Preface x
1 Collisional Phenomena, Cross Sections, and Rates 1
1.1 Electronic and Nuclear Motions in Collisional Phenomena 1
1.2 Collisional Cross Sections 4
1.2.1 Definition of a Cross Section 4
1.2.2 Conservation Laws 5
1.2.3 Collisions in the Center of Mass Frame 5
1.2.4 Classification of Collision Processes 7
1.3 Quantal Description of Collisions 8
1.3.1 Time-Dependent Quantal States 8
1.3.2 Time-Independent Steady States 10
1.4 Examples of Physical Systems and Phenomena 11
1.4.1 Overview of Phenomena 11
1.4.2 Electronically Adiabatic Heavy-Particle Collisions 12
1.4.3 Electronically Diabatic Heavy-Particle Collisions 19
1.4.4 Electron and Photon Scattering by Molecules 22
1.5 Transport, Energy Relaxation, and Reaction Rates in Gases 26
1.6 Concepts and Methods in the Quantal Modelling 27
References 27
2 Elastic Collisions 33
2.1 Elastic Collision Cross Sections 34
2.1.1 Classical Mechanics Treatment 34
2.1.2 Quantal Treatment 39
2.1.3 Scattering Resonances and Quantal Exchange Symmetry 41
2.2 Integral Equation and Approximations 43
2.2.1 Green Functions and Integral Equations 43
2.2.2 Born Expansion 45
2.2.3 Eikonal Wavefunctions and Semiclassical Treatments 45
2.3 Partial-Wave Analysis 52
2.3.1 Radial Wavefunctions and Phase Shifts 52
2.3.2 Exceptional Cases 56
2.3.3 Radial Integral Equations 57
2.3.4 Resonance Energies and Angular Momenta 59
2.4 Numerical Methods for Scattering 61
2.4.1 Numerical Procedures for Differential Equations 61
2.4.2 Numerical Procedures for Integral Equations 63
2.4.2.1 Stepwise Solutions of the Volterra Integral Equation 63
2.4.2.2 Piecewise Solutions 64
2.4.2.3 Expansion of Amplitude Densities 65
2.4.2.4 Improvements by a Variational Procedure 66
2.4.3 Numerical Procedures for Semiclassical Scattering 67
2.4.4 Extraction of Potential Functions by Inversion 69
2.5 Examples 70
2.5.1 Atom–Atom Collisions 70
2.5.2 Electron Scattering by Atoms 71
References 72
3 Inelastic Collisions: Dynamics 75
3.1 Inelastic Collision Cross Sections 76
3.1.1 Kinematics of Inelastic Collisions 76
3.1.2 Classical Dynamics Treatment 79
3.1.3 Quantal Treatment 81
3.1.4 Semiclassical Treatment 83
3.1.4.1 Eikonal Description 83
3.1.4.2 Trajectories Description 87
3.2 Coupled-Channel Equations 89
3.2.1 Differential Equations and Boundary Conditions 89
3.2.2 Integral Equations 89
3.2.3 Partial-Wave Expansion 91
3.3 Matrix Form of Partial-Wave Equations 93
3.3.1 Differential Equations and Boundary Conditions 93
3.3.2 Integral Equations 95
3.4 Collisions Involving Two Coupled Channels 98
3.4.1 Two Open Channels 98
3.4.2 Resonances for One Open- and One Closed-Channel 99
3.5 Distorted-Waves Treatment 101
3.5.1 General Distortion Potential 101
3.5.2 Multichannel Distorted Partial-Wave Treatment 103
3.6 Optical Potential Models 105
3.6.1 Physical Models of Optical Potentials 105
3.6.2 Basis-Set Partitioning Method 107
References 108
4 Inelastic Collisions: Adiabatic Energy Transfer 111
4.1 Adiabatic Energy Transfer Cross Sections 112
4.1.1 Kinematics of Adiabatic Energy Transfer 112
4.1.2 Quantal and Semiclassical Equations for Rovibrational Energy Transfer 113
4.1.2.1 Differential Equations and Boundary Conditions 113
4.1.2.2 Integral Equations 116
4.1.2.3 Distorted-Wave Approximation 118
4.1.2.4 Semiclassical Equations 120
4.2 Numerical Methods 122
4.2.1 Step-by-Step Propagation of Wavefunctions 122
4.2.2 Piecewise Propagation 123
4.2.3 Propagation of Amplitude Densities 126
4.2.4 Variational Calculation of Scattering Amplitudes 126
4.2.5 Integration of Semiclassical Coupled Equations 128
4.3 Electronically Adiabatic Rotational Transitions 130
4.3.1 Expansion in a Basis Set of Rotor States 130
4.3.2 Helicity Representation and the Body-Fixed Frame 134
4.3.3 Atom–Polyatomic and Molecule–Molecule Collisions 136
4.4 T–R Transfer Calculations and Comparisons with Experimental Values 139
4.4.1 Approximate Treatments 139
4.4.1.1 Distorted-Wave Approximation 139
4.4.1.2 Coupled-States and Infinite-Order Sudden Approximations 140
4.4.1.3 Semiclassical Dynamics Treatments 141
4.4.2 Numerical Results 142
4.4.2.1 Numerical Results for Atom–Diatom Collisions 142
4.4.2.2 Numerical Results for Diatom–Diatom Collisions 145
4.5 Translational–Rotational–Vibrational (T–R–V) Transfer 146
4.5.1 Landau–Teller Model of T–V Transfer 146
4.5.2 Atom–Diatomic T–R–V Transfer 148
4.5.3 T–R–V Transfer Involving Polyatomic Molecules 150
4.5.4 Rates of Energy Transfer 153
4.5.4.1 Reduced Dimensionality and Optical Potential Treatments 153
4.5.4.2 Thermal Rates from Cross Sections 154
References 154
5 Electronically Diabatic Collisions 159
5.1 Expansion in an Electronic Basis Set 160
5.2 Electronic Representations 167
5.2.1 Adiabatic Representation and Momentum Couplings 167
5.2.2 Nonadiabatic Representations 168
5.2.3 Two-State Case 170
5.2.4 Fixed-Nuclei, Adiabatic, and Condon Approximations 171
5.2.5 Approximate Nonadiabatic Coupling for Same-Symmetry States 175
5.3 Collisional Coupling of Molecular Electronic States 176
5.3.1 Quantal Transition Amplitudes and Cross Sections 176
5.3.2 Non-Crossing Rule 180
5.3.3 Quantal Cross Sections of Atom–Atom Collisions 182
5.3.4 Short-Wavelength Approximation 185
5.4 Semiclassical Description 190
5.4.1 Eikonal Wavefunctions 190
5.4.2 Calculation of Nonadiabatic Dynamics for Coupled Electronic States 199
5.4.3 Short-Wavelength Approximation and Two-State Models 204
5.4.4 First-Principles Dynamics Treatments 210
5.4.4.1 Nonadiabatic Multistate Dynamics 210
5.4.4.2 First-Principles Eik/TDHF Treatment 215
5.4.4.3 Nuclear-Electronic Dynamics Treatment 217
5.5 Electronic Rearrangement for Several Interatomic Variables 218
5.5.1 Potential Energy Surfaces and Their Couplings: Multistate Cases 218
5.5.2 Crossings in Several Dimensions: Conical Intersections and Seams 219
5.5.3 Geometrical Phase and Generalizations 225
References 227
6 Reactive Collisions 233
6.1 Arrangement Channels and Coordinate Transformations 234
6.1.1 A Three-Atom System 234
6.1.2 Curvilinear and Hyper-spherical Coordinates 236
6.1.3 Arrangements and Potential Energies 237
6.2 Classical Reaction Dynamics 239
6.2.1 Atomic Rearrangements 239
6.2.2 Reaction Probabilities 240
6.3 Quantal Theory of Adiabatic Reactions 243
6.3.1 Wavemechanical Treatment 243
6.3.2 Integro-differential Equations for Reaction Amplitudes 246
6.3.3 Coupled Arrangement Channels Method 250
6.3.4 Rearrangements in a Three-atom System 251
6.4 Calculation of Adiabatic Reaction Cross Sections 252
6.4.1 Approximation of Distorted Waves for Rearrangement Collisions 252
6.4.2 Variational Calculations with Expansions in Normalizable Functions 256
6.4.3 Treatments Using Matching of Wavefunctions 260
6.4.4 Treatments Using Hyperspherical Coordinates 264
6.5 Electronically Diabatic Reactions 266
6.5.1 Quantal Treatment 266
6.5.2 Eikonal and Semiclassical Treatments 272
6.6 First-Principles (Ab initio) Treatments of Reactive Collisions 273
6.6.1 Time-dependent Eikonal Method 273
6.6.2 Wavepacket-spawning Transitions 275
6.6.2.1 Coherent-paths Treatment 276
6.7 Reduced Dimensionality, Optical-potential, and Machine-learning Treatments 277
6.7.1 Reduced Dimensionality Models 277
6.7.2 Optical Potentials Treatment 278
6.7.3 Machine-learning Procedures 279
References 281
7 The Quantum Scattering Operator and the Statistical Density Operator 287
7.1 Scattering Operators and Transition Rates 287
7.1.1 Time-dependent Treatment 287
7.1.2 Time-independent Treatment 289
7.1.3 Rates of Change of Observables 292
7.2 Partitioning the Space of State Wavefunctions 293
7.2.1 Effective Hamiltonian Operators 293
7.2.2 Optical Potentials and Scattering Resonances 295
7.3 Many-Atom Scattering Operators 297
7.3.1 Reactions in Three-atom Systems 297
7.3.2 Multiple Scattering 299
7.3.3 Coupled Arrangement-channel Effective-Hamiltonian Equations 300
7.4 Density Operator Treatments 302
7.4.1 Equation of Motion for the Density Operator 302
7.4.2 Partitioning of the Density Operator 304
7.4.3 Quantum-Classical Density Operator 306
7.4.3.1 Eikonal Representation 306
7.4.3.2 Wigner-Transform Treatment 307
7.5 Density Operator Treatments for Reactive Collisions 314
References 314
Index 319
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