Laser–Induced Control of Condensed Phase Electron Transfer
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Laser–Induced Control of Condensed Phase Electron Transfer
Rob D. Coalson, Dept. of Chemistry, Univ. of PittsburghYuri Dakhnovskii, Dept. of Physics, Univ. of Wyoming
Deborah G. Evans, Dept. of Chemistry, Univ. of New MexicoVassily Lubchenko, Dept. of Chemistry, M.I.T.
H3N
H3N
Ru N
NH3NH3
NH3
Ru
NC
CNCN
CN
CNC
[polar Solvent]
+3 +2
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Tunneling in A 2-State Systemq
tropolone
|L> |R>
q
V(x)Proton Transfer
GaA
s
GaA
s
AlG
aAs
q
|L> |R> V(x)
e-
q
Electron TransportIn Solids
Multi (Double) QuantumWell Structure
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1. . . . . N
e-
|D> |A>AtomicOrbitals
Ru+2 Ru+3
|D> |A>
Donor Acceptor
Molecular Electron Transfer
Equivalent 2-state model
For any of these systems: |(t)> = cD(t) |D> + cA(t) |A>
where HAA
HDA
HAD
HDD
cA
cD
= iħ ddt
cA
cD
1
For a Symmetric Tunneling System: HAA= HDD= 0; HAD= HAD = (< 0)
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0.5
1
cos x( )( )2
x
Given initial preparation in |D>, for a symmetric system:
PD(t) = 1 – PA(t) = cos2(t) =
t2
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Now, apply an electric field
|D> |A>
e-
E
0R2
R2
Q: How does this modify the Hamiltonian??
A: It modifies the site energies according to “ - • E ”
Thus: H =HAA
HDD
- EeoR/2
-eoR/20
0
Permanent dipole moment of A
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Analysis in the case of time-dependent E(t) = EocosotConsider first the symmetric case:
i ddt
cA
cD
=0
0
- Eocosot
1 0
0 -1
cA
cD
Permanent dipole moment difference
Letting: cA(t) = e i a sinot cAI(t) ; cD(t) = e i a sinot cD
I(t)
Where: a = Eo
(ħ)o[ N.B.: 0Eocosot'dt' = ]t Eo
osinot
Thus, Interaction Picture S.E. reads:
cAI
cDI
=0
e2i a sinot 0
e-2i a sinot cAI
cDI
i ddt
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Now note: e i b sint = Jm(b) e i mt
m=-
So: e2i a sinot = Jm(2a) e i mot m=-
·Jo(2a) , for /o << 1RWA
Thus, the shuttle frequency is renormalized to |Jo(2a)|
00
a2.5 5.5
[ <<1 ]
NB: Trapping or localization occurs at certain Eo values!
[Grossman - Hänggi, Dakhnovskii – Metiu]
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Add coupling to a condensed phase environment
H = T ˆˆ 10
01
+ VD(x)
-VA(x) Eocosot1 0
0 -1
Nuclear coordinatekinetic energy
Nuclear coordinate Field couples only to2-level system
VD(x)VA(x)
D(x,t)A(x,t)
x
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Now: T+VD(x)T+VA(x)
- Eocosot1 0
0 -1
D(x,t)
A(x,t)
=D(x,t)
A(x,t)i d
dt
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Construction of (Diabatic) Potential Energy functions for Polar ET Systems:
VD(x)
VA(x)
A
A D
D
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A few features of classical Nonadiabatic ET Theory [Marcus, Levich-Doganadze…]
T+V1(x)T+V2(x)
H = ˆ
ˆHamiltonian
non-adiabatic couplingmatrix element
kinetic E ofnuclear coordinates
(diabatic) nuclear coord. potential for electronic state 2
1(x,t)
2(x,t) |(t)> = States
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Given initial preparation in electronic state 1 (and assuming nuclear coordinates are equilibrated on V1(x))
x
1
2
1(x,0) Then, P2(t) = fraction of molecules in electronic state 2
= < 2(x,t)| 2(x,t)> k12t
where k12 = (Golden Rule) rate constant
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x
2
1
Er
For the “backwards” Reaction: k21 = 2 ( )ErkT
½e
-(Er+ )2/4Er kBT
In classical Marcus (Levich-Doganadze) theory, k12 is determined by 3 molecular parameters: , Er,
k12 = 2 ( )ErkT
½e
-(Er- )2/4Er kBT
w/ Er = “Reorganization Energy” ; = “Reaction Heat”
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To obtain electronic state populations at arbitrary times, solve kinetic [“Master”] Eqns.:
dP1(t)/dt = - k12P1(t) + k21P2(t)
dP2(t)/dt = k12P1(t) - k21P2(t)
Note that long-time asymptotic [“Equilibrium”] distributions are then given by:
Keq =P2()P1()
k12
k21
= e /kBT=
for Marcus formula rate constants
N.B. Marcus theory for nonadiabatic ET reactions works experimentally. See: Closs & Miller, Science 240, 440 (1988)
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Control of Rate Constants in Polar Electronic Transfer ReactionsVia an Applied cw Electric Field
The Hamiltonian is:
x
VA
VD
H = ˆ
ˆ ˆ
ˆ ˆ
hD
hA
0
0+
0
0
+ 12Eocosot
1
-1
0
0
The forward rate constant is: [Y. Dakhnovskii, J. Chem. Phys. 100, 6492 (1994)kDA = 2 Jm(a) ·
m=-
dt e i mot tr {Re
0
D e e-i hA t ˆ i hD t }
a = 2 12 Eo/ ħo
2
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kDA = Jm(212Eo/ħo) •m=-
2( )
ErkT
½
e-(Er – + ħmo)2/4Er kBT
2
4
+ AD
Rate constants in presence of cw E-field
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D
A
ħo
0 5x
Ener
gy
200
100
0
-200
-100
300
J-1
J0
J1
2
2
2
Schematically:
In polar electron transfer reactions, for Reorganization Energy Er ħo (the quantum of applied laser field)
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0 1 2 3 4 5 6 7 8 9 100
0.2
0.4
0.6
0.8
1
a
Jo(a)
J1(a)
J2(a)
2
2
2
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Dramatic perturbations of the “one-way” rate constants may be obtained by varying the laser field intensity:
[Activationless reaction, Er=1eV] Forward rate constant
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Dakhnovskii and RDC showed how this property can be used to control Equilibrium Constants with an applied cw field:
“bias”
D
A
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Results for activationless reaction:
eq
=
P D(0
0)P A
(00)
Y. Dakhnovskii and RDC, J. Chem. Phys. 103, 2908 (1995)
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Evans, RDC, Dakhnovskii & Kim,PRL 75, 3649 (95)
= 100 cm -1
Activationless ET
Er = = ħo = 1eV
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REALITY CHECK on coherent control of mixed valence ET reactions in polar media
“·E” (1)
E
Orientational averaging will reduce magnitude of desired effects
[Lock ET system in place w/ thin polymer films]
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(2) Dielectric breakdown of medium?
To achieve resonance effects for = 34DEr = ħo = 1eV Electric field 107 V/cmGiant 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]
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(4) ħ0 1eV ; intense fields (multiphoton) excitation to higher energy states in the ET molecule, which are not considered in the present 2-state model.
ħ
ħ
ħ
ħ
1eV
1eV
1eV
1eV
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Absolute Negative Conductance in Semiconductor Superlattice
-Keay et al., Phys. Rev. Lett. 75, 4102 (1995)
- +
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Absolute Negative Conductance in Semiconductor Superlattice
-Keay et al., Phys. Rev. Lett. 75, 4102 (1995)
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Immobilized long-range Intramolecular Electron Transfer Complex:
A
D
TetheredAlkane Chain molecule
(Inert) Substrate
E
k
(ħ)
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Light Absorption by Mixed Valence ET Complexes in Polar Solvents:
Ru+2
Ru+3
Solvent
ħ
ħ
ħ
ħ
ħ
ħ
Incident Photons
Transmitted Photons
PhotonsIncident #Photons Absorbed #)(Ksection cross Absorption abs
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Absorption Cross Section Formula:
(1) ),(12 dt
aEdE
kabs
(2) ),( 2)(2
1)(1 t
UntUn
dtaEd eqeq
TkEmE
TkEmE
aJmTkEt
U
Br
r
Br
r
mm
Br
4exp
4exp
(3) )(4
22
1
22
12
2,1
Ea 2
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FCF
PhotonsIncident #Photons Absorbed # )( Spectrum AbsorptionHush
212
L
L
MarcusGaussian
permanentdipole momentdifference
moment" dipole transition" effective w/
res12
Δ
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Barrierless
E2
Hush Absorption=
PRL 77, 2917 (1996)J. Chem. Phys. 105, 9441 (1996)
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Stimulated Emission using two incoherent lasers
J. Chem. Phys. 109, 691 (1998)
absorption emission
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J. Chem. Phys. 109, 691 (1998)