By Baer M., Billing G.D. (eds.)

ISBN-10: 0471433462

ISBN-13: 9780471433460

A distinct issues quantity at the function of degenerate states within the top sequence on chemical physicsEdited by means of Nobel Prize-winner Ilya Prigogine and well known authority Stuart A. Rice, the Advances in Chemical Physics sequence offers a discussion board for severe, authoritative reviews in each region of the self-discipline. In a layout that encourages the expression of person issues of view, specialists within the box current entire analyses of topics of curiosity. This stand-alone, unique themes quantity, edited via Gert D. Billing of the collage of Copenhagen and Michael Baer of the Soreq Nuclear learn heart in Yavne, Israel, experiences fresh advances at the position of degenerate states in chemistry. quantity 124 collects cutting edge papers on "Complex States of straightforward Molecular Systems," "Electron Nuclear Dynamics," "Conical Intersections and the Spin-Orbit Interaction," and lots of extra similar themes. Advances in Chemical Physics is still the optimum venue for shows of latest findings in its box.

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**Additional resources for Advances in Chemical Physics. The Role of Degenerate States in Chemistry**

**Sample text**

Tensorial gauge fields, 250–252 Linear combinations of atomic orbitals (LCAO), direct molecular dynamics, complete active space self-consistent field (CASSCF) technique, non-adiabatic systems, 4–5–411 Linear coupling approximation, geometric phase theory, 3 Jahn-Teller effect, 18–20 Linear triatomic molecules, Renner-Teller effect: singlet state vibronic coupling, 598–600 vibronic/spin-orbit coupling, 600–605 Line integral techniques: adiabatic-to-diabatic transformation matrix, 50–57 quasidiabatic framework, 53–57 single-valued diabatic potentials and topological matrix, 50–53 non-adiabatic coupling: three-state molecular system, sign flip derivation, 73–77 783 two-state molecular system and isotopic analogues, 108–109 C2H-molecule: (1,2) and (2,3) conical intersections, 111–112 Lithium compounds: direct molecular dynamics, ab initio multiple spawning, 413–414 permutational symmetry: adiabatic states, conical intersections: invariant operators, 735–737 Jahn-Teller theorem, 733–735 antilinear operator properties, 721–723 degenerate/near-degenerate vibration levels, 728–733 degenerate states chemistry, xiii electronic wave function, 680–682 energy functional form, 737–738 GBO approximation and geometric phase, two-dimensional Hilbert space model, 718–721 geometric phase theory, single-surface nuclear dynamics, 30–31 group theoretical issues, 668–674 nuclear spin function, 678–680 phase-change rule, 451–453 rotational wave function, 683–687 rovibronic/vibronic wave functions, 682– 683 2 S systems: alkali metal trimers, 712–713 dynamic Jahn-Teller and geometric phase effects, 698–711 electron/nuclear spin effects, 711–712 1 H3 isotopomers, 713–717 nonadiabatic coupling effects, 711 potential energy surfaces, 692–694 static Jahn-Teller effect, 694–698 theoretical background, 660–661 time-dependent Schro¨ dinger equation, 723–728 total molecular wave function, 661–668, 674–678 vibrational wave function, 687–692 Local harmonic approximation (LHA), direct molecular dynamics, Gaussian wavepacket propagation, 378–381 Local hyperspherical surface functions (LHSFs), electronic states, triatomic quantum reaction dynamics, partial wave expansion, 315–317 784 subject index Localized molecular orbital/generalized valence bond (LMO/GVB) method, direct molecular dynamics, ab initio multiple spawning (AIMS), 413–414 Longuet-Higgins phase-change rule: conical intersections: chemical reaction, 446–453 pericyclic reactions, 447–450 pi-bond reactions, 452–453 sigma bond reactions, 452 comparison with other techniques, 487– 493 loop construction, 441–446 dynamic phase properties, 210 loop construction: cyclopentadienyl cation (CPDC), 467–472 cyclopentadienyl radical (CPDR), 464–467 Jahn-Teller theorem, 461–472 non-adiabatic coupling, 148–168 geometric phase effect, two-dimensional two-surface system, 148–157 quasi-Jahn-Teller model, scattering calculation, 150–155 historical background, 145–148 Jahn-Teller systems, 119–122 theoretical background, 42–44 three-particle reactive system, 157–168 D þ H2 reaction: quasiclassical trajectory (QCT) calculation, 160–163 semiclassical calculation, 163–167 H þ D2 reaction, quasiclassical trajectory calculation, 167–168 permutational symmetry, 1H3 isotopomers, 717 theoretical background, 434–435 Loop construction: conical intersections, photochemical systems, 453–460 four-electron systems, 455–458 larger four-electron systems, 458–459 multielectron systems, 459–460 three-electron systems, 455 phase-change rule and, 441–446 coordinate properties, 443–446 qualitative molecular photochemistry, 472– 482 ammonia, 480–481 benzene derivatives, 479–480 butadiene, 474–479 cyclooctatetraene (COT), 482 cyclooctene isomerization, 473–474 ethylene, 472–473 inorganic complexes, 481–482 theoretical background, 434–435 LSTH potential energy parameters: non-adiabatic coupling, quasiclassical trajectory (QCT) calculation: H þ D2 reaction, 167–168 three-particle reactive system, D þ H2 reaction, 160–163 semiclassical calculation, D þ H2 reaction, 166–167 Manifold approximation, non-adiabatic coupling, line integral conditions, adiabatic-to-diabatic transformation matrix, 53 Marcus theory, electron nuclear dynamics (END), intramolecular electron transfer, 349–351 Maslov index, molecular systems, 212 Mass polarization effect, electronic state adiabatic representation, Born-Huang expansion, 287–289 Matrix elements, Renner-Teller effect, triatomic molecules, 594–598 Maxwell equation, non-adiabatic coupling, pseudomagnetic field, 97 Minimal diabatic potential matrix, non-adiabatic coupling, 81–89 Minimal models, Renner-Teller effect, triatomic molecules, 615–618 Minimal residuals (MINRES) filter diagonalization, permutational symmetry: dynamic Jahn-Teller and geometric phase effects, 699–711 theoretical background, 660–661 Minimum energy method (MEM), direct molecular dynamics, Gaussian wavepacket propagation, 379–381 Minimum energy path (MEP), direct molecular dynamics, theoretical background, 358– 361 Mixed-state trajectory: conical intersection research, 495–496 direct molecular dynamics: Ehrenfest dynamics, 396–399 error sources, 403–404 subject index molecular mechanics valence bond (MMVB), 411 Mixing angle, non-adiabatic coupling, two-state molecular system, H3 molecule, 104– 109 Mo¨ bius strip, phase-change rule: ammonia and chiral systems, 457–458 general bond reactions, 452–453 pericyclic reactions, 448–450 pi bond reactions, 452–453 sigma bond reactions, 452 Modulus-phase formalism, molecular systems, 205 component amplitude analysis, 214–215, 217–218 Lagrangean properties: Dirac electrons, 266–268 topological phase, 270–272 Lagrangean-density correction term, 269– 270 nearly nonrelativistic limit, 268–269 nonrelativistic electron, 263–265 nonrelativistic/relativistic cases, 262–263 potential fluid dynamics and quantum mechanics, 265–266 spinor phases, 272 Molecular dynamics: adiabatic molecular dynamics, 362–381 Gaussian wavepacket propagation, 377– 381 initial condition selection, 373–377 nuclear Schro¨ dinger equation, 363–373 conical intersection location, 491–492 degenerate states chemistry, xii–xiii direct molecular dynamics, theoretical background, 356–362 geometric phase theory, single-surface nuclear dynamics, vector-potential, molecular Aharonovo-Bohm effect, 25–31 Molecular-fixed coordinates, crude BornOppenheimer approximation, hydrogen molecule, Hamiltonian equation, 514– 516 Molecular mechanics (MM) potentials, direct molecular dynamics: complete active space self-consistent field (CASSCF) technique, non-adiabatic systems, 406–411 theoretical background, 359–361 785 Molecular mechanics valence bond (MMVB): conical intersection location, 489–490 direct molecular dynamics: complete active space self-consistent field (CASSCF) technique, non-adiabatic systems, 406–411 theoretical background, 359–361 Molecular orbital-conical intersection (MO-CI): Longuet-Higgins phase-change rule, cyclopentadienyl radical (CPDR), 464–467 two-state systems, 438 Molecular orbital (MO) theory: conical intersection research, 493–496 crude Born-Oppenheimer approximation, hydrogen molecule, minimum basis set calculation, 548–550 direct molecular dynamics: ab initio multiple spawning (AIMS), 413–414 AM1 Hamiltonian, 415 complete active space self-consistent field (CASSCF) technique, non-adiabatic systems, 405–411 nuclear motion Schro¨ dinger equation, 372–373 phase-change rule: chemical reactions, 450–453 cyclopentadienyl cation (CPDC), 467–472 Molecular systems: analytic theory, component amplitudes, 214–233 Cauchy-integral method, 219–220 cyclic wave functions, 224–228 modulus and phase, 214–215 modulus-phase relations, 217–218 near-adiabatic limit, 220–224 reciprocal relations, 215–217, 232–233 wave packets, 228–232 electron nuclear dynamics (END), 337–351 final-state analysis, 342–349 intramolecular electron transfer, 349–351 reactive collisions, 338–342 four-state molecular system, non-adiabatic coupling: quantization, 60–62 Wigner rotation/adiabatic-to-diabatic transformation matrices, 92 786 subject index Molecular systems: (Continued) modulus-phase formalism, Lagrangean properties: Dirac electrons, 266–268 topological phase, 270–272 Lagrangean-density correction term, 269– 270 nearly nonrelativistic limit, 268–269 nonrelativistic electron, 263–265 nonrelativistic/relativistic cases, 262–263 potential fluid dynamics and quantum mechanics, 265–266 spinor phases, 272 multiple degeneracy non-linearities, 233–249 adiabatic-to-diabatic transformation, 241– 242 component phase continuous tracing, 236– 241 conical intersection pairing, 235–236 direct integration, 242–243 experimental phase probing, 248–249 Jahn-Teller/Renner-Teller coupling effects, 243–248 complex representation, 243–244 generalized Renner-Teller coupling, 247 off-diagonal coupling, 246–247 off-diagonal element squaring, 245–246 phase factors, 205–214 quantum theory and, 198–205 three-state molecular system, non-adiabatic coupling: minimal diabatic potential matrix, noninteracting conical intersections, 81–89 numerical study, 134–137 extended Born-Oppenheimer equations, 174–175 quantization, 59–60 extended Born-Oppenheimer equations, 173–174 sign flip derivation, 73–77 strongly coupled (2,3) and (3,4) conical intersections, ‘‘real’’ three-state systems, 113–117 theoretical-numeric approach, 101–103 Wigner rotation/adiabatic-to-diabatic transformation matrices, 92 two-state molecular system, non-adiabatic coupling: Herzberg-Longuet-Higgins phase, 185 quantization, 58–59 ‘‘real’’ system properties, 104–112 C2H-molecule: (1,2) and (2,3) conical intersections, 109–112 C2H-molecule: (1,2) and (2,3) conical intersections, ‘‘real’’ two-state systems, 109–112 H3 system and isotopic analogues, 103– 109 single conical intersection solution, 97–101 Wigner rotation/adiabatic-to-diabatic transformation matrices, 92 Yang-Mills fields: alternative derivation, 254–255 curl condition, 252–253 future implications, 255–257 Hamiltonian formalism, observability in, 259–261 nuclear Lagrangean equation, 249–250 pure vs.

Tensorial gauge fields, 251–252 tensorial field vanishing criteria, 257–259 untruncated Hilbert space, 253–254 Momentum operator, non-adiabatic coupling, Longuet-Higgins phase-based treatment, three-particle reactive system, 157–168 MORBID Hamiltonian, Renner-Teller effect, triatomic molecules, benchmark handling, 621–623 Morse oscillator: non-adiabatic coupling: quantum dressed classical mechanics, 179 quasiclassical trajectory (QCT) calculation, three-particle reactive system, D þ H2 reaction, 160–163 semiclassical calculation, D þ H2 reaction, 164–167 Renner-Teller effect, triatomic molecules, benchmark handling, 622–623 Morse potentials, direct molecular dynamics, Gaussian wavepacket propagation, 378– 383 Mulliken population, electron nuclear dynamics (END), intramolecular electron transfer, 349–351 Multiconfiguration self-consistent field (MCSCF) technique, direct molecular dynamics: complete active space self-consistent field (CASSCF) technique, non-adiabatic systems, 404–411 subject index theoretical background, 358–361 vibronic coupling, diabatic representation, 385–386 Multiconfiguration time-dependent Hartree (MCTDH) method, direct molecular dynamics: Gaussian wavepacket propagation, 380–381 nuclear motion Schro¨ dinger equation, 364– 373 theoretical background, 357–361 Multidegenerate conditions: molecular system non-linearities, 233–249 adiabatic-to-diabatic transformation, 241– 242 component phase continuous tracing, 236– 241 conical intersection pairing, 235–236 direct integration, 242–243 experimental phase probing, 248–249 Jahn-Teller/Renner-Teller coupling effects, 243–248 complex representation, 243–244 generalized Renner-Teller coupling, 247 off-diagonal coupling, 246–247 off-diagonal element squaring, 245–246 non-adiabatic coupling, 80–81 Wigner rotation/adiabatic-to-diabatic transformation matrices, 91–92 Multiple independent spawning (MIS), direct molecular dynamics, non-adiabatic coupling, 402 Multiple spawning, direct molecular dynamics: ab initio multiple spawning, 411–414 non-adiabatic coupling, 399–402 Multivalued matrix elements, non-adiabatic coupling: adiabatic-to-diabatic transformation matrix, 126–132 Herzberg-Longuet-Higgins phase, Jahn-Teller model, 185–186 minimal diabatic potential matrix, 83–89 Mystery band, direct molecular dynamics, vibronic coupling, 381–382 Na3F2 cluster, direct molecular dynamics, semiempirical studies, 415 Near-adiabatic limit, molecular systems, component amplitude analysis, 220–224 Near-degenerate states, permutational symmetry, vibrational levels, 728–733 787 Neumann boundary conditions, electronic states, adiabatic-to-diabatic transformation, two-state system, 304–309 Newton-Raphson equation, conical intersection location: locations, 565 orthogonal coordinates, 567 Non-Abelian theory, molecular systems, Yang-Mills fields: nuclear Lagrangean, 250 pure vs.

See Direct molecular dynamics Oosterhoff correlation diagram, conical intersection research, 494–496 790 subject index Open-path phase: molecular systems, multidegenerate nonlinear coupling, 242–243 properties, 210 Operator definitions, phase properties, 206–207 Optical phases, properties, 206–207 Orbital overlap mechanism, phase-change rule, chemical reactions, 450–453 Orthogonal transformation matrix: conical intersections, spin-orbit interaction: invariant parameters, 574–576 seam loci, 576–578 molecular systems, 204–205 non-adiabatic coupling: adiabatic-to-diabatic transformation, 122–123 Longuet-Higgins phase-based treatment, two-dimensional two-surface system, scattering calculation, 151–155 two-state molecular system, H3 molecule, 104–109 Orthonormalization: electron nuclear dynamics (END), molecular systems, final-state analysis, 343–349 permutational symmetry, GBO approximation/geometric phase, Hilbert space model, 719–721 Out-of-phase states: conical intersection, two-state systems, 438 loop construction, benzene molecules, 479–481 phase-change rule, pericyclic reactions, 448– 450 phase inverting reactions, 496–499 quantitative photochemical analysis, 485–487 Overlap integrals, crude Born-Oppenheimer approximation, angular-momentumadopted Gaussian matrix elements, 518–519 Pairing approximation, phase inverting reactions, 499 Pancharatnam phase, properties, 206 Parabolical insertions, non-adiabatic coupling, topological spin, 70–73 Parallel transported eigenstates, geometric phase theory, 10–11 Partial wave expansion, electronic states, triatomic quantum reaction dynamics, 312–317 Pauli principle: conical intersections: phase-change rule, chemical reaction, 446–453 pericyclic reactions, 447–450 pi-bond reactions, 452–453 sigma bond reactions, 452 two-state chemical reactions, 436–438 degenerate states chemistry, xii–xiii loop construction, coodinate properties, 443–446 permutational symmetry, rotational wave function, 685–687 Pauli spin matrices, geometric phase theory, eigenvector evolution, 14–17 Pegg-Barnett operators, phase properties, 207–208 Pericyclic reactions, phase-change rule, 447–450 Permutational symmetry: adiabatic states, conical intersections: invariant operators, 735–737 Jahn-Teller theorem, 733–735 antilinear operator properties, 721–723 degenerate/near-degenerate vibration levels, 728–733 degenerate states chemistry, xiii electronic wave function, 680–682 energy functional form, 737–738 GBO approximation and geometric phase, two-dimensional Hilbert space model, 718–721 geometric phase theory, single-surface nuclear dynamics, 30–31 group theoretical issues, 668–674 nuclear spin function, 678–680 phase-change rule, 451–453 rotational wave function, 683–687 rovibronic/vibronic wave functions, 682–683 2 S systems: alkali metal trimers, 712–713 dynamic Jahn-Teller and geometric phase effects, 698–711 electron/nuclear spin effects, 711–712 1 H3 isotopomers, 713–717 nonadiabatic coupling effects, 711 potential energy surfaces, 692–694 static Jahn-Teller effect, 694–698 theoretical background, 660–661 subject index time-dependent Schro¨ dinger equation, 723–728 total molecular wave function, 661–668, 674–678 vibrational wave function, 687–692 Perturbation theory: conical intersections: location, 488–489 spin-orbit interaction, 559, 561–563 time-reversal symmetry, 563–564 crude Born-Oppenheimer approximation, basic principles, 510–512 electronic states, quantum reaction dynamics, 285–286 non-adiabatic coupling, two-state molecular system, single conical intersection solution, 97–101 permutational symmetry, total molecular wave function, 665–668 Renner-Teller effect: tetraatomic molecules: Á electronic states, 647–653 Å electronic states, 641–646 triatomic molecules, minimal models, 615–618 Petelin-Kiselev (PK) model, Renner-Teller effect, tetraatomic molecules, 625–633 Å electronic states, 634–640 Phase-change rule.

### Advances in Chemical Physics. The Role of Degenerate States in Chemistry by Baer M., Billing G.D. (eds.)

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