Molecular Symmetry

1. Symmetry Elements and Operations.

1.1 Introduction.

1.2 Symmetry Elements and Operations.

1.3 Examples of the Impact of Geometric Symmetry on Chemistry.

1.4 Summary.

1.5 Self-Test Questions.

Further Reading.

2. More Symmetry Operations and Products of Operations.

2.1 Introduction.

2.2 Background to Point Groups.

2.3 Closed Groups and New Operations.

2.4 Properties of Symmetry Operations.

2.5 Chirality and Symmetry.

2.6 Summary.

2.7 Completed Multiplication Tables.

2.8 Self-Test Questions.

3. The Point Groups Used with Molecules.

3.1 Introduction.

3.2 Molecular Classification Using Symmetry Operations.

3.3 Constructing Reference Models with Idealized Symmetry.

3.4 The Nonaxial Groups: Cs,Ci,C1.

3.5 The Cyclic Groups: Cn, Sn.

3.6 Axial Groups Containing Mirror Planes: Cnh and Cnv.

3.6.1 Examples of Axial Groups Containing Mirror Planes: Cnh and Cnv.

3.7 Axial Groups with Multiple Rotation Axes: Dn, Dnd and Dnh.

3.8 Special Groups for Linear Molecules: Cìv and Dìh.

3.9 The Cubic Groups: Td and Oh.

3.10 Assigning Point Groups to Molecules.

3.11 Example Point Group Assignments.

3.12 Self-Test Questions.

4. Point Group Representations, Matrices and Basis Sets.

4.1 Introduction.

4.2 Symmetry Representations and Characters.

4.3 Multiplication Tables for Character Representations.

4.4 Matrices and Symmetry Operations.

4.5 Diagonal and Off-Diagonal Matrix Elements.

4.6 The Trace of a Matrix as the Character for an Operation.

4.7 Noninteger Characters.

4.8 Reducible Representations.

4.9 Classes of Operations.

4.10 Degenerate Irreducible Representations.

4.11 The Labelling of Irreducible Representations.

4.12 Summary.

4.13 Completed Tables.

4.14 Self-Test Questions.

Further Reading.

5. Reducible and Irreducible Representations.

5.1 Introduction.

5.2 Irreducible Representations and Molecular Vibrations.

5.3 Finding Reducible Representations.

5.4 Properties of Point Groups and Irreducible Representations.

5.5 The Reduction Formula.

5.6 A Complete Set of Vibrational Modes for H2O.

5.7 Choosing the Basis Set.

5.8 The d-Orbitals in Common Transition Metal Complex Geometries.

5.9 Linear Molecules: Groups of Infinite Order.

5.10 Summary.

5.11 Self-Test Questions.

6. Applications in Vibrational Spectroscopy.

6.1 Introduction.

6.2 Selection Rules.

6.3 General Approach to Analysing Vibrational Spectroscopy.

6.4 Symmetry-Adapted Linear Combinations.

6.5 Normalization.

6.6 The Projector Operator Method.

6.7 Linking Results for Symmetry-Inequivalent Sets of Atoms.

6.8 Additional Examples.

6.9 Summary.

6.10 Self-Test Questions.

Further Reading.

7. Symmetry in Chemical Bonding.

7.1 Introduction.

7.2 Bond Energies.

7.3 The Relative Energies of Hydrogen-Like Atomic Orbitals.

7.4 The Molecules Formed by Other Second-Row Elements with Hydrogen.

7.5 The Second-Row Diatomic Molecules.

7.6 More Complex Polyatomic Molecules.

7.7 Metal Complexes.

7.8 Summary.

7.9 Self-Test Questions.

Further Reading.

Appendices.

Appendix 1. H2O Models for Identifying the Results of Symmetry Operation Products.

Appendix 2. Assignment of Chiral Centre Handedness using Cahn-Ingold-Prelog Rules.

Appendix 3. Model of a Tetrahedron and the Related Cube.

Appendix 4. Model of an Octahedron.

Appendix 5. Matrices and Determinants.

Appendix 6. The Mathematical Background to Infrared Selection Rules.

Appendix 7. The Franck-Condon Principle.

Appendix 8. Classical Treatment of Stokes/Anti-Stokes Absorption.

Appendix 9. The Atomic Orbitals of Hydrogen.

Appendix 10. The Origin of Chemical Bonding in H+2.

Appendix 11. H2O Molecular Orbital Calculation in C2v Symmetry.

Appendix 12. Character Tables.

Index.