage together would show c symmetry, across the plane between the object animaland its mirror image (Figure 1.4). This plane together with that through the

Sec 1.1] Introduction: Looking for Symmetry 3

Doberrnann, and her mirror image, combine to give another symmetry element ,along the line of intersection of the two symmetry planes . We shall consider laterthe combinations of symmetry elements.

Often, the apparent symmetry of an object may not be exact, as we see if wepursue the illustrations in Figure 1.3 a little further . The Doberrnann, beautifulanimal that she is, if scrutinized carefully will be seen not to have perfect csymmetry; again, only the outline of the urn conforms to mirror symmetry. In amolecule, the atoms may vibrate anisotropically, that is, with differing amplitudes ofvibration in different directions ; this anisotropy could perturb the exact c symmetrydepicted by the molecular model.

Under a microscope, even the most perfect-looking real crystals can be seen tohave minute flaws that are not in accord with the symmetry of the conceptually

F

iTo110

001

100

III 110

110- 100

Fig.I.3 Examples of reflection symmetry: (a) The Dobennann, Vijentor Seal of Approval atValmara (c vertical); (b) Grecian urn (o vertical) ; (c) 3-Chloroflurobenzene molecule (c inthe molecular plane) ; (d) Crystal of potassium tetrathionate; o vertical, relating faces withMiller indices!l) (hkl) and (h k l) .

4 Symmetry Everywhere [Ch. I

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

perfect crystals shown by drawings such as Figures 1.1 and 1.3d. Then, if weconsider internal symmetry, common alum KAI(S04h.12 H20, for example, whichcrystallizes as octahedra, has an internal symmetry that is of a lesser degreee thanthat of an octahedron.

1.1.2 Symmetry in extended patternsIf we seek examples of symmetry around us, we soon encounter It III repeatingpatterns, as well as in finite bodies. Consider the tiled floor or the brick wallillustrated by Figure 1.5. Examine such structures at your leisure, but do not be toocritical 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 andcrystals, conceptual.

Each of the patterns in Figure 1.5 contains a motif, a tile or a brick, and amechanism for repeating it in a regular manner. Ideally, the symmetry of repetitionimplies infinite extent , because the indistinguishability of the object before and aftera 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 wallis a part , just as the stacking of the unit cells of a crystal is limited by its faces . In

Sec 1.2] What do we mean by Symmetry? 5

both examples, we may utilize satisfactorily the symmetry rules appropriate toinfinite patterns provided that size of the object under examination is very largecompared to the size of the repeating unit itself.

Real molecules and chemical structures, then, rarely have the perfection ascribedto them by the geometrical illustrations to which we are accustomed. Nevertheless,we shall find it both important and rewarding to apply symmetry principles to themas though they were perfect, and so build up a symmetry description of both finitebodies 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 testfor symmetry may depend upon the nature of the examining probe used. Forexample, the crystal structure of metallic chromium may be represented by the body-centred cubic unit cell shown in Figure 1.6a, as derived from an X-ray diffractionanalysis 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. Chromiumhas the electronic configuration (lS)2 (2S)2 (Zp)" (3S)2 (3p)6 (3d)5 (4S)I, and theunpaired electrons in this species are responsible for its paramagnetic property. If acrystal of chromium is examined by neutron diffraction, the same positions arefound for the atoms. However, the direction of the magnetic moment of the atom atthe centre of the unit cell is opposite to that of the atoms at the corners (Figure1.6b). X-rays are diffracted by the electronic structure of atoms, but neutrondiffraction arises both by scattering from the atomic nuclei and by magneticinteractions between the neutrons and the unpaired electrons of the atoms. Themagnetic structure of chromium is based on a primitive (pseudo-body-centred) cubicunit cell, so it is evident that symmetry under examination by neutrons can differfrom that under examination by X-rays.

In this book, we shall take as a practical definition of symmetry that property ofabody (or pattern) by which the body (or pattern) can be brought from an initialspatial position to another, indistinguishable position by means of a certainoperation, known as a symmetry operation. These operations and the results of theiractions 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-raydiffraction, (b) from neutron diffraction. The arrows represent the directions of the magneticmoments associated with the unpaired electrons in the atoms.

6 Symmetry Everywhere [Ch.I

1.3 SYMMETRY THROUGHOUT SCIENCEThe manifestations of symmetry can be observed in many areas of science and,indeed, throughout nature; they are not confined to the study of molecules andcrystals. In botany, for example, the symmetry inherent in the structures of flowersand reproductive systems is used as a means of classifying plants, and so plays afundamental role in plant taxonomy. In chemistry, symmetry is encountered instudying individual atoms, molecules and crystals. Curiously, however, althoughcrystals 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 limitationson symmetry in crystals will emerge when we study this topic in a later chapter.

Symmetry arises also in mathematics and physics. Consider the equationr = 16. (1.1)

The roots of (1.1) are X = 2 and X = 2i, and we can see immediately that thesesolutions 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 theSchrodinger equation for the hydrogen atom, or of the equation for the harmonicoscillator. 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 convertsX 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 asolution of (1.2). If, instead of reflection symmetry, we apply to (1.3) a translationalsymmetry that converts X into X + t, where t is a constant, we would find thatalthough 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, forexample, between the limits X = Y2, can be represented by a series of sine andcosine terms known as a Fourier series:

yh=oo

Ao

+ 2 LAh cos (2rrhX) + Bh sin(2rrhX),h=l

(1.5)

where A o is a constant. A typical cosine term, as in Figure 1.7, shows symmetricbehaviour (Y-c = Yc ) with respect to the origin: it is equivalent to a reflection of thecurve 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 oddfunction, as it is antisymmetric (y_. = -Y.) about the origin; the curve is mapped onto itself by a rotation of 1800 (twofold rotation) about the pointX= Y= O.

Sec 1.3]

-~

Symmetry throughout Science

y axis

7

Fig. 1.7 Curve of cos(27thx): h = 2; -\I, S x S +\1,. Reflection of the curve across the line x = 0leaves the curve indistinguishable from its initial state; the function is even.

y axis

y,

Fig. 1.8 Curve of sin(27thx): h = 2; -\I, S x S +Y.. Rotation of the curve about the point x = y= 0 by 1800 leaves the curve indistinguishable from its initial state; the function is odd.

EXAMPLE 1.1. Isj(x) = [x3 cos(x) - x] an even or an odd function?We need to evaluate the function at a few point around zero:

X

-0.2-0.1

3[X cos (X) - Xl

0.1920.099

3X [X c cs tx ) - Xl

0.2 -0.1920.1 -0.0990 Zero

Evidently, the functionj(x) is odd.

EXAMPLE 1.2. The electron density p(x) in Rutile, Ti02, projected along the x axis, can beexpressed by the Fourier series (1.5). We use the X-ray crystallographic data below to

8 Symmetry Everywhere [eh.l

compute p(x), conveniently at intervals of 1/32, from 0/32 to 8/32 only: the function is even,and is reflected across the lines at x =1/4, 1/2 and 3/4.

o18.0

23.9

47.3

65.9

80.5

103.0

121.0

Bi; is zero for all values of h observed experimentally because of the synunetry of thestructure. Forming the sum, we obtain

x

p(x)o61.2

136.3

29.9

36.7

42.4

511.5

624.1

717.5

810.0

The function may be plotted and extended to 1/2, 3/4, or the complete repeat period of unity.The titanium atoms, represented by the highest peak, lie at x = 0, 1/2 and 1 (equivalent to 0by translation), and the oxygen atoms at x, (\I, - x), (\I, + x) and (1 - x), where x, from thegraph, is 0.19. Ideal line peak profiles for the atomic positions would be obtained only with avery large nurnber'" ofAll data.

Finally here, we consider the framework of a cube constructed from twelve identicalI ohm resistors, as shown in Figure 1.9. Let an electrical circuit include the paththrough the points A and G, which lie on a (threefold) symmetry axis of the cube.The planes ACGE, ADGF and ABGH are all rr planes, of the type that we havealready discussed. We can use the symmetry properties of the cube to determine theeffective resistance of the cube to a current I flowing along a path from A to G. Thesymmetry equivalence of the three paths emanating from A and of the three pathsconverging at G requires that the currents in AB, AD, AE, CG, FG and HG are allequal to 1/3, flowing in the directions shown by the arrows. The c symmetryrequires that the currents in EF and EH are the same, so that each is equal to 1/6; itfollows that the currents through BF and DH are also equal to 1/6. A similarargument applies to the paths BC and DC. Thus, the effective resistance of the cubefor a path from A to G is 1/3 + 1/6 + 1/3, or 5/6, ohm.

Fig. 1.9 Framework of a cube formed by twelve identical 1 ohm resistors; the current I flowsthrough the network from A to G

Problems 9

1.4 HOW DO WE APPROACH SYMMETRYSymmetry, then, is a feature of both scientific and everyday life[2-61. In the followingchapters, we shall study the symmetry of chemical species and the applications ofsymmetry principles in chemistry. However, before embarking on these topics, weshall have to spend some time sharpening our notions of molecular symmetry, andin acquiring the requisite descriptive and manipulative tools.

We have seen that symmetry may be made manifest through both geometricaldrawings and mathematical equations. The choice of approach is dictated largely bythe application under consideration. On the one hand, when we are consideringsymmetry in relation to chemical bonding or molecular vibrations, the techniquesthat evolve through group theory are the more appropriate. On the other hand, inthe study of crystals and crystal structure it may be enlightening sometimes to use amore illustrative procedure. Some topics, such as the derivation of point groups, canbe studied readily by both methods: we shall try to make the best choice for eachapplication.

PROBLEMS 11.1 What symmetry is common to the following two-dimensional figures: (a) theemblem of the National Westminster Bank pic, (b) the emblem of the Mercedes-Benz car, and (c) the molecular skeleton of cyanuric triazide (1,3,5-triazidotriazine)? Is there any other symmetry present in any of these objects?

a b c

1.2 Find the following objects in the home, or elsewhere, and study their symmetry.Report the numbers and nature of the o planes and symmetry axes present.

(a) Plain cup;(b) Rectangular plain table;(c) Plain glass tumbler;(d) Inner tray of matchbox;(e) Round pencil, sharpened conically;

(f) Plain brick, with plane faces;(g) Round pencil, unsharpened;(h) Gaming die;(i) Chair;(j) Single primrose floret.

1.3 Study the patterns of the tiled floor and brick wall shown by Figures 1.5a and1.5b. Illustrate each pattern by a number (preferably a minimum) of pointsneccessary to represent it, where each such point has a constant location in thepattern motif, such as its top, left-hand comer. Indicate relative dimensions, asappropriate.

10 Symmetry Everywhere [eh.l

1.4 Twelve 1 ohm resistors are connected so as to form the outline of a regularoctahedron, which has the same symmetry as a cube. An electric circuit iscompleted across a pair of opposite apices of the octahedron. Use the symmetry ofthe octahedron to determine the effective resistance of a path through the octahedralnetwork.

1.5 State the even or odd nature of the symmetry of the following functions of avariable X: (a)xB; (b) sin2(X); (c) (I/X) sin(X); (d)Xcos2(X); (e)Xtan(X).

1.6 Write in upper case those letters of the alphabet that cannot exhibit symmetry.The letters should be treated as two-dimensional, and your answer could dependupon how you form the letters.

2 Symmetry operations and symmetry elements Our torments also may in length of time Become our elements. John Milton (1608-1674): Paradise Lost

2.1 INTRODUCTION: THE TOOLS OF SYMMETRY In order that the concept of symmetry shall be generally useful, it is necessary to develop precisely the tools of symmetry, the symmetry operations and symmetry elements appropriate to finite bodies which, for our purposes, are mainly chemical molecules. Then, as a prerequisite to group theory and its applications to chemistry, we shall consider some of the basic manipulations of vectors and matrices that can be used to simpllfy the discussion of symmetry operations and their combinations.

There exist two important notations for symmetry, and both of them are in general use. In studying the symmetry of molecules and the applications of group theory in chemistry, we shall make use of the Schonilies notation, as is customary. When we come to consider the symmetry of the extended patterns of atomic arrangements in crystals, the Hermann-Mauguin notation is always to be preferred. Once we have become familiar with symmetry concepts in the first of these notations, the Hermann-Mauguin notation will produce little difficulty.

2.2 DEFINING SYMMETRY OPERATIONS, ELEMENTS AND

We follow our statement of symmetry in Section 1.2, and define a symmetry operation as an action that moves a body into a position that is indistinguishable fiom its initial position: it is the action of a symmetry operation that reveals the symmetry inherent in a body. A symmetry operation may be considered to take place with respect to a symmetry element. A symmetry element is a geometrical entity, a plane, a line or a point, which is associated with its corresponding symmetry operation. It is preferable not to say that a symmetry element generates symmetry operations in a body: a body may or may not possess symmetry; if symmetry is present, that symmetry is revealed through a symmetry operation, and with that operation we may associate the corresponding symmetry element.

OPERATORS

2.2.1 Operators and their properties An operator is, in general, the symbol for an operation that changes one function into another. Thus, if we write

0 ( h 2 + x ) = 4 x + 1, (2.1)

12 Symmetry operations and symmetry elements [Ch. 2

0 is an operator acting on the function 2x2 + x; in this example it is the differential operator ( ). There are many such operators and a particular case is that of the d

linear operator. An operator 0 is linear if, for any functionJ

Okf= k(OJ),

o(ri +h 1 = of; + Of, where k is a constant, and if

d

dx wheref; a n d h are two functions. Evidently, - ( ) is a linear operator but In( ), for

example, is not. Linear operators possess several important properties, as follow: The sum of two linear operators O1 and O2 acting on a given functionfis given by

(2.4) (01 + 0 2 Y= O d + 0 2 f ; the parenthetical expresion may be calculated first, if appropriate.

The product of two linear operators follows the rule

010f= OI(O2J). (2.5)

01 ( 0 2 + 0 3 ) = 0 1 0 2 + 0 1 0 3 , (2.6)

01 ( 0 2 0 3 ) = (oioz)03. (2.7)

Linear operators follow the distributive law

and the associative law,

d d br dx

EXAMPLE2.1. Le tOI= - ( ) , 0 2 = x 2 ( ) , 0 3 = ?( ) , 0 4 = 2 ( ) , k = 2 , f i = X 3 - k + 1,

fi = 2.x - 3. Then, from the foregoing: (a) ~ f i = 3x2 - 2;

(c) 0 1 cfi + f2 ) = of; + O h = (3x2 - 2) + (4x) = 3x2 + 4x - 2; (b) 0 1 kfi = k(O& = k(3x2 - 2) = 6x2 - 4;

(d) (01 + 03s = Of; + Oji = (3x2 - 2) + 6~ = 3x2 + 6~ - 2; (e) 010ji = 01 (02fi) = 01 (xs - 2x3 + x z ) = 5x4 - 6xz + 2x. Note that

0 2 0 f ; = 0 2 (Of; ) = 0 2 (3x2 - 2) = 3x4 - 2x2 f OlOji but

0104 = 01 (04) = 01(Zu - 4~ + 2 ) = 6 x 2 - 4 = 040fi.

Except for special cases two operators do not commute, that is,

oio, f o,o,. (2.8) 2.2.2 Symmetry elements In Figure 2.1 we may imagine a symmetry element which is an axis of fourfold rotational symmetry normal to the square (a), passing through its centre; we

Sec. 2.21 Defining Symmetry Operations 13

(a)

symbolize this element as C4. The italic letters on the figure are used to monitor the motion of the square about the axis, and should not be regarded as a part of it. The operator C4 determines the operation, also symbolized by C4 , that is carried out. In words, C4 (square) = square rotated anticlockwise by 90 about the symmetry element C4 .

Strictly, symmetry elements are conceptual, but it is convenient to accord them a sense of reality, and they may be considered to connect all parts of a body into a number of symmetrically related sets. Frequently, different symmetry operations correspond to one and the same symmetry element. We identifl the combination of the two operations (a) to (c) in Figure 2.1 as Ca and, similarly, (a) to (d) may be

written C: . Thus, we may regard C f and C: as either multiple-step operations of C4 or single-step operations in their own right: but all are contained within the symmetry of the square, and are associated with the single symmetry element C4.

2.2.3 Rotation symmetry We consider next the individual symmetry elements of finite bodes, specifically molecules, and firstly the rotation axis. We have used this symmetry element in the discussion of Figure 2.1, and we define it now formally. A molecule is said to

a b

d

T c4 d a

c4 - C

C b

b C

a

C d

b a

Fig. 2.1 Illustration of fourfold rotational symmetry: the C4 rotation axis is normal to the square and passes through its centre; the symbol at the centre represents the C4 axis. Successive C4 operations move the square through the sequence (a) -+ (b) -+ (c) -+ (d) -+ (a)+(b) ... . exhibit an n-fold rotational symmetry axis, symbol C,,, if a rotation of (36Oh)O about that axis brings the molecule into an orientation indistinguishable from that before

14 Symmetry operations and symmetry elements [Ch. 2

exhibit an n-fold rotational symmetry axis, symbol C,, if a rotation of (36Oh)O about that axis brings the molecule into an orientation indistinguishable from that before the operation. We use now the word orientation instead of the more general term position because, as we shall see, no symmetry operation on a finite body produces any translational motion of that body.

In principle, the value of n can range from unity to infinity, and several different values are found for molecules. Figure 2.2 shows the fivefold symmetry of nitrosylcyclopentadienylnickel. The value of infinity for n is found in linear molecules; thus, iodine monochloride, IC1, has a C, axis along the length of the molecule.

1 I

Fig 2.2 Stereoscopic illustration of the molecule of nitrosylcyclopentadienylnickel, (CsH5)NONi. The vertical axis is C5, and there are five vertical cr planes, each passing through the Ni, N and 0 atoms and one >CH group.

I I I

I

I

I

I

I

I

I

I

Fig. 2.3 Cube, showing a (vertical) Cd axis along which the cube may be imagined to be compressed to a square.

In two dimensions, the rotation axis, strictly, collapses to a poinf of rotation. Imagine compressing a cube in a direction normal to a face until it becomes a square. The C4 axis along the direction of compression, Figure 2.3, would become a point at the centre of the square, as in Figure 2.1. However, we retain the notation C, in two-dimensional symmetry elements; they are, after all, conceptual.

Sec 2.21 Defining Symmetry Operations 15

Sign of rotation We distinguish between clockwise and anticlockwise rotations in the following manner: the fourfold rotation operation, for example, illustrated by Figure 2.1, is symbolized as C4 (sometimes C 4 ) for an anticlockwise rotation of the object or of a vector within it, and C i1 (sometimes C 4 ), the inverse of C4, for the corresponding clockwise movement. Thus, in the given example, the positional result of C: is equivalent to that of C i1 . We may note en passant that the convention adopted here for rotation is the same as that used for the sign of an angular momentum vector resolved along the z reference axis in a species.

2.2.4 Reflection symmetry A reflection symmetry plane, symbol o, is said to be present in a molecule If it divides that molecule into halves that are related to each other as an object is to its mirror image. The operation of reflection, unlike rotation, cannot be performed physically on a body, but if it could, the body would be indistinguishable before and after the reflection. In combination with a rotation axis, it is necessary to distingwsh between a reflection plane normal to the rotation axis, o h (h = horizontal), and one containing that axis, o,(v = vertical). In Figure 2.2, five o, planes are present: each contains the vertical C, axis, and passes through a >C-H group, the centre of the opposite C 4 bond, and the Ni, N and 0 atoms.

In two dimensions, reflection may be said to take place across a line: it remains symbolized as o, but the subscripts h and v become superfluous. Thus, in Figure 2.1 we could draw four CY reflection lines, all passing through the centre of the square. We may note that a reflection line will result from projecting a three- dimensional figure, with a o symmetry plane, on to a plane that is normal to the o plane in the object. The reader is invited to draw a projection of the nitrosylcyclopentadienyl molecule in the plane of the cyclopentadienyl ring, and to mark in the o lines.

2.2.5 Roto-reflection symmetry A molecule contains an n-fold roto-reflection axis (also called an alternating axis), symbol S,,, if it is brought into an orientation indistinguishable from its original orientation by means of a rotation of (36Oh)O about that axis, followed by reflection across a plane normal to the axis, the two movements constituting a single symmetry operation. It is important to note that the reflection plane used here may not be a symmetry (reflection) plane of the molecule itselJ For example, Figure 2.4 is a stereoview of the dihydrogenphosphate ion, IH,P04]-, which exhibits the symmetry element S4, but does not, itself, possess reflection symmetry. However, in the tetracyanonickelate(I1) ion, Ni[CN4]*-, Figure 2.5, the S4 axis is normal to a (3 plane, the plane of the ion itself.

16 Symmetry operations and symmetry elements [Ch. 2

Fig. 2.4 Stereoscopic illustration of the dihydrogenphosphate ion, [H2PO4]-, as in crystalline potassium dihydrogen phosphate; circles in order of increasing size represent H, 0 and P atoms. Two of the four hydrogen atom sites on each ion are occupied in a statistical manner throughout the structure: we may regard the sites as occupied by four half-hydrogen atoms per ion. The only symmetry element here is S4 , normal to the plane of the drawing.

A

Fig.2.5 Stereoscopic diagram of the tetracyanonickelate(II) ion, Ni[CN4I2-; circles in order of increasing size represent C, N and Ni atoms. The axis normal to the drawing is C4, with S4 collinear. The molecular plane is bh and the centre of the molecule is an inversion centre, i.

The combined actions of the S, and o h symmetry elements result in a C4 axis coincident with S4: but the important point here is that the plane involved in the S4 operation is now also a symmetry plane of the species.

In general, we note that for a finite body containing an S, axis but no other symmetry element, a 0 reflection plane does not coexist as a symmetry element within that body i f n is an even number. An equivalent element to S1 is a (J plane normal to the direction of Sl .

2.2.6 Inversion symmetry The inversion operation i acts through a point in a molecule, its centre of symmetry, and consists in taking every part of the molecule in a straight line through that point to an equal distance on the opposite side of it: it results in an inversion of its configuration. The species illustrated by Figure 2.5 contains the element i , as does

Sec. 2.21 Defining Symmetry Operations 17

the molecule of dibenzyl, Figure 2.6. The operation i is equivalent to S2, but the former designation is preferred for this symmetry operation. In two dimensions i degrades to a twofold rotation operation about a point.

We may choose to arrange all symmetry operations of molecules under two headings, proper rotations C, and improper rotations S,. However, it is conventional to use both elements (J and I , rather than S, and S,, respectively, in discussing molecular symmetry.

2.2.7 Identity symmetry The identity operation, symbol E (Ger. Einheit = unity), consists effectively in doing nothing to the object. Alternatively, we may regard it as a C1 rotation about any axis (C,) through a body. All molecules possess identity symmetry; some, such as CHFClBr, Figure 2.7, show no other symmetry. The E operation, although apparently trivial, is fundamental to group theory. With reference to Figure 2.1, C , and C in general, is equivalent to identity.

2.2.8 Roto-inversion symmetry The roto-inversion axis is not a part of the Schonflies notation, but we describe it

Fig. 2.6 Stereoscopic illustration of the structure of the molecule of dibenzyl, (C&IsCH2)2; circles in order of increasing size represent H and C atoms. An inversion centre lies halfway along the central C-C bond.

Fig 2.7 Stereoscopic illustration of the molecule of fluorochlorobromomethane, CHFCBr, an example of identity symmetry; circles in order of increasing size represent H, C, F, C1 and Br atoms.

18 Symmetry operations and symmetry elements [Ch. 2

here as it will be needed when we consider crystal symmetry within the Hermann- Mauguin symmetry notation. Like the roto-reflection axis, it is a single symmetry operation consisting of two movements, a rotation of (36O/n)O about the ;axis followed by inversion (Section 2.2.6) through a point on the i axis; this point is a centre of symmetry only when i is an odd integer. Figure 2.4 shows the symmetry element 4 : evidently, it is equivalent to the symmetry element S4, but the operation 4 is equivalent to the operation S i , with the same sense of rotation in each case. An equivalence between S, and ;is not general. We note also that the point of inversion on the ; axis is also the origin of the reference axes (see Section 2.3), for reasons that we discuss in Section 3.3.

To facilitate an understanding of the symmetry operation S4 (and i), instructions are given in Appendix 1 for constructing a model that possesses this symmetry. As we may show from the next chapter, for a body that has a single symmetry element S, , the following relationships hold for the corresponding operation (n 2 1).

-

For n odd: S, = Combination of C, and Oh; For n even: S4n-2 = Combination of CZn-, and i;

S4n No equivalence

It is sometimes stated that S4 is equivalent to the combination of C4 and Oh. While this statement provides a way of looking at the symmetry operation S4, it is implicit then that the body in question possesses also the symmetry element (3h and so contains a symmetry operation higher than S4 (see also Section 2.2.5 and Problem 3.6). Further discussions on symmmetry may be found in ~tandard~ literature .

2.3 SETTING UP REFERENCE AXES It is convenient to discuss molecules and their symmetry by reference to right- handed axes that are mutually perpendicular (Figure 2.8); such axes are termed orthogonal (see also Section 4.2. l) , and the sequence x -+ y -+ z simulates a right- handed screw movement.

The selection of the orientation of reference axes within a molecule is somewhat arbitrary: the molecule knows nothing about the axes we have set up in Figure 2.8. We shall adopt a convention that is common, albeit not universal. The z reference axis is aligned with the principal axis, that is, the rotation axis of highest degree : if there are two or more axes of that degree, z is chosen so as to intersect the maximum number of atoms. If the molecule is planar and z lies in that plane, the x axis lies normal to the plane, that is, the molecular plane is the yz plane. If the molecule is planar and z is normal to that plane, then y lies in the plane and, preferably, passes through the maximum number of atoms. In each case the x axis is perpendicular to both y and z, as in Figure 2.8.

Sec 2.41

A

z-axis

y-axis

Relationship of Symmetry to Chirality

Fig. 2.8 Orthogonal reference axes: f l y =y"z = z"x = 90"

19

Fig 2.9 Stereoview of the trans-tetranitrodiamminocobaltate@) ion, [Co(NO2)4(NH2)2]-; circles in order of increasing size represent N, 0, N H 2 and Co species. The N H 2 groups are in free rotation, and their effective shape is spherical.

A molecule may exhibit symmetry axes of more than one degree. Figure 2.9 illustrates the trans-tetranitrodiamminocobaltate(II1) ion, which shows both C4 and C, symmetry axes: the principal axis is C4, and z is aligned with this axis. How would the x and y axes be set for this molecule? We may note that when the axes are not orthogonal, the usual notation for the interaxial angles is y^z = a, zAx = p and xAy = y. Any other orientation of the orthogonal axes could have been made, but a common sense choice leads normally to the simplest manipulations.

2.4 RELATIONSHIP OF SYMMETRY TO CHIRALITY IN MOLECULES We noted in Section 2.2.6 that all symmetry operations with which we are concerned here are either proper rotations C, or improper rotations S,.. Chiral molecules, that is, those with one asymmetric centre or more, such as lactic acid CH3C*H(OH)C02H, exhibit optical activity: they rotate the plane of polarization of plane-polarized light. The necessary and sac ien t condition for a molecule to be optically active is that it cannot be superimposed on to its mirror image. When this condition holds, the molecule exists in two forms known as enantiomers. Superimposability depends upon symmetry. A molecule with an S, axis is always superimposable on to its mirror image, as the following argument shows.

20 Symmetry operations and symmetry elements [Ch. 2

Whatever the orientation of a given molecule it can have only one mirror image. If the molecule possesses an S,, axis we may choose, arbitrarily, that the image plane coincides with the reflecting plane associated with the S,, symmetry element. From the discussions in Sections 2.2.5 and 2.2.8, if n is an odd integer, then the reflecting plane exists as a symmetry element, and the molecule is superimposable on to its mirror image. If n is even and q, does not exist in the molecule, then the operation S, does not lead immediately to a superimposable mirror image. However, if the whole molecule is then rotated by ( 3 6 O / n ) O , the molecule and its mirror image are superimposable. An alternative way of looking at this situation is that because S,, introduces a change-of-hand, even when there is no CT symmetry plane present, the molecule contains its own mirror image.

Molecules that exhibit only C, symmetry are often termed dissymmetric: an asymmetric molecule has no symmetry, so that chiral molecular species are those that are either asymmetric or dissymmetric.

2.5 A BRTEF LOOK AT VECTORS AND MATRICES In this section, we describe some of the elementary operations with vectors and matrices that can be used to add a degree of conciseness and elegance to the manipulation of symmetry operations. In chemistry, vectors and matrices tend not to be among the more popular topics for study; indeed, a significant amount of chemistry can be studied quite satisfactorily without them.

A vector differs from an ordinary number, or scalar, merely by having a direction in space; a matrix is a collection of numbers that can be manipulated en bloc. With a little practice, we shall gain a familiarity that will render the study of symmetry and group theory remarkably straightforward.

2.5.1 Sum, difference and scalar (dot) product of two vectors Let rl and r2 , Figure 2.10, be any two vectors from an origin 0. Their difference r2 - rl is the vector roc, which may be represented also by the vector from A to B, rm. The magnitude roc (= rm) is obtained by forming the dot product of roc (from 0 to C) with itself, and expanding the resulting expression algebraically, noting that a dot product rl -rJ is dejned by

(2.9) rr -rJ = rl rJ cos(rlArJ),

where rlA rj is the angle between rl and r, ; here, rlAr, = 0 so that

2 2 rZoc = roc -roc = (r2 - r,)-(r2 - rl) = r , + r - 2rl r2 cOs(8). (2.10) It may be noted that r2 cos(8) is the projection of r2 on to the direction of rl so that the product of rl and r2 cos(0) acts along the direction of r l . We may recognize (2.10) as an expression of the extension of Pythagoras's theorem to the obtuse- angled triangle OAB.

Any vector r from the origin of orthogonal axes to a point x, y, z may be written as

r = x i + y j + z k , (2.11)

Sec 2.51 A Brief Look at Vectors and Matrices 21

where i, j and k are vectors of unit magnitude (unit vectors) along the x, y and z axes, and x, y and z are the coordinates of the termination of the vector r, or the

C R

Fig 2.10 Vectors rl and rz &om a common origin 0.

lengths of the projections of the vector on to the axes, in the same order. From (2.9), we have

(2.12) 3 = x2 + y2 + z2. Unit vectors on orthogonal axes obey the rule, from (2.9):

i.j = 6,, (2.13)

where 6, is the Kronecker delta. This notation is simply shorthand for saying that i.j = 1 when i = j but is zero otherwise, and applies to all pairs of i, j and k.

The dot product relationship can be used for very straightforward calculation of bond lengths and bond angles in a crystal structure, as we show in Example 2.2 below.

2.5.2 Vector (cross) product of two vectors The vector product (cross product) of two vectors rl and r2 is dejined by

rl x r2 = rl r2k sin(rlAr2 ), (2.14)

where k is a unit vector perpendicular to the plane of rl and r2 , and directed such that rl , rz and k form a right-handed set of directions (like x, y and z in Figure 2.8). We should note here that whereas rl -rz = rz -rl and is a scalar, rl x r2 = -rz x rl and remains a vector. An important application of (2.14) arises in calculating the volume of a parallelepipedon.

EXAMPLE 2.2. The x , y and z coordinates of the hydrogen atoms in the water molecule are given, in order, as -0.024, 0.093, 0.000 nm and 0.096, 0.000, 0.000 nm, with respect to oxygen at the origin of orthogonal axes. We calculate the bond angle H6H. The 0-H bond lengths are clearly 0.096 nm. From (2.9),

(-0.024i + 0.093j + Ok).(0.096i + Oj + Ok) = 0.096' COS(HOH) COS(HOH) = (-0.024 x 0.096)/0.096'

whence H6H = 104.5'. We note here that the same general equations can be employed where the reference axes are not orthogonal.

EXAMPLE 2.3. A general parallelepipedon is characterized by the parameters a, b, c, a, j3 and y, where the edges a, b and c are parallel to the x, y and z axes, respectively. We need a general expression for its volume V. Now, V = area of base x perpendicular height: b x c is a

22 Symmetry operations and symmetry elements [Ch. 2

V 2 =

vector of magnitude bc sin(a), the area of the base, along the direction of k, perpendicular to the bc plane, and the perpendicular height is ak. Hence, V = a(b x c), or one of its cyclic permutations, that is, b.(c x a) or c.(a x b).

a a + a a + a a a b + a b + a b a c + a c + a c 1 1 2 2 3 3 1 1 2 2 3 3 1 1 2 2 3 3

1 1 2 2 3 3 2 2 3 3

1 1 2 2 3 3 1 1 2 2 3 3 1 1 2 2 3 3

b a +b a +b a blbl+b b +b b blcl+b2c2+b3c3 I c a +c a +c a c b +c b +c b c c +c c +c c

Let a = ali + a2 j + a3 k, where al, a2 and a3 are the direction cosines of a with respect to the x, y and z axes, respectively (see Appendix 2). Similar equations may be written for b and c. Using (2.9) and (2.14), and substituting and expanding

V = a(b x c) = (ali + azj + a3k).(bmk - blcj + bzclk - bzc3i + b3cj - bmi).

The right-hand side is the expansion of the determinant (see Section 2.5.3)

2 v =

a a 1 2

v =

a*a a*b a*c

boa b*b b.c . c*a cob C e C

I 2 c3 I since rows and columns of a determinant can be interchanged without affecting its numerical value, we can write

a a a 1 2 3 2

bl b2 b3 v =

c c c 1 2 3

a bl c

a b3 c

1 1

a 2 b2 c2

3 3

Multiplying the two determinants (see Section 2.5.3) leads to

Sec. 2.51 A Brief Look at Vectors and Matrices 23

which simplifies to V = abc[l - cos (a) - cos (p) - cos (y) + 2cos(a)cos(~)cos(y)]

2.5.3 Manipulating determinants and matrices We consider next those operations with determinants and matrices that will be of importance in our discussion and use of group theory in the ensuing chapters. For those readers who may wish for more detail on this subject, suitable material is indicated in the section on References and Selected Reading.

Matrices and determinants A matrix is a rectangular array of terms av (also known as elements) that may be combined with other matrices according to certain rules. We may illustrate a general matrix A by the equation

A =

a a . . . . . . a a a a a . . . . . . a a a a . . . . . . a

11 12 13 in

21 22 23 2n

31 32 33 3n

a a a . . . . . . a mi m2 m3 m

(2.15)

The vertical sets of terms are called coluuins (i increasing downwards), and the horizontal sets are rows (j increasing left to right), so that the general term av lies i terms down a n d j terms across in the matrix; the size of the matrix is m x n. The terms aii form the diagonal of the matrix, and the sum of these terms is known as the trace x, or character, of the matrix. A one-column matrix is a vector, so that (2.11) may be written as

(2.16)

We shall be concerned almost always with square matrices, that is, those for which m = n.

It is important to distinguish between a matrix and a determinant, the latter being an array of terms of a square matrix that represents a sum of certain products of the terms. Lfa 2 x 2 matrixA is represented by

(2.17)

then the corresponding determinant is given as det(A) by

24

A21 = (-1)2+1...i

Symmetry operations and symmetry elements

a a a ; 1 1 12 13

............. a ............ a ........ :21 22 23

a a

a a 11 12

21 22 det(A) =

a l l a12 O13

det (A) = aZ1 an aU 0 3 1 a32 a33

and its numerical value is defined as

all a22 - a21 a12.

= UIIAII + a21A21 + a31A31.

,

[Ch. 2

(2.18)

(2.19)

Cofactors In general, the value of a determinant is obtained by forming the sum of the product of each term in a given column (or row) with its cofactor. The cofactor A, of a term a, is the determinant of one order lower, obtained by striking out the row and column that intersect in a, , the minor M , of a,, multiplied by (-1)l; Thus, for a third order determinant A, Azl would be given by

a a I a31 32 33 which, from (2.18)-( 2.19), becomes

A21 = -(a12 a33 a32 a13 1;

= - a a

a a 12 13

32 33 ( 2 . 2 0 )

(2.21)

the unsigned 2 x 2 determinant in (2.20) would be symbolized MZl. For a 3 x 3 matrix A, given by

a a a ( 2 . 2 2 ) 12 13

31 32 33

(2.23)

(2.24)

The matrix A with each term replaced by the corresponding cofactor may be denoted by A, the cofactor matrix of A.

A determinant of any order can be decomposed systematically in this manner, and it is clear from the foregoing that a determinant can be evaluated only for a matrix that is square. The value of a determinant is not altered if the rows and columns are interchanged. Thus, if A is modified to A by interchanging the rows and columns, then

Sec. 2.51

1 0 - 1 2 0 1 0 2 1 1 - 1 0 2 1 0 3

A =

A Brief Look at Vectors and Matrices

*

25

(2.25)

i e t ( A ) = (1)

and it is easy to show that det(A) = det( A ) . (2.26)

The matrix A is known as the transpose of matrix A : the terms have been interchanged, or transposed, across the diagonal, or the line of a,, terms. If for any matrix a, = a,, for all i andj, the matrix is termed symmetric. A symmetric matrix and its transpose are equal, that is, a, = a,, for all i andj, whereupon we would then writeA =A.

0 1 0 1 0 2 0 1 2 1 -1 o + (-1) 1 1 0 - (2) 1 1 -1 1 0 3 2 1 3 2 1 0

EXAMPLE 2.4. determinant

We illustrate (2 .20H2.24) by finding the value of the fourth-order

- -: 1 } = -3 + 2 + 3 - 2 + 4 + 4 = 8 . Addition and subtraction of matrices The sum or difference of two matrices is obtained by takmg the sum or difference of similar terms; thus, the general term c , of the sum or difference of two matrices A and B is given by

(2.27) Q y f b, = C,J and the result for the two matrices in their entirety is written as

A f B = C . (2.28)

It should be evident that matrix addition is commutative, that is, A + B = B + A , (2.29)

26 Symmetry operations and symmetry elements

but with subtraction, generally

A - B f B - A .

[Ch.