Theory and Application of Quantum Molecular Dynamics

Preface......Page 8
Contents......Page 12
1.1 Adiabatic Representation 1......Page 20
1.2 Born-Oppenheimer Approximation 3......Page 22
1.3 Hellmann-Feynman Theory 5......Page 24
1.4 Diabatic Representation 6......Page 25
1.5 Transformation Between Representations 8......Page 27
1.6 Crossing of Adiabatic Potentials 10......Page 29
2.1.1 Hartree-Fock Equation 13......Page 32
2.1.2 Restricted Hartree-Fock (RHF) 18......Page 37
2.1.3 Unrestricted Hartree-Fock (UHF) 19......Page 38
2.1.4 Koopmans\' Theorem 20......Page 39
2.1.5 SCF Solution to HF Equation 21......Page 40
2.2.1 Multiconfiguration 23......Page 42
2.2.2 Perturbation Methods 24......Page 43
3.1.1 Harmonic Potential 27......Page 46
3.1.3 General Potential 29......Page 48
3.1.4 Discrete Variable Representation 31......Page 50
3.1.5 Gaussian Basis Functions 33......Page 52
3.2.1 Diatomic Molecules 36......Page 55
3.2.2 Triatomic Molecules 37......Page 56
3.2.3 Tetraatomic Molecules 41......Page 60
3.2.4 Bound State Calculation of (HF)2 45......Page 64
3.3.1 Autocorrelation Function 50......Page 69
3.3.2 Energy Spectrum 51......Page 70
3.3.3 Spectrum of H02 56......Page 75
4.1.1 Miller Operator 59......Page 78
4.1.2 Scattering Operator 63......Page 82
4.2.1 Green\'s Function 64......Page 83
4.2.2 Lippmann-Schwinger Equation 66......Page 85
4.2.3 The S Matrix 67......Page 86
4.2.4 Distorted Wave 71......Page 90
4.3.1 Radial SchrSdinger Equation 74......Page 93
4.3.2 Free Radial Functions 75......Page 94
4.3.3 Radial Green\'s Function 76......Page 95
4.3.4 Scattering Phase Shift 78......Page 97
4.3.5 Scattering Cross Section 82......Page 101
4.4.1 Coupled Channel Equations 85......Page 104
4.4.2 Multichannel Green\'s Function 86......Page 105
4.4.3 S, T and K Matrices 88......Page 107
4.4.4 Scattering Cross Section 89......Page 108
4.4.5 R Matrix Method 95......Page 114
4.5.1 Partitioning of the Hamiltonian 99......Page 118
4.5.2 Scattering Matrix and Cross Section 100......Page 119
4.5.3 Jacobi Coordinates 104......Page 123
4.5.4 A Note on Reactive Scattering 106......Page 125
5.1 Introduction 107......Page 126
5.2 Distorted-Wave Born Approximation 108......Page 127
5.3 Hyperspherical Coordinate Approach 109......Page 128
5.4.1 Multi-arrangement Expansion of Wavefunction 111......Page 130
5.4.2 Coupled Arrangement Integral Equation 112......Page 131
5.4.3 Algebraic Method 115......Page 134
5.4.4 Schwinger Variational Method 118......Page 137
5.4.5 Kohn Variational Method 120......Page 139
5.5 Atom-Diatom Reactive Scattering 127......Page 146
5.5.1 Hamiltonian and Basis Set 128......Page 147
5.5.2 Calculation of Direct Matrix Elements 129......Page 148
5.5.3 Calculation of Exchange Matrix Elements 131......Page 150
5.5.4 H + H2 Reaction 134......Page 153
5.5.5 F + H2 Reaction 140......Page 159
6.1 Introduction 145......Page 164
6.2 Representations 146......Page 165
6.3.1 Finite Difference Method 147......Page 166
6.3.2 Split-Operator Method 148......Page 167
6.3.3 Chebychev Polynomial Method 149......Page 168
6.3.4 Gaussian Wavepackets 151......Page 170
6.4 Application to Reactive Scattering 153......Page 172
6.4.1 Extraction of Scattering Information 154......Page 173
6.4.2 Reactive Flux and Total Reaction Probability 155......Page 174
6.4.3 Use of Absorbing Potentials 157......Page 176
6.5.1 Hamiltonian and Wavepacket Propagation 158......Page 177
6.5.2 Reaction of H + 02 160......Page 179
6.6.1 Beyond Triatomic Systems 163......Page 182
6.6.2 Hamiltonian and Basis Functions 165......Page 184
6.6.3 Treatment for Spectator Bonds 169......Page 188
6.6.4 Reaction of H2 + OH 170......Page 189
6.6.5 Reaction of HO + CO 175......Page 194
7.1 Introduction 179......Page 198
7.2.1 Time-independent Expressions of S Matrix Elements 180......Page 199
7.2.2 Time-dependent Expressions of S Matrix Elements 182......Page 201
7.2.3 State-to-State Reactive Scattering of H + 02 187......Page 206
7.3 RPD Approach to State-to-State Reactions 189......Page 208
7.3.1 The RPD Equations 191......Page 210
7.3.2 Reactant Component Wavefunction 192......Page 211
7.3.3 Product Component Wavefunction 193......Page 212
7.3.5 A Collocation Quadrature Scheme 195......Page 214
7.4.1 Transformation of Jacobi Coordinates 196......Page 215
7.4.2 Numerical Test for H + H2 Reaction 197......Page 216
8.1.1 Introduction 201......Page 220
8.1.2 Phase Shift and Time Delay 202......Page 221
8.1.3 Lifetime Matrix 204......Page 223
8.1.4 Implication of Resonance in Cross Sections 205......Page 224
8.2.1 Complex Symmetric Operator 206......Page 225
8.2.2 Projection Operators and Feshbach Partitioning 207......Page 226
8.2.3 Resonance States 210......Page 229
8.2.4 Decay of Resonance 213......Page 232
8.2.5 Calculation of Resonance 215......Page 234
8.3.1 Cumulative Reaction Probability 216......Page 235
8.3.2 Relation to Transition State Theory 218......Page 237
9.1 Introduction 223......Page 241
9.2 Time-Dependent Perturbation Theory 224......Page 243
9.3.1 Vector Potentials in Quantum Mechanics 227......Page 246
9.3.2 Quantized Radiation Field 230......Page 249
9.3.3 One Photon Processes 232......Page 251
9.3.4 Two Photon Processes: Light Scattering 234......Page 253
9.4.1 Half Collision Dynamics 238......Page 257
9.4.2 Theoretical Treatment 239......Page 258
9.4.3 Dissociation of H20 241......Page 260
9.4.4 Dissociation of H202 242......Page 261
9.5.1 Introduction 246......Page 265
9.5.2 Time-dependent Golden Rule Treatment 248......Page 267
9.5.3 HeCl2(u) -► He + Cla(« - 1) 250......Page 269
9.5.4 D2HF(u = 1) -> D2+ HF(i; = 0) 252......Page 271
9.6.1 Single Arrangement 254......Page 273
9.6.2 Multi-arrangement 257......Page 2736
9.6.3 Time-Dependent Flux Calculation 258......Page 277
9.7.1 Theoretical Formulation 261......Page 280
10.1 Introduction 265......Page 284
10.2.1 Dissociative Adsorption on Metal Surfaces 266......Page 285
10.2.2 Flat Surface Model 267......Page 286
10.2.3 Fixed Site Corrugated Surface Model 270......Page 289
10.2.4 A Selection Rule for Homonuclear Diatoms 271......Page 290
10.2.5 Fully Corrugated Surface Model 273......Page 292
10.3.1 Background 275......Page 294
10.3.2 Potential Energy Surface for H2/Cu(lll) 276......Page 295
10.3.3 Results of Quantum Dynamics Studies 278......Page 297
11.1.1 General Formalism 285......Page 303
11.1.2 Turning Points and Connection Formulae 287......Page 306
11.2.1 Comparison Equations 290......Page 309
11.2.2 Applications of the Uniform Approximation 292......Page 311
11.2.3 Langer Modification 299......Page 318
11.3.1 Phase Shift in Elastic Scattering 301......Page 320
11.3.2 Elastic Cross Sections 302......Page 321
11.4.1 Stationary Phase Approximation 305......Page 324
11.4.2 Path Integral Representation of Quantum Mechanics 306......Page 325
11.4.3 Stationary Phase Approximation of the Path Integral 309......Page 328
11.4.4 Initial Value Representation 311......Page 330
A.1 Hermite Polynomials 315......Page 334
A.2 Legendre Polynomials 317......Page 336
A.3 Spherical Harmonics 318......Page 337
A.4 Chebychev Polynomials 320......Page 339
A.5 Spherical Bessel Functions 321......Page 340
A.6 Useful Mathematical Formulae 323......Page 342
Appendix B Gaussian Quadrature 325......Page 343
Appendix C Clebsch-Gordon Coefficients 329......Page 348
D.1 Coordinate Rotations and Angular Momentum 333......Page 352
D.2 Rotation Matrix 334......Page 353
D.3 Transformation Between SF and BF Basis 337......Page 356
D.4 Total Angular Momentum in the BF Frame 340......Page 359
Bibliography 345......Page 364
Index 359......Page 378