Introduction to Symmetry and Group Theory for Chemists

133 Pages·2004·0.631 MB·English

by  Arthur M. Lesk

INTRODUCTION TO SYMMETRY AND GROUP THEORY
FOR CHEMISTS
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Introduction to Symmetry and
Group Theory for Chemists
by
Arthur M. Lesk
KLUWER ACADEMIC PUBLISHERS
NEW YORK,BOSTON, DORDRECHT, LONDON, MOSCOW
eBookISBN: 1-4020-2151-8
Print ISBN: 1-4020-2150-X
©2004 Springer Science + Business Media, Inc.
Print ©2004 Kluwer Academic Publishers
Dordrecht
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Contents
Preface ix
1. THE RELATIONSHIP BETWEEN GROUP THEORY AND
CHEMISTRY 1
1.1 Introduction 1
1.2 Applicationsofgrouptheory 1
2. SYMMETRY 3
2.1 Abridgefromgeometrytoarithmetic 3
2.2 Classifyingsymmetryoperations 3
2.3 Fullanalysisofthesymmetryofthewatermolecule:Introduction
tonotation 4
2.4 Productsofcoveringoperations: multiplicationtables 8
2.5 Whatisagroup? 8
3. GROUPTHEORY 11
3.1 Definitionofagroup 11
3.2 Subgroups 12
3.3 Examplesofgroups 13
4. POINTGROUPS–THESYMMETRYGROUPSOFSMALL
MOLECULES 15
4.1 Introduction 15
4.2 Axesofrotation: C 15
n
4.3 Mirrorplanes: σ 16
4.4 Stereographicprojectiondiagrams 17
4.5 Inversion: i 19
4.6 Rotatoryreflections,orimproperrotations,S 19
n
vi INTRODUCTIONTOSYMMETRYANDGROUPTHEORYFORCHEMISTS
4.7 Catalogue raisonée of the common point groups: symbols,
molecularexamples,andmacroscopicexamples 20
5. INTRODUCTIONTOLINEARALGEBRA 23
5.1 Introduction 23
5.2 Systemsofcoordinates 23
5.3 Vectors 24
5.4 Normorlengthofavector 25
5.5 Anglesandinnerproducts 25
5.6 Generalizationstondimensions 26
5.7 Orthogonalityandnormality 26
5.8 Lineartransformationsandmatrices 27
5.9 Successivetransformations;matrixmultiplication 30
5.10 Theeffectonamatrixofachangeincoordinatesystem 31
5.11 Orthogonaltransformations 33
5.12 Tracesanddeterminants 34
5.13 Matrixrepresentationofsymmetrygroups 36
6. GROUPREPRESENTATIONSANDCHARACTERTABLES 39
6.1 Introduction 39
6.2 Grouprepresentations 41
6.3 Charactertables 46
6.4 Propertiesofcharactertables 48
6.5 Calculationswithcharactertables 49
7. MOLECULARVIBRATIONS 53
7.1 Introduction 53
7.2 Classicaldescriptionofmolecularvibrations 54
7.3 Eigenvalueproblems 57
7.4 Determinationofthesymmetriesofthenormalmodes 60
7.5 Useofinternalcoordinates 63
8. ELECTRONICSTRUCTUREOF
ATOMSANDMOLECULES 65
8.1 Thequantum-mechanicalbackground 65
8.2 Symmetrypropertiesofwavefunctions 69
8.3 Molecularwavefunctions 71
8.3.1 Propertiesoftheexactwavefunctions 72
Contents vii
8.3.2 TheHartree-Fockapproximation 73
8.3.3 The Linear Combination of Atomic Orbitals (LCAO)
approximation 75
8.4 Expectationvaluesandthevariationtheorem 76
9. SYMMETRYPROPERTIESOFMOLECULARORBITALS 83
9.1 Diatomicmolecules 83
9.2 Triatomicmolecule–Walshdiagrams 87
9.3 MolecularorbitalsforthebentAH2 molecule(C2v) 87
9.4 MolecularorbitalsforthelinearAH2 molecule(D∞h) 89
9.5 Correlationoforbitalsbetweenbentandlineargeometries 90
10.SPECTROSCOPYANDSELECTIONRULES 93
10.1 Introduction 93
10.2 Therelationshipbetweensymmetrypropertiesandthevanishing
ofmatrixelements 93
10.3 Thedirect-productrepresentation 94
10.4 Selectionrulesinspectroscopy 97
10.4.1 Electronictransitions 98
10.4.2 Vibrationaltransitions 100
11.MOLECULARORBITALTHEORYOF
PLANARCONJUGATEDMOLECULES 103
11.1 Introduction 103
11.2 TheLCAO–MOdescriptionofpyridine 104
11.3 Distributionofmolecularorbitalsamongsymmetryspecies 107
11.4 TheHückelapproximation 108
11.5 Projectionoperators 110
11.6 Generalpropertiesofprojectionoperators 114
Conclusion 119
Index 121
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Preface
This book is based on a one-semester course for advanced undergraduates
specializing in physical chemistry. I am aware that the mathematical training
of most science majors is more heavily weighted towards analysis – typic-
ally calculus anddifferential equations –than towards algebra. Butitremains
my conviction that the basic ideas and applications of group theory are not
onlyvital,butnotdifficulttolearn,eventhoughaformalmathematicalsetting
withemphasisonrigorandcompletenessisnottheplacewheremostchemists
wouldfeelmostcomfortable inlearningthem.
Thepresentationhereisshort,andlimitedtothoseaspectsofsymmetryand
group theory that are directly useful in interpreting molecular structure and
spectroscopy. Nevertheless I hope that the reader will begin to sense some of
thebeautyofthesubject. Symmetryisattheheartofourunderstanding ofthe
physical lawsof nature. If areader ishappy withwhat appears in this book, I
must count this a success. But if the book motivates a reader to move deeper
intothesubject, Ishallbegratified.
Cambridge,
January2004