Preface p. xiii
Abbreviations p. xvii
Mathematical Introduction p. 1
Hilbert Space p. 1
Eigenfunction and Electronic Nonadiabatic Coupling Term p. 1
Abelian and Non-Abelian Curl Equations p. 3
Abelian and Non-Abelian Divergence Equations p. 6
Hilbert Subspace p. 8
Vectorial First-Order Differential Equation and Line Integral p. 11
Vectorial First-Order Differential Equation p. 12
Study of Abelian Case p. 12
Study of Non-Abelian Case p. 13
Orthogonality p. 14
Integral Equation p. 14
Integral Equation along an Open Contour p. 15
Integral Equation along a Closed Contour p. 16
Solution of Differential Vector Equation p. 20
Summary and Conclusions p. 23
Problem p. 23
References p. 25
Born-Oppenheimer Approach: Diabatization and Topological Matrix p. 26
Time-Independent Treatment p. 26
Adiabatic Representation p. 26
Diabatic Representation p. 28
Adiabatic-to-Diabatic Transformation p. 30
Transformation for Electronic Basis Sets p. 30
Transformation for Nuclear Wavefunctions p. 33
Implications Due to Adiabatic-to-Diabatic Transformation p. 34
Final Comments p. 38
Application of Complex Eigenfunctions p. 39
Introducing Time-Independent Phase Factors p. 39
Adiabatic Schrodinger Equation p. 39
Adiabatic-to-Diabatic Transformation p. 40
Introducing Time-Dependent Phase Factors p. 41
Time-Dependent Treatment p. 43
Time-Dependent Perturbative Approach p. 43
Time-Dependent Nonperturbative Approach p. 45
Adiabatic Time-Dependent Electronic Basis Set p. 45
Adiabatic Time-Dependent Nuclear Schrodinger Equation p. 46
Time-Dependent Adiabatic-to-Diabatic Transformation p. 47
Summary p. 49
Problem p. 50
Appendixes p. 51
Dressed Nonadiabatic Coupling Matrix [tau] p. 51
Analyticity of Adiabatic-to-Diabatic Transformation Matrix A in Spacetime Configuration p. 52
References p. 54
Model Studies p. 58
Treatment of Analytical Models p. 58
Two-State Systems p. 59
Adiabatic-to-Diabatic Transformation Matrix p. 59
Topological (D) Matrix p. 60
The Diabatic Potential Matrix p. 61
Three-State Systems p. 62
Adiabatic-to-Diabatic Transformation Matrix p. 62
Topological Matrix p. 63
Four-State Systems p. 64
Adiabatic-to-Diabatic Transformation Matrix p. 64
Topological Matrix p. 65
Comments Related to General Case p. 66
Study of 2 x 2 Diabatic Potential Matrix and Related Topics p. 67
Treatment of General Case p. 67
The Jahn-Teller Model p. 70
Elliptic Jahn-Teller Model p. 72
Distribution of Conical Intersections and Diabatic Potential Matrix p. 73
Adiabatic-to-Diabatic Transformation Matrix and Wigner Rotation Matrix p. 75
Wigner Rotation Matrices p. 76
Adiabatic-to-Diabatic Transformation Matrix and Wigner d[superscript j] Matrix p. 77
Problem p. 79
References p. 82
Studies of Molecular Systems p. 84
Introductory Comments p. 84
Theoretical Background p. 85
Quantization of Nonadiabatic Coupling Matrix: Study of Ab Initio Molecular Systems p. 87
Two-State Quasiquantization p. 87
{{H[subscript 2],H}} System p. 87
{{H[subscript 2],O}} System p. 91
{{C[subscript 2],H[subscript 2]}} System p. 92
Multistate Quasiquantization p. 96
{{H[subscript 2],H}} System p. 96
{{C[subscript 2],H}} System p. 99
References p. 102
Degeneracy Points and Born-Oppenheimer Coupling Terms as Poles p. 105
Relation between Born-Oppenheimer Coupling Terms and Degeneracy Points p. 105
Construction of Hilbert Subspace p. 108
Sign Flips of Electronic Eigenfunctions p. 109
Two-State Hilbert Subspace p. 109
Three-State Hilbert Subspace p. 110
General Hilbert Subspace p. 114
Multidegeneracy Point p. 120
General Approach p. 120
Model Studies p. 121
Topological Spin p. 122
Extended Euler Matrix as a Model for Adiabatic-to-Diabatic Transformation Matrix p. 125
Introductory Comments p. 125
Two-Dimensional Case p. 126
Three-Dimensional Case p. 127
Multidimensional Case p. 130
Quantization of [tau] Matrix and its Relation to Size of Configuration Space: Mathieu Equation as a Case of Study p. 131
Mathieu Equation and Its Eigenfunctions p. 131
Nonadiabatic Coupling Matrix ([tau]) and Topological Matrix (D) p. 133
Problems p. 134
References p. 136
Molecular Field p. 139
Solenoid as a Model for the Seam p. 139
Two-State (Abelian) System p. 141
Nonadiabatic Coupling Term as a Vector Potential p. 141
Two-State Curl Equation p. 143
(Extended) Stokes Theorem p. 144
Application of Stokes Theorem for Several Two-State Conical Intersections p. 146
Application of Vector Algebra to Calculate the Field of a Two-State Hilbert Space p. 147
A Numerical Example: Study of {{H[subscript 2],Na}} System p. 149
A Short Summary p. 151
Multistate Hilbert Subspace p. 151
Non-Abelian Stokes Theorem p. 151
The Curl-Divergence Equations p. 154
Three-State Hilbert Subspace p. 155
Derivation of Poisson Equations p. 157
Strange Matrix Element and Gauge Transformation p. 159
A Numerical Study of {{H[subscript 2],H}} System p. 160
Introductory Comments p. 160
Introducing Fourier Expansion p. 160
Introducing Boundary Conditions p. 161
Numerical Results p. 162
Multistate Hilbert Subspace: Breakup of Nonadiabatic Coupling Matrix p. 162
Pseudomagnetic Field p. 167
Quantization of Pseudomagnetic Field along the Seam p. 168
Non-Abelian Magnetic Fields p. 168
Problems p. 168
References p. 172
Open Phase and Berry Phase for Molecular Systems p. 175
Studies of Ab Initio Systems p. 175
Introductory Comments p. 175
Open Phase and Berry Phase for Singlevalued Eigenfunctions: Berry's Approach p. 176
Open Phase and Berry Phase for Multivalued Eigenfunctions: Present Approach p. 177
Derivation of Time-Dependent Equation p. 177
Treatment of Adiabatic Case p. 179
Treatment of Nonadiabatic (General) Case p. 181
{{H[subscript 2],H}} System as a Case Study p. 183
Phase-Modulus Relations for an External Cyclic Time-Dependent Field p. 187
Derivation of Reciprocal Relations p. 187
Mathieu Equation p. 189
Time-Dependent Schrodinger Equations p. 189
Numerical Study of Topological Phase p. 191
Short Summary p. 194
Problem p. 194
References p. 195
Extended Born-Oppenheimer Approximations p. 197
Introductory Comments p. 197
Born-Oppenheimer Approximation as Applied to a Multistate Model System p. 199
Extended Approximate Born-Oppenheimer Equation p. 199
Gauge Invariance Condition for Approximate Born-Oppenheimer Equations p. 201
Multistate Born-Oppenheimer Approximation: Energy Considerations to Reduce Dimensions of Diabatic Potential Matrix p. 201
Introductory Comments p. 201
Derivation of Reduced Diabatic Potential Matrix p. 202
Application of Extended Euler Matrix: Introducing the N-State Adiabatic-to-Diabatic Transformation Angle p. 206
Introductory Comments p. 206
Derivation of Adiabatic-to-Diabatic Transformation Angle p. 206
Two-State Diabatic Potential Energy Matrix p. 210
Derivation of Diabatic Potential Matrix p. 210
A Numerical Study of [Delta] W Matrix Elements p. 211
A Different Approach: Helmholtz Decomposition p. 214
A Numerical Study of a Reactive (Exchange) Scattering Two-Coordinate Model p. 214
Basic Equations p. 214
A Two-Coordinate Reactive (Exchange) Model p. 216
Numerical Results and Discussion p. 217
Problem p. 220
References p. 221
A Summary p. 224
Index p. 227