The Relativistic Jahn-Teller Effect

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1 The Relativistic Jahn-Teller Effect Wolfgang Domcke Technical University of Munich Leonid V. Poluyanov Russian Academy of Sciences, Moscow JT Conference Fribourg 010 1

2 Motivation Science 33, 489 (009) JT Conference Fribourg 010

3 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 JT Conference Fribourg 010 3

4 Spin-Orbit Coupling spin-orbit coupling is a relativistic effect Dirac - Coulomb - Breit Hamiltonian exact oneelectron kinematics Nonrelativistic Coulomb int. retardation and magnetic interaction W. Heisenberg, Z. Physik 39, 514 (196) W. Pauli, Z. Physik 43, 601 (197) G. Breit, Phys. Rev. 34, 553 (199) Reduction from four to two components Breit-Pauli operator: JT Conference Fribourg 010 4

5 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 JT Conference Fribourg 010 5

6 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 (C 3v, D 3h ): spin-free: relativistic (C' 3v, D' 3h ): E modes are JT active (E x E JT effect) tetragonal groups (D d, C 4v, D 4h ): E modes are JT active relativistic (D' d, C' 4v, D' 4h ) : spin-free: B 1, B modes are JT active in E states (E x B JT effect) B 1, B and E modes are JT active in E states (E x (B + E ) JT effect) JT Conference Fribourg 010 6

7 Spin-free and relativistically generalized JT selection rules tetrahedral (T d ) and octahedral (O h ) 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 G 3/ states (G 3/ x (T + E ) JT effect) JT Conference Fribourg 010 7

8 Spin-free and relativistically generalized JT selection rules linear molecules (C v, D h ): spin-free: relativistic (C' v, D' h ) : Renner effect, no first-order JT effect The bending mode ( ) is JT active through relativistic forces JT Conference Fribourg 010 8

9 The E x E JT Hamiltonian in trigonal systems equilateral triatomic system (D 3h ) adiabatic potentials (g = 0): (g = 0) linear and quadratic JT coupling Mexican Hat JT Conference Fribourg 010 9

10 The E x E Jahn-Teller Hamiltonian with spin-orbit coupling in trigonal systems adiabatic potentials: unitary transformation: JT Conference Fribourg

11 The E x B Jahn-Teller Hamiltonian in tetragonal systems D d four-atomic system (e. g. AlO 4 ) vibrational modes: Γ = A 1 + B 1 + B + E E state: adiabatic potentials: linear E x B JT coupling elliptical Mexican Hat JT Conference Fribourg

12 The relativistic E x (B + E ) JT Hamiltonian in tetragonal systems E mode E state unitary transformation: adiabatic potentials: JT Conference Fribourg 010 1

13 The electrostatic T x T Hamiltonian The in tetrahedral sytems JT Conference Fribourg

14 The T x T spin-orbit vibronic Hamiltonian electronic basis states JT Conference Fribourg

15 The T x T Jahn-Teller Hamiltonian L. V. Poluyanov and W. Domcke, Chem. Phys., in press JT Conference Fribourg

16 The T (G 3/ ) x T Hamiltonian adiabatic potentials: four-dimensional Mexican Hat JT Conference Fribourg

17 The E (G 3/ ) 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 JT Conference Fribourg

18 P 4 + As 4 + L. S. Wang et al., JCP 93, 6318 (1990) JT Conference Fribourg

19 X 4+ tetrahedral radial cations of group V elements (X = P, As, Sb, Bi) T mode T T E E CASSCF, nonrelativistic T T CASSCF, relativistic E E JT Conference Fribourg

20 X 4+ tetrahedral radial cations of group V elements (X = P, As, Sb, Bi) E mode T T Sb 4 + E Bi 4 + E CASSCF, nonrelativistic E T T CASSCF, relativistic E E JT Conference Fribourg 010 0

21 Relativistic and Electrostatic JT Coupling Parameters of P 4+, As 4+, Sb 4+, Bi 4 + D. Opalka, M. Segado, L. V. Poluyanov and W. Domcke, PRA 81, (010) JT Conference Fribourg 010 1

22 Linear Molecules: The Renner Effect isolated state =1) E. Renner, Z. Physik 9, 17 (1934) Q e 1 x i y i Q x iq y diabatic electronic basis states degenerate bending mode W + H 1 T N 1 H RT W - H RT E c e i c e E i c : RT coupling constant symmetry properties: H, j 0 j i i K l is a good quantum number JT Conference Fribourg 010

23 Relativistic Π Renner Hamiltonian H 1 SO T N 1 H RT 4 SO H RT E c e de,3 / i i 0 c e E i,1/ 0 de i de E 0 c e i,1/ i 0 de c e E i,3 / i d : linear spin-orbit vibronic coupling (SO-VC) constant E 3/ -E,1/ = SO splitting Symmetry properties of H SO RT, J 0 SO H, Tˆ 0 RT z SO H RT J i z z ˆ 0 1 T cˆc 1 0 L. V. Poluyanov and W. Domcke Chem. Phys. 301, 111 (004) JT Conference Fribourg 010 3

24 Sears Resonances in GeCH ground state of GeCH complexity of the spectrum: 1. Renner-Teller effect. strong spin-orbit coupling 3. Fermi resonances S.-G. He et al. J. Chem. Phys. 119, (003) JT Conference Fribourg 010 4

25 Sears Resonances in GeCH Parameter Fitted Value Expt. Fit Value Calculated Value (cm -1 ) (cm -1 ) , State Obs. energy in cm -1 Obs. energy -Calc. energy in cm -1 With SO-VC Without SO-VC d (cm -1 ) S. Mishra, V. Vallet, L. V. Poluyanov, W. Domcke, J. Chem. Phys. 13, (005) c c JT Conference Fribourg 010 5

26 Motivation Science 33, 489 (009) JT Conference Fribourg 010 6

27 High-order expansion of T x (T + T ) JT potential-energy surfaces in tetrahedral systems combine JT theory with invariant theory of polynomials application: CH 4 + T bending: up to 1th order; T stretching: up to 8th order D. Opalka and W. Domcke, JCP 13, (010) cuts along the bending modes: see Poster P0 (Daniel Opalka) ab initio data: full-valence CASSCF plus internally contracted MRCI JT Conference Fribourg 010 7

28 High-order expansion of T x E JT potential-energy surfaces in tetrahedral systems expansion up to 10th order in the E mode application: CH 4 + a b D. Opalka and W. Domcke, CPL 494, 134 (010) ab initio data: full-valence CASSCF plus internally contracted MRCI JT Conference Fribourg 010 8

29 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 M E 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 JT Conference Fribourg 010 9

30 Coworkers Leonid V. Poluyanov Institute of Chemical Physics, Academy of Sciences, Moscow Sabyashachi Mishra TUM (PhD student, now University of Zürich) Padmabati Mondal TUM (PhD student) see Poster P16 Daniel Opalka TUM (PhD student) see Poster P JT Conference Fribourg