Quantum Mechanics in Chemistry

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ISBN: 9780486420035 | 0486420035
Cover: Paperback
Copyright: 1/28/2002

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Advanced graduate-level text looks at symmetry, rotations, and angular momentum addition; introduces basic formalism of time-dependent quantum mechanics and occupation number representations; focuses on scattering theory; uses its concepts to develop basic theories of chemical reaction rates, and more. Problems appear at end of each chapter; solutions included.

Preface

xvii

Review of Basic Concepts in Quantum Mechanics

(20)

Fundamental Definitions

(2)

Eigenvalues and Eigenfunctions

(2)

Approximate Methods

(2)

Time-Independent Perturbation Theory

(1)

Variational Theory

(1)

Raising and Lowering Operators

(4)

Harmonic Oscillator

(2)

Angular Momentum Operators

(2)

Two-Body Problems

(2)

Relative-Motion Schrodinger Equation

(1)

Hydrogen Atom

(1)

Electronic Structure of Atoms and Molecules

(8)

Many-Electron Hamiltonian; Born-Oppenheimer Approximation

(2)

Pauli Principle; Hartree-Fock Theory

(2)

LCAO-MO-SCF

(1)

Electronic Structure Methods

(1)

Bibliography for Chapter 1

(1)

Problems for Chapter 1

(2)

Symmetry Considerations: Point Groups and Electronic Structure

(22)

Group Theory for Point Groups

(6)

Symmetry Operations

(1)

Representations

(1)

Similarity Transformations

(1)

Irreducible Representations

(1)

Character Tables

(1)

Direct (or Tensor) Products

(1)

Clebsch-Gordan Series

(1)

Applications of Group Theory to Quantum Mechanics

(9)

Symmetry-Adapted Linear Combinations

(1)

Constructions of SALCs

(2)

Huckel Theory Applications

(3)

Shortcuts

(1)

Energy Eigenvalues

(2)

Symmetry Properties of Many-Electron Wavefunctions

(7)

Many-Electron Configurations and Terms

(1)

Application to C3H+3

(1)

Incorporation of the Pauli Principle

(2)

Optical Spectrum: Dipole Selection Rules

(1)

Bibliography for Chapter 2

(1)

Problems for Chapter 2

(3)

Symmetry Considerations: Continuous Groups and Rotations

(14)

Introduction

(1)

Continuous Groups; The Electronic Structure of Linear Molecules

(7)

Two-Dimensional Rotation Group

(2)

Cxv Group

(1)

Dxh Group

(1)

Simple Example: O2

(4)

Three-Dimensional Rotation Group; Angular Momentum Addition

(7)

Angular Momentum Addition; Clebsch-Gordan Coefficients

(1)

Properties of Clebsch-Gordan Coefficients

(2)

Worked Examples

(1)

3-j and Higher Symbols

(1)

Bibliography for Chapter 3

(1)

Problems for Chapter 3

(2)

Time-Dependent Quantum Mechanics

(21)

Introduction

(1)

Time-Dependent Schrodinger Equation: Basis-Set Solution

(2)

Time-Dependent Perturbation Theory

(6)

First-Order Time-Dependent Perturbation Theory

(1)

Example: Collision-Induced Excitation of a Diatomic Molecule

(4)

Second-Order Perturbation Theory

(1)

Simplifications and Extensions to Higher Order

(1)

Time-Ordering Operators

(1)

Representations in Quantum Mechanics

(5)

Schrodinger Representation

(1)

Heisenberg Representation

(2)

Interaction Representation

(1)

Transition Probabilities per Unit Time

(8)

Perturbation Theory for a Constant Interaction Potential

(2)

Fermi's Golden Rule

(1)

State-to-State Form of Fermi's Golden Rule

(2)

Treatment of Periodic Interactions

Bibliography for Chapter 4

(1)

Problems for Chapter 4

(3)

Interaction of Radiation with Matter

(33)

Introduction

(1)

Electromagnetic Fields

(4)

Vector Potentials and Wave Equations

(2)

Plane Waves

(1)

Energy and Photon Number Density

(1)

Interaction between Matter and Field

(4)

Classical Theory

(1)

Derivation of Classical Hamiltonian

(1)

Quantum Hamiltonian for a Particle in an Electromagnetic Field

(3)

Absorption and Emission of Light

(13)

Application of Fermi's Golden Rule

(1)

Dipole Approximation

(1)

Photon Density of States

(1)

Emission Rate

(2)

Absorption Rate

(1)

Einstein A and B Coefficients

(2)

Oscillator Strengths

(1)

Electric Quadrupole, Magnetic Dipole Mechanisms

(3)

Molecular Transitions: Frank-Condon Factors

(4)

Light Scattering

(12)

Qualitative Description of Light Scattering

(1)

Simplification of Electric Dipole Interaction

(1)

Interaction between Field and Induced Dipole; Two-Photon Process

100

(2)

Raman Scattering

102

(2)

Evaluation of αkm: Kramers-Heisenberg Formula

104

(4)

Bibliography for Chapter 5

108

(1)

Problems for Chapter 5

108

(3)

Occupation Number Representations

111

(42)

Introduction

111

(1)

Occupation Number Representation for Harmonic Molecular Vibrations and Quantized Radiation Fields

112

(7)

Single Harmonic Oscillator

112

(1)

Normal Modes

113

(2)

Quantized Radiation Fields

115

(2)

Coupling of Radiation to Matter Using Second Quantization

117

(1)

Applications of Fermi's Golden Rule

118

(1)

Occupation Number Representations for Electrons

119

(13)

Fermion Creation and Destruction Operators

119

(2)

Slater Determinants and Electron Creation Operators

121

(4)

Manipulation of Fermion Operators; Commutators and Anticommutators

125

(1)

Arbitrary Electronic Operators in the Occupation Number Representation

126

(6)

Fermion Field Operators and Second Quantization

132

(3)

Molecular Electronic Structure: Model Hamiltonians and Occupation Number Representations

135

(7)

Model Hamiltonians: Basis Set and Matrix Elements

135

(2)

Noninteracting Electrons: Huckel, Extended Huckel, and Free-Electron Models

137

(2)

Molecular Orbitals for Noninteracting Electrons

139

(3)

Treatment of Interacting Electrons

142

(11)

Model Hamiltonians

142

(3)

Self-Consistent Field (SCF) Solution

145

(2)

Example: SCF Solution for the Two-Center, Two-Orbital Problem

147

(1)

Bibliography for Chapter 6

148

(1)

Problems for Chapter 6

149

(4)

Quantum Scattering Theory

153

(29)

Introduction

153

(1)

One-Dimensional Scattering

154

(13)

Introduction

154

(1)

Wavepackets in One Dimension

155

(2)

Wavepackets for the Complete Scattering Problem

157

(1)

Fluxes and Probabilities

158

(1)

Time-Independent Approach to Scattering

159

(2)

Scattering Matrix

161

(2)

Green's Functions for Scattering

163

(3)

Born Approximation

166

(1)

Semiclassical Theory

167

(8)

WKB Approximation

167

(6)

Semiclassical Wavepackets

173

(2)

Scattering in Three Dimensions

175

(7)

Bibliography for Chapter 7

178

(1)

Problems for Chapter 7

179

(3)

Theories of Reaction Rates

182

(17)

Introduction

182

(1)

Rate Constants for Bimolecular Reactions: Cumulative Reaction Probabilities

182

(2)

Transition-State Theory

184

(1)

RRKM Theory

185

(1)

Formal Expression for Rates in Terms of Flux Operators

186

(3)

Additional Expressions for Rate Constant

189

(3)

Flux-Flux Autocorrelation Functions

192

(2)

Evaluation of Propagator Matrix Elements Using Path Integrals

194

(5)

Bibliography for Chapter 8

197

(1)

Problems for Chapter 8

197

(2)

Time-Dependent Approach to Spectroscopy: Electronic, Vibrational, and Rotational Spectra

199

(32)

Introduction

199

(1)

Thermal Averages and Imaginary Time Propagation

200

(1)

Electronic Spectra from Time Correlation Functions

201

(1)

Electronic Spectra: Time Development of the Correlation Functions

202

(7)

Optical Spectra with Narrow-Line or Wide-Line Excitations: Time-Dependent Picture

209

(5)

Vibrational Spectra: Correlation Function Approach

214

(6)

Rotational, Raman, and Magnetic Resonance Spectra

220

(5)

Motional Narrowing and Stochastic Motion

225

(6)

Bibliography for Chapter 9

227

(1)

Problems for Chapter 9

227

(4)

Correlation Functions and Dynamical Processes: Nonadiabatic Intramolecular Electron Transfer

231

(31)

Introduction

231

(1)

Electron Transfer: Some Generalities

232

(5)

Molecular Crystal Model

237

(3)

Rate Processes with Vibronic Coupling: Canonical Transformations and Franck-Condon Factors

240

(7)

Steepest-Descents Evaluation of Franck-Condon Behavior: Energy Sharing

247

(4)

Solvent Contributions to Electron Transfer Rates

251

(2)

Electron Transfer Reactions: Qualitative Remarks

253

(9)

Bibliography for Chapter 10

256

(1)

Problems for Chapter 10

256

(6)

Density Matrices

262

(53)

Introduction

262

(1)

Density Operators and Density Matrices: Definitions and Averages

263

(3)

Wavefunctions and the Pure State Density Operator

263

(1)

Density Operators for Mixed States

264

(2)

Representations and Equations of Motion

266

(1)

Example: Spin-1/2 Particles

267

(5)

Representation of Density Matrix Using Pauli Spin Matrices

269

(1)

Crude Description of Dephasing in Two-Site Systems

269

(3)

Reduced Density Matrices

272

(5)

Reduced Density Matrices and Electronic Structure

273

(2)

Hartree-Fock Density Matrices; Natural Orbitals

275

(2)

Reduced Density Matrices for Dynamical Statistical Systems

277

(21)

Reduced Density Matrices for Subsystems

278

(3)

Zero-Order Density Matrix: Equilibrium

281

(1)

First-Order Expressions for the Density Matrix: Linear Response Theory

281

(3)

Second-Order Response and the Density Matrix: Redfield Equations and Relaxation Processes

284

(5)

Example: Relaxation and Dephasing in a Simple Spin System

289

(6)

Second-Order Corrections to the Density: Application to Molecular Nonlinear Optics

295

(3)

Higher-Order Corrections to the Density Matrix: Pulsed Spectroscopy

298

(17)

Spins: Rotations and Angular Momentum

299

(1)

Rotating Frame Transformation

300

(1)

Simple Pulse Experiment: Carr-Purcell Spin Echo for Uncoupled Spins

301

(3)

Average Hamiltonian Theory for Multiple-Pulse NMR

304

(6)

Bibliography for Chapter 11

310

(1)

Problems for Chapter 11

311

(4)

Appendix A: Dirac Delta Function

315

(2)

Appendix B: Laplace Transforms

317

(2)

Appendix C: Solutions to Problems

319

(36)

Index

355

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Quantum Mechanics in Chemistry

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