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

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

A unique subject matters quantity at the position of degenerate states within the prime sequence on chemical physicsEdited through Nobel Prize-winner Ilya Prigogine and popular authority Stuart A. Rice, the Advances in Chemical Physics sequence offers a discussion board for serious, authoritative reviews in each region of the self-discipline. In a structure that encourages the expression of person issues of view, specialists within the box current accomplished analyses of matters of curiosity. This stand-alone, detailed themes quantity, edited by way of Gert D. Billing of the college of Copenhagen and Michael Baer of the Soreq Nuclear study heart in Yavne, Israel, reviews fresh advances at the function of degenerate states in chemistry. quantity 124 collects leading edge papers on "Complex States of straightforward Molecular Systems," "Electron Nuclear Dynamics," "Conical Intersections and the Spin-Orbit Interaction," and lots of extra similar issues. Advances in Chemical Physics is still the most excellent venue for displays of recent findings in its box.

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See also Quantum reaction dynamics electron nuclear dynamics (END), timedependent variational principle (TDVP), general reactions, 334–337 geometric phase theory: quadratic Jahn-Teller effect, 22–23 single-surface nuclear dynamics, 23–31 molecular Aharonov-Bohm effect, vector-potential theory, 25–31 vibronic multiplet ordering, 24–25 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, 678–682 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, 713–717 789 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 Nuclear Lagrangean equation, molecular systems, Yang-Mills fields, 249–250, 255–257 Nuclear motion Schro¨ dinger equation: direct molecular dynamics, 363–373 vibronic coupling, adiabatic effects, 382–384 electronic states: adiabatic representation, 289–290 adiabatic-to-diabatic transformation, 293–295 diabatization matrix, 296–300 diabatic representation, 292–293 triatomic quantum reaction dynamics, partial wave expansion, 313–317 principles of, 417–420 Nuclear spin function, permutational symmetry, 678–680, 711–712 Nuclei subsystems, permutational symmetry, total molecular wave function, 677–678 Off-diagonal elements: adiabatic-to-diabatic transformation matrix, quantization, 67 conical intersection location, 488–489 multidegenerate nonlinearity: generalized coupling, 246–247 squaring-off method, 245–246 permutational symmetry, total molecular wave function, 666–668 One-dimensional representations: conical intersections, spin-orbit coupling, 558–559 Renner-Teller effect: theoretical principles, 585–586 triatomic molecules, pragmatic models, 620–621 On-the-fly molecular dynamics.

Tensorial gauge fields, 250–253 Non-adiabatic coupling: adiabatic-to-diabatic transformation matrix analyticity, 123–126 derivation, 47–48 historical background, 40–44 line integral approach, 50–57 quasidiabatic framework, 53–57 single-valued diabatic potentials and topological matrix, 50–53 orthogonality, 122–123 quantization, 63–67 single/multivaluedness, 126–132 solution conditions, 48–50 Wigner rotation matrix and, 89–92 conical intersections: Born-Oppenheimer approximation, matrix elements, 186–191 coordinate origin removal, 137–138 extended Born-Oppenheimer equations: closed path matrix quantization, 171– 173 theoretical principles, 144–148 three-state matrix quantization, 173–174 three-state system analysis, 174–175 Herzberg-Longuet-Higgins phase-based treatment, Jahn-Teller model, 185–186 Jahn-Teller systems, Longuet-Higgins phase, 119–122 Longuet-Higgins phase-based treatment, 148–168 geometric phase effect, two-dimensional two-surface system, 148–157 three-particle reactive system, 157–168 quantum dressed classical mechanics, 177– 183 geometric phase effect, 180–183 vector potential formulation, 191–196 curl condition, Yang-Mills field, 92–97 pseudomagnetic field, 95–96 788 subject index Non-adiabatic coupling: (Continued) vector potential theory, 93–95 diabatic potential matrix, minimal conditions, 81–89 noninteracting conical intersections, 85–89 diabatic representation, 132–134 direct molecular dynamics: ab initio multiple spawning, 411–414 CASSCF techniques, 404–411 direct dynamics, 410–411 MMVB method, 406–410 Ehrenfest dynamics, 395–397 Gaussian wavepackets and multiple spawning, 399–402 mixed techniques, 403–404 semiempirical studies, 414–415 theoretical background, 356–362 trajectory surface hopping, 397–399 vibronic effects, 381–393 adiabatic properties, 382–384 conical intersections, 386–389 diabatic properties, 384–386 Hamiltonian model, 389–393 geometric phase theory, 2–3 sign flip interpretation, 77–80 historical background, 40–44 Jahn-Teller model, Longuet-Higgins phase, 119–122 molecular systems, 203–205 Yang-Mills fields, nuclear Lagrangean, 249–250 multidegenerate case, 80–81 nuclear motion Schro¨ dinger equation, principles of, 419–420 permutational symmetry, 711 quantization: general case techniques, 63–67 model systems, 57–63 extensions, 62–63 four-state case, 60–62 three-state case, 59–60 two-state system, 58–59 sub-Hilbert space construction, 67–69 sub-sub-Hilbert space construction, 69–70 theoretic-numerical approach: three-state system in plane, 101–103 two-state system in plane: conical intersection distribution solution, 101 single conical intersection solution, 97–101 three-state molecular systems: numerical study, 134–137 sign flip derivation, 73–77 strongly coupled (2,3) and (3,4) conical intersections, ‘‘real’’ three-state systems, 113–117 theoretic-numerical in plane, 101–103 topological spin, 70–73 two-state molecular systems: C2H-molecule: (1,2) and (2,3) conical intersections, ‘‘real’’ two-state systems, 109–112 H3 system and isotopic analogues, ‘‘real’’ systems, 103–109 theoretic-numerical approach, in-plane systems: conical intersection distribution solution, 101 single conical intersection solution, 97– 101 Noncrossing rule, geometric phase theory, 2 Nondemolition measurements, phase interference, 207 Nonlinear coupling, multidegenerate conditions: higher order coupling, complex representations, 243–244 molecular systems, 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 Nonlinear molecules: permutational symmetry: electronic wave function, 681–682 static Jahn-Teller effect, 696–698 vibrational wave function, 688–692 Renner-Teller effect, 606–610 Nonrelativistic states: conical intersections, spin-orbit interaction, seam loci, 573–574 molecular systems, modulus-phase formalism: subject index electron configuration, 263–265 nearly nonrelativistic limit, 268–269 theoretical background, 262–263 Nonremovable couplings, electronic states, adiabatic-to-diabatic transformation, two-state systems, 301–309 Nonvanishing matrix elements, crude BornOppenheimer approximation, hydrogen molecule, minimum basis set calculation, 546–550 Normalization factor, angular-momentumadopted Gaussian matrix elements, crude Born-Oppenheimer approximation, 517 Nuclear dynamics.

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.

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