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

Fermi’s golden rule

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

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

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.

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.

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

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

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.

Physically meaningful quantity Near k: density of states

Physically meaningful quantity Using

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

Transition dipole moment Electronic transition dipole moment

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

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

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

Einstein coefficient and oscillator strength In atomic units:

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

non-adiabatic transition probabilities

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

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

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

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

Zener approximation: x E

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

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

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?

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

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

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

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

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

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.

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