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Symmetry-Adapted Tensorial Formalism to Model Rovibrational and Rovibronic Molecular Spectra

Symmetry-Adapted Tensorial Formalism to Model Rovibrational and Rovibronic Molecular Spectra. Vincent BOUDON Laboratoire de Physique de l’Université de Bourgogne – CNRS UMR 5027 9 Av. A. Savary, BP 47870, F-21078 DIJON, FRANCE E-mail : Vincent.Boudon@u-bourgogne.fr

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Symmetry-Adapted Tensorial Formalism to Model Rovibrational and Rovibronic Molecular Spectra

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  1. Symmetry-Adapted Tensorial Formalism to Model Rovibrational and Rovibronic Molecular Spectra Vincent BOUDON Laboratoire de Physique de l’Université de Bourgogne – CNRS UMR 5027 9 Av. A. Savary, BP 47870, F-21078 DIJON, FRANCE E-mail : Vincent.Boudon@u-bourgogne.fr Web : http://www.u-bourgogne.fr/LPUB/tSM.html Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  2. Contents • Introduction & general ideas • Symmetry adaptation • Rovibrational spectroscopy • Spherical tops: CH4, SF6, … • Quasi-spherical tops • Other symmetric and asymmetric tops • Rovibronic spectroscopy • Jahn-Teller effect, (ro)vibronic couplings, … • Electronic operators • Application to some open-shell systems • Conclusion & perspectives Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  3. I. Introduction & general ideas Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  4. Why tensorial formalism ? • Take molecular symmetry into account • Simplify the problem (block diagonalization, …) • Also consider approximate symmetries • Systematic development of rovibrational/rovibronic interactions, for any polyad scheme • Effective Hamiltonian and transition moments construction • Global analyses Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  5. II. Symmetry adaptation Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  6. z z z Case of a symmetric top Quantum number = irreducible representation of a group Cv symmetrization: Wang basis C3v symmetrization: use of projection methods Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  7. Sphere Molecule G Point groupG (or GS) Lie group O(3) (or SU(2)CI) Spherical tops: the O(3) G group chain Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  8. Rank / O(3) symmetry (irrep) O(3) z-axis projection / component Symmetry / G irrep G Component Multiplicity index What do all these indexes mean ? Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  9. Example 1: “Octahedral harmonic” of rank 4 Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  10. > 0 < 0 Antisymmetric function Example 2: Rank 3 harmonic, A2 symmetry Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  11. The idea consists in diagonalizing a typical octahedral (or tetrahedral) splitting term: In the standard |j,m> basis this amounts to diagonalize the matrix: The eigenvectors lead to the G matrix; the eigenvalues are oriented Clebsch-Gordan coefficients: G matrix: Principle of the calculation Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  12. • Build symmetry-adapted tensorial operators: Construction of Hamiltonian and transition-moment operators • Calculate symmetry-adapted coupling coefficients: 3j-p (p = nCs) 3j-m (Wigner) Coupling of operators, calculation of matrix elements Use of the G matrix Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  13. • In the O(3)  G group chain: Matrix element (p = nCs) Reduced matrix element « physical part » Group-dependant phase factor Coupling coefficient « geometric part » • In the G group: Matrix elements: the Wigner-Eckart theorem • Recoupling formulas: Using 6C, 9C, 12C coefficients, etc. Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  14. ~ ~ ~ ~ Quasi-spherical tops: Reorientation Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  15. S4 C3 III. Rovibrational spectroscopy Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  16. • Rotation, recursive construction of Moret-Bailly & Zhilinskií: • Vibration, construction of Champion: Rotational & vibrational operators Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  17. Polyad structure P3 • Systematic tensorial development Vibration Rotation P2 • Coupled rovibrational basis P1 • Effective Hamiltonian and vibrational extrapolation P0 Effective tensorial Hamiltonian Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  18. Dipole moment Direction cosines tensor Rovibrational operators Stone coefficients Parameters • Polarizability Transition moments Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  19. S4 C4 C3 C3 Spherical tops Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  20.  Global fit  Polyad Pn: The polyads of CH4 Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  21. Recent spherical top analyses Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  22. The 2+4 combination band of SF6 Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  23. Quasi-spherical tops: SO2F2 and SF5Cl Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  24. X2Y4 molecules Example: Ethylene, C2H4 • XY3Z molecules Examples: CH3D, CH3Cl, … Other molecules Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  25. IV. Rovibronic spectroscopy Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  26. The problem: Degenerate electronic states • Open-shell molecules have degenerate electronic states. We only consider rovibronic transitions inside a single isolated degenerate electronic state. • Transition-metal hexafluorides (ReF6, IrF6, NpF6, …), hexacarbonyles (V(CO)6, …), radicals (CH3O, CH3S, …), etc have a degenerate electronic ground state. • In this case, the Born-Oppenheimer approximation is no more valid. There are complex rovibronic couplings (Jahn-Teller, …). • Molecules with an odd number of electrons have half-integer angular momenta: use of spinorial representations. Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  27. Degenerate electronic state : sum restricted to the [] degenerate states Modified Born-Oppenheimer approximation • Inclusion of non-adiabatic interactions among the [] multiplet • Non-adiabatic interactions with other states neglected Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  28. Electronic energy = 0 Electronic operators After some rearrangements: ! Q0 is usually not an equilibrium configuration Hermann Arthur JAHN (1907 – 1979) Edward TELLER (1908 – 2003) The Jahn-Teller « effect »  Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  29. Infinite matrices  truncation • HJT is non perturbative ! ! EE problem : linear JT levels Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  30. 45,000  45,000 for J = 28.5 only ! Example of the G’gF1u problem (ReF6) If we include the molecular rotation … … the problem becomes intractable ! Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  31. Ge = E’ :Je = 1/2, operators • Ge = F :Je = 1, operators • Ge = G’ :Je = 3/2, operators Constructing electronic operators Electronic stateGElectronic angular momentumJe Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  32. Rovibronic effective Hamiltonian Electronic Vibration Rotation • Coupled rovibronic basis • Rovibronic transition moments Dipole moment: And similarly for the polarizability … Rovibronic operators Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  33. J. Chem. Phys. 114, 10773–10779 (2001) Threefold degenerate electronic state The 6 (C–O stretch) band of V(CO)6 Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  34. • Vectorial (standard) representations: • Projective representations: • Spinorial representations: Projective representations that allow to symmetrizeSU(2)  CI representations (spin states) into a subgroup G  “ Group ” GS Half-integer states: spinorial representations Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  35. S Example: The Oh “group” Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  36. 129 electrons • Strong spin orbit coupling • Half-integer angular momenta (29430 cm-1) 2Eg G’g(b) (d)1 (5015 cm-1) E’2g(a) 2F2g G’g(X) >> Re6+ Voct Hso (0 cm-1) Rhenium hexafluoride Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  37. n3 n2 + n6 E’1 E’2 G’ G’ The n3 band of ReF6 Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  38. Complex irreps: S C3v and its spinorial representations: C3v Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  39. S Cv and its spinorial representations: Cv Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  40. Spin Orbit The ground electronic state of CH3O Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  41. One order 0 non-trivial operator for the ground state: Electronic operators for CH3O Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  42. 6 non-trivial operators up to order 2 for v = 1: Presumably 3 main contributions : t1, t3 and t5 Case of a doubly degenerate vibration Vibronic operators for CH3O Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  43. ~ 62 cm-1 Spin-Vib. + Orb.-Vib. + …  JT Vibration Spin-orbit Vibronic levels for an E-mode of CH3O Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  44. Coupled rovibronic basis: Spin Orbit Vibrational Electronic Rotational Vibronic Rovibronic Rovibrational operators Rovibronic operators and basis for CH3O Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  45. Comparison with the “usual” approach • E electronic state associated to the L = 1, KL= 1 effective quantum numbers through symmetry reasons only (KL does not need to be identified to ) • Separate JT calculation replaced by tensorial operators built on powers of L and S • Global spin-orbital-vibrational-rotational calculation • All vibronic levels in a given polyad considered as a whole • Method based on symmetry (construction of invariants in a group chain); the link to the “usual” physical (JT) problem is not straightforward Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  46. V. Conclusion & perspectives Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  47. Future developments • Rovibronic transitions between different electronic states • General rovibronic model • Analytic derivation of the effective Hamiltonian and transition moments from an ab initio potential energy surface • Analytic contact transformations • Cf. work of Vl. Tyuterev (Reims) on triatomics Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  48. Programs STDS& Co. Spherical Top Data System www.u-bourgogne.fr/LPUB/shTDS.html • Molecular parameter database • Calculation and analysis programs • XTDS : Java interface Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

  49. Acknowledgments • M. Rotger, A. El Hilali, M. Loëte, N. Zvereva-Loëte, Ch. Wenger, J.-P. Champion, F. Michelot (Dijon) • M. Rey (Reims) • D. Sadovskií, B. Zhilinskií (Dunkerque) • M. Quack et al. (Zürich) • … Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

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