Symmetry (eBook)

Front Cover 1
Symmetry: An Introduction to Group Theory and its Applications 4
Copyright Page 5
Table of Contents 10
THE INTERNATIONAL ENCYCLOPEDIA OF PHYSICAL CHEMISTRY AND CHEMICAL PHYSICS 7
INTRODUCTION 8
PREFACE 14
CHAPTER 1. GROUPS 16
1.1 Symbols and the group property 16
1.2 Definition of a group 21
1.3 The multiplication table 22
1.4 Powers, products, generators 24
1.5 Subgroups, cosets, classes 26
1.6 Invariant subgroups. The factor group 28
1.7 Homomorphisms and isomorphisms 29
1.8 Elementary concept of a representation 31
1.9 The direct product 33
1.10 The algebra of a group 34
CHAPTER 2. LATTICES AND VECTOR SPACES 37
2.1 Lattices. One dimension 37
2.2 Lattices. Two and three dimensions 40
2.3 Vector spaces 42
2.4 n-Dimensional space. Basis vectors 43
2.5 Components and basis changes 46
2.6 Mappings and similarity transformation 48
2.7 Representations. Equivalence 53
2.8 Length and angle. The metric 56
2.9 Unitary transformations 62
2.10 Matrix elements as scalar products 64
CHAPTER 3. POINT AND SPACE GROUPS 69
3.1 Symmetry operations as orthogonal transformations 69
3.2 The axial point groups 74
3.3 The tetrahedral and octahedral point groups 84
3.4 Compatibility of symmetry operations 90
3.5 Symmetry of crystal lattices 93
3.6 Derivation of space groups 100
CHAPTER 4. REPRESENTATIONS OF POINT AND TRANSLATION GROUPS 106
4.1 Matrices for point group operations 106
4.2 Nomenclature. Representations 110
4.3 Translation groups. Representations and reciprocal space 120
CHAPTER 5. IRREDUCIBLE REPRESENTATIONS 124
5.1 Reducibility. Nature of the problem 124
5.2 Reduction and complete reduction. Basic theorems 125
5.3 The orthogonality relation 131
5.4 Group characters 136
5.5 The regular representation 139
5.6 The number of distinct irreducible representations 140
5.7 Reduction of representations 141
5.8 Idempotents and projection operators 146
5.9 The direct product 148
CHAPTER 6. APPLICATIONS INVOLVING ALGEBRAIC FORMS 155
6.1 Nature of applications 155
6.2 Invariant forms. Symmetry restrictions 156
6.3 Principal axes. The eigenvalue problem 162
6.4 Symmetry considerations 165
6.5 Symmetry classification of molecular vibrations 166
6.6 Symmetry coordinates in vibration theory 174
CHAPTER 7. APPLICATIONS INVOLVING FUNCTIONS AND OPERATORS 181
7.1 Transformation of functions 181
7.2 Functions of Cartesian coordinates 185
7.3 Operator equations. Invariance 189
7.4 Symmetry and the eigenvalue problem 196
7.5 Approximation methods. Symmetry functions 202
7.6 Symmetry functions by projection 205
7.7 Symmetry functions and equivalent functions 210
7.8 Determination of equivalent functions 212
CHAPTER 8. APPLICATIONS INVOLVING TENSORS AND TENSOR OPERATORS 218
8.1 Scalar, vector and tensor properties 218
8.2 Significance of the metric 221
8.3 Tensor properties. Symmetry restrictions 223
8.4 Symmetric and antisymmetric tensors 226
8.5 Tensor fields. Tensor operators 233
8.6 Matrix elements of tensor operators 239
8.7 Detennination of coupling coefficients 246
APPENDIX 1: Representations carried by harmonic functions 250
APPENDIX 2: Alternative bases for cubic groups 256
INDEX 260