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SFB 450 seminary: wave packet dynamics & relaxation
Jan. 21, 2003
Arthur Hotzel, FU Berlin
density matrix representation, relaxation: energy dissipation and pure dephasing
Liouville representation:
d
i
L H , R
dt
L = Liouville tensor, R = relaxation tensor (4th order tensors)
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
relaxation of an harmonic oscillator:
2 2 m 2
q q0 2
H
2
2 q
2
"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 ~ perturbation q2:
energy relaxation by 2-quantum steps
pure dephasing
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)
enol
enol
enol/enol
keto
energy
keto
keto/enol
enol/keto
Q
(scaffold coord.)
q
(proton coordinate)
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)
Dynamics without dissipation
second Born-Oppenheimer approximation:
pure enol(A)/keto(B) eigenstates:
(Q, q, r ) (Q) (Q, q) (Q, q, r )
: scaffold, : proton, : electrons
total Hamiltonian:
pure enol/keto Hamiltonian:
enol-keto coupling:
H H 0 W (Q, q, r )
H 0 E
' ' ' W dQ * ' ' ' (Q) Wpr'.'el. (Q) (Q)
with
Wpr'.'el. (Q) dq * ' ' (Q, q ) Wel'. (Q, q ) (Q, q)
and
Wel'. (Q, q) d 3n r *' (Q, q, r ) W (Q, q, r ) (Q, q, r )
consider only first electronically excited enol and keto singlet states
assume electronic coupling independent of nuclear coordinates:
el.
el.
Wel'. (Q, q) Wketo
517 meV
,enol (Q, q) : W
Proton wave functions
furthermore, consider only proton vibrational ground states ( = ' = 0):
FCPR(Q) dq 0*, keto(Q, q) 0,enol (Q, q) protonic Frank Condon factor
W ' ', '0, keto W , 0,enol W el. dQ * ', keto(Q) ,enol (Q) FCPR(Q)
enol
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)
enol
keto
keto
energy
Q
(scaffold coord.)
q
(proton coordinate)
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 ħ = 335 meV).
Keto and enol scaffold equilibrium positions are shifted by 0.077 Å with respect to
each other.
Eigenenergies of scaffold vibrational states without enol-keto coupling:
enol :
keto :
2
E ' W ' 1 43.4 meV
2
E W 1
, ' 0, 1, 2, 3,
enol keto basis transformation:
FCSC ', * ', keto ,enol
exothermic ity
Enol-keto coupling
W ' ', ' 0, keto W , 0,enol W el. dQ * ', keto(Q) ,enol (Q) FCPR(Q)
approximate FCPR(Q) by parabola:
FCPR(Q) F 0 F1 Q F 2 Q 2
express Q in terms of creation/annihilation operators of vibrational scaffold states
(enol basis):
0
0
0 1
2 0
1 0
Q
0
2 0
3
2W M
3 0
0 0
W ', W el. FCSC F 0 F1 Q F 2 Q 2 ',
FCSC ', * ', keto ,enol
Total Hamiltonian in the enol/keto basis
enol states, α 0 ,1, 2 ,
keto states, α ' 0 ,1, 2 ,
H
W 1
2
W
, '
W†',
W ' 1 43.4 meV
2
enol states, α 0,1, 2,
keto states, α' 0,1, 2,
Q[Å]
initial state (energy basis)
pure keto states
H eigenstates
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.
pure enol states
initial state (enol basis)
Eigenstates of H (considering enol/keto states = 0, ..., 9, ' = 0, ..., 9):
energy [amu Å2 ps-2]
Wavefunction dynamics without dissipation
diagonalize H: eigenvalues Hk, eigenvectors k
express initial state in terms of eigenstates of H
propagate for time t:
i Hk t
k exp
k
transfer back into enol/keto basis
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]
Dissipation
We consider random perturbation of the form (in enol/keto basis):
enol states, α 0 ,1, 2 , enol states, α 0 ,1, 2 ,
0
1
0
0
~
G
1
0
2
0
0
2
0
3
0
0
0
3
0
0
0
1
0
0
1
0
2
0
0
2
0
3
0
0
3
0
enol states, α 0 ,1, 2 ,
keto states, α ' 0,1, 2,
Random perturbation proportional to scaffold elongation from equilibrium
(Q - Q0) in the enol and keto states.
Relaxation tensor
~
G G
Make basis transformation to eigensystem of H:
We assume short correlation time tc of random correlation:
1
tc
Hi H j
i, j
spectral density of autocorrel ation function of random perturbati on const.
relaxation tensor (in eigenvecto r basis of H ) :
Glj Gik
Glj Gik
f
Rijkl
2 1 exp H i H k kT 1 exp H j H l
kT
Glr Grj
Gir Grk
ik
1
exp
H
H
kT
1
exp
H
H
kT
r
k
r
l
r
r
jl
We take f = 200 ps
Wavepacket dynamics with dissipation
= density matrix in eigenvector basis of H
dij
dt
Lijkl kl
kl
i Hi H j
ij Rijkl kl
kl
L = Liouville tensor
treat as 2n 2 -dimensional vector (n = 10 = number of included scaffold vibrational
states in the enol- and keto electronic states)
treat L as 2n 2 2n 2 -matrix
diagonalize L: eigenvalues Lk, eigenvectors sk
express initial state (t = 0) in terms of eigenstates of L
propagate for time t:
s k exp Lk t s k
transfer back into eigenvector basis of H and then into enol/keto basis
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]