Symmetry and group theory in chemistry ( PDFDrive )

Symmetry operations operators in a group Spectroscopic term symbol Irreducible representation; matrix; general constant; A-face centred unit cell; member of a group ijth term of cofactor

Symmetry and Group Theory

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 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

Symmetry and Group Theory

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 o q g e n 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

A l 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 = 2 4 ) ; 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 <111>;

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 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

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

Symmetry operations (operators) in a group

Spectroscopic term symbol

Irreducible representation; matrix; general constant; A-face centred unit cell; member of a group

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

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

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

J

species

constant; Miller index along y axis

J K-’)

xvi List of symbols

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

List of symbols xvii

T Tetrahedral (cubic) (point) group

t Translation vector

t

U

[ Direction in a lattice

<uvw> Form of directions [UVB'I

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

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

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

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

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-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 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

three-Fig1.1 Potassiumhydrogen bistrichloroacetate (CbC02)2HK

2 Symmetry Everywhere [eh.l

Fig1.2 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 dimensionsshown 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 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

two-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 chlorofluorobenzene and a crystal of potassium tetrathionate Yet each is anexample of reflection symmetry: a (mirror) symmetry plane, symbolo (Ger.Spiegel

3-=mirror), can be imagined for each entity, dividing it into halves that are related as

an object is to its mirror image

Ifit were possible to perform physically the operation of reflecting the halves of anobject across the symmetry plane dividing them, then the whole object would appearunchanged after the operation.Ifwe view the Doberma~ from the side its mirrorsymmetry 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 herimage together would show c symmetry, across the plane between the object animal

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 ifwepursue 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

4 Symmetry Everywhere [Ch I

Fig 1.4 Vijentor Seal of Approval at Valmara: object and mirror image relationship acros s avertical (Jsymmetry plane From a three-dimensional point of view, there are three symmetryelements here: the(Jplane just discussed, the (Jplane shown byFigure 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 patterns

If 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 after

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

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

to form a brick wall is limited by the terminations of the building of which the wall

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 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 Ifacrystal 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 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-raydiffraction, (b) from neutron diffraction The arrows represent the directions of the magneticmoments associated with the unpaired electronsin the atoms

6 Symmetry Everywhere [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 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 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

The roots of (1.1) areX= ±2 andX = ±2i, and we can see immediately that thesesolutions have a symmetrical distribution about zero The differential equation

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 harmonicoscillator The general solution for (1.2) may be written as

where A and B also are constants If we consider a reflection symmetry that converts

Xinto-X,then the solution of (1.2) would become

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, forexample, between the limits X = ±Y2, can be represented by a series of sine andcosine terms known as a Fourier series:

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.

Fig 1.7 Curve of cos(27thx): h=2;-\I,Sx S +\1,.Reflection of the curve across the linex=0

leaves the curve indistinguishable from its initial state; the function is even.

y axis

y,

Fig 1.8 Curve of sin(27thx):h=2;-\I, SxS+Y Rotation of the curve about the point x=y

=0 by 1800

leaves the curve indistinguishable from its initial state; the function isodd.

EXAMPLE 1.1 Isj(x)= [x 3 cos(x) -x] an even or an odd function?

We need to evaluate the function at a few point around zero:

Evidently, the functionj(x) is odd

EXAMPLE 1.2 The electron density p(x)in Rutile, Ti02, projected along thexaxis, 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 atx=1/4, 1/2 and 3/4.

12 1.0

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

by translation), and the oxygen atoms atx, (\I, - x), (\I,+x) and(1 -x), wherex, from the graph, is 0.19 Ideal line peak profiles for the atomic positions would be obtained only with a very large nurnber'" ofAlldata.

Finally here, we consider the framework of a cube constructed from twelve identical

I ohm resistors, as shown in Figure 1.9 Let an electrical circuit include the paththrough the pointsA and G, which lie on a (threefold) symmetry axis of the cube.The planes ACGE, ADGFand ABGHare 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 currentIflowing along a path fromAto G Thesymmetry equivalence of the three paths emanating fromA and of the three pathsconverging at G requires that the currents inAB, AD, AE, CG, FG andHG are allequal to 1/3, flowing in the directions shown by the arrows The c symmetryrequires that the currents inEFandEHare the same, so that each is equal to1/6;itfollows that the currents through BF and DH are also equal to 1/6. A similarargument applies to the pathsBCandDC. Thus, the effective resistance of the cubefor a path fromA 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 currentI flows

Problems 9

1.4 HOW DO WE APPROACH SYMMETRY

Symmetry, 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, and

in 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, can

be studied readily by both methods: we shall try to make the best choice for eachapplication

PROBLEMS 1

1.1 What symmetry is common to the following two-dimensional figures: (a) the

emblem of the National Westminster Bank pic, (b) the emblem of the 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?

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 avariableX: (a)xB; (b)sin2(X); (c)(I/X) sin(X);(d)Xcos 2(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

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 d ( ) There are many such operators and a particular case is that of the

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

Okf= k(OJ),

where k is a constant, and if

the parenthetical expresion may be calculated first, if appropriate

The product of two linear operators follows the rule

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,

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

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

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

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

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 4bond, 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 S 4 , 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

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

a x i s 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 s a c i e n 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

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

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),

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