Ultrafast processes in molecules

1. Ultrafast processes in molecules V – Finding conical intersections Mario Barbatti barbatti@kofo.mpg.de

2. Where are the conical intersections? formamide pyridone Antol et al. JCP 127, 234303 (2007)

3. Primitive conical intersections

5. Conical intersections: Twisted-pyramidalized Barbatti et al. PCCP 10, 482 (2008)

6. Conical intersections in rings: Stretched-bipyramidalized

7. CH2NH2+ MXS The biradical character Aminopyrimidine MXS

8. p1* p2 S0 ~ (p2)2 S1 ~ (p2)1(p1*)1 The biradical character

9. One step back: single p-bonds p*2 CH2NH2+ CH2CH2 pp* pp* sp* p2 p2 p*2 CH2CHF CH2SiH2 pp* p*2 pp* p2 p2 Barbatti et al. PCCP 10, 482 (2008)

10. p*2 C2H4 pp* p2 One step back: single p-bonds b

11. One step back: single p-bonds The energy gap at 90° depends on the electronegativity difference (d) along the bond. Michl and Bonačić-Koutecký, Electronic Aspects of Organic Photochem. 1990

12. One step back: single p-bonds • d depends on: • substituents • solvation • other nuclear coordinates For a large molecule is always possible to find an adequate geometric configuration that sets d to the intersection value.

13. Urocanic acid • Major UVB absorber in skin • Photoaging • UV-induced immunosuppression

14. Finding conical intersections • Conventional geometry optimization: • Minimize: f(R) = EJ • Conical intersection optimization: • Minimize: f(R) = EJ • Subject to: EJ – EI = 0 • HIJ = 0 • Three basic algorithms: • Penalty function (Ciminelli, Granucci, and Persico, 2004; MOPAC) • Gradient projection (Beapark, Robb, and Schlegel 1994; GAUSSIAN) • Lagrange-Newton (Manaa and Yarkony, 1993; COLUMBUS) Keal et al., Theor. Chem. Acc. 118, 837 (2007)

15. This term minimizes the energy average This term (penalty) minimizes the energy difference Penalty function Function to be optimized: Recommended values for the constants: c1 = 5 (kcal.mol-1)-1 c2 = 5 kcal.mol-1

16. Gradient DE2 E E E2 E2 E1 Projection of gradient of EJ E1 Rparallel Rx Rperpend Rx Gradient projection method Minimize in the branching space: EJ - EI Minimize in the intersection space: EJ

17. Minimize energy difference along the branching space Minimize energy along the intersection space Gradient projection method Gradient used in the optimization procedure: Constants: c1 > 0 0 < c2 1

18. Suppose that L was determined at x0 and l0. If L(x,l) is quadratic, it will have a minimum (or maximum) at [x1 = x0 + Dx, l1 = l0 + Dl], where Dx and Dl are given by: Lagrange-Newton Method A simple example: Optimization of f(x) Subject to r(x) = k Lagrangian function:

19. Lagrange-Newton Method

20. Lagrange-Newton Method Solving this system of equations for Dx and Dl will allow to find the extreme of L at (x1,l1). If L is not quadratic, repeat the procedure iteratively until converge the result.

21. allows for geometric restrictions restricts energy difference to 0 minimizes energy of one state restricts non-diagonal Hamiltonian terms to 0 Lagrange-Newton Method In the case of conical intersections, Lagrangian function to be optimized:

22. Expanding the Lagrangian to the second order, the following set of equations is obtained: Compare with the simple one-dimensional example: Lagrange-Newton Method Lagrangian function to be optimized:

23. Expanding the Lagrangian to the second order, the following set of equations is obtained: Solve these equations for Update Repeat until converge. Lagrange-Newton Method Lagrangian function to be optimized:

24. Comparison of methods LN is the most efficient in terms of optimization procedure. GP is also a good method. Robb’s group is developing higher-order optimization based on this method. PF is still worth using when h is not available. Keal et al., Theor. Chem. Acc. 118, 837 (2007)

25. Crossing of states with different multiplicities Example: thymine Serrano-Pérez et al., J. Phys. Chem. B 111, 11880 (2007)

26. Crossing of states with different multiplicities Lagrangian function to be optimized: Now the equations are: Different from intersections between states with the same multiplicity, when different multiplicities are involved the branching space is one dimensional.

27. Three-states conical intersections Example: cytosine Kistler and Matsika, J. Chem. Phys. 128, 215102 (2008)

28. Conical intersections between three states Lagrangian function to be optimized: This leads to the following set of equations to be solved: Matsika and Yarkony, J. Chem. Phys. 117, 6907 (2002)

29. Fast H elimination Slow H elimination Example of application: photochemistry of imidazole Devine et al. J. Chem. Phys. 125, 184302 (2006)

30. Fast H elimination Slow H elimination Example of application: photochemistry of imidazole Fast H elimination:ps* dissociative state Slow H elimination: dissociation of the hot ground state formed by internal conversion How are the conical intersections in imidazole? Devine et al. J. Chem. Phys. 125, 184302 (2006)

31. Predicting conical intersections: Imidazole

32. Barbatti et al., J. Chem. Phys. 130, 034305 (2009)

33. It is not a minimum on the crossing seam, it is a maximum! Crossing seam Geometry-restricted optimization (dihedral angles kept constant)

34. Pathways to the intersections

35. At a certain excitation energy: 1. Which reaction path is the most important for the excited-state relaxation? 2. How long does this relaxation take? 3. Which products are formed?

36. Time evolution

38. Next lecture • Transition probabilities Contact barbatti@kofo.mpg.de