1. Ultrafast processes in molecules
VI – Transition probabilities
Mario Barbatti

2. Fermi’s golden rule

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

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

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

6. An approximate way to solve the differential equation
Guess the “0-order” solution:
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.

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

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. Physically meaningful quantity
Near k: density of states

11. Physically meaningful quantity
Using

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

13. Transition dipole moment
Electronic 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. Einstein coefficient and lifetimeR E
If DE21 = 4.65 eV and f21 = ,
what is the lifetime of the excited state?
Converting to nanoseconds:

19. non-adiabatic transition probabilities

20. 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 = +∞? x E
H11
H22 E2 E1
H12

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

22. Derivation of Landau-Zener formula
Multiply by at left and integrate
In the deduction it was used:

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

24. Zener approximation: x E

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

26. The probability of finding the system in state 2 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. 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?
0.43
0.57

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

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

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

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
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