Symmetry of Intramolecular Quantum Dynamics (De Gruyter Studies in Mathematical Physics Book 11)

The main goal of this book is to give a systematic description of intramolecular quantum dynamics on the basis of only the symmetry principles. In this respect, the book has no analogs in the world literature. This approach does not introduce a configuration space of the molecular system in explicit form at all and, consequently, does not consider in explicit form the wave functions of the coordinates of this space. However, precisely because of its deep philosophical and technical difference this approach is the only possible for the solution of many topical problems of the internal dynamics of molecules. The obtained models lead to a simple, purely algebraic, scheme of calculation and are rigorous in the sense that their correctness is limited only to the correct choice of symmetry of the internal dynamics. The book is basically intended for scientists working in the field of molecular spectroscopy, quantum and structural chemistry. The reader is not supposed to know the apparatus of group representation theory needed for application of symmetry methods in quantum intramolecular dynamics since the first part of the book is dedicated to it. Preface 5 I Foundations of the mathematical apparatus 15 1 Basic concepts of group theory 17 1.1 The group postulates 17 1.2 Subgroup, direct product of groups, isomorphism, and homomorphism 20 1.3 Cosets. Semidirect product of groups 21 1.4 Conjugacy classes 23 2 Basic concepts of group representation theory 24 2.1 Linear vector spaces 24 2.2 Operators in configuration and function spaces 27 2.3 Representations of groups 29 2.4 Characters. Decomposition of reducible representations 31 2.5 Direct product of representations. Symmetric power 34 2.6 The Clebsch-Gordan coefficients 37 2.7 Basis functions of irreducible representations 39 2.8 Irreducible tensor operators. The Wigner-Eckart theorem 42 3 The permutation group 45 3.1 Operations in the permutation group. Classes 45 3.2 Irreducible representations. The Young diagrams and tableaux 47 3.3 Basis functions of irreducible representations 49 3.4 The conjugate representation 51 4 Continuous groups 53 4.1 Compact Lie groups 53 4.2 Lie group of linear transformations 55 4.3 Lie algebra. Three-dimensional rotation group 56 4.4 Irreducible representations of a three-dimensional rotation group 60 5 Point groups 65 5.1 Operations in point groups 65 5.2 Discrete axial groups 67 5.3 Cubic groups. Icosahedral groups 68 5.4 Continuous axial groups 70 6 Dynamic groups 72 6.1 Invariant dynamic groups 72 6.2 Noninvariant dynamic groups 73 II Qualitative intramolecular quantum dynamics 79 7 The philosophy of using the symmetry properties of internal dynamics 81 7.1 Symmetry groups of internal dynamics 81 7.2 Significance of the analysis of symmetry properties 87 7.3 On the domain of the point group 90 7.4 The chain of symmetry groups 95 7.5 The concept of the coordinate spin 97 7.6 The influence of numerical methods on the overall description 100 7.7 Conclusions 100 8 Internal dynamics of rigid molecules 103 8.1 Nonlinear molecules without inversion center 103 8.2 Nonlinear molecules with inversion center 115 8.3 Linear molecules 122 8.4 Description of quasidegenerate vibronic states 129 8.5 Conclusions 133 9 Molecules with torsional transitions of the exchange type 137 9.1 Extended point groups. Intermediate configuration 137 9.2 Methanol molecule CH3OH 143 9.3 Ethane molecule C2H6 149 9.4 The molecules of complex hydrides LiBH4 and NaBH4 154 9.5 The molecules of dimethyl ether (CH3)2O and acetone (CH3)2CO 162 9.6 Conclusions 169 10 Molecules with pseudorotations of the exchange type 171 10.1 Extended point groups 171 10.2 Cyclobutane molecule C4H8 173 10.3 Molecules of the XPF4 type 180 10.4 Phosphorus pentafluoride molecule PF5 185 10.5 The separation of internal motions 193 10.6 Conclusions 195 11 Molecules with transitions of the nonexchange type between equivalent configurations 196 11.1 Extended point groups 196 11.2 The ammonia molecule NH3 199 11.3 The peroxide molecule HOOH 203 11.4 The hydrazine molecule N2H4 209 11.5 Conclusions 218 12 On the meaning of the Born-Oppenheimer Approximation 219 12.1 Nondegenerate electronic states 220 12.2 Degenerate electronic states 221 12.3 Internal geometric symmetry of the Hamiltonian 224 12.4 Definition of the rotational motion 230 12.5 Selection of physically meaningful states 233 12.6 Symmetry methods in the description of intramolecular dynamics 237 12.7 Geometric symmetry and definitions of nonrigid motions 240 12.8 Nuclear statistical weights 243 12.9 Conclusions 244 13 Molecules with transitions of the exchange and nonexchange types between equivalent configurations 249 13.1 Extended point groups 249 13.2 Methanol molecule CH3OH 253 13.3 Methylamine molecule CH3NH2 259 13.4 Cyclopentane molecule C5H10 264 13.5 Conclusions 269 14 On the construction of extended point groups 270 14.1 Hydrogen fluoride dimer (HF)2 270 14.2 Ionic complexes ArH+3 andArD+3 275 14.3 Carbocation C2H+3 282 14.4 Conclusions 291 15 Nonrigid molecular systems with continuous axial symmetry groups 292 15.1 Systems of the HCN/HNC type 293 15.2 Complexes of the XCO type 296 15.3 Nonrigid water molecules H2O 297 15.4 Conclusions 304 16 Molecules with different isomeric forms in a single electronic state 305 16.1 Distorted molecular systems 305 16.2 The methanol molecule CH2DOH 307 16.3 The ethane molecule CH2D-CH2D 311 16.4 The ethanol molecule CH3CH2OH 316 16.5 The cyclobutane 1,1-d2 molecule 321 16.6 The tetrahydrofuran molecule C4H8O 329 16.7 Conclusions 341 17 Molecules with different isomeric forms in different electronic states 343 17.1 Formaldehyde molecule H2CO 343 17.2 The ethylene molecule CH2-CD2 350 17.3 Conclusions 362 18 Algebraic models of the global description of molecular spectrum 363 18.1 The rigid water molecule H2O 364 18.2 The nonrigid methanol molecule CH3OH 368 18.3 The nonrigid water molecule H2O 373 18.4 Conclusions 375 19 Description of the Zeeman and Stark effects 377 19.1 External field and symmetry of the stationary states 377 19.2 The Zeeman effect in the case of a rigid molecule 378 19.3 The Zeeman effect in the case of a nonrigid molecule 381 19.4 The Stark effect in the case of a rigid molecule 382 19.5 The Stark effect in the case of a nonrigid molecule 387 19.6 Conclusions 391 20 Additional remarks on describing intramolecular dynamics 393 20.1 The nonrigid trimethylborane molecule B(CH3)3 393 20.2 Hyperfine interactions in the methane molecule CH4 408 20.3 Parity violation effects in molecules with stereoisomers 410 20.4 Parity violation effects in the molecules without stereoisomers 412 20.5 Conclusions 415 Conclusion 417 Appendices 419 Bibliography 431 Index 437 The main goal of this book is to give a systematic description of intramolecular quantum dynamics on the basis of only the symmetry principles. In this respect, the book has no analogs in the world literature. The obtained models lead to a simple, purely algebraic, scheme of calculation and are rigorous in the sense that their correctness is limited only to the correct choice of symmetry of the internal dynamics. The book is basically intended for scientists working in the field of molecular spectroscopy, quantum and structural chemistry. Alexander V. Burenin, Institute of Applied Physics, Russian Academy of Sciences, Russia