Femtochemistry: A theoretical overview Mario Barbatti VI – Transition probabilities This lecture can be downloaded at lecture6.ppt

2 Fermi’s golden rule

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

4 Derivation of Fermi’s Golden Rule See Fermi‘s Golden Rule paper at the course homepage Time-dependent formulation H 0 – Non-perturbated Hamiltonian H p – 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 1 st -order approximation: Use the 1 st -order to get the 2 nd -order approximation and so on.

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

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

9 Transition probability In this derivation for constant perturbation, only transitions with  ~ 0 take place. If the perturbation oscillates harmonically (like a photon),  ≠ 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 Fermi’s Golden Rule: photons and molecules Transition rate: H 0 – Non-perturbated molecular Hamiltonian – Light-matter perturbation Hamiltonian

13 Transition dipole moment 0 Electronic transition dipole moment

14 Einstein coefficients Rate of absorption i → k Einstein coefficient B for absorption - degeneracy of state n (see Einstein coefficients text in the course homepage)

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 lifetime R E If  E 21 = 4.65 eV and f 21 = , what is the lifetime of the excited state? Converting to nanoseconds:

19 non-adiabatic transition probabilities

20 Non-adiabatic transitions 0 x E 0 H 11 H 22 E2E2 E1E1 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 =  ∞ ? H 12

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: (i) (ii) Solving (i) for a 2 and taking the derivative: (iii) Substituting (iii) in (ii):

24 Zener approximation: x 0 E 0

25 Problem: Find a 2 (+∞) subject to the initial condition a 2 (  ∞) = 1. The solution is: (The complete derivation is in the paper on Landau-Zener in the course homepage) 0 x E 0

26 The probability of finding the system in state 1 is: The probability of finding the system in state 2 is: P nad P ad P nad

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

29 0 x E 0 0 x E 0 v v For the same H 12, Landau-Zener predicts: Non-adiabatic (diabatic) Adiabatic H 12

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

31 0 x E 0 0 x E 0 For the same  0 (H 12 ), Nikitin predicts: Non-adiabatic (diabatic) Adiabatic vv

32 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.

33 Next lecture Quantum dynamics methods Contact This lecture can be downloaded at lecture6.ppt