Mathematical Methods for Ab Initio Quantum Chemistry Nice October 20–21, 2006 Symmetry-Adapted Tensorial Formalism to Model Rovibrational and Rovibronic Molecular Spectra Vincent BOUDON Laboratoire de Physique de l’Université de Bourgogne – CNRS UMR Av. A. Savary, BP 47870, F DIJON, FRANCE Web :

Mathematical Methods for Ab Initio Quantum Chemistry Nice October 20–21, 2006 Contents I.Introduction & general ideas II.Symmetry adaptation III.Rovibrational spectroscopy  Spherical tops: CH 4, SF 6, …  Quasi-spherical tops  Other symmetric and asymmetric tops IV.Rovibronic spectroscopy  Jahn-Teller effect, (ro)vibronic couplings, …  Electronic operators  Application to some open-shell systems V.Conclusion & perspectives

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

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

Mathematical Methods for Ab Initio Quantum Chemistry Nice October 20–21, 2006 Case of a symmetric top Quantum number = irreducible representation of a group zz C  v symmetrization: Wang basis C 3v symmetrization: use of projection methods z

Mathematical Methods for Ab Initio Quantum Chemistry Nice October 20–21, 2006 Spherical tops: the O(3)  G group chain G Sphere Lie group O(3) (or SU(2)  C I ) Molecule Point group G (or G S )

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

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

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

Mathematical Methods for Ab Initio Quantum Chemistry Nice October 20–21, 2006 G matrix: Principle of the calculation 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:

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

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

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

Mathematical Methods for Ab Initio Quantum Chemistry Nice October 20–21, 2006 III. Rovibrational spectroscopy S4S4 C3C3

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

Mathematical Methods for Ab Initio Quantum Chemistry Nice October 20–21, 2006 Effective tensorial Hamiltonian Systematic tensorial development Effective Hamiltonian and vibrational extrapolation Coupled rovibrational basis Polyad structure P0P0 P1P1 P2P2 P3P3 Rotation Vibration

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

Mathematical Methods for Ab Initio Quantum Chemistry Nice October 20–21, 2006 Spherical tops S4S4 C3C3 C3C3 C4C4

Mathematical Methods for Ab Initio Quantum Chemistry Nice October 20–21, 2006 The polyads of CH 4  Global fit  Polyad P n :

Mathematical Methods for Ab Initio Quantum Chemistry Nice October 20–21, 2006 Recent spherical top analyses 12 CH 4, 13 CH 4, 12 CD 4 Analysis of high polyads, intensities GeH 4, GeD 4, GeF 4 Fundamental bands (isotopic samples) P 4 3 stretching band 32 SF 6, 34 SF 6 First vibrational levels, hot bands SeF 6, WF 6 Fundamental bands Mo(CO) 6 6 (C  O stretch) band

Mathematical Methods for Ab Initio Quantum Chemistry Nice October 20–21, 2006 The combination band of SF 6

Mathematical Methods for Ab Initio Quantum Chemistry Nice October 20–21, 2006 Quasi-spherical tops: SO 2 F 2 and SF 5 Cl

Mathematical Methods for Ab Initio Quantum Chemistry Nice October 20–21, 2006 Other molecules X 2 Y 4 molecules Example: Ethylene, C 2 H 4 XY 3 Z molecules Examples: CH 3 D, CH 3 Cl, …

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

Mathematical Methods for Ab Initio Quantum Chemistry Nice October 20–21, 2006 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 (ReF 6, IrF 6, NpF 6, …), hexacarbonyles (V(CO) 6, …), radicals (CH 3 O, CH 3 S, …), 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 Modified Born-Oppenheimer approximation Degenerate electronic state  : sum restricted to the [  ] degenerate states 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 The Jahn-Teller « effect » After some rearrangements: ! Q 0 is usually not an equilibrium configuration Electronic energy = 0Electronic operators Hermann Arthur JAHN (1907 – 1979) Edward TELLER (1908 – 2003)

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

Mathematical Methods for Ab Initio Quantum Chemistry Nice October 20–21, 2006 If we include the molecular rotation … Example of the G’ g  F 1u problem (ReF 6 ) 45,000  45,000 for J = 28.5 only ! 45,000  45,000 for J = 28.5 only ! … the problem becomes intractable !

Mathematical Methods for Ab Initio Quantum Chemistry Nice October 20–21, 2006 Constructing electronic operators Electronic state   Electronic angular momentum J e   e = E’:J e = 1/2,operators  e = F:J e = 1,operators   e = G’:J e = 3/2,operators

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

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

Mathematical Methods for Ab Initio Quantum Chemistry Nice October 20–21, 2006 Half-integer states: spinorial representations Vectorial (standard) representations: Projective representations: Spinorial representations: Projective representations that allow to symmetrize SU(2)  C I representations (spin states) into a subgroup G  “ Group ” G S

Mathematical Methods for Ab Initio Quantum Chemistry Nice October 20–21, 2006 Example: The O h “group” S S

Mathematical Methods for Ab Initio Quantum Chemistry Nice October 20–21, 2006 Rhenium hexafluoride (d) 1 2 F 2g 2Eg2Eg G’ g (b) G’ g (X) E’ 2g (a) Re 6+ V oct H so >> (0 cm -1 ) (5015 cm -1 ) (29430 cm -1 ) 129 electrons Strong spin orbit coupling Half-integer angular momenta

Mathematical Methods for Ab Initio Quantum Chemistry Nice October 20–21, 2006 E’1E’1 E’2E’2 G’G’G’G’ The 3 band of ReF 6

Mathematical Methods for Ab Initio Quantum Chemistry Nice October 20–21, 2006 C 3v and its spinorial representations: C 3v S S Complex irreps:

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

Mathematical Methods for Ab Initio Quantum Chemistry Nice October 20–21, 2006 The ground electronic state of CH 3 O SpinOrbit

Mathematical Methods for Ab Initio Quantum Chemistry Nice October 20–21, 2006 Electronic operators for CH 3 O One order 0 non-trivial operator for the ground state:

Mathematical Methods for Ab Initio Quantum Chemistry Nice October 20–21, 2006 Vibronic operators for CH 3 O Case of a doubly degenerate vibration 6 non-trivial operators up to order 2 for v = 1: Presumably 3 main contributions : t 1, t 3 and t 5

Mathematical Methods for Ab Initio Quantum Chemistry Nice October 20–21, 2006 Vibronic levels for an E-mode of CH 3 O VibrationSpin-orbit Spin-Vib. + Orb.-Vib. + …  JT ~ 62 cm -1

Mathematical Methods for Ab Initio Quantum Chemistry Nice October 20–21, 2006 Rovibronic operators and basis for CH 3 O Coupled rovibronic basis: SpinOrbit Vibrational Electronic RotationalVibronic Rovibronic Rovibrational operators

Mathematical Methods for Ab Initio Quantum Chemistry Nice October 20–21, 2006 Comparison with the “usual” approach E electronic state associated to the L = 1, K L = 1 effective quantum numbers through symmetry reasons only (K L 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 V. Conclusion & perspectives

Mathematical Methods for Ab Initio Quantum Chemistry Nice October 20–21, 2006 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 Programs STDS & Co. Spherical Top Data System Molecular parameter database Calculation and analysis programs XTDS : Java interface

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