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

density matrix representation, relaxation: energy dissipation and pure dephasing

Liouville representation:

L = Liouville tensor, R = relaxation tensor (4th order tensors)

RHiL

dtd ,

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

202

2

22

22qqm

qH

relaxation of an harmonic oscillator:

"random" variation of equilibrium position q0 ~ perturbation q: energy dissipation by 1-quantum stepsno pure dephasing (diagonal elements of perturbation vanish)

"random" variation of eigenfrequency ~ perturbation q2: energy relaxation by 2-quantum stepspure dephasing

Coupled harmonic oscillators:Excited state intramolecular proton transfer (ESIPT)SFB 450 seminary, Jan. 21, 2003 Arthur Hotzel, FU Berlin

enol/enol

keto/enol enol/keto

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)

q(proton coordinate)

Q(scaffo ld coord.)

ener

gy

enolketo

enolketo

Dynamics without dissipation second Born-Oppenheimer approximation:

pure enol(A)/keto(B) eigenstates:electrons: proton,: scaffold,:

),,(),()(),,( rqQqQQrqQ

consider only first electronically excited enol and keto singlet states

total Hamiltonian:pure enol/keto Hamiltonian:

EH

rqQWHH

0

0 ),,(

enol-keto coupling:

),,(),,(),,(),(

),(),(),()(

)()()(

*'

3.'

.'

*''

..''

..''

*''''''

rqQrqQWrqQrdqQW

qQqQWqQdqQW

QQWQdQW

nel

elelpr

elpr

and

with

meV517:),(),( ..,

.' elel

enolketoel WqQWqQW

assume electronic coupling independent of nuclear coordinates:

Proton wave functions

furthermore, consider only proton vibrational ground states ( = ' = 0):

)()()(

),(),()(

,*

,'.

,0,,0',''

,0*,0

QFCPRQQdQWWW

qQqQdqQFCPR

enolketoel

enolketo

enolketo

factor CondonFrankprotonic

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)q

(proton coordinate)

Q(scaffo ld coord.)

ener

gy

enolketo

enolketo

Scaffold vibrational states

Effective scaffold potentials are harmonic potentials with vibrational energy ħ = 14.6 meV, reduced mass M = 47.8 amu (proton vibrational energy ħ = 335 meV).

enol keto basis transformation:

enolketoFCSC ,*

,','

Keto and enol scaffold equilibrium positions are shifted by 0.077 Å with respect to each other.

,3,2,1,0',

4.4321''2

1

ityexothermicmeV:keto

:enol

E

E

Eigenenergies of scaffold vibrational states without enol-keto coupling:

Enol-keto coupling

enolketo

el

FCSC

QFQFFFCSCWW

MQ

,*

,','

,'2.

,' 210

03003020

02010010

2

express Q in terms of creation/annihilation operators of vibrational scaffold states(enol basis):

)()()( ,*

,'.

,0,,0','' QFCPRQQdQWWW enolketoel

enolketo

approximate FCPR(Q) by parabola:

2210)( QFQFFQFCPR

Total Hamiltonian in the enol/keto basis

,,,α,,,α

W

WH

†

,,,α,,,α

210'210

4.4321'

21

',

,'

210'210

states, keto states, enol

meV

states, keto states, enol

Eigenstates of H (considering enol/keto states = 0, ..., 9, ' = 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.

Q[Å]

ener

gy [a

mu

Å2 p

s-2]

initi

al s

tate

(eno

l bas

is)

pure

eno

l sta

tes

pure

ket

o st

ates

initi

al s

tate

(ene

rgy

bas

is)

H e

igen

stat

es

Wavefunction dynamics without dissipation

transfer back into enol/keto basis

diagonalize H: eigenvalues Hk, eigenvectors k

express initial state in terms of eigenstates of H

kk

ktHi

exp

propagate for time t:

blue: projection onto enol basisred: projection onto keto basis

elapsed time [oscillation periods = 0.283 ps]energy

(reduced enol/keto Hamiltonian)[amu Å2 ps-2]

Q[Å]

Dissipation We consider random perturbation of the form (in enol/keto basis):

,,,α

,,,α

G

,,,α,,,α

210'

210

03003020

02010010

0

003003020

02010010

~

210210

states, keto

states, enol

states, enol states, enol

Random perturbation proportional to scaffold elongation from equilibrium (Q - Q0) in the enol and keto states.

Relaxation tensor

Make basis transformation to eigensystem of H:

We assume short correlation time c of random correlation:

r lr

rjlrik

r kr

rkirjl

lj

iklj

ki

ikljijkl

ji

c

kTHHGG

kTHHGG

kTHHGG

kTHHGGfR

H

jiHH

exp1exp1

exp1exp1

,1

2

:) of basis reigenvecto (in tensor relaxation

const.onperturbati random of function ationautocorrel ofdensity spectral

GG ~

We take f = 200 ps

Wavepacket dynamics with dissipation

= density matrix in eigenvector basis of H

L = Liouville tensor

kl

klijklijji

klklijkl

ij RHHi

Ldtd

22n treat as -dimensional vector (n = 10 = number of included scaffold vibrational states in the enol- and keto electronic states)

22 22 nn treat L as -matrix

diagonalize L: eigenvalues Lk, eigenvectors k

express initial state (t = 0) in terms of eigenstates of L

kkk tL exp propagate for time t:

transfer back into eigenvector basis of H and then into enol/keto basis

blue: projection onto enol basisred: projection onto keto basis

energy(reduced enol/keto Hamiltonian)

[amu Å2 ps-2]elapsed time [oscillation periods = 0.283 ps]

Q[Å]