Introduction to Group Theory with Applications

Preface

Acknowledgments

Chapter 1 Symmetry Operations

1-1 Introduction

1-2 Point Symmetry Operations

1-3 The Stereographic Projection

1-4 The 32 Crystallographic Point Groups

1-5 Related Considerations

1-6 Space Group Example

Notes

Problems

Chapter 2 Group Concepts

2-1 Introduction

2-2 Definition of a Group

2-3 Symmetry Operations Form a Group

2-4 Related Group Concepts

2-5 Isomorphism and Homomorphism

2-6 Special Kinds of Groups

2-7 More Involved Group Concepts (including a Factor Group of a Space Group)

Appendix to Chapter 2

Notes

Problems

Chapter 3 Matrix Representations of Finite Groups

3-1 Introduction

3-2 Representations

3-3 Irreducible Representations

3-4 Representations of a Factor Group

Appendix to Chapter 3

Notes

Problems

Chapter 4 Characters of Matrix Representations of Finite Groups

4-1 Properties of Characters of Irreducible Representations

4-2 Character Tables

4-3 Reduction of a Reducible Representation

4-4 Basis Functions

4-5 Examples—Neumann Principle

4-6 Atomic Positions

4-7 The Hamiltonian

Appendix to Chapter 4

Notes

Problems

Chapter 5 Vibrations of Molecules and Crystals

5-1 3N Degrees of Freedom

5-2 General Considerations

5-3 Number and Type of Normal Modes for Molecules

5-4 Internal Coordinates

5-5 Crystals

5-6 Eigenvectors and Symmetry Adapted Vectors

5-7 Projection Operators

5-8 Projection Operators Applied to Normal Coordinates

Notes

Problems

Chapter 6 Normal Modes (Direct Product and Selection Rules)

6-1 Direct Product of Irreducible Representations

6-2 Vibrational Wave Function

6-3 Selection Rules—Infrared and Raman

6-4 Molecular Approximations (Site Symmetry and Davydov Splitting)

Notes

Problems

Chapter 7 Quantum Mechanics

7-1 Atomic Wave Functions

7-2 Transformation of Functions

7-3 Eigenfunctions as Basis Functions

7-4 Proper Rotations and Angular Momentum

7-5 Perturbations

7-6 Matrix Elements (Selection Rules)

7-7 General Secular Equation Problem

Appendix to Chapter 7

Notes

Problems

Chapter 8 Crystal Field Theory (and Atomic Physics)

8-1 Rotations in Terms of Euler Angles

8-2 Representations of the Full Rotation Group

8-3 Reduction of Symmetry

8-4 Energy Level Diagrams (Correlation Diagrams)

8-5 Crystal Double Groups

8-6 Correlation Diagrams including Double Groups

8-7 Other Crystal Field Effects

Appendix to Chapter 8

Notes

Problems

Chapter 9 Hybrid Functions

9-1 Introduction

9-2 Simple Hybrid Functions and Bonding

9-3 Tetrahedral Hybridization

9-4 Other Hybrid Functions

9-5 π-Hybrid Functions

9-6 Comment on Hybrid Orbitals (Slater Determinant)

Notes

Problems

Chapter 10 Molecular Orbital Theory

10-1 Hydrogen Molecular Ion

10-2 Simple MO Theory

10-3 Transition Metal Complexes

10-4 LCAO-MO of π-Electrons in Conjugated Hydrocarbons

10-5 Woodward-Hoffman Rules

Notes

Problems

Chapter 11 Symmetry of Crystal Lattices

11-1 The Real Affine Group

11-2 Space Group

11-3 Translational Lattice

11-4 International Tables for X-Ray Crystallography, International Notation, etc.

11-5 Magnetic Groups (Color Groups)

Notes

Problems

Chapter 12 Band Theory of Solids

12-1 Translational Symmetry

12-2 Symmorphic Space Groups

12-3 Nonsymmorphic Space Groups

12-4 Spin-Orbit Effects on Bands

12-5 Time Reversal Symmetry

Notes

Problems

Chapter 13 The Full Rotation Group

13-1 The Homomorphism between the Special Unitary Group in Two Dimensions, SU(2), and the Three-Dimension Rotation Group

13-2 Irreducible Representations of SU(2)

13-3 Wigner Coefficients

13-4 Irreducible Tensor Operators

13-5 The Wigner-Eckart Theorem

13-6 Survey of 3j and Racah Coefficients

Notes

Problems

Appendix 1 Crystal Systems

Appendix 2 The 32 Point Groups

Appendix 3 Character Tables

Appendix 4 Space Groups

Appendix 5 Matrices, Vector Spaces, and Linear Operators

Appendix 6 Direct Product Tables

Appendix 7 Correlation Tables

Appendix 8 Spherical Harmonics

Appendix 9 Tanable-Sugano Diagrams

Appendix 10 Double Group Character Tables

Bibliography

Index