Ultrafast processes in molecules

1. Ultrafast processes in molecules IX – Surface hopping implementation Mario Barbatti barbatti@kofo.mpg.de

2. Surface hopping Wave packet propagation Surfacehoppingpropagation • Tully, Preston, JCP 55, 562 (1971)

3. the semi-classical propagation

4. Semi-classical dynamics Adiabatic basis Diabatic basis

5. Semi-classical dynamics Any standard method can be used in the integration of the Newton’s equations. A goodoneistheVelocityVerlet Foreachnucleusm: • Swope et al. J. Chem. Phys. 76, 637 (1982)

6. Time-step • Schlick, Barth and Mandziuk, Annu. Rev. Biophys. Struct. 26, 181 (1997)

7. Time-step • Time step should not be larger than 1 fs (1/10v). • Dt = 0.5 fs assures a good level of conservation of energy. • Exceptions: • Dynamics close to the conical intersection may require 0.25 fs • Dissociation processes may require even smaller time steps

8. Semi-classical TDSE SC-TDSE The SC-TDSE is solved with standard methods (Unitary Propagator, Adams Moulton 6th-order, Butcher 5th-oder)

9. Time-step for the SC-TDSE Dt = 0.5 fs Dt/ms . . . h(t) h(t+Dt)

10. Decoherence Uncorrected Butatriene cation

11. Decoherence TDSE E Q SC-TDSE E Q • Schwartz, Bittner, Prezhdo, Rossky, J. Chem. Phys. 104, 5942 (1996) • Zhu, Jasper, Truhlar, J. Chem. Phys. 120, 5543 (2004) • Granucci, Persico, Zoccante, J. Chem. Phys. 133, 134111 (2010)

12. Decoherence Decoherenceis introduced ad hoc by correcting the time dependent coefficients: • Granucciand Persico, J. Chem. Phys. 126, 134114 (2007)

13. Decoherence Corrected Uncorrected Butatriene cation

14. Hop probability: Landau-Zener 0.57 0.43

15. Hop probability: fewest-switches • Tully, J Chem Phys 93, 1061 (1990)

16. Hopping algorithm A hopping will take place if two conditions are satisfied: 1) A uniformly selected random number rt in the [0, 1] interval is such that 2) The energy gap between the final and initial states satisfies

17. Frustrated hopping Forbidden hop Total energy E R • Howtotreatsuch situations: • Reject all classicallyforbiddenhopandkeepthemomentum • Reject all classicallyforbiddenhopandinvertthemomentum • Usethe time uncertaintyprincipletosearchfor a pointwherethehopisallowed • Jasper, Stechmann, Truhlar, J. Chem. Phys. 116, 5424 (2002)

18. Adjustment after hopping E R Total energy KN(t) KN(t+Dt) After hop, what are the new nuclear velocities? • Redistribute the energy excess equally among all degrees • Adjust velocities components in the direction of the nonadiabatic coupling h12 • Adjust velocities components in the direction of the difference gradient vector g12 • Pechukas, Phys. Rev. 181, 174 (1969) • Fabiano, Keal, Thiel, Chem. Phys. 349, 334 (2008)

19. Initial conditions: semiclassical spectrum • Wigner Distribution for harmonic oscillator • For each Rn in {R,P} • Compute cross section • Compare to experiments • Barbatti, Aquino, Lischka, PCCP 12, 4959 (2010) • Barbatti,PCCP 13, 4686 (2011)

20. Initial conditions: semiclassical spectrum {R,P}

21. surface hopping: panoramic view

22. Surface hopping: fewest-switches • Solve SE for Rc • Solve Newton’s equation on one surface • Integrate the SC-TDSE • Compute transition probability • Decide surface for next time step Step 1 is the computational bottleneck • If it hops, then adjust momentum • Repeat procedure until the end of the trajectory • Compute many trajectories

23. Surface hopping: advantages…

24. Surface hopping: …and disadvantages

25. Surface hopping benchmarks & reviews

26. Surface hopping applications • See References in: • Barbatti, WIREs: Comp. Mol. Sci. 1, 620 (2011)

27. Surface hopping generalization • Tully, Faraday Discuss. 110, 407 (1998) • Herman, J. Chem. Phys. 103, 8081 (1995)

28. the newton-x program

29. Newton-X (www.newtonx.org) Barbatti, Granucci, Ruckenbauer, Plasser, Crespo-Otero, Pittner, Persico, Lischka NEWTON-X: A Package for Newtonian Dynamics Close to the Crossing Seam (2007-2013) • Barbatti, Granucci, Persico, Ruckenbauer, Vazdar, Eckert-Maksic, Lischka, J PhotochemPhotobiol A 190, 228 (2007)

30. Newton-X (www.newtonx.org)

31. NEWTON-X: nxinp ------------------------------------------ NEWTON-X Newton dynamics close to the crossing seam ------------------------------------------ MAIN MENU 1. GENERATE INITIAL CONDITIONS 2. SET BASIC INPUT 3. SET GENERAL OPTIONS 4. SET NONADIABATIC DYNAMICS 5. GENERATE TRAJECTORIES 6. SET STATISTICAL ANALYSIS 7. EXIT Select one option (1-7):

32. NEWTON-X: nxinp ------------------------------------------ NEWTON-X Newton dynamics close to the crossing seam ------------------------------------------ SET BASIC OPTIONS nat: Number of atoms. There is no value attributed to nat Enter the value of nat : 6 Setting nat = 6 nstat: Number of states. The current value of nstat is: 2 Enter the new value of nstat : 3 Setting nstat = 3 nstatdyn: Initial state (1 - ground state). The current value of nstatdyn is: 2 Enter the new value of nstatdyn : 2 Setting nstatdyn = 2 prog: Quantum chemistry program and method 0 - ANALYTICAL MODEL 1 - COLUMBUS 2.0 - TURBOMOLE RI-CC2 2.1 - TURBOMOLE TD-DFT The current value of prog is: 1 Enter the new value of prog : 1

33. Newton-X: file structure

34. Three ways to go FSSH needs coefficients c: 1) Through: A) First order nonadiabatic couplings (NAC) Explicit evaluation of Only few methods (MCSCF, MRCI) B) CI overlap (OVL) Numerical evaluation of Any method allowing to get a CI-like wavefunction (TDDFT, ADC(2), …) • Hammes-Schiffer and Tully, J Chem Phys 101, 4657 (1994) • Pittner, Lischka, MB, ChemPhys356, 147 (2009)

35. Three ways to go FSSH needs coefficients c: 2) Through: C) Local diabatization (LD) Löwdinorthogonalization of CI overlap matrix S Any method allowing to get a CI-like wavefunction (TDDFT, ADC(2), …) • Granucci, Toniolo, Persico, J Chem Phys 114, 10608 (2001) • Plasser, Granucci, Pittner, MB, Persico, Lischka, J Chem Phys 137, 22A314 (2012)

36. Three ways to go • Plasser, Granucci, Pittner, MB, Persico, Lischka, J Chem Phys 137, 22A314 (2012)

37. The nuclear Ensemble Method

38. Urocanicacid hn t67 trans cis • Review: Gibbs, Tye, Norval, PhotochemPhotobiolSci7, 655 (2008)

39. UA’s anomalous photophysics hn UVC UVB UVA Ftrans→cis (289 nm) = 0.08 Ftrans→cis (302 nm) = 0.31 t67 Ftrans→cis (313 nm) = 0.49 trans cis

40. Tautomeric effects in UA Exp. Theor. • Barbatti,PCCP 13, 4686 (2011)

41. Absorption cross sections First order of the time-dependent perturbation theory The problemcanberecast in the time domain: • Sakurai (1994) Modern Quantum Mechanics • Tannor, Heller, J Chem Phys 77, 202 (1982)

42. Extinction coefficients Relation between cross section (cm2) and extinction coefficient (M-1cm-1)

43. Overlaps The core of the method is to compute the overlap needed to integrate Tannor and Heller proposed a analytical solution based on harmonic oscillator

44. Post-Condon But we want to go beyond the Condon approximation. Starting from Tannor-Heller equation: The expansion to second order is This motivates to introduce the following functional: • Crespo-Otero, Barbatti, TheorChemAcc131, 1237 (2012)

45. Integration

46. Monte-Carlo Integration If we have a ground state distribution of points: We can integrate the cross section by Monte-Carlo and get:

47. Sampling the ground state Using a Wigner distribution to a harmonic oscillator, we can sample the ground state: Another way of sampling is to run a very long trajectory in the ground state and pick points from it. But be careful with the right temperature!

48. Pros and Cons • Pros: • Easy to use • Clear conceptual basis • Absolute heights • Absolute widths • Post Condon Approximation • Dark vibronic bands • Implemented in Newton-X • Cons: • No vibrational resolution • No non-adiabatic info • No info on excited state wavefunction • One arbitrary parameter

49. Example: Benzene TD-CAM-B3LYP/TZVP

50. Dark bands, vibronic coupling azomethane • Szalay, Aquino, Barbatti, Lischka, Chem Phys 380, 9 (2011)