Introductory Group Theory

1 Symmetry in General.- 1-1. Introduction.- 1-2. Definition of Symmetry.- 1-3. Symmetry in Our Culture.- 1-4. Symmetry in Nature.- 1-5. Symmetry in Science.- 1-6. References.- 2 Symmetry in Chemistry.- 2-1. Introduction.- 2-2. Symmetry Elements.- 2-3. Point Groups.- 2-4. Rules for Classification of Molecules into Point Groups.- 2-5. Problems.- 2-6. References.- 2-7. Bibliography.- 3 Elementary Mathematics.- 3-1. Definition of a Group.- 3-2. Finite and Infinite Groups.- 3-3. Subgroups.- 3-4. Classes.- 3-5. Definition of a Matrix.- 3-6. Multiplication of Matrices.- 3-7. Transpose of a Matrix.- 3-8. Representation of Groups.- 3-9. Problems.- 3-10. References.- 4 The Character Table.- 4-1. Introduction.- 4-2. Types of Representations.- 4-3. Character Tables for the More Common Point Groups.- 4-4. Problem.- 5 Derivation of Selection Rules.- 5-1. Derivation of Selection Rules for Nonlinear Molecules of the Most Common Point Groups.- 5-2. The Td Point Group.- 5-3. The D4h Point Group.- 5-4. The Oh Point Group.- 5-5. The D3h Point Group.- 5-6. The C3v Point Group.- 5-7. The C2v Point Group.- 5-8. Derivation of Selection Rules for Linear Molecules.- 5-9. Problem.- 5-10. References.- 6 Potential Force Fields.- 6-1. The General Quadratic Potential Function.- 6-2. The Central Force Field.- 6-3. The Valence Force Field.- 6-4. The Generalized Valence Force Field.- 6-5. The Urey-Bradley Function.- 6-6. References.- 7 The Normal Coordinate Treatment for Molecules with C2v, C3v, Td, and Oh Symmetry.- 7-1. Procedure Necessary in the NCT Method.- 7-2. Normal Coordinate Treatment of H2O (C2v Symmetry).- 7-3. Normal Coordinate Treatment of NH3(C3v Symmetry).- 7-4. Normal Coordinate Treatment of CH4(Td Symmetry).- 7-5. Normal Coordinate Treatment of UF6(OhSymmetry).- 7-6. Some Results of NCT of Molecules.- 7-7. The Product Rule.- 7-8. The Sum Rule.- 7-9. Summary.- 7-10. Problems.- 7-11. References.- 8 Applications of Group Theory.- 8-1. Introduction.- 8-2. Procedure Used in Determing the Structure of a Molecule.- 8-3. Examples Illustrating the Use of Group Theory in Determining Molecular Structure.- 8-4. Practice Problem.- 8-5. Selection of Rules for A3 to A8 Molecules.- 8-6. Forbidden Vibrations for Several Point Groups.- 8-7. Problems.- 8-8. References.- Appendices.- 1. Conversion Table for Wavelength and Wave Number Units.- 3. Diagrams of Normal Vibrations for Common Point Groups.- 4. Calculation of Thermodynamic Functions for Vibrational-Rotational Spectra.- 6. General Method of Obtaining Molecular Symmetry Coordinates.- 7. Conversion Chart for Various Forms of Energy and Velocities.