Ultrafast processes in molecules

1. Ultrafast processes in molecules IV – Non-crossing rule and conical intersections Mario Barbatti barbatti@kofo.mpg.de

2. The non-crossing rule “For diatomics, the potential energy curves of the electronic states of the same symmetry species cannot cross as the internuclear distance is varied.” von Neumann and Wigner, Z. Phyzik 30, 467 (1929) Teller, JCP 41, 109 (1937)

3. The energies are given by Simple argument Suppose a two-level molecule whose electronic Hamiltonian is H(R), where R is the internuclear coordinate. Given a basis of unknown orthogonal functions f1 and f2, we want to solve the Schrödinger equation Check it!

4. Simple argument

5. Simple argument (i) (ii) It is unlikely (but not impossible) that by varying the unique parameter R conditions (i) and (ii) will be simultaneously satisfied.

6. Rigorous proof Naqvi and Brown, IJQC 6, 271 (1972)

7. Conical intersections What does happen if the molecule has more than one degree of freedom? • In diatomics the unique parameter R is not enough to satisfy the two conditions for crossing. • In polyatomics there are 3N-6 internal coordinates!

8. The energies are given by Conical intersections Suppose a two-level molecule whose electronic Hamiltonian is H(R), where R are the nuclear coordinates. Given a basis of unknown orthogonal functions f1 and f2, we want to solve the Schrödinger equation

9. A degeneracy at Rx will happen if In general, two independent coordinates are necessary to tune these conditions. Conical intersections In a more compact way: where and

10. Expansion in first order around Rx for S: Conical intersections In a more compact way: where and

11. In first order around Rx each of these terms are: And the energies in a point RX + R are in first order: Conical intersections In a more compact way: where and

12. Conical intersections Writting then

13. Conical intersections Atchity, Xantheas, and Ruedenberg, J. Chem. Phys. 95, 1862 (1991)

14. Linear approximation fails E E2 E1 Rperpend Rx Crossing seam Conical intersections What does happen if the molecule is distorted along a direction that is perpendicular to g and f?

15. Linear approximation fails E E2 E1 Rparallel Rx Crossing seam Conical intersections What does happen if the molecule is distorted along a direction that is parallel to g or f?

16. Branching space • Startingattheconicalintersection, geometricaldisplacement in the „branchingspace“ liftsthedegeneracylinearly. • The branchingspaceisthe plane definedbythevectorsgandf. • Geometricaldisplacementsalongtheother 3N-8 internalcoordinateskeepthedegeneracy (in first order). These coordinatespaceiscalled „seam“ or „intersection“ space. Note that Non-adiabatic coupling vector For this reason the branching space is also referred as g-h space. See the proof, e.g., in Hu at al. J. Chem. Phys. 127, 064103 2007 (Eqs. 2 and 3)

17. Why are non-adiabatic coupling vectors important? • The coupling vectors define one of the directions of the branching space around the conical intersections, which is important for the localization of these points of degeneracy.

18. Conical intersections are not rare “When one encounters a local minimum (along a path) of the gap between two potential energy surfaces, almost always it is the shoulder of a conical intersection. Conical intersections are not rare; true avoided intersections are much less likely.” E E e << 1 a.u. R R r ~ O(1) is the density of zeros in the Hel matrix. Truhlar and Mead, Phys. Rev. A 68, 032501 (2003)

19. Energy R Crossing seam Minimum on the crossing seam (MXS) Conical intersections are connected

20. q b Crossing seam in ethylene C3V H-migration Ethylidene Pyramidalized Barbatti, Paier and Lischka, J. Chem. Phys.121, 11614 (2004)

21. Example of dynamics results

22. t ~ 100-140 fs Ethylidene MXS 11% ~7.6 eV 23% 60% Pyram. MXS H-migration Torsion + Pyramid.

23. It can be rewritten as a general cone equation (Yarkony, JCP 114, 2601 (2001)): pitch parameter tilt parameters asymmetry parameter Conical intersections are distorted

24. Example: pyrrole Energy h01 g01

25. Photoproduct depends on the direction that the molecule leaves the intersection

26. Example: protonated Schiff Base Q r In water In gas phase Burghardt, Cederbaum, and Hynes, Faraday. Discuss. 127, 395 (2004)

27. Ruckenbauer, Barbatti, Niller, and Lischka, JPCA 2009

29. gasphase water

30. Bersuker, The Jahn Teller effect, 2006 Yarkony, Rev. Mod. Phys. 68, 985 (1996) Intersections don’t need to be conical!

31. Next lecture • Finding conical intersections Contact barbatti@kofo.mpg.de