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

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

A different themes quantity at the function of degenerate states within the prime 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 presents a discussion board for severe, authoritative reviews in each quarter of the self-discipline. In a structure that encourages the expression of person issues of view, specialists within the box current accomplished analyses of topics of curiosity. This stand-alone, targeted themes quantity, edited by way of Gert D. Billing of the college of Copenhagen and Michael Baer of the Soreq Nuclear learn heart in Yavne, Israel, experiences contemporary advances at the function of degenerate states in chemistry. quantity 124 collects cutting edge papers on "Complex States of easy 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 efficient venue for shows of recent findings in its box.

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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, 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.

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.

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