1. Femtochemistry: A theoretical overview Mario Barbatti mario.barbatti@univie.ac.at VIII – Surface hopping implementation This lecture can be downloaded at http://homepage.univie.ac.at/mario.barbatti/femtochem.html lecture8.ppt

2. MQCD methods: surface hopping Use the energy gradient to update the nuclear geometry according to the Newton`s Eq. 2 For the new nuclear geometry (only!), solve the TDSE and correct classical solution by performing a hopping if necessary.3 Go back to step 1 and repeat the procedure until the end of the trajectory.4 Repeat procedure for a large number of trajectories to have the “classical wave packet”.5 For a fixed nuclear geometry, solve time- independent Schrödinger Eq. for electrons. Get the energy gradient and the couplings1

3. Q E Transition probability is evaluated at each time step Classical nuclear motion on the on-the-fly BO surface Tully, J. Chem. Phys. 93, 1061 (1990) A stochastic algorithm decides on which surface the molecule will continue MQCD methods: surface hopping

4. 4 Newton equations A good one is the Velocity Verlet For each nucleus I Any standard method can be used in the integration of the Newton equations. Swope et al. JCP 76, 637 (1982)

5. 5 Schlick, Barth and Mandziuk, Annu. Rev. Biophys. Struct. 26, 181 (1997). Time-step for the classical equations

6. 6 Time step should not be larger than 1 fs (1/10v).  t = 0.5 fs assures a good level of conservation of energy most of time. Exceptions: Dynamics close to the conical intersection may require 0.25 fs Dissociation processes may require even smaller time steps

7. 7 On the-fly approach: (Dis)advantages Advantages: It is not need to get the complete surface. Only that regions spanned during the dynamics It dispenses interpolation, extrapolation and fitting schemesDisadvantages: Time-expensive dynamics No non-local effects (tunneling) Preparation of surfacedynamicsConventional On-the-fly Total time

8. 8 R(t), v(t) t+  t, R(t+  t), v(t+  t/2) For R(t+  t) compute: E k, E l,  E k, h kl v(t+  t) a kk, P k  l (t+  t) Initial condition generation Statistical analysis On-the-fly outputs

9. 9 Quantum equations Time-derivative couplingNon-adiabatic coupling vector with (adiabatic basis)

10. 10 Semi-classical time-dependent Schrödinger equation R c is the classical Newtonian geometry The SC-TDSE is solved with standard methods (Unitary Propagator, Adams Moulton 6 th -order, Butcher 5 th -oder) SC-TDSE

11. 11 h(t)h(t) h(t+t)h(t+t) t/mst/ms...  t = 0.5 fs Time-step for the SC-TDSE

12. 12 Alternative way to deal with the TDSE Time derivative coupling Non-adiabatic coupling vector When using methods for which h is not available, it is possible to compute the time-derivative coupling numerically. Is to compute the time-derivative coupling numerically faster than to compute the non-adiabatic coupling vector analytically?

13. 13 Problems... Butatriene cation Uncorrected Average adiabatic population in certain state is not equal to the fraction of trajectories (occupation) in that state.

14. 14 Decoherence Q E Q E TDSE SC-TDSE Granucci and Persico, J. Chem. Phys. 126, 134114 (2007)

15. 15 Decoherence Because in the SC-TDSE the “wave-packet” split among the several electronic surface is kept correlated by the coordinate R c, the time propagation is fully coherent. Decoherence is introduced ad hoc by correcting the time dependent coefficients: Granucci and Persico, J. Chem. Phys. 126, 134114 (2007).

16. 16 Decoherence Butatriene cation Uncorrected Corrected Decoherence is introduced ad hoc by correcting the time- dependent coefficients  k

17. 17 Density Matrix Before continuing, it is useful to define the density matrix: (adiabatic population)

18. 18 Fewest switches: two states Population in S 2 : Trajectories in S 2 : Minimum number of hoppings that keeps the correct number of trajectories: Probability of hopping 0 1 P2→1P2→1

19. 19 Fewest switches: several states Example: Three states Only the fraction of derivative connected to the particular transition 0 1 P3→2P3→2 P 3→2 +P 3→1

20. 20 Fewest switches model for transition probability Tully, J. Chem. Phys. 93, 1061 (1990) Hammer-Schiffer and Tully, J. Chem. Phys. 101, 4657 (1994)

21. 21 R E Total energy Forbidden hop Forbidden hop makes the classical statistical distributions deviate from the quantum populations. How to treat them: Reject all classically forbidden hop and keep the momentum. Reject all classically forbidden hop and invert the momentum. Use the time uncertainty to search for a point in which the hop is allowed (Jasper et al. 116 5424 (2002)). Forbidden hops

22. 22 Adjustment of momentum after hopping R E KN(t)KN(t) KN(t+t)KN(t+t) Total energy 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 vector h 12 Adjust velocities components in the direction of the difference gradient vector g 12 Adjust velocities in the direction

23. a package for Newtonian dynamics close to the crossing seam M. Barbatti, G. Granucci, M. Ruckenbauer, J. Pittner, M. Persico, H. Lischka Dynamics General-purpose On-the-fly excited-state molecular dynamics Non-adiabatic methods (surface hopping) N EWTON -X C OLUMBUS C OLUMBUS(MCSCF, MRCI) T URBOMOLE(TD-DFT, CC2) G AUSSIAN(CASSCF) D FTB(TD-DFTB) + T INKER (QM/MM) Interfaces to Freeware and open source www.univie.ac.at/newtonx

24. 24 NX aims Easy and practical of using: just make the inputs and start the simulations; monitor partial results on-the-fly; get relevant summary of results at the end; Robust: if the input is right, the job will run: in case of error, messages must guide the user to fix the problem; Flexible: some different case to study or new method to implement? It should be easy to change the code; Open source: NX is the first MQCD-oriented program freely available and opened to the community.

25. 25 NX input facility: 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):

26. 26 NX input facility: 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

27. 27 Conclusions

28. 28 Basic processes Ground State R*R* h 1 P*P* Photoproduct Excited State Photoproduct Potential energy h 2 h 3 Non-adiabatic photochemistry Photophysics Adiabatic photochemistry Reactant Nuclear coordinates

29. 29 Phase  ik Dynamics control Newton’s equations Hopping probability or Average potential Quantum chemistry and MQCD methods

30. 30 Methods allowing for excited-state calculations: Present situation of quantum chemistry methods

31. 31 Methods allowing non-adiabatic coupling calculations: AdvantagesDisadvantages MRCIAb initio Well defined hierarchy of approximations Too expensive MCSCFAb initio Well defined hierarchy of approximations Still quite expensive Orbital rotations Bad vertical energies TD-DFTLow computational costSingle reference Semi-empiricalLow computational costUnreliable surfaces

32. Hybrid strategy Wavepacket < 0.1 ps Generation of initial conditions from wavepacket propagation MQCD < 1 ps Non-uniform distribution of initial conditions along the crossing seam Classical < 10 ps

33. 33 The Almost Ten Commandments Thou shalt use dynamics only for what is really necessary Test thy inputs and outputs before submitting thy real production jobs Thou shalt NOT let computers running alone without checking what is going on Thou shalt NOT start dynamics without a very careful investigation of the theoretical level Thou art a limited and biased being, therefore, thou shalt NOT rely on thy eyes to perform analysis

34. 34 StepMethodology 1. Find PES minima and conical intersections Geometry and MXS search 2. Determine the reaction paths Linear interpolation paths; minimum energy paths 3. Determine mechanisms, lifetimes and quantum yields Dynamics simulation The three steps to Success

35. Outlook The main challenges in the field are: Number of atoms Simulation time Modeling quality

36. Outlook ONIOM: Beapark et al, J. Phys. Chem. A 112, 7286 (2008) QM/MM: Virshup et al. J. Phys. Chem. B 113, 3280 (2009) Hybrid methods allow much larger systems FOMO/AM1: Granucci et al., J. Chem. Phys. 114, 10608 (2001) OM2: Fabiano et al., Chem. Phys. 349, 334 (2008) ROKS/CPMD: Doltsinis and Marx, Phys. Rev. Lett. 88, 166402 (2002) Semiempirical methods for QM part FISH: Bonačić-Koutecký/Mitrić, Phys. Rev. Letters, in press 2009 Inclusion of electric field in the dynamics. Pump-Probe simulations Metadynamics: Laio and Parrinelo, PNAS 99, 12562 (2002) Beyond few picoseconds Adiabatic hopping: Fabiano et al., Chem. Phys. 351, 111 (2008) Faster surface-hopping algorithms

37. Contact mario.barbatti@univie.ac.at This lecture can be downloaded at http://homepage.univie.ac.at/mario.barbatti/femtochem.html lecture8.ppt