Femtochemistry: A theoretical overview Mario Barbatti V – Finding conical intersections This lecture can be downloaded at lecture5.ppt

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

3 Conical intersectionStructureExamples TwistedPolar substituted ethylenes (CH 2 NH 2 + ) PSB3, PSB4 HBT Twisted-pyramidalizedEthylene 6-membered rings (aminopyrimidine) 4MCF Stilbene Stretched- bipyramidalized Polar substituted ethylenes Formamide 5-membered rings (pyrrole, imidazole) H-migration/carbeneEthylidene Cyclohexene Out-of-plane OFormamide Rings with carbonyl groups (pyridone, cytosine, thymine) Bond breakingHeteroaromatic rings (pyrrole, adenine, thiophene, furan, imidazole) Proton transferWatson-Crick base pairs Primitive conical intersections

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

6 Conical intersections in rings: Stretched-bipyramidalized

7 The biradical character Aminopyrimidine MXS CH 2 NH 2 + MXS

8 The biradical character 22 1*1* S 0 ~ (  2 ) 2 S 1 ~ (  2 ) 1 (  1 * ) 1

9 One step back: single  -bonds Barbatti et al. PCCP 10, 482 (2008) 22   2 CH 2 SiH 2 22   2 CH 2 CH 2 22   CH 2 NH 2 + 22   2 CH 2 CHF

10 One step back: single  -bonds 22   2 C2H4C2H4C2H4C2H4 

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

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

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

14 Finding conical intersections 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) Conical intersection optimization: Minimize: f(R) = E J Subject to: E J – E I = 0 H IJ = 0 Keal et al., Theor. Chem. Acc. 118, 837 (2007) Conventional geometry optimization: Minimize: f(R) = E J

15 Penalty function Function to be optimized: This term minimizes the energy average Recommended values for the constants: c 1 = 5 (kcal.mol -1 ) -1 c 2 = 5 kcal.mol -1 This term (penalty) minimizes the energy difference

16 Gradient projection method E R perpend RxRx E1E1 E2E2 E R parallel RxRx E1E1 E2E2 Minimize in the branching space: Minimize in the intersection space: E J - E I EJEJ Gradient  E 2 Projection of gradient of E J

17 Gradient projection method Gradient used in the optimization procedure: Constants: c 1 > 0 0 < c 2  1 Minimize energy difference along the branching space Minimize energy along the intersection space

18 Lagrange-Newton Method A simple example: Optimization of f(x) Subject to  (x) = k Lagrangian function: Suppose that L was determined at x 0 and 0. If L(x, ) is quadratic, it will have a minimum (or maximum) at [x 1 = x 0 +  x, 1 = 0 +  ], where  x and  are given by:

19 Lagrange-Newton Method

20 Lagrange-Newton Method Solving this system of equations for  x and  will allow to find the extreme of L at (x 1, 1 ). If L is not quadratic, repeat the procedure iteratively until converge the result.

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

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

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

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

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

31 Predicting conical intersections: Imidazole

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

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

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

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