Symmetry and group theory in chemistry

423 Pages·17.627 MB·English

by  Mark Ladd , foreword by Lord Lewis.

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 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 hs research
contributions in the areas of crystallography and solid-state chemistry.
Mark Ladd is the author, or co-author, of many books: Analytical
Chemistry, Radiochemistry, Physical Chemistry, Direct Methods in
Crystallography, Structure Determination by X-ray Crystallography (now
in its third edition), Structure and Bonding in Solid-state Chemistry,
Symmetry in Molecules and Crystals, and Chemical Bonding in Solids and
Fluids, the last three with Ellis Horwood Limited. His Introduction to
Physical Chemistry (Cambridge University Press) is now in its third edition.
He has published over one hundred research papers in crystallography and in
the energetics and solubility of ionic compounds, and he has recently retired
from his position as Reader in the Department of Chemistry at Surrey
University.
His other activities include music: he plays the viola and the double bass in
orchestral and chamber ensembles, and has performed the solo double bass
parts in the Serenata Notturna by Mozart and the Carnival of Animals by
Saint-Saens. He has been an exhibitor, breeder and judge of Dobermanns,
and has trained Dobermanns in obedience. He has written the successful book
Dobermanns: An Owner s Companion, published by the Crowood Press
and, under licence, by Howell Book House, New York. Currently, he is
engaged, in conjunction with the Torch Trust, in the computer transcription
of Bibles into braille in several African languages, and has completed the
whole of the Chichewa (Malawi) Bible.
Mark Ladd is married with two sons, one is a Professor in the Department
of Chemical Engineering at the University of Florida in Gainsville, and the
other is the vicar of St Luke’s Anglican Church in Brickett Wood, St Albans.
He lives in Farnham, Surrey, with his wife and one Dobermann.
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 p ublication may be reproduced, stored in a
retrieval system, or transmitted, in any form or by any means, electronic,
mechanical, photocopying, recording, or otherwise, without the permission of
Horwood Publishing, International Publishers, Coll House, Westergate, Chichester,
West Sussex, England
0M . Ladd, 1998
British Library Cataloguing in Publication Data
A catalogue record of this book is available from the British Library
ISBN 1-898563-39-X
Printed in Great Britain by Martins Printing Group, Bodmin, Cornwall
Table of contents
Foreword .................................................................................. ................. v-VI
Preface ................................................................................................ ... vii
.....
List of symbols ............................................................................................ .~ii-mi
1 Symmetry everywhere ...................................................................................... 1
1.1 Introduction: Looking for symmetry. ............................................................. 1
1.1.1 Symmetry in finite bodies ................................... .................2
1.1.2 Symmetry in extended patterns .................................................................. 4
1.2 What do we mean by symmetry. .................................................................... 5
1.3 Symmetry throughout science ....................................................................... 6
1.4 How do we approach symmetry
Problems 1 .................................................................
2 Symmetry operations and symmetry elements ............................... 11
2.1 Introduction: The tools of symmetry
2.2 Defining symmetry operations, ele
.................... 11
...................... 13
Sign of rotation ............................ ....................................... 15
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 .....................................
Matrices and determinants; Cofactors; Addition and subtraction
of matrices; Multiplication of matrices; Inversion of matrices;
Orthogonality ............................... ..............................
2.5.4 Eigenvalues and eigenvectors... ................................................................ .28
Diagonalization; Similarity transformation; Jacobi diagonalization ..... 30-3 1
2.5.5 Blockdiagonal and other special matrices ....................................... 33
Adjoint and complex conjugate matrices;
matrix; Unitary matrix ......................................................................... 34-35
........................................ .................................. 35
3 Group theory and point groups ............. .................................. 38
3.1 Introduction: Groups and group the0 .................................. 38
3.2 What is group theory ......................... ..................................... ..38
3.2.1 Group postulates ............................................ ................................ 38
Closure; Laws of co
Inverse member ........... .................................... ................. 38-39
3.2.2 General group definitions... ......................... .............................
3.2.3 Group multiplication tables . ................................
3.2.4 Subgroups and cos
3.2.5 Symmetry classes and conjugates ........ ...............................
3.3 Defining, deriving
3.3.1 Deriving point groups.. ............................ ................................... 46
Euler's construction ...............................................
3.3.2 Building up the ............................ 52
..........................................
......................... 59
Problems 3 ................................................................ ............................... 67
4 Representations and character tables .........................................
4.1 Introduction: What is a representation ............ ........................... 72
4.1.1 Representations on position vectors .......................................
4.1.2 Representations on basis vectors ......................... ............................ 75
4.1.3 Representations on atom vectors.. ..... ............................................. 77
Unshifted-atom contributions to a re
4.1.4 Representations on functions... .............. ................................ 82
4.1.5 Representations on direct product functions .............................
4.2 A first look at character tables.. .................... .......................... 86
4.2.1 Orthonormality ............. ........................................... 87
4.2.2 Notation for irreducible representations ........ ............................. 88
Complex characters ...................................................................... 89
4.3 The great orthogonality theorem ..................... ............................ 90
4.4 How to reduce a reducible representation ......................................... 94
4.5 Constructing a character table.. .......................... ................................ 96
4.6 How we have used the direct product .............. ............................ 103
Problems 4 ............................................................ ............................ 104
5 Group theory and wavefunctions. .......................... .............................. 108
5.1 Introduction: Using the Schrodinger equation ............................... 108
5.2 Wavefunctions and the Hamiltonian operator.. .......................................... ,109
5.2.1 Properties of wavefunctions ................................ ................. 110
5.3 A further excursion into function space.. ............................................
5.3.1 Defining operators in function space ....................................................... 112
5.4 Using operators with direct products ......................................... 115
5.5 When do integrals vanish ................................ ............................... 117
5.6 Setting up symmetry-adapted linear combinati
5.6.1 Deriving and using projection operators.. ..... ............................... 119
5.6.2 Deriving symmetry-adapted orbitals for the carbonate ion
Generating a second function for a degenerate representation
5.6.3 Handling complex characters ...........................................
Problems 5 ................................................... ...................................... 128
6 Group theory and chemical bonding ............................................................. 130
6.1 Introduction: molecular orbitals .................................
Classlfylng molecular orbitals by symmetry ...............
6.2 Setting up LCAO approximations.. ........................................................... .13 1
Function of the Schrijdinger equation ........ ....................................... 132
Introducing the variation principle .......
6.2.1 Defining overlap integrals... ............................ .............................. 134
6.2.2 Defining Coulomb and resonance inte .............................. 134
Continuing with the variation principle ..............................
6.2.3 Applying the LCAO method to the oqgen molecule... ........................... .137
6.2.4 Bonding and antibonding molecular orbitals and notation ....................... 140
Total bond order ................................................ ...........1 42
6.3 P-electron approximations ...... .............................................................. 142
6.3.1 Using the Huckel molecular- ............................. 143
Benzene. ................................................................................................. 144
6.3.2 Further features of the Huckel molecular-orbital theory.. ........................ .149
ll-Bond order ................................................ ........................ 149
Free valence .......................................................................................... .15 1
Charge distribution ................................................................ 152
6.3.3 Altemant and nonaltemant hydrocarbons ..... ..................................... 152
Methylenecyclobutene; methylenecyclopropene ...................................... 15 3
6 4 4 Huckel's 4n + 2 rule .......................................... .................1 56
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
6.4 Schemes for hybridization: water methane ............. ...... 167
6.4.1 Symmetrical hybrids ............................ ....................................... 169
Walsh diagrams ...................................
Further study of methane ..................... ....................................... 173
6.5 Photoelectron spectroscopy ..........................................................
Sulfur hexafluoride. ................................... ............................... 178
6.6 Cyclization and correlation ....................
6.7 Group theory and transition-metal compounds... ...... .................. 186
6.7.1 Electronic structure and term symbols. .......................................
Russell-Saunders coupling.. .. ..................................................... 188
6.7.2 How energy levels are split in a crystal field.. ..... .................... 192
Weak fields and strong fields ...............
6.7.3 Correlation diagrams in 0, and Tds ymmetry ......................................... .197
'Holes' in d orbitals ..................................... ........................... 203
6.7.4 Ligand-field theory .... ...................................................... .....2 05
Spectral properties ............................ ............................................... 211
Problems 6 .................................................................... ................... 217
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 ................................... .............................. 222
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 ............................................. ................................... 230
Polarization of Raman spectra .. .............................................
7.4 Classlflmg vibrational modes ...................
7.4.1 Combination bands, overtone bands and Fermi resonance... .........
7.4.2 Using correlation tables with vibrational spectra ..................................... 239
7.4.3 Carbon &oxide as an example of a linear molecule ........................
7.5 Vibrations in gases and solids .................................... ...................... 241
7.6 Electron transitions in chemical species... .........................................
7.6.1 Electron spin... ...............................................................................
7.6.2 Electron transitions among degenerate states ......................................... .243
7.6.3 Electron transitions in transition-metal compounds,.. ..................
Problems 7 ....... ..................................................................
8 Group theory and crystal symmetry ............................................................. 248
8.1 Introduction: two levels of crystal symmetry ...........
8.2 Crystal systems and crystal classes. .................. .....................
8.3 Why another symmetry notation ...................... ................................ .249
8.4 What is a lattice ........................................................................................ .25 2
8.4.1 Defining and choosing unit cells ............................ ................ 253
8.4.2 Why only fourteen Bravais lattices ......................................................... .256
8.4.3 Lattice rotational symmetry degrees are 1, 2, 3, 4 and 6 ....
8.4.4 Translation unit cells ...................................... ................................ 261
8.4.5 Wigner-Seitz cells. ............................................................
8.5 Translation groups ......... ................. ............................... .263
8.6 Space groups ........................................................................
8.6.1 Symmorphic space groups... ................................................................... ,265
Glide planes and screw axes ......................... ................................ 269
8.6.2 And nonsymmorphic space groups... .................................. ............. 272
Monoclinic nonsymmorphic space groups... ...... ............................... ,272
Orthorhombic nonsymmorphic space groups .............................
Some useful rules; Tetragonal nonsymmorp
8.7 Applications of space groups ..........................
Naphthalene; Biphenyl; Two cubic structures
3 8.8 What is a factor group .................................
8.8.1 Simple factor-group analysis of iron(I1) su
8.8.2 Site-group analysis... .................................... ...................... 284
Factor-group method for potassium chro
Problems 8 ....................................................................................................... 285
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 ................3 19
A1 1.1 Character tables .. .......................................................... ....3 19
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 ..
A11.2 Correlation tables .............................................. .................3 37
Groups C (n = 2-4, 6). Groups C (n = 2
Groups D , T and 0 .............................. .................... 337
A1 1.3 Multiplication properties of irreducible r 337
General rules; Subscripts on A and B; Doubly-degenerate
representations; Triply-degenerate representations; Linear
groups; Direct products of spin multiplicities .......................................