Nonlinearity and chaos in molecular vibrations

Front Cover......Page 1
Nonlinearity and Chaos in Molecular Vibrations......Page 4
Copyright Page......Page 5
Contents......Page 10
Preface......Page 6
1.1 Normal modes......Page 16
1.2 Morse oscillator......Page 19
1.3 Second quantized operators......Page 22
1.4 Algebraic Hamiltonians......Page 26
References......Page 29
2.1 Continuous groups......Page 30
2.2 Coset spaces......Page 32
2.3 Dynamical applications......Page 35
2.4 Algebraic difference of molecular vibrational and electronic properties......Page 36
2.5 Explicit expressions......Page 37
2.6 Heisenberg's correspondence......Page 46
References......Page 48
3.1 Universality of chaos......Page 49
3.2 One-dimensional map......Page 51
3.4 KAM theorem......Page 54
3.5 Poincare surface of section......Page 56
3.6 kicked rotor......Page 57
3.7 Geometric and dynamical aspects of chaos......Page 59
References......Page 61
4.1 Coupling of two Morse oscillators......Page 62
4.2 su(2) algebraic properties of a two-mode system......Page 63
BookmarkTitle:......Page 64
4.4 Relation between Heisenberg's correspondence and the coset representation......Page 66
4.6 Dynamical analysis......Page 67
References......Page 72
5.2 Coset representatives of SU(1,1)/U(1)1 <8> SU(1,1)/U(2)2 for two-mode system......Page 73
5.3 Contrast of su( 1,1) to su(2)......Page 75
5.4 Numerical simulation......Page 77
References......Page 79
6.1 Breaking of su(3) algebra......Page 80
6.2 Numerical simulation......Page 84
6.3 su(3) represented Fermi resonance......Page 90
6.4 Dynamics under strong Fermi resonance......Page 93
6.5 Semiclassical fixed point structure......Page 96
References......Page 101
7.1 su(3) algebraic method......Page 102
7.2 Fitting of the coefficients......Page 106
7.3 Dynamical properties......Page 107
7.4 Coset potential......Page 109
7.5 Statistical interpretation of locality and normality......Page 112
7.6 Spontaneous symmetry breaking of identical modes......Page 114
7.7 Global symmetry and antisymmetry......Page 116
7.8 Action transfer coefficient......Page 117
7.9 Relaxational probability......Page 119
7.10 Action localization......Page 120
References......Page 124
Appendices......Page 125
8.2 Coset space representation of molecular rotation......Page 130
8.3 Quantum-classical transition......Page 131
8.4 su(2) x h4 coupling......Page 133
8.5 Regular and chaotic motions......Page 134
References......Page 136
9.1 Pendulum......Page 137
9.2 Resonance......Page 138
9.3 Molecular highly excited vibration......Page 142
References......Page 147
10.1 Periodic and quasiperiodic motions......Page 148
10.2 Sine circle map......Page 150
10.3 Resonance overlap and birth of chaos......Page 152
10.4 Coincidence of chaotic and barrier regions......Page 155
References......Page 157
11.2 Fractal dimension......Page 158
11.3 Multifractal......Page 161
11.4 f (a ) function......Page 162
11.6 Fractal of eigencoefficients......Page 166
11.7 Multifractal of eigencoefficients......Page 169
11.8 Self-similarity of eigencoefficients......Page 172
11.9 Fractal significance of eigencoefficients......Page 173
References......Page 174
12.1 Introduction......Page 175
12.2 Empirical CH bend Hamiltonian......Page 176
12.3 Second quantization representation of Heff......Page 177
12.4 su(2) x su(2) represented CH bend motion......Page 178
12.5 Coset space representation......Page 179
12.6 Dynamics......Page 180
12.7 Modes of CH bend motion......Page 183
12.8 Geometric interpretation of vibrational angular momentum......Page 189
12.9 Reduced Hamiltonian of CH bend motion......Page 190
12.10 Mode characters......Page 192
12.11 Modes of CH bend motion......Page 193
12.12 su(2) origin of precessional mode......Page 195
References......Page 199
13.1 Lyapunov exponent......Page 200
13.2 Important concepts of a Lyapunov exponent......Page 206
13.3 Nonergodicity of CH bend motion......Page 208
References......Page 215
Appendices......Page 216
14.1 Chaotic motion of DCN......Page 220
14.2 Periodic trajectories......Page 222
14.3 Chaotic motion originating from the D-C stretch......Page 230
References......Page 232
15.1 Introduction. algebraic method......Page 233
15.2 Diabatic correlation, formal quantum numbers and ordering of levels......Page 235
15.3 Acetylene case......Page 238
15.4 Background of diabatic correlation......Page 241
15.5 Approximately conserved quantum numbers......Page 243
15.6 DCN case......Page 246
15.7 Difference between approximate and formal quantum numbers......Page 250
BookmarkTitle:......Page 251
15.9 Lyapunov exponents......Page 253
References......Page 257
16.1 Classical analogues of LCAO of one-electronic system......Page 258
16.2 Hamiltonian of one electron in multiple sites: the coset representation......Page 259
16.3 Analogy with Hckel MO......Page 260
16.4 Dynamical interpretation of HMO......Page 262
16.5 Anderson localization......Page 264
16.6 Hammett equation......Page 266
16.7 Two-electronic correlation in Hckel system......Page 268
References......Page 271
17.1 Introduction......Page 272
17.2 Hamiltonian for one electron in multiple sites......Page 273
17.3 Quantization: the least of the averaged Lyapunov exponents......Page 275
17.4 Quantization of H2O vibration......Page 278
17.6 Action integrals of periodic trajectories......Page 279
17.7 Retrieval of low quantal levels......Page 284
17.8 Conclusion......Page 287
References......Page 288
18.2 Construction of the H function for vibrational relaxation......Page 289
18.3 Resonances in H2O and DCN vibration......Page 290
References......Page 293
19.1 The Dixon dip......Page 294
19.3 Dixon dips in the systems of HenonHeiles and quartic potentials......Page 295
A. H2O system......Page 297
B. DCN system: overlapping of resonances and chaos......Page 299
References......Page 304
20.1 Chaos in dissociation......Page 305
20.2 Chaos in the transitional states of bend motion......Page 306
20.3 HCN, HNC and the delocalized state......Page 308
20.4 The Lyapuov exponent for transitional chaos......Page 311
References......Page 315
Index......Page 316