Table of Contents

1 Mathematical Foundations 1

1.1 Matrices and Systems of Linear Equations 1

1.2 Properties of Eigenvalues and Eigenvectors 6

1.3 Variational Approximations 10

1.4 Atomic Units 15

1.5 The Electron Distribution in Molecules 17

1.6 Exchange-overlap Densities and the Chemical Bond 19

Part 1: Short-range Interactions 27

2 The Chemical Bond 29

2.1 An Elementary Molecular Orbital Model 30

2.2 Bond Energies and Pauli Repulsions in Homonuclear Diatomics 34

2.2.1 The Hydrogen Molecular Ion H₂⁺ (N=1) 35

2.2.2 The Hydrogen Molecule H₂(N=2) 35

2.2.3 The Helium Molecular Ion He₂⁺ (N=3) 35

2.2.4 The Helium Molecule He₂ (N=4) 36

2.3 Multiple Bonds 37

2.3.1 σ²π² Description of the Double Bond 38

2.3.2 B₁²B₂² Bent (or Banana) Description of the Double Bond 40

2.3.3 Hybridization Effects 42

2.3.4 Triple Bonds 46

2.4 The Three-centre Double Bond in Diborane 47

2.5 The Heteropolar Bond 49

2.6 Stereochemistry of Polyatomic Molecules 55

2.6.1 The Molecular Orbital Model of Directed Valency 55

2.6.2 Analysis of the MO Bond Energy 58

2.7 sp-Hybridization Effects in First-row Hydrides 60

2.7.1 The Methane Molecule 61

2.7.2 The Hydrogen Fluoride Molecule 64

2.7.3 The Water Molecule 75

2.7.4 The Ammonia Molecule 87

2.8 Delocalized Bonds 96

2.8.1 The Ethylene Molecule 98

2.8.2 The Allyl Radical 98

2.8.3 The Butadiene Molecule 100

2.8.4 The Cyclobutadiene Molecule 102

2.8.5 The Benzene Molecule 104

2.9 Appendices 108

2.9.1 The Second Derivative of the Hückel Energy 108

2.9.2 The Set of Three Coulson Orthogonal Hybrids 109

2.9.3 Calculation of Coefficients of Real MOs for Benzene 110

3 An Introduction to Bonding in Solids 119

3.1 The Linear Polyene Chain 120

3.1.1 Butadiene N = 4 122

3.2 The Closed Polyene Chain 123

3.2.1 Benzene N = 6 126

3.3 A Model for the One-dimensional Crystal 131

3.4 Electronic Bands in Crystals 133

3.5 Insulators, Conductors, Semiconductors and Superconductors 138

3.6 Appendix: The Trigonometric Identity 143

Part 2: Long-Range Interactions 145

4 The van der Waals Bond 147

4.1 Introduction 147

4.2 Elements of Rayleigh–Schrödinger (RS) Perturbation Theory 149

4.3 Molecular Interactions 151

4.3.1 Non-expanded Energy Corrections up to Second Order 152

4.3.2 Expanded Energy Corrections up to Second Order 153

4.4 The Two-state Model of Long-range Interactions 157

4.5 The van der Waals Interactions 159

4.5.1 Atom–Atom Dispersion 161

4.5.2 Atom–Linear Molecule Dispersion 162

4.5.3 Atom–Linear Dipolar Molecule¹⁰ Induction 163

4.6 The C₆ Dispersion Coefficient for the H–H Interaction 165

4.7 The van der Waals Bond 167

4.8 The Keesom Interaction 169

5 The Hydrogen Bond 177

5.1 A Molecular Orbital Model of the Hydrogen Bond 178

5.2 Electrostatic Interactions and the Hydrogen Bond 179

5.2.1 The Hydrogen Fluoride Dimer (HF)₂ 182

5.2.2 The Water Dimer (H₂O)₂ 185

5.3 The Electrostatic Model of the Hydrogen Bond 186

5.4 The Rg–HF Heterodimers 197

References 201

Author Index 209

Subject Index 213