Femtochemistry: A theoretical overview

1. Femtochemistry: A theoretical overview IV – Non-crossing rule and conical intersections Mario Barbatti mario.barbatti@univie.ac.at This lecture can be downloaded at http://homepage.univie.ac.at/mario.barbatti/femtochem.html lecture4.ppt

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. E E2 = E0 + dE2 E1 = E0 + dE1 R0 R R+dR To have a “crossing” not only the degeneracy condition is necessary. It is also need: Rigorous proof Suppose that the degeneracy occurs at R0: Naqvi and Brown, IJQC 6, 271 (1972)

7. Small displacement from R0 to R0 + dR: Neglecting terms in d2: Multiply by to the left and integrate over the nuclear coordinates: Prove it! Rigorous proof Schrödinger equation for the first state:

8. Because of the second condition Therefore Expanding in Taylor to the first order Rigorous proof After repeating the same steps for the second state:

9. (b) If the states have different symmetries, (ii) is trivially satisfied because: (for any R) Rigorous proof Having a crossing between two states requires two conditions: (i) degeneracy (ii) crossing It is unlikely (but not impossible) that by varying the unique parameter R conditions (i) and (ii) will be simultaneously satisfied. • If only (i) is satisfied it is not a crossing, but a complete state degeneracy for any R.

10. 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!

11. 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

12. 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

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

14. 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

15. Conical intersections Writting then

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

17. 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?

18. 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?

19. Branching space • Starting at the conical intersection, geometrical displacement in the „branching space“ lifts the degeneracy linearly. • The branching space is the plane defined by the vectors g and f. • Geometrical displacements along the other 3N-8 internal coordinates keeps the degeneracy (in first order). These coordinate space is called „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)

20. 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.

21. 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)

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

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

24. Example of dynamics results

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

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

27. Example: pyrrole Energy h01 g01

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

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

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

32. gasphase water

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

34. Next lecture • Finding conical intersections Contact mario.barbatti@univie.ac.at This lecture can be downloaded at http://homepage.univie.ac.at/mario.barbatti/femtochem.html lecture4.ppt