Ultrafast processes in molecules

1. Ultrafast processes in molecules VI – Transition probabilities Mario Barbatti barbatti@kofo.mpg.de

2. Fermi’s golden rule

3. Transition rate: Quantum levels of the non-perturbed system Perturbation is applied Transition is induced Fermi’s Golden Rule

4. which solves: and Derivation of Fermi’s Golden Rule Time-dependent formulation H0 – Non-perturbated Hamiltonian Hp – Perturbation Hamiltonian

5. Multiply by at left and integrate Prove it! Derivation of Fermi’s Golden Rule Note that the non-perturbated Hamiltonian is supposed non-dependent on time.

6. Use this guess to solve the equation and to get the 1st-order approximation: Use the 1st-order to get the 2nd-order approximation and so on. An approximate way to solve the differential equation Guess the “0-order” solution:

7. Suppose the simplified perturbation: Constant between 0 and t Otherwise 0 t t 0 First order approximation Guess the “0-order” solution:

8. First order approximation Between 0 and t Otherwise It was used:

9. Transition probability In this derivation for constant perturbation, only transitions with w ~ 0 take place. If the perturbation oscillates harmonically (like a photon), w≠ 0 can occur. The final result for the Fermi’s Golden Rule is still the same.

10. Near k: density of states Physically meaningful quantity

11. Physically meaningful quantity Using

12. Fermi’s Golden Rule: photons and molecules H0 – Non-perturbated molecular Hamiltonian – Light-matter perturbation Hamiltonian Transition rate:

13. 0 Electronic transition dipole moment Transition dipole moment

14. Einstein coefficients Rate of absorption i→k Einstein coefficient B for absorption - degeneracy of state n

15. Einstein coefficients Rate of stimulated emission k→i Einstein coefficient B for stimulated emission

16. Einstein coefficients Rate spontaneous decay k→i Einstein coefficient A for spontaneous emission

17. Einstein coefficient and oscillator strength In atomic units:

18. E Converting to nanoseconds: R Einstein coefficient and lifetime If DE21 = 4.65 eV and f21 = -0.015, what is the lifetime of the excited state?

19. non-adiabatic transition probabilities

20. E E2 H22 0 H11 E1 x 0 Non-adiabatic transitions Problem: if the molecule prepared in state 2 at x = -∞ moves through a region of crossing, what is the probability of ending in state 1 at x = +∞? H12

21. 1. Landau-Zener 2. Demkov / Rosen-Zener 3. Nikitin Models for non-adiabatic transitions 4. Bradauk; 5. Delos-Thorson; 6. …

22. Multiplybyatleftandintegrate In the deduction it was used: Derivation of Landau-Zener formula

23. Solving (i) for a2 and taking the derivative: (iii) Since there are only two states: (i) (ii) Substituting (iii) in (ii):

24. E 0 x 0 Zener approximation:

25. E 0 x 0 Problem: Find a2(+∞) subject to the initial condition a2(-∞) = 1. The solution is:

26. The probability of finding the system in state 2 is: The probability of finding the system in state 1 is: Pad Pnad Pnad Pad

27. Example: In trajectory in the graph, what are the probability of the molecule to remain in the ps* state or to change to the closed shell state?

28. 0.57 0.43 Example: In trajectory in the graph, what are the probability of the molecule to remain in the ps* state or to change to the closed shell state?

29. For the same H12, Landau-Zener predicts: Adiabatic Non-adiabatic (diabatic) E E v v H12 0 0 x x 0 0

30. v v For the same , Rosen-Zener predicts: Adiabatic Non-adiabatic (but not diabatic!) E E 0 0 x x 0 0

31. For the same w0 (H12), Nikitin predicts: Adiabatic Non-adiabatic (diabatic) E E v v 0 0 x x 0 0

32. Marcus Theory AB → A+B- l 2HAB DG0 A-B A+-B-

33. The problem with the previous formulations is that they only predict the total probability at the end of the process. If we want to perform dynamics, it is necessary to have the instantaneous probability.

34. Next lecture Organic photovoltaics Contact barbatti@kofo.mpg.de