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

ISBN-10: 0471433462

ISBN-13: 9780471433460

A distinct themes quantity at the position of degenerate states within the major sequence on chemical physicsEdited by way 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 zone 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, distinct issues quantity, edited by way of Gert D. Billing of the collage of Copenhagen and Michael Baer of the Soreq Nuclear examine middle in Yavne, Israel, studies fresh advances at the function of degenerate states in chemistry. quantity 124 collects leading edge papers on "Complex States of easy Molecular Systems," "Electron Nuclear Dynamics," "Conical Intersections and the Spin-Orbit Interaction," and lots of extra comparable subject matters. Advances in Chemical Physics is still the greatest venue for shows of recent findings in its box.

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780 subject index Herzberg-Longuet-Higgins phase: (Continued) quantum dressed classical mechanics, 177–183 geometric phase effect, 180–183 theoretical background, 177–180 permutational symmetry: dynamic Jahn-Teller and geometric phase effects, 698–711 total molecular wave function, 667–668 Hilbert space. See also Full-Hilbert space; SubHilbert space; Sub-sub-Hilbert space Berry’s phase, 209–210 molecular systems, Yang-Mills fields, untruncated Hilbert space, 253–254 non-adiabatic coupling: adiabatic-to-diabatic transformation matrix, quasidiabatic framework, 54–56 Born-Oppenheimer approximation, 189– 191 Born-Oppenheimer-Huang equation, 44– 45 extended Born-Oppenheimer equations, 168–171 theoretical background, 42–44 permutational symmetry, GBO approximation/geometric phase, Hilbert space model, 718–721 phase properties, operators, 207–208 quantum theory, 199 1 H2 molecule, permutational symmetry, rotational wave function, 686–687 1 H3 molecule, permutational symmetry, isotopomers, 713–717 H3 molecule, permutational symmetry: 1 H3 isotopomers, 713–717 potential energy surfaces, 692–694 Homonuclear molecules, permutational symmetry: electronic wave function, 680–682 nuclear spin function, 679–680 rovibronic/vibronic wave functions, 682–683 vibrational wave function, 687–692 Hougen, Bunker, and Johns (HBJ) configuration, Renner-Teller effect: tetraatomic molecules, Hamiltonian equations, 626–628 triatomic molecules, 614–615 pragmatic models, 619–621 Hu¨ ckel’s 4n þ 2 rule: conical intersections, two-state chemical reactions, 436–438 phase change rule: ammonia and chiral systems, 457–458 orbital overlap, 451–452 pericyclic reactions, 448–450 pi bond reactions, 452–453 Hund’s coupling, permutational symmetry, rotational wave function, 684–687 Hydrodynamic theory, direct molecular dynamics, trajectory ‘‘swarms,’’ 421– 422 Hydrogen molecules: crude Born-Oppenheimer approximation: Hamiltonian equation, 512–516 minimum basis set calculation, 542–550 nuclei interaction integrals, 527 H3 molecule: Longuet-Higgins phase-change rule, loop construction, 463–472 phase-change rule, 443–446 two-state system: adiabatic-to-diabatic transformation, 301–309 non-adiabatic coupling, 104–109 H4 molecule, phase-change rule, 443–446 permutational symmetry, total molecular wave function, 675–678 Hyperspherical coordinates: electronic states: adiabatic-to-diabatic transformation, twostate system, 302–309 triatomic quantum reaction dynamics, 310–312 non-adiabatic coupling: Longuet-Higgins phase-based treatment, three-particle reactive system, 158–168 semiclassical calculation, D þ H2 reaction, 164–167 two-state molecular system, H3 molecule, 106–109 vector potential formulation, 191–194 permutational symmetry: potential energy surfaces, 693–694 total molecular wave function, 668 Independent Gaussian approximation (IGA), direct molecular dynamics, Gaussian wavepacket propagation, 379–383 subject index Infinite-order sudden approximation (IOSA), electron nuclear dynamics (END), molecular systems, 345–349 Initial relaxation direction (IRD), direct molecular dynamics, theoretical background, 359–361 Inorganic compounds, loop construction, photochemical reactions, 481–482 In-phase states: conical intersection, two-state systems, 438 phase-change rule, pericyclic reactions, 448– 450 Integral properties, crude Born-Oppenheimer approximation: angular-momentum-adopted Gaussian matrix elements: nuclei interaction, 519–527 overlap integrals, 518–519 equations for, 551–555 Interference effects: molecular systems, 211 phase properties, 206–207 quantum theory, 200 Intraanchor reactions, conical intersection, twostate systems, 437–438 Intramolecular electron transfer, electron nuclear dynamics (END), 349–351 Intrinsic reaction coordinate (IRC), direct molecular dynamics, theoretical background, 358–361 Invariant operators, permutational symmetry, conical intersection, adiabatic state, 735–737 Irreducible representations (IRREPs), permutational symmetry: degenerate/near-degenerate vibrational levels, 728–733 electronic wave function, 681–682 group theoretical properties, 669–674 invariant operators, 735–737 nuclear spin function, 678–680 time-dependent equations, 727–728 total molecular wave function, 667–668 vibrational wave function, 688–692 Isomerization reactions: loop construction: benzene molecules, 479–481 cyclooctenes, 473–474 ethylene photolysis, 472–473 phase-change rules, loop construction, 456 781 quantitative photochemical analysis, 482–487 ‘‘Isomorfic Hamiltonian,’’ Renner-Teller effect, triatomic molecules, 618 Isotopomers, permutational symmetry: alklali metal trimers, 712–713 1 H3 molecule, 713–717 vibrational wave function, 689–692 Jacobi coordinates: electronic state adiabatic representation, Born-Huang expansion, 286–289 electronic states, triatomic quantum reaction dynamics, 310–312 non-adiabatic coupling, vector potential formulation, 191–194 Jahn-Teller effect: canonical intersection, Herzberg-LonguetHiggins theorem, historical background, 144–148 conical intersection location, 489 degenerate states chemistry, x–xiii direct molecular dynamics: conical intersections, 388–389 vibronic coupling, 381–382, 391–393 geometric phase theory: conical intersections, 5–8 E Â E problem, 17–23 linear Jahn-Teller effect, 18–20 principles of, 2–4 quadratic Jahn-Teller effect, 22–23 spin-orbit coupling, 2E state, 20–22 single-surface nuclear dynamics, vectorpotential, molecular Aharonovo-Bohm effect, 28–31 Longuet-Higgins phase-change rule, loop construction, 461–472 multidegenerate nonlinear coupling: E Â E problem, 233–234, 238–241 higher order coupling, 243–248 complex representation, 243–244 interpretation, 248 nonlinear diagonal elements, 247 off-diagonal coupling, 246–247 off-diagonal squaring, 245–246 non-adiabatic coupling: Herzberg-Longuet-Higgins phase, 185–186 Longuet-Higgins phase, 119–122 two-dimensional two-surface system, quasi-Jahn-Teller scattering calculation, 150–155 782 subject index Jahn-Teller effect: (Continued) theoretical background, 41–44 topological spin insertion, 70–73 two-state molecular system, 58–59 permutational symmetry: conical intersection, adiabatic state, 733–735 dynamic effect, 698–711 electron/nuclear spin function, 712 1 H3 isotopomers, 713–717 potential energy surfaces, 692–694 static effect, 694–698 phase properties, 209 Jaynes-Cummings model, phase properties, 206 Jungen-Merer (JM) pragmatic model, RennerTeller effect, triatomic molecules, 619–621 benchmark handling, 621–623 Kekule´ structure: conical intersections, two-state chemical reactions, 436–438 phase-change rule, permutational mechanism, 451–453 Kinetic energy operator (KEO): crude Born-Oppenheimer approximation, basic principles, 507–512 direct molecular dynamics: theoretical background, 360–361 trajectory ‘‘swarms,’’ 420–422 vibronic coupling Hamiltonian, 390–393 electronic states: adiabatic representation, Born-Huang expansion, 287–289 triatomic quantum reaction dynamics, 311–312 non-adiabatic coupling: Born-Oppenheimer approximation, 187– 191 historical background, 145–148 Longuet-Higgins phase-based treatment: semiclassical calculation, D þ H2 reaction, 164–167 three-particle reactive system, 158–168 two-dimensional two-surface system, 149–157 nuclear motion Schro¨ dinger equation, 418–420 Renner-Teller effect: tetraatomic molecules: Å electronic states, 638–640 vibronic coupling, 628–631 triatomic molecules, 594–598 Hamiltonian equations, 612–615 pragmatic models, 620–621 Kramers doublets, geometric phase theory: linear Jahn-Teller effect, 20–22 spin-orbit coupling, 20–22 Kramers-Kronig reciprocity, wave function analycity, 201–205 Kramers’ theorem: conical intersections, spin-orbit interaction, 561 degenerate states chemistry, xiii geometric phase theory, conical intersections, 6–8 permutational symmetry, 712 group theoretical properties, 669–674 rotational wave function, 684–687 Kronecker delta, molecular systems, Yang-Mills fields, nuclear Lagrangean, 249–250 Lagrangian density: electron nuclear dynamics (END), timedependent variational principle (TDVP), 327–328 basic ansatz, 330–333 molecular systems: modulus-phase formalism: correction term, 269–270 Dirac electrons, 266–268 topological phase, 270–272 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 Yang-Mills fields, 249–250, 255–257 Lagrangian multiplier, conical intersection location, 488–489, 565 Laguerre polynomials, Renner-Teller effect, triatomic molecules, 589–598 Lanczos reduction: direct molecular dynamics, nuclear motion Schro¨ dinger equation, 364–373 non-adiabatic coupling, Longuet-Higgins phase-based treatment: subject index semiclassical calculation, D þ H2 reaction, 164–167 two-dimensional two-surface system, scattering calculation, 152–155 Landau-Zener model: direct molecular dynamics: dependency properties, 415–416 trajectory surface hopping, 397–399 non-adiabatic coupling: sub/sub-sub-Hilbert construction, 67–70 topological spin insertion, 70–73 Laplace transform: electronic state adiabatic representation, Born-Huang expansion, 286–289 permutational symmetry, total molecular wave function, 664–668 Legendre polynomials: permutational symmetry, degenerate/neardegenerate vibrational levels, 732–733 Renner-Teller effect, triatomic molecules, benchmark handling, 622–623 Legendre wave function, non-adiabatic coupling, semiclassical calculation, D þ H2 reaction, 164–167 Lie groups, molecular systems, Yang-Mills fields: nuclear Lagrangean, 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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