分子振动：代数与非线性方法（英文版）

Contents
Preface
1 Pendulum Dynamics 1
1.1 Pendulum dynamics 1
1.2 Morse oscillator 3
1.3 Hamilt.on's equations of motioii 6
1.4 PenduluIn dynamics as the basic unit for resonance 7
1.5 Standard map and KAM theorem 9
1.6 Conclusion 11
References 11
2 Algebraic Approach to Vibrational Dynamics 13
2.1 The algebraic Hamiltonian 13
2.2 Heisenberg's cpondence and coset representation 15
2.3 An example: The H20 case 16
2.4 su(2) dynamical properties 19
Reference 22
Appendix: The derivation of raising and lowering operators 23
3 Chaos 25
3.1 Definition and Lyapunov exponent: Tent map 25
3.2 Lyapunov exponent in Hamiltonian system 28
3.3 Period 3 route to chaos 28
3.4 Resonance overlapping and sine circle mapping 29
3.5 The case study of DCN 32
3.5.1 The chaotic motion 32
3.5.2 Periodic trajectories 34
3.5.3 Chaotic motion originating from the D-C stretching 42
References 44
Appendix: Calculation of the maximal Lyapunov exponent 44
4 C-H Bending Motion of Acetylene47
4.1 Introduction 47
4.2 Empirical C-H bending Hamiltonian 48
4.3 Second quantization representation of Hea 49
4.4 su(2)O su(2) represented C-H bending motion 50
4.5 Coset representation 52
4.6 Modes of C-H bending motioii 52
4.7 Reduced Hamiltonian of CH bending motion 60
4.8 su(2) origin of precessional mode 66
4.9 Nonergodicity of C-H bending motion 68
4.10 Int.ramolecular vibrational relaxation 74
References 77
5 Assignments and Classification of Vibrational Manifolds79
5.1 Formaldehyde case 79
5.2 Diabatic correlation， formal quantum number and level reconstruction 81
5.3 Acetylene case 85
5.4 Background of diabatic correlation 88
5.5 Approximately conserved quantum number 91
5.6 DCN case 94
5.7 Density p in the coset space 98
5.8 Lyapunov exponent analysis 100
References 102
6 Dixon Dip103
6.1 Significance of level spacings 103
6.2 Dixon dip 103
6.3 Dixon dips in the systems of Henon-Heiles and quartic potentials 104
6.4 Destruction of Dixon dip under multiple resonances 106
6.5 Dixon dip and chaos 113
References 115
7 Quantization by Lyapunov Exponent and Periodic rDajectories 117
7.1 Introduction 117
7.2 Hamiltonian for one electron in mult.iple sites 118
7.3 Quantization: The least averaged Lyapunov exponent 120
7.4 Quantization of H20 vibration 123
7.5 Action integrals of periodic trajectories: The DCN case 125
7.6 Retrieval of low quantal levels of DCN 128
7.7 Quantization of Henon-Heiles system 131
7.8 Quantal correspondence in the classical AKPsystem 138
7.9 A comment 142
References 142
8 Dynamlcs of DCO/HCO and Dynamical Barrier Due to Extremely Irrational Couplings 145
8.1 The coset Hamiltonian of DCO 145
8.2 State dynamics of DCO 148
8.3 Contrast of the dynamical potentials of D-C and C-O stretchings 152
8.4 The HCO case 155
8.5 Comparison of the dynamical potentials 157
8.6 A comment: The IVR role of bending motion 157
8.7 Dynamical barrier due to extreInely irrational couplings: The role of bending motion 159
References 165
9 Dynamical Potential Analysis for HCP， DCP， N20， HOCI and HOBr 167
9.1 Introduction 167
9.2 The coset represented Hamiltonian of HCP 168
9.3 Dynamical potentials and state properties inferred by action population 170
9.4 State classification and quantal environments 175
9.5 Localized bending mode 178
9.6 The condition for localized mode 182
9.7 0n the HPC formation 182
9.8 The fixed point structure 183
9.9 DCP Hamiltonian 183
9.10 Dynamical similarity between DCP and HCP 188
9.11 N20 dynamics 191
9.12 The cases of HOCl and HOBr 199
9.13 A comment 214
References 215
Appendix 215
10 Chaos in the rlyansition State Induced by the Bending Motion219
10.1 Chaos in the transition state 219
10.2 The cases of HCN， HNC and the transition state 221
10.3 Lyapunov exponent analysis 224
10.4 Statistical analysis of the level spacing distribution 226
10.5 Dixon dip analysis 227
10.6 Coupling of pendulum and harmonic oscillator 227
10.7 A comment 230
References 230
Appendix: Author-s Publications Related to this Monograph 231