Molecular symmetry

Preface

1. Symmetry Elements and Operations

1.1. Introduction

1.2. Symmetry Elements and Operations

1.2.1. Proper Rotations: Cn

1.2.2. The Plane of Symmetry: σ

1.2.3. The Inversion Centre: i

1.3. Examples of the Impact of Geometric Symmetry on Chemistry

1.3.1. Oxygen Transfer via Metal Porphyrins

1.3.2. Nuclear Magnetic Resonance: Chemical Equivalence

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

2.3.1. Products of Operations

2.3.2. Fixed Symmetry Elements

2.3.3. The Final Missing Operation, Improper Rotations: Sn

2.3.4. Equivalences for Improper Rotation Operations

2.4. Properties of Symmetry Operations

2.4.1. Equivalent Operations and Equivalent Atoms

2.4.2. The Inverse of an Operation

2.4.3. The Order of the Product; Operations that Commute

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

3.4.1. Examples of Molecules for the Nonaxial Groups: Cs, Ci, C1

3.5. The Cyclic Groups: Cn, Sn

3.5.1. Examples of Molecules for the Cyclic Groups: Cn, Sn

3.6. Axial Groups Containing Mirror Planes: Cnh and Cnv

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

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

3.7.1. Examples of Axial Groups with Multiple Rotation Axes: Dn, Dnd and Dnh

3.8. Special Groups for Linear Molecules: C&infty;v and D&infty;h

3.9. The Cubic Groups: Td and Oh

3.10. Assigning Point Groups to Molecules

3.11. Example Point Group Assignments

3.11.1. Example 1: Conformations of Cyclohexane

3.11.2. Example 2: Six-Coordinate Metal Complexes

3.12. Self-Test Questions

4. Point Group Representations, Matrices and Basis Sets

4.1. Introduction

4.2. Symmetry Representations and Characters

4.2.1. Water, H2O, C2v

4.2.2. Direct Products

4.3. Multiplication Tables for Character Representations

4.4. Matrices and Symmetry Operations

4.5. Diagonal and Off-Diagonal Matrix Elements

4.5.1. Ammonia, NH3, C3v

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

4.7. Noninteger Characters

4.7.1. Boron Trifluoride, BF3, D3h

4.8. Reducible Representations

4.8.1. Water, H2O, C2v

4.9. Classes of Operations

4.9.1. [Ni(CN)4]2-, D4h

4.10. Degenerate Irreducible Representations

4.10.1. Ammonia, NH3, C3v

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

5.5.1. Applying the Reduction Formula

5.6. A Complete Set of Vibrational Modes for H2O

5.7. Choosing the Basis Set

5.7.1. Carbonyl Stretching Modes of [Fe(CO)5], D3h

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

5.8.1. Square Planar, D4h

5.8.2. Tetrahedral, Td

5.8.3. Octahedral, Oh

5.8.4. Trigonal Bipyramidal, D3h

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

6.2.1. Infrared Spectroscopy

6.2.2. Infrared Absorption and the Greenhouse Gases

6.2.3. Interstellar H2

6.2.4. Raman Spectroscopy

6.2.5. Comparison of Infrared and Raman Selection Rules

6.3. General Approach to Analysing Vibrational Spectroscopy

6.3.1. Example: the C-H Stretch Bands of 1,4-Difluorobenzene

6.4. Symmetry-Adapted Linear Combinations

6.5. Normalization

6.6. The Projection Operator Method

6.6.1. Projection Operator Applied to the C-H Stretches of 1,4-Difluorobenzene

6.6.2. The Projection Operator and Degenerate Representations

6.7. Linking Results for Symmetry-Inequivalent Sets of Atoms

6.7.1. Sets of Atoms Differing in Mass or Chemical Bond Strength

6.8. Additional Examples

6.8.1. Benzene, D6h

6.8.2. The fac and mer Isomers of Transition Metal Complexes

6.9. Summary

6.10. Self-Test Questions

Further Reading

7. Symmetry in Chemical Bonding

7.1. Introduction

7.1.1. Wave Phenomena and Interference

7.1.2. The Born Interpretation of the Wavefunction

7.2. Bond Energies

7.2.1. The Symmetry-Adapted Linear Combinations for the Molecular Orbitals of H2+ and H2

7.2.2. The Chemical Bond Energy from Molecular Orbitals

7.2.3. The Molecular Orbital Energy

7.2.4. Bond Order

7.3. The Relative Energies of Hydrogen-Like Atomic Orbitals

7.3.1. Radial Behaviour of Atomic Orbitals

7.3.2. The Relative Energies of Atomic Orbitals in Different Elements

7.3.3. The Relative Energies of Atomic Orbitals from Electronegativity

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

7.4.1. BeH2, Beryllium Hydride

7.4.2. BH3, Boron Hydride

7.4.3. CH4, Methane

7.4.4. NH3, Ammonia

7.4.5. H2O, Water

7.5. The Second-Row Diatomic Molecules

7.5.1. Homonuclear Diatomics

7.5.2. Heteronuclear Diatomics of Second-Row Elements

7.6. More Complex Polyatomic Molecules

7.6.1. Ethene

7.7. Metal Complexes

7.7.1. Complexes Containing σ-Donor Ligands

7.7.2. The JahnTeller Effect

7.7.3. Complexes Containing Ligand Orbitals of π-Symmetry

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 CahnIngoldPrelog Rules

Appendix 3. Model of a Tetrahedron and the Related Cube

Appendix 4. Model of an Octahedron

Appendix 5. Matrices and Determinants

A5.1. Matrices as Representations of Symmetry Operators

A5.1.1. Products of Matrices

A5.1.2. Products of Matrices, Expressed as Summations

A5.2. Matrices for Solving Sets of Linear Equations

Further Reading

Appendix 6. The Mathematical Background to Infrared Selection Rules

A6.1. Model Based on Classical Mechanics

A6.2. Model Based on Quantum Mechanics

A6.3. Excited Vibrational States

A6.4. Vibrational Modes for Polyatomic Molecules

A6.5. Generalization to Arbitrary Transitions

A6.6. Summary of Selection Rules

Further Reading

Appendix 7. The FranckCondon Principle

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

Appendix 9. The Atomic Orbitals of Hydrogen

A9.1. Choice of Coordinate System

A9.2. Separation of Variables

A9.3. The Angular Equation

A9.4. Physical Interpretation of the Angular Equation Solutions

A9.5. Angular Momentum

A9.6. The Radial Equation

A9.7. The Complete Atomic Orbitals

A9.8. Expectation Values

A9.9. Real Combinations to Form the Familiar Atomic Orbitals

A9.10. Cartesian Forms of the Real Angular Functions

A9.11. Endnote on Imaginary Numbers

Further Reading

Appendix 10. The Origin of Chemical Bonding in H2+

A10.1. Chemical Bond Formation

A10.2. H Atom and H+ Cation

A10.3. The Virial Theorem

A10.4. H2+ Molecule

A10.5. Choice of Coordinate System for H2+: Cylindrical Polar Coordinates

A10.6. H2+: the Electron Kinetic Energy

A10.7. H2+: the Electronic Potential Energy

A10.8. The Chemical Bond Formation Energy Based on Rigid Atomic Orbitals

A10.9. Optimal Radial Decay of Molecular Orbitals

Further Reading

Appendix 11. H2O Molecular Orbital Calculation in C2v Symmetry

Further Reading

Appendix 12. Character Tables

A12.1. Non-Axial Groups

A12.2. Axial Groups

A12.2.1. Cn Groups

A12.2.2. Sn Groups

A12.2.3. Cnv Groups

A12.2.4. Cnh Groups

A12.2.5. Dn Groups

A12.2.6. Dnd Groups

A12.2.7. Dnh Groups

A12.3. Cubic Groups

A12.3.1. Tetrahedral, Td

A12.3.2. Rotational Subgroup of Td, T

A12.3.3. Octahedral, Oh

A12.3.4. Rotational Subgroup of Oh, O

A12.4. Groups for Linear Molecules

Index