Advanced Molecular Dynamics and Chemical

A comprehensive, in-depth presentation of theoretical underpinnings and mathematical techniques This is the first book of its kind to combine all the theories of molecular reaction dynamics and chemical kinetics in a single source. It provides a sophisticated treatment of the material that functions both as a professional reference and a high-level text for PhD and postdoctoral researchers. Advanced Molecular Dynamics and Chemical Kinetics offers exceptional, in-depth coverage and includes a complete discussion of the theoretical as well as mathematical presentation of techniques. It features relevant exercises as well as comprehensive coverage of: * Second Quantization * Semiclassical Theory * Quantum Theory of Reaction Rates * Feynman Path Integrals * Wavepacket Propagation and Grid Methods * Photodissociation * Molecular Properties of Solvated Molecules * Quantum Model for Electron Transfer * Electron Transfer Coupling Elements * Proton Transfer Reactions in Solution This is the ideal reference for seasoned professionals in molecular reaction dynamics as well as for younger researchers who may want to enter the field or simply wish to learn more about it. Also available: Introduction to Molecular Dynamics and Chemical Kinetics Gert D. Billing and Kurt V. Mikkelsen

GERT D. BILLING is Professor of Theoretical Chemistry at the University of Copenhagen. He serves as referee and sits on the editorial boards of numerous professional journals and has been a visiting professor at many universities and research institutions worldwide. Professor Billing is the author or coauthor of over 250 published papers and 150 conference presentations as well as Wiley's Introduction to Molecular Dynamics and Chemical Kinetics.<br> <br> . MIKKELSEN is Associate Professor in the Department of Chemistry, University of Copenhagen, Denmark. He has been an invited lecturer at dozens of universities and conferences in Europe and the United States and has published 80 papers in international journals as well as coauthoring Introduction to Molecular Dynamics and Chemical Kinetics.

Preface

xi

1 Introduction

1

(4)

2 Second Quantization

5

(30)

2.1 Applications of Second Quantization

14

(21)

2.1.1 The Linearly Forced Harmonic Oscillator

14

(6)

2.1.2 Coherent States

20

(3)

2.1.3 Energy Transfer in Polyatomic Molecules

23

(5)

2.1.4 Energy Transfer to Solids

28

(7)

3 Effective Hamiltonian Approaches

35

(12)

3.1 Gaussian Wavepackets

36

(3)

3.2 The Classical Path Equations

39

(2)

3.3 Corrections to the SCF Approach

41

(2)

3.4 Wave Operator Approach

43

(4)

4 Semiclassical Theories

47

(16)

4.1 Feynman Path Integrals

47

(6)

4.1.1 The Short-Time Propagator

51

(1)

4.1.2 The Classical Limit

52

(1)

4.2 Semiclassical Feynman Path Theory

53

(2)

4.3 Classical S-Matrix Theory

55

(6)

4.4 Resolution in Eigenstates and Statistical Mechanics

61

(2)

5 Wavepacket Propagation

63

(18)

5.1 Fourier Transform Method

63

(1)

5.2 DVR Method

64

(3)

5.3 Time Propagation

67

(5)

5.4 Bohm Trajectories

72

(1)

5.5 Tunneling of Wavepackets

73

(5)

5.6 Absorbing Boundaries

78

(3)

6 Potential Energy Surfaces

81

(6)

6.1 The Born-Oppenheimer Separation

82

(2)

6.2 Surface Characteristics

84

(3)

7 The Reaction Path Method

87

(8)

7.1 Reaction Path Constraints

89

(6)

8 Variational Transition State Theory

95

(12)

8.1 Tunneling

101

(3)

8.2 The Transmission Factor

104

(3)

9 Quantum Theory for Rate Constants

107

(9)

9.1 The Parabolic Barrier Problem

112

(1)

9.2 Recombination Reactions

113

(3)

10 Statistical and Phase Space Methods

116

(11)

10.1 Prior Distributions and Surprisals

119

(4)

10.2 Stochastic Methods

123

(4)

10.2.1 The Forced Oscillator

124

(3)

11 Photodissociation

127

(14)

11.1 Strong-Field Limit

131

(5)

11.1.1 Asymptotic Projection

134

(2)

11.2 The PRGM Method

136

(5)

12 Density Operators

141

(7)

12.1 Properties of Density Operators

142

(1)

12.2 Quantum Statistical Mechanics

143

(3)

12.3 Subsystems Within a System

146

(2)

13 Evolution of a Total System

148

(5)

14 Nonequilibrium Solvation

153

(17)

14.1 The Multiconfigurational Self-Consistent Reaction Field Method

156

(12)

14.1.1 The Homogeneous Dielectric Continuum Semiclassical Solvent Model

156

(6)

14.1.2 Implementation

162

(2)

14.1.3 Optimization Procedure

164

(4)

14.2 Numerical Examples

168

(2)

15 Molecular Properties of Solvated Molecules

170

(24)

15.1 Definitions

171

(4)

15.2 Screening of the Externally Applied Field. Local Field Corrections

175

(3)

15.3 Response Functions in the MCSCF Approximation

178

(1)

15.4 A Reaction Field Response Method

179

(12)

15.4.1 Ehrenfest's Equation of Motion

185

(2)

15.4.2 Contribution of Nonequilibrium Solvation Terms to Linear Response Equations

187

(4)

15.5 Numerical Examples

191

(3)

16 Magnetic Properties of Solvated Molecules

194

(9)

16.1 Introduction

194

(1)

16.2 Method

195

(6)

16.3 Numerical Examples

201

(2)

17 Quantum Model for Electron Transfer

203

(20)

17.1 Introduction

203

(5)

17.1.1 Rate Constant for Electron Transfer in the Quantum Mechanical Regime

207

(1)

17.2 The Model Hamiltonian for the System

208

(8)

17.2.1 The Statistical Density Operator

209

(1)

17.2.2 The Dielectric Medium

209

(1)

17.2.3 The Solvent Model Hamiltonian

210

(1)

17.2.4 The Statistical Density Operator for the Solvent

211

(1)

17.2.5 Variational Selection of Initial State

212

(1)

17.2.6 The Hamiltonian for the Intramolecular Degrees of Freedom of the Molecular System

213

(2)

17.2.7 The Statistical Density Operator for the Molecular System

215

(1)

17.3 The Time-Dependent Equation of Motion and the Electron Transfer Probability

216

(2)

17.4 Numerical Examples

218

(5)

18 Electron Transfer Coupling Elements

223

(13)

18.1 Introduction

223

(1)

18.2 Theoretical Methods

224

(7)

18.2.1 Numerical Examples

229

(2)

18.3 Methods for Electron Coupling Elements in Proteins

231

(5)

18.3.1 Numerical Examples

234

(2)

19 Electron Transfer Reactions Coupled to a Quantum Mechanical Radiation Field

236

(14)

19.1 The Model

238

(1)

19.2 The Hamiltonian

239

(3)

19.3 The Time-Dependent Statistical Density Operator

242

(3)

19.4 The Equation of Motion

245

(1)

19.5 Discussion of the Model

246

(4)

20 Proton Transfer Reactions in Solution

250

(7)

20.1 The Hamiltonian

251

(6)

Appendix A

257

(1)

Appendix B

258

(2)

Appendix C: Answers to Exercises

260

(13)

Bibliography

273

(12)

Index

285

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Advanced Molecular Dynamics and Chemical Kinetics

Summary

Author Biography

Table of Contents

Supplemental Materials