Jahn-Teller Intersections of Potential Energy Surfaces: A Paradigm for Nonadiabatic Excited-State Dynamics

Transcription

1 Jahn-Teller Intersections of Potential Energy Surfaces: A Paradigm for Nonadiabatic Excited-State Dynamics Horst Köppel Horst.Koeppel@pci.uni-heidelberg.de Theoretische Chemie Physikalisch-Chemisches Institut der Universität Heidelberg

2 Overview A tutorial: E e Jahn-Teller effect Relation to B-O approximation and conical intersections Triatomic hydrogen Cyclopropane radical cation Allene radical cation Reference: H. K., L. Cederbaum and S. Mahapatra, Theory of the Jahn-Teller Effect, in: Handbook of High-Resolution Spectroscopy (M. Quack and F. Merkt, Eds.) Wiley, 2010

3 The Jahn-Teller theorem... stability and degeneracy are not possible simultaneously unless the molecule is a linear one... [Proc. Roy. Soc. A 161, 220 (1937)] Structural aspects, global minimum of (JT or pseudo-jt) distorted potential energy surfaces, see: I.B. Bersuker, various books and publications, and in: The Jahn-Teller Effect (H. K., D. Yarkony and H. Barentzen, Eds.) Springer Series in Chemical Physics, 2009, p. 3. Elegant proof: A. Ceulemans, same volume, p. 25.

4 The Jahn-Teller theorem... stability and degeneracy are not possible simultaneously unless the molecule is a linear one... [Proc. Roy. Soc. A 161, 220 (1937)] Principle of proof: linear terms (in Q asym ) of potential energy matrix Group theory: symmetry selection rule: [Γ el 2 ] Γ vib Three representative cases: Linear molecules [E k2 ] = A 1 + E 2k no JTE! Tetragonal point groups [E 2 ] = A 1 + B 1 + B 2 E b JTE Trigonal point groups [E 2 ] = A 1 + E E e JTE

5 Hamiltonian and PES for E e JT effect

6 Hamiltonian and PES for E e JT effect for ρ 0 (Quadratic coupling g=0). Singular derivative (nonadiabatic) couplings at conical intersections of PES Paradigm for nonadiabatic excited-state dynamics Historically first example: JT intersections

7 Quantum treatment of E e JT dynamics Proc. Royal Soc. (London) A 244, 1 (1958) First quantum treatment of nonadiabatic motion at a conical intersection

8 Model spectra for linear E e JT coupling Arrangement of figure: Energy always increases to the right (vertical axes: intensity / coordinate ρ) Only linear part of rotationally symm. PES is shown; ground state parabolic E=V=0 always in the figure centre; energy spacing matches between lowest 2 panels. FC-type transition indicated by arrow. Line spectrum computed; envelope by convolution with Lorentzians.

9 Model spectra for linear E e JT coupling Broadening of upper-surface vibrational levels through nonadiabatic coupling (~ no change below conical intersection). Requires sufficiently strong coupling. Note regularity due to cylindrical symmetry. See also: J. Slonczewski and V. Moruzzi, Physics 3, 237 (1967) M. O Brien and S. Evangelou, Sol. State Comm. 36, 29 (1980) H.K., L. Cederbaum and W. Domcke, Mol. Phys. 41, 669 (1980)

10 Further issues (1): quadratic coupling g 0 Note: rotational symmetry is lost, mixing of different j-values. k= 5 g=0.2 k= 10 [Spectra from A. Thiel and H. K., J. Chem. Phys. 110, 9371 (1999)]

11 Further issues (2): multi-mode effects Most molecules possess several JT active modes: H= H 1 + H 2 + H 3 + H with [H i, H j ] 0 (i j) Multi-mode spectrum not by convolution (complex)! Multi-mode potential energy surfaces [from: E. Haller, H. K. et al, J. Chem. Phys. 78, 1359 (1983)]

12 Further issues (2): multi-mode effects Most molecules possess several JT active modes: H= H 1 + H 2 + H 3 + H with [H i, H j ] 0 (i j) Multi-mode spectrum not by convolution (complex)! Multi-mode potential energy surfaces [from: E. Haller, H. K. et al, J. Chem. Phys. 78, 1359 (1983)]

13 Generalization: The (E+A) e pseudo-jt Effect

14 Generalization: The (E+A) e pseudo-jt Effect Profound impact of geometric phase on low-energy vibronic motion around a conical intersection. From: J. Schön and H. K., Chem. Phys. Lett. 231, 55 (1994) For related work, see: H. Koizumi and I.B. Bersuker, Phys. Rev. Lett. 83, 3009 (1999)

15 Other related scenarios The (E e + A) e pseudo - JT effect The (E b+ A) e pseudo - JT effect The Σ - Π vibronic coupling problem Bilinear couplings, multimode effects... See: H. K., L. Cederbaum and S. Mahapatra, Theory of the Jahn-Teller Effect, in: Handbook of High-Resolution Spectroscopy (M. Quack and F. Merkt, Eds.) Wiley, 2010 Tutorial on the E e JT effect: P. Tregenna-Piggott and M. Riley, in: The Jahn-Teller Effect (H. K., D. Yarkony and H. Barentzen, Eds.) Springer Series in Chemical Physics, 2009, p. 371

16 Triatomic hydrogen Three H atoms at equilateral triangle: 2 E (D 3h ) H 3 not a stable entity (repulsive GS surface, H + H 2 scattering) Plethora of work (Porter.., Miller.., Kuppermann.. ) Low-order Taylor series inapplicable; use linear JT model for ADT angle (regularized diab. states) V 1,2 from Varandas et al, J. Chem. Phys. 86, 6258 (1987)

17 R γ r

18 H 3 Extremely broad resonances (also breathing mode)

19 (Similar to experimental estimates)

20 The allene radical cation (C 3 H 4+ ) Perpendicular CH 2 planes: D 2d point group Γ el = (E + ) E + B 2 ; Γ vib = 3a 1 b 1 3b 2 4e (also: Woywod & Domcke) Vibronic problem is [E (b 1 + b 2 )+ B] e multi-mode JTE + PJTE Ab initio computation of coupling constants (OVGF with cc-pvtz basis) Dynamical calculations with 10 modes (time-independent) and all 15 modes (time-dependent, with MCTDH algorithm) Further reference: S. Mahapatra, L. Cederbaum and H. K., J. Chem. Phys. 111, (1999); J. Phys. Chem. A 105, 5567 (2001)

21 Cuts through potential energy surfaces 1e MO of allene e modes a 1 + b 2 modes of C 3 H 4 +

22 PE spectra and electronic populations (a) 15 modes w. MCTDH (b) Baltzer et al. (1995) (c) 10 modes w. Lanczos Low-energy vibronic line structure: localized geometrical change

23 PE spectra and electronic populations Higher energy (B state): very irregular discrete line structure! C 3 H 4+ after broadband excitation to B 2

24 The cyclopropane radical cation (C 3 H 6+ ) C-atoms form equilateral triangle: D 3h point group Γ el = E + E ; Γ vib = 3a 1 a 2 4e a 1 2a 2 3e Vibronic problem is [E e + E e ] e multi-mode JTE + PJTE Ab initio computation of coupling constants (OVGF with cc-pvtz basis) Dynamical calculations with 11 modes (time-independent) and 14 modes (time-dependent, with MCTDH algorithm) Further reference: S. Mahapatra, T. S. Venkatesan, H. K. et al, J. Phys. Chem. A 108, 2256 (2004); 111, 1746 (2007).

25 Cuts through potential energy surfaces CP, mode 2 JT- induced inter-state E E intersections

26 First and second PE spectral bands Holland et al. (2002) LVC QVC QVC / PJT 1st / 2nd band (theory) Composite band (exp./theory)

27 Conclusions & outlook Basic characteristics of JT conical intersections: E e JTE Triatomic hydrogen: extremely fast transitions / broad resonances Allene radical cation: broad bands, highly irregular line structure Cyclopropane radical cation: PJT intersections by double JTE Wave-packet propagation: MCTDH (H.-D. Meyer) Similar dynamics: Abelian points groups (F-Bz +, H 2 CO) Current extensions: photochemistry (furan, pyrrole..) Thanks to all co-workers, and for your attention!