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

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

3. 3 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  1 and  2, we want to solve the Schrödinger equation Check it! The energies are given by

4. 4 Simple argument

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

6. 6 Rigorous proof Naqvi and Brown, IJQC 6, 271 (1972) Suppose that the degeneracy occurs at R 0 : R0R0 E R R+RR+R E 1 = E 0 +  E 1 E 2 = E 0 +  E 2 To have a “crossing” not only the degeneracy condition is necessary. It is also need:

7. 7 Rigorous proof Schrödinger equation for the first state: Small displacement from R 0 to R 0 +  R: Neglecting terms in  2 : Multiply by to the left and integrate over the nuclear coordinates: Prove it!

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

9. 9 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. (a)If only (i) is satisfied it is not a crossing, but a complete state degeneracy for any R. (for any R) (b) If the states have different symmetries, (ii) is trivially satisfied because:

10. 10 Conical intersections In diatomics the unique parameter R is not enough to satisfy the two conditions for crossing. In polyatomics there are 3N-6 internal coordinates! What does happen if the molecule has more than one degree of freedom?

11. 11 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  1 and  2, we want to solve the Schrödinger equation The energies are given by

12. 12 Conical intersections In a more compact way: where and A degeneracy at R x will happen if In general, two independent coordinates are necessary to tune these conditions.

13. 13 Conical intersections In a more compact way: where and Expansion in first order around R x for  :

14. 14 Conical intersections In a more compact way: where and In first order around R x each of these terms are: And the energies in a point R X + R are in first order:

15. 15 Conical intersections Writting then

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

17. 17 Conical intersections What does happen if the molecule is distorted along a direction that is perpendicular to g and f? Linear approximation fails Crossing seam E R perpend RxRx E1E1 E2E2

18. 18 Conical intersections What does happen if the molecule is distorted along a direction that is parallel to g or f? Linear approximation fails Crossing seam E R parallel RxRx E1E1 E2E2

19. 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. 20 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. Why are non-adiabatic coupling vectors important?

21. 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.” Truhlar and Mead, Phys. Rev. A 68, 032501 (2003) R E  << 1 a.u. R E  ~ O(1) is the density of zeros in the H el matrix.

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

23. Ethylidene Pyramidalized H-migration Barbatti, Paier and Lischka, J. Chem. Phys. 121, 11614 (2004)   C 3V Crossing seam in ethylene

24. Example of dynamics results

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

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

27. 27 h 01 g 01 Energy Example: pyrrole

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

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

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

31. 31

32. 32 gasphasewater

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

34. 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