SFB 450 seminary: wave packet dynamics & relaxation Jan. 21, 2003 Arthur Hotzel, FU Berlin

1.  Liouville representation: L = Liouville tensor, R = relaxation tensor (4th order tensors) relaxation of an harmonic oscillator: SFB 450 seminary: wave packet dynamics & relaxation Jan. 21, 2003 Arthur Hotzel, FU Berlin density matrix representation, relaxation: energy dissipation and pure dephasing  Redfield theory I: relaxation due to "random" perturbation  relaxation rates given by spectral densities of the autocorrelation function of the perturbation needs ad-hoc correction for finite temperature  Redfield theory II: relaxation due to coupling to bath which is in thermal equilibrium  gives correct temperature dependence  "random" variation of equilibrium position q0 ~ perturbation  q: energy dissipation by 1-quantum steps no pure dephasing (diagonal elements of perturbation vanish)  "random" variation of eigenfrequency w ~ perturbation  q2: energy relaxation by 2-quantum steps pure dephasing

2. Coupled harmonic oscillators: Excited state intramolecular proton transfer (ESIPT) SFB 450 seminary, Jan. 21, 2003 Arthur Hotzel, FU Berlin 2,5-bis(2-benzoxazolyl)-hydroquinone (BBXHQ) proton transfer in the first excited state (singlet), enol (A)  keto (B)  high-frequency proton oscillation around equilibrium positions A, B (coordinate q)  proton site-site distance modulated by low-frequency scaffold mode (coordinate Q)

3. second Born-Oppenheimer approximation: pure enol(A)/keto(B) eigenstates: total Hamiltonian:pure enol/keto Hamiltonian: enol-keto coupling: assume electronic coupling independent of nuclear coordinates: Dynamics without dissipation  consider only first electronically excited enol and keto singlet states

4. furthermore, consider only proton vibrational ground states (b = b' = 0):  protonic Frank-Condon factor FCPR depends strongly on scaffold coordinate Q: FCPR(Q) = 0.006 at left-hand classical turning point of scaffold vibrational ground state (enol) FCPR(Q) = 0.081 at right-hand classical turning point of scaffold vibrational ground state (enol) Proton wave functions

5.  Eigenenergies of scaffold vibrational states without enol-keto coupling:  enol  keto basis transformation: Scaffold vibrational states  Effective scaffold potentials are harmonic potentials with vibrational energy ħW = 14.6 meV, reduced mass M = 47.8 amu (proton vibrational energy ħw = 335 meV).  Keto and enol scaffold equilibrium positions are shifted by 0.077 Å with respect to each other.

6. approximate FCPR(Q) by parabola:  express Q in terms of creation/annihilation operators of vibrational scaffold states(enol basis): Enol-keto coupling

7. Total Hamiltonian in the enol/keto basis  Eigenstates of H (considering enol/keto states a = 0, ..., 9, a' = 0, ..., 9):  Initial state: Excitation from molecular ground state with delta pulse; scaffold ground state equilibrium position shifted by 0.077 Å with respect to electronically excited enol state. H eigenstates pure enol states initial state (enol basis) pure keto states initial state (energy basis) energy [amu Å2 ps-2] Q[Å]

8.  diagonalize H: eigenvalues Hk, eigenvectors  propagate for time t: Wavefunction dynamics without dissipation  express initial state in terms of eigenstates of H  transfer back into enol/keto basis

9. blue: projection onto enol basis red: projection onto keto basis energy (reduced enol/keto Hamiltonian) [amu Å2 ps-2] elapsed time [oscillation periods = 0.283 ps] Q[Å]

10. Dissipation  We consider random perturbation of the form (in enol/keto basis):  Random perturbation proportional to scaffold elongation from equilibrium (Q - Q0) in the enol and keto states.

11. Relaxation tensor  Make basis transformation to eigensystem of H:  We assume short correlation time tc of random correlation:  We take f = 200 ps

12. r = density matrix in eigenvector basis of H L = Liouville tensor  treat r as -dimensional vector (n = 10 = number of included scaffold vibrational states in the enol- and keto electronic states)  treat L as -matrix  propagate for time t: Wavepacket dynamics with dissipation diagonalize L: eigenvalues Lk, eigenvectors sk  express initial state r(t = 0) in terms of eigenstates of L  transfer back into eigenvector basis of H and then into enol/keto basis

13. blue: projection onto enol basis red: projection onto keto basis energy (reduced enol/keto Hamiltonian) [amu Å2 ps-2] elapsed time [oscillation periods = 0.283 ps] Q[Å]