The Relativistic Jahn-Teller Effect

Transcription

1 The Relativistic Jahn-Teller Effect Wolfgang Domcke Technical University of Munich Leonid V. Poluyanov Russian Academy of Sciences, Moscow 1

2 The Jahn-Teller Theorem: A configuration of a polyatomic molecule for an electronic state having orbital degeneracy cannot be stable with respect to all displacements of the nuclei unless in the original configuration the nuclei all lie on a straight line H. A. Jahn and E. Teller, Proc. Roy. Soc. A, 161, 0 (1937) T h e The E x E Jahn-Teller Effect: Example: X3 molecule (D3h symmetry) Degenerate electronic state: Ψx,Ψy (E) Degenerate vibrational modes: Qx, Qy (E) Mexican hat dynamic JT effect static JT effect J a h n T e

3 The non-born-oppenheimer coupling operator: The topological phase ( Berry phase ): γn(c ) = π adiabatic electronic wave functions are 4 π - periodic 3

4 Adiabatic and diabatic representations (i) adiabatic representation: (floating electronic basis) two electronic states: (ii) diabatic representation: (smooth electronic basis) adiabatic representation: Hel is diagonal, TK is nondiagonal (Λnm) diabatic representation: Hel is nondiagonal (Unm), TK is diagonal 4

5 Motivation Science 33, 489 (009) 5

6 Vibronic Coupling R. Renner 1934 H. A. Jahn and E. Teller 1937 W. Moffit and W. Thorson 1957 H. C. Longuet-Higgins et al R. L. Fulton and M. Gouterman 1961 (i) Representation of the nonrelativistic electronic Hamiltonian in a (quasi)diabatic basis (ii) Taylor expansion of the Hamiltonian in normal coordinates at a suitable reference geometry (usually up to second order) (iii) Symmetry selection rules (in the point group) for the matrix elements 6

7 Spin-Orbit Coupling spin-orbit coupling is a relativistic effect Dirac - Coulomb - Breit Hamiltonian exact oneelectron kinematics Nonrelativistic Coulomb int. W. Heisenberg, Z. Physik 39, 514 (196) W. Pauli, Z. Physik 43, 601 (197) G. Breit, Phys. Rev. 34, 553 (199) retardation and magnetic interaction Reduction from four to two components Breit-Pauli operator: 7

8 Spin-Orbit Vibronic Coupling (i) Representation of the Breit-Pauli spin-orbit operator in a (nonrelativistic) quasidiabatic electronic basis (ii) Taylor expansion of the spin-orbit operator in normal coordinates at a suitable reference geometry (iii) Symmetry selection rules (in the spin double group) for the matrix elements 8

9 Spin-free and relativistically generalized JT selection rules H. A. Jahn and E. Teller, Proc. Roy. Soc. A 161, 0 (1937) H. A. Jahn, Proc. Roy. Soc. A 164, 117 (1938) trigonal groups (C3v, D3h ): relativistic (C'3v, D'3h ): spin-free: E modes are JT active (E x E JT effect ) tetragonal groups (Dd, C4v, D4h ): E modes are JT active relativistic (D'd, C'4v, D'4h ) : spin-free: B1, B modes are JT active in E states (E x B JT effect) B1, B and E modes are JT active in E states (E x (B + E ) JT effect) 9

10 Spin-free and relativistically generalized JT selection rules tetrahedral (Td ) and octahedral (Oh ) groups : spin-free: relativistic (T'd, O'h ): E modes are JT active in E states (E x E JT effect) T and E modes are JT active in T states (T x (T+E ) JT effect) E and T modes are JT active in G3/ states (G3/ x (T + E ) JT effect) 10

11 Spin-free and relativistically generalized JT selection rules linear molecules (C v, D h ): spin-free: Renner effect, no first-order JT effect relativistic (C' v, D' h ) : The bending mode (Π ) is JT active through relativistic forces 11

12 The E x E JT Hamiltonian in trigonal systems equilateral triatomic system (D3h) adiabatic potentials (g = 0): (g = 0) linear and quadratic JT coupling Mexican Hat 1

13 The Breit-Pauli operator for the equilateral triangle (D3h)), example: the equilateral triangle (D3h), single unpaired electron symmetry operations of the spin double group: for example: 13

14 The E x E Jahn-Teller Hamiltonian with spin-orbit coupling in trigonal systems adiabatic potentials: unitary transformation: 14

15 Transition-Metal Trifluorides: CASSCF Adiabatic Potentials and Vibronic Spectra MnF3 without SO coupling E' 5 with SO coupling 15

16 Transition-Metal Trifluorides: CASSCF Adiabatic Potentials and Vibronic Spectra without SO coupling CoF3 E' 5 with SO coupling 16

17 The E x B Jahn-Teller Hamiltonian in tetragonal systems Dd four-atomic system (e. g. AlO4) vibrational modes: Γ = A1 + B1 + B + E E state: linear E x B JT coupling adiabatic potentials: elliptical Mexican Hat 17

18 The relativistic E x (B + E ) JT Hamiltonian in tetragonal systems E mode E state unitary transformation: adiabatic potentials: 18

19 The electrostatic T x T Hamiltonian in tetrahedral sytems The

20 The T x T spin-orbit vibronic Hamiltonian electronic basis states 0

21 The T x T Jahn-Teller Hamiltonian L. V. Poluyanov and W. Domcke Chem. Phys. 374, 86 (010) 1

22 The T (G3/ ) x T Hamiltonian adiabatic potentials: four-dimensional Mexican Hat

23 The geometric phases of the T(G3/) x T Hamiltonian adiabatic electronic eigenvectors: Berry phase: plane of integration contour: 3

24 The E (G3/ ) x T Hamiltonian There is no electrostatic vibronic coupling involving the T mode in E states E x T Jahn-Teller coupling is a purely relativistic effect electronic basis states: adiabatic potentials: L. V. Poluyanov and W. Domcke, JCP 19, 410 (008) four-dimensional Mexican Hat 4

25 Sb4+ As4+ L. S. Wang et al., JCP 93, 6318 (1990) 5

26 X4+ tetrahedral radial cations of group V elements (X = P, As, Sb, Bi) T mode T T CASSCF, nonrelativistic E T E E CASSCF, relativistic T E 6

27 X4+ tetrahedral radial cations of group V elements (X = P, As, Sb, Bi) E mode T Sb4+ Bi4+ T E T E CASSCF, nonrelativistic E E CASSCF, relativistic T E 7

28 Relativistic and Electrostatic JT Coupling Parameters of P4+, As4+, Sb4+, Bi4+ D. Opalka, M. Segado, L. V. Poluyanov and W. Domcke, PRA 81, (010) 8

29 Photoelectron spectrum of As4 (T x T) relativistic zeroth order nonrelativistic relativistic first order 9

30 Photoelectron spectrum of Sb4 (T x T) relativistic zeroth order nonrelativistic relativistic first order 30

31 Conclusions Relativistic JT coupling is significant for systems containing heavy atoms. There exists a largely unexplored world of vibronic-coupling phenomena beyond the nonrelativistic approximation, for example: (i) relativistic pseudo-jahn-teller effect in ME states of tetragonal systems (ii) relativistic Jahn-Teller effect in E and T states of tetrahedral and octahedral systems. (iii) relativistic pseudo-jahn-teller effect in spatially degenerate states of linear molecules Jahn-Teller theory is the paradigm for more general conical intersections. 31

32 Coworkers Leonid V. Poluyanov Sabyashachi Mishra Institute of Chemical Physics, Academy of Sciences, Moscow TUM (PhD student, now University of Zürich) TUM (PhD student) Padmabati Mondal Daniel Opalka TUM (PhD student) 3