Femtochemistry: A theoretical overview

1. Femtochemistry: A theoretical overview VII – Methods in quantum dynamics Mario Barbatti mario.barbatti@univie.ac.at This lecture can be downloaded at http://homepage.univie.ac.at/mario.barbatti/femtochem.html lecture7.ppt

2. Multiply by yi at left and integrate in the electronic coordinates Time dependent Schrödinger equation for the nuclei Quantum dynamics nuclear wave function

3. Wave-packet propagation Expand nuclear wave function in a basis of nk functions fk for each coordinate Rk: Wavepacket program: http://page.mi.fu-berlin.de/burkhard/WavePacket/ Kosloff, J. Phys. Chem. 92, 2087 (1988)

5. Multiconfiguration time-dependent Hartree (MCTDH) method Variational method leads to equations of motion for A and f. Conventional: 1 – 4 degrees of freedom MCTDH: 4 -12 degrees of freedom Heidelberg MCDTH program: http://www.pci.uni-heidelberg.de/tc/usr/mctdh/ Meyer and Worth, Theor. Chem. Acc. 109, 251 (2003)

6. UV absorption spectrum of pyrazine (24 degrees of freedom!) Exp. MCTDH Raab, et al. J. Chem. Phys. 110, 936 (1999)

7. Multiple spawning dynamics Nuclear wavefunction is expanded in a Gaussian basis set centered at the classical trajectory Note that the number of gaussian functions (Ni) depends on time. Ben-Nun, Quenneville, Martínez, J. Phys. Chem. A 104, 5161 (2000)

8. and with Multiple spawning dynamics

9. Multiple spawning dynamics The non-adiabatic coupling within Hij is computed and monitored along the trajectory. When it become larger than some pre-define threshold, new gaussian functions are created (spawned).

10. Multiple spawning dynamics Saddle point approximation

11. S0 S1

12. QM (FOMO-AM1)/MM Virshup et al. J. Phys. Chem. B 113, 3280 (2009)

13. E E1 t’ R’ t R Global nature of the nuclear wavefunction

14. E E1 t’ R’ t R Global nature of the nuclear wavefunction

15. Within this approximation, the nuclear wavefunction is local: it does not depend on the wavefunction values at other positions of the space • This opens two possibilities: • On-the-fly approaches (global PESs are no more needed) • Classical independent trajectories approximations However, because of the non-adiabatic coupling between different electronic states, the problem cannot be reduced to the Newton’s equations We use, therefore, Mixed Quantum-Classical Dynamics approaches (MQCD)

16. y x Mixed Quantum-Classical Dynamics Is the classical position

17. Nuclear motion: Mixed Quantum-Classical Dynamics Is the classical position Multiply at left by d(R-Rc)1/2 and integrate in R.

18. With (adiabatic basis) which solves: Prove! Remember we assumed Weighted average over all gradients Ehrenfest (Average Field) Dynamics Meyer and Miller, J. Chem. Phys. 70, 3214 (1979)

19. Average surface tci E t 0

20. Ehrenfest dynamics fails for dissociation Energy pp* 10 fs ns* ps* cs N-H dissociation

21. 0.57 0.43 Landau-Zener predicts the populations at t = ∞ In this case, Ehrenfest dynamics would predict the dissociation on a surface that is ~half/half S0 and S1, which does not correspond to the truth.

22. Decoherence The problem with the Ehrenfest dynamics is the lack of decoherence. The non-diagonal terms should quickly go to zero because of the coupling among the several degrees of freedom. The approximation ci(R(t)) ~ ci(t) does not describe this behavior adequately. Ad hoc corrections may be imposed. Zhu, Jasper and Truhlar, J. Chem. Phys. 120, 5543 (2004)

23. Mixed Quantum-Classical Dynamics Is the classical position Nuclear motion:

24. Surface hopping dynamics Dynamics runs always on a single surface (diabatic or adiabatic). Every time step a stochastic algorithm decides based on the non-adiabatic transition probabilities on which surface the molecule will stay. The wavepacket information is recovered repeating the procedure for a large number of independent trajectories. Because the dynamics runs on a single surface, the decoherence problem is largely reduced (but not eliminated). Tully, J. Chem. Phys. 93, 1061 (1990)

25. E tci t 0

26. Comparison between methods Tully, Faraday Discuss. 110, 407 (1998). surface-hopping (diabatic) surface-hopping (adiabatic) mean-field wave-packet Landau-Zener Burant and Tully, JCP 112, 6097 (2000)

27. Ethylene Barbatti, Granucci, Persico, Lischka, CPL401, 276 (2005). Butatriene cation Worth, hunt and Robb, JPCA 127, 621 (2003). Oscillation patterns are not necessarily quantum interferences

28. Hierarchy of methods Quantum Wave packet (MCTDH) Multiple spawning MQCD dynamics Classical

29. Next lecture: • Implementation of surface hopping dynamics Contact mario.barbatti@univie.ac.at This lecture can be downloaded at http://homepage.univie.ac.at/mario.barbatti/femtochem.html lecture7.ppt