Transcript Laser–Induced Control of Condensed Phase Electron Transfer

Laser–Induced Control of
Condensed Phase Electron Transfer
Rob D. Coalson, Dept. of Chemistry, Univ. of Pittsburgh
Yuri Dakhnovskii, Dept. of Physics, Univ. of Wyoming
Deborah G. Evans, Dept. of Chemistry, Univ. of New Mexico
Vassily Lubchenko, Dept. of Chemistry, M.I.T.
NH3
H3N
+3
Ru
H3N
CN
NH3
NH3
CN
N
+2
Ru
C
NC
[polar Solvent]
CN
CN
Tunneling in A 2-State System
q
|L>
|R>
V(x)
tropolone
q
q
Proton
Transfer
Multi (Double) Quantum
Well Structure
e-
GaAs
AlGaAs
GaAs
|R>
|L>
q
Electron
V(x) Transport
In Solids
1. . . . . N
Ru+2
Atomic
Orbitals
Ru+3
e|D>
|A>
|D>
Molecular Electron
Transfer
|A>
Acceptor
Donor

Equivalent 2-state model
For any of these systems: |(t)> = cD(t) |D> + cA(t) |A>
where
HAA HAD
cA
HDA HDD
cD
1
=
cA
d
iħ
dt c
D
For a Symmetric Tunneling System: HAA= HDD= 0; HAD= HAD =  (< 0)
Given initial preparation in |D>, for a symmetric system:
1
PD(t) = 1 – PA(t) = cos2(t) =
( cos ( x) )
2 0.5

2
x
t

e-
Now, apply an electric field
E
|D>
|A>
R
2
0 R
2
Q: How does this modify the Hamiltonian??
A: It modifies the site energies according to “ - • E ”
Permanent dipole moment of A
Thus: H

=
HAA


HDD
-E
eoR/2
0
0
-eoR/2
Analysis in the case of time-dependent E(t) = Eocosot
Consider first the symmetric case:
i d cA
dt
cD
0 
=
 0
- E cos t
o
o
1 0
cA
0 -1
cD
Permanent dipole moment difference
Letting: cA(t) = e i a sinot cAI(t) ; cD(t) = e i a sinot cDI(t)
Eo
Where: a =
(ħ)o
t
[ N.B.:  E cos t'dt' =
0 o
o
Eo
sin
t
]
o
o
Thus, Interaction Picture S.E. reads:
I
c
d
A
i
=
dt
I
cD
0
e2i a sinot 
e-2i a sinot 
0
cAI
cDI
Now note:
So:
e
i b sint

= m=-
 Jm(b) e i mt

 e2i a sinot = m=-
 Jm(2a) e i mot
 ·Jo(2a) , for /o << 1
RWA
Thus, the shuttle frequency is renormalized to |Jo(2a)|

NB: Trapping or localization
occurs at certain Eo values!
0
[ 
0
2.5
<<1 ]
5.5
a
[Grossman - Hänggi,
Dakhnovskii – Metiu]
Add coupling to a condensed phase environment
ˆ = Tˆ
H
1 0
0 1
+
Nuclear coordinate
kinetic energy
VD(x) 
 VA(x)
- Eocosot
D(x,t)
0
0 -1
Nuclear coordinate
VD(x)
1
VA(x)
A(x,t)
x
Field couples only to
2-level system
Now:
T+VD(x) 

T+VA(x)
=
D(x,t)
d
i
dt A(x,t)
- E cos t
o
o
0
D(x,t)
0 -1
A(x,t)
1
Construction of (Diabatic) Potential Energy functions for
Polar ET Systems:
A
D
VD(x)
A
D
VA(x)
A few features of classical Nonadiabatic ET Theory [Marcus,
Levich-Doganadze…]
Ĥ =
ˆT+V1(x)

non-adiabatic coupling
matrix element

ˆT+V2(x)
Hamiltonian
kinetic E of
nuclear coordinates
|(t)> =
1(x,t)
2(x,t)
(diabatic) nuclear coord. potential
for electronic state 2
States
Given initial preparation in electronic state 1 (and assuming
nuclear coordinates are equilibrated on V1(x))
1
2
1(x,0)
x
Then, P2(t) = fraction of molecules
in electronic state 2
= < 2(x,t)| 2(x,t)>  k12t
where k12 = (Golden Rule) rate constant
In classical Marcus (Levich-Doganadze) theory, k12 is determined
by 3 molecular parameters: , Er, 
k12 = 2
 ½ -(Er- )2/4Er kBT
e
ErkT
( )
w/ Er = “Reorganization Energy” ;  = “Reaction Heat”
1
2

Er
x
 ½ -(Er+ )2/4Er kBT
e
ErkT
( )
For the “backwards” Reaction: k21 = 2
To obtain electronic state populations at arbitrary times, solve
kinetic [“Master”] Eqns.:
dP1(t)/dt = - k12P1(t) + k21P2(t)
dP2(t)/dt = k12P1(t) - k21P2(t)
Note that long-time asymptotic [“Equilibrium”] distributions are
then given by:
Keq
P2()
k12
=
=
P1()
k21
= e /kBT
for Marcus formula rate constants
N.B. Marcus theory for nonadiabatic ET reactions works
experimentally.
See: Closs & Miller, Science 240, 440 (1988)
Control of Rate Constants in Polar Electronic Transfer Reactions
Via an Applied cw Electric Field
The Hamiltonian is:
VD
VA
ˆ =
H
x
ˆhD
0
0
ˆhA
+
0


0
+ 12Eocosot
1
0
0
-1
The forward rate constant is: [Y. Dakhnovskii, J. Chem. Phys.
100, 6492 (1994)


2
Re
2
i ˆhD t }
kDA =   Jm(a) ·
i mot tr {D -iˆhA t
dt
e
ˆ
e
e
 0
m=-


a = 2 12 Eo/ ħo
2
kDA =
4
AD
 ½
ErkT
( )


m=-
2
Jm(212Eo/ħo) •
e
-(Er –+  + ħmo)2/4Er kBT
Rate constants in presence of cw E-field
300
D
200
2
J1
Schematically:
Energy
100
A
2
J0
0
-100
2
J-1
ħo
-200
0
5
x
In polar electron transfer reactions, for
Reorganization Energy Er  ħo (the quantum of applied
laser field)
1
0.8
2
Jo(a)
0.6
2
J1(a)
0.4
2
J2(a)
0.2
0
0 1
2
3
4
5 6
a
7
8 9 10
Dramatic perturbations of the “one-way” rate constants may be
obtained by varying the laser field intensity:
[Activationless reaction, Er=1eV]
Forward rate
constant
Dakhnovskii and RDC showed how this property can be
used to control Equilibrium Constants with an applied cw field:
D
A
“bias”

PA(00)
eq =
PD(00)
Results for activationless reaction:
Y. Dakhnovskii and RDC, J. Chem. Phys. 103, 2908 (1995)
Activationless ET
 = 100 cm -1
Er =  = ħo = 1eV
Evans, RDC, Dakhnovskii & Kim,
PRL 75, 3649 (95)
REALITY CHECK on coherent control of mixed valence ET
reactions in polar media
 
(1) “·E”
E

Orientational averaging will
reduce magnitude of desired effects
[Lock ET system in place w/ thin polymer films]
(2) Dielectric breakdown of medium?
To achieve resonance effects for  = 34D
Er = ħo = 1eV
Electric field  107 V/cm
Giant dipole ET complex, solvent w/ reduced Er,
pulsed laser reduce likelihood of catastrophe]
(3) Direct coupling of E(t) to polar solvent


[Dipole moment of solvent molecules << Dipole moment
of giant ET complex]
(4) ħ0  1eV ; intense fields  (multiphoton) excitation to higher
energy states in the ET molecule, which are not considered in
the present 2-state model.
ħ
1eV
ħ
1eV
ħ
1eV
ħ
1eV
Absolute Negative Conductance in Semiconductor Superlattice
-
+
-Keay et al., Phys. Rev. Lett. 75, 4102 (1995)
Absolute Negative Conductance in Semiconductor Superlattice
-Keay et al., Phys. Rev. Lett. 75, 4102 (1995)
Immobilized long-range Intramolecular Electron Transfer Complex:
D
Tethered
Alkane
Chain
molecule

A
E
(ħ)

k
(Inert) Substrate
Light Absorption by Mixed Valence
ET Complexes in Polar Solvents:
Transmitted
Photons
Ru+3
Incident
Photons
Ru+2
Solvent
Absorption cross section K abs ( ) 
# Absorbed Photons
# Incident Photons
Absorption Cross Section Formula:
k abs
1 d E ( a ,  )
 2
E
dt
d E ( a ,  )
( eq ) U 1
( eq ) U 2
 n1 
 n2 
dt
t
t
U1, 2


t
4
2
  


 E r k BT 
1
(1)
(2)
2 
2
m


J
  m (a) 
m 1
  Er    m 2 
 Er    m 2 
  exp  

exp  



4
E
k
T
4
E
k
T
 
r B
r B



a  2 E 
(3)
# Absorbed Photons
Hush Absorption Spectrum   ( L ) 
# Incident Photons
  L 212 FCF
Marcus
Gaussian
Δ
w/ 12  effective " transition dipole moment"   
 res
 
permanent
dipole moment
difference
Barrierless
Hush Absorption
= 2  E

PRL 77, 2917 (1996)
J. Chem. Phys. 105, 9441 (1996)
absorption
emission
Stimulated Emission using two incoherent lasers
J. Chem. Phys. 109, 691 (1998)
J. Chem. Phys. 109, 691 (1998)