Introduction to Quantum Mechanics in Chemistry

This book serves as a self-study guide to familiarize users with the crucial language of modern chemistry science. It provides a background of electronic structure programs, and includes worked examples in problem solving and computer exercises. For computational chemists, materials scientists, and chemical engineers who want to learn more about their field without unnecessary complexity, detail, or formalism.

Preface

ix

Introduction and Background to Quantum Mechanics

1

(22)

Aim of Theoretical Chemistry

1

(1)

Key Concepts from Classical Physics

2

(1)

Classical Mechanics

3

(5)

Classical Wave Theory

8

(4)

Early History of Quantum Mechanics

12

(1)

Particle Nature of Light

12

(2)

Wave Nature of Particles

14

(1)

Uncertainty Principle

15

(1)

Discovery of Quantum Mechanics

16

(1)

Concepts in Quantum Mechanics

17

(6)

Quantum Theory

23

(15)

Postulates of Quantum Mechanics

23

(1)

Definition of Ψ and |Ψ|2

23

(2)

Operators

25

(3)

Time-dependent and Time-independent Schrodinger Equations

28

(2)

Eigenvalues

30

(1)

Expectation Values

31

(1)

Properties of the Time-Independent Schrodinger Eigenfunctions

32

(6)

Particle-in-Box Models

38

(14)

Particle in a One-Dimensional Box

38

(6)

Particle in a Two-Dimensional Box

44

(3)

Particle in a Three-Dimensional Box

47

(1)

Free-Electron Molecular Orbital Model

48

(4)

Rigid-Rotor Models and Angular-Momentum Eigenstates

52

(19)

Motions of a Diatomic Molecule: Separation of the Center of Mass

52

(1)

Rigid-Rotor Model in Two Dimensions

53

(3)

Three-Dimensional Rigid Rotor

56

(4)

Spherical harmonics

60

(1)

Rotational Spectra

61

(2)

Angular Momentum

63

(2)

Dirac Notation

65

(1)

Raising and Lowering Angular-Momentum Operators

66

(5)

Molecular Vibrations and Time-Independent Perturbation Theory

71

(17)

Diatomic Molecule Vibrations

71

(2)

Raising and Lowering Operators for the Harmonic Oscillator

73

(5)

Polyatomic Molecule Vibrations

78

(4)

Time-Independent Perturbation Theory

82

(2)

Examples

84

(4)

The Hydrogen Atom

88

(15)

The Schrodinger Equation

88

(2)

Radial Solutions and Eigenvalues

90

(1)

Energy Eigenvalues; Spectroscopy of the H Atom

91

(8)

Properties of Hydrogen and Hydrogenlike Wavefunctions

Atomic Units

99

(4)

The Helium Atom

103

(7)

Schrodinger Equation

103

(1)

Independent-Particle Model

104

(1)

The Variational Method

105

(3)

Better Wavefunctions

108

(2)

Electron Spin and the Pauli Principle

110

(17)

Electron Spin

110

(2)

The Pauli Principle

112

(1)

He-Atom Wavefunctions, Including Spin

113

(3)

Excited State of He

116

(3)

Energies of He(1s2s) States

119

(2)

Interaction of Electron Spin with Magnetic Fields

121

(2)

EPR and NMR

123

(4)

Many-Electron Atoms

127

(18)

Many-Electron Hamiltonian and Schrodinger Equation

127

(1)

Slater Determinants

128

(2)

Hartree Method

130

(3)

Hartree-Fock Method

133

(2)

Koopmans' Theorem

135

(1)

Electron Correlation

135

(1)

Constants of the Motion

136

(1)

Angular-Momentum Operators for Many-Electron Atoms

137

(2)

Relativistic Effects

139

(6)

Homonuclear Diatomic Molecules

145

(17)

Hydrogen Molecular Ion: Born-Oppenheimer Approximation

145

(3)

LCAO-MO Treatment of H2+

148

(3)

Other H2+ States

151

(2)

Electronic Structure of Homonuclear Diatomics

153

(2)

Electronic Structure of H2: Molecular Orbital and Valence Bond Wavefunctions

155

(3)

Improvements to MO and VB Results for H2

158

(4)

Ab Initio and Density Functional Methods

162

(17)

LCAO-MO-SCF Theory for Molecules

162

(3)

Atomic Orbitals

165

(3)

Hartree-Fock Calculations

168

(3)

Beyond Hartree-Fock

171

(2)

Density Functional Theory Methods

173

(6)

Semiempirical Methods

179

(15)

Huckel Model

180

(8)

Extended Huckel Method

188

(1)

PPP Method

189

(1)

NDO Methods

189

(5)

Applications of Group Theory

194

(24)

Group Theory for Point Groups

194

(8)

Applications of Group Theory to Molecular Quantum Mechanics

202

(7)

Symmetry Properties of Many-Electron Wavefunctions

209

(3)

Symmetry Properties of Molecular Vibrations

212

(6)

Applications of Electronic Structure Theory

218

(17)

Potential-Energy Functions

219

(3)

Optimized Geometries and Frequencies

222

(2)

IR Spectra

224

(1)

Barriers to Reaction

225

(2)

Excited States

227

(2)

Molecular Clusters

229

(2)

Remarks on Other Methods

231

(4)

Time Dependence and Spectroscopy

235

(16)

Transition Probabilities and the Golden Rule

235

(3)

Electronic Spectroscopy of Molecules

238

(7)

Vibrational (Infrared) Spectroscopy

245

(6)

Appendices

251

(18)

Appendix A Mathematical Background

252

(4)

A.1 Complex numbers

252

(1)

A.2 A Tiny Bit of Differential Equations

253

(1)

A.3 Matrices, Determinants, and Eigenvalues

254

(2)

Appendix B Two-Electron Repulsion Integral

256

(2)

Appendix C Character Tables

258

(8)

Appendix D Atomic Units, Energy Conversion Factors, and Physical Constants

266

(3)

Solutions to Odd-Numbered Problems

269

(28)

Index

297

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Introduction to Quantum Mechanics in Chemistry

Summary

Table of Contents

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