Transcript Introduction to electron transport in molecular systems

A. Nitzan, Tel Aviv University
ELECTRON TRANSFER AND
TRANSMISSION IN MOLECULES
AND MOLECULAR JUNCTIONS
AEC, Grenoble, Sept 2005
Lecture 2
Grenoble Sept 2005
(1) Relaxation and reactions in
condensed molecular systems
•Kinetic models
•Transition state theory
•Kramers theory and its extensions
•Low, high and intermediate
friction regimes
•Diffusion controlled reactions
Coming March 2006
Chapter 13-15
Molecular vibrational relaxation
2
kVR ~
 1 2 
 c

D 

e
1
kVR ~ e
 c  1 /  D 
D
Frequency dependent friction



iifi
t if tˆ
kkf fi i~~ dtedte

(ˆt(0)
)Fˆ (0) constant
F ( tFˆ)F
T
T
 t
MARKOVIAN LIMIT
WIDE BAND
APPROXIMATION
1
D
Dielectric solvation
C153 / Formamide (295 K)
Relative Emission Intensity
CF3
450
N
O
O
q=0
q=+e
q=+e
a
b
c
Born solvation energy
500
550
Wavelength / nm
600
 q2  
1
  1    1  2eV
 2a    s 
(for a molecular charge)
Continuum dielectric theory of solvation
e
L  D
s
WATER:
D=10 ps
L=125 fs
Electron solvation
C153 / Formamide (295 K)
Relative Emission Intensity
CF3
450
N
500
550
O
O
600
Wavelength / nm
Quantum solvation
(1) Increase in the kinetic energy (localization) – seems NOT to affect
dynamics
(2) Non-adiabatic solvation (several electronic states involved)
Activated rate processes
EB
reaction
coordinate
KRAMERS THEORY:
Low friction limit
High friction limit
Transition State
theory
k

0 J B e  EB / kBT
kBT
0 B  EB / kBT
B
k
e
 kTST
2

0  EB / kBT
kTST 
e
2
Diffusion
controlled
rates
k  4 DR
kBT
D
m
The physics of transition state rates
B
Assume:
(1) Equilibrium in the well
EB
0
reaction
coordinate
(2) Every trajectory on the barrier that goes out makes it

kTST   d v v P ( x B , v )  v f  P  x B 
a
b
0
THIS IS AN UPPER BOUND ON THE ACTUAL RATE!
XB
diabatic
Quantum barrier

crossing:
kTST   d v v P ( x B , v ) Pab  v 
0
2
1
1
XB
Adiabatic
PART B
Electron transfer
Grenoble Sept 2005
(2) Electron transfer
processes
•Simple models
•Marcus theory
•The reorganization energy
•Adiabatic and non-adiabatic
limits
•Solvent controlled reactions
•Bridge assisted electron transfer
Coming March 2006 •Coherent and incoherent
transfer
Chapter 16
•Electrode processes
Theory of Electron Transfer



Activation energy
Transition probability
Rate – Transition state theory
Transition rate

kTST   d v v P ( x B , v ) Pab  v 
0
 Rate – Solvent controlled
NOTE: “solvent controlled” is the term used in this field for
the Kramers low friction limit.
Electron transfer in polar media
•Electron are much faster than nuclei
• Electronic transitions take place in fixed nuclear
configurations
• Electronic energy needs to be conserved during the
change in electronic charge density
q=0
a
Electronic
transition
q=+e
b
q=+e
c
Nuclear
relaxation
Electron transfer
q=0
q=1
q=1
q=0
Nuclear
motion
Nuclear
motion
q=0
q=1
q=1
q=0
Electron transition takes place in unstable nuclear
configurations obtained via thermal fluctuations
Electron transfer
a
b
energy

EA
Ea
E
Eb
Xa
Xtr
Xb
Solvent
polarization
coordinate
Transition state theory of electron
Alternatively –
transfer
solvent control
Adiabatic and non-adiabatic ET processes
E
Landau-Zener
problem
E2(R)
Ea(R)

k   dRR P ( R* , R ) Pba ( R )
Vab
0
2

 2 | Va ,b | 

Pba ( R)  1  exp  

R

F



 R R*
Eb(R)
E1(R)
*
R
t=0
R
t
k NA 
 K
2
| Va ,b |2
F
e  EA
R  R*
(For diabatic surfaces (1/2)KR2)
Solvent controlled electron transfer
Correlation between the fluorescence lifetime and the longitudinal dielectric
relaxation time, of 6-N-(4-methylphenylamino-2-naphthalene-sulfon-N,Ndimethylamide) (TNSDMA) and 4-N,N-dimethylaminobenzonitrile (DMAB) in
linear alcohol solvents. The fluorescence signal is used to monitor an
electron transfer process that precedes it. The line is drawn with a slope of
1. (From E. M. Kosower and D. Huppert, Ann. Rev. Phys. Chem. 37, 127
(1986))
Electron transfer – Marcus theory
(0) (0)
qA qB
(1) (1)
q=0q A qB
q=1
  D  4
E  D  4 P
P  Pe  Pn
e  1
Pe 
E
4
 s  e
Pn 
E
4
q=0
D   sE
(0)
(1)
(1)
q(0)

q

q

q
B
B
A
A
q=1
q=0
We are interested in changes in solvent
configuration that take place at
constant solute charge distribution 
They have the following characteristics:
(1) Pn fluctuates because of thermal
motion of solvent nuclei.
q=1
q=1
q=0
(2) Pe , as a fast variable, satisfies the
equilibrium relationship
(3) D = constant (depends on  only)
Note that the relations E = D-4P;
P=Pn + Pe are always satisfied per
definition, however D  sE. (the latter
equality holds only at equilibrium).
Electron transfer – Marcus theory
(0) (0)
qA qB
(1) (1)
 q A qB
  D  4
E  D  4 P
P  Pe  Pn
e  1
Pe 
E
4
 s  e
Pn 
E
4
D   sE
(0)
(1)
(1)
q(0)

q

q

q
B
B
A
A
(0)
  0   q(0)
q
A B
Free energy
associated with a
nonequilibrium
fluctuation of Pn
“reaction coordinate” that
characterizes the nuclear
polarization
q
The Marcus parabolas
q  0  q ( 1  0 )
Use q as a reaction coordinate. It defines the state of the
medium that will be in equilibrium with the charge
distribution q. Marcus calculated the free energy (as
function of q) of the solvent when it reaches this state in the
systems q =0 and q=1.
W0 (q )  E0  q
2
W1 (q )  E1    1  q 
 1
1  1
1
1
    


  e  s   2 RA 2 RB RAB
 2
 q

2
Electron transfer: Activation energy
W0 (q )  E0  q 2
W1 (q )  E1    1  q 
a
2
b
energy

EA
Ea
E
Eb
[( Eb  Ea )   ]2
EA 
4
 1
1  1
1
1
    


  e  s   2 RA 2 RB RAB
qa qtr
 2
 q

qb
Reorganization
energy
Activation energy
Electron transfer: Effect of Driving
(=energy gap)
Experimental confirmation of the
inverted regime
Marcus papers
1955-6
Miller et al,
JACS(1984)
Marcus Nobel
Prize: 1992
Electron transfer – the coupling
2
ket ~ Vab e

 Eab   
4 kBT
• From Quantum Chemical Calculations
•The Mulliken-Hush formula VDA 
• Bridge mediated electron transfer
max 12
eRDA
Bridge assisted electron transfer
B2
B1
V12
D
2
1
B3
V23
3
EB
A
V3A
VD1
A
D
N
Hˆ  E D D D   E j j
j 1
E j  E B , V j , j 1
j  EA A A
 VD1 D 1  V1 D 1 D  V AN A N  VNA N

N 1
 V j , j 1
j 1
j
j  1  V j , j 1 j  1 j

A
EB  ED / A
  c D D  c A A   j c j j
 ED  E

 V1 D

0

0



0

VD1
E1  E
V21
0
0
V12
0
0
E2  E V23
V32
EN  E
0
0
 E1  E

 V21
 0


 0

  cD 
 
  c1 
  c2 
   0
0  
VNA   c N 
  
EA  E   cA 
0
0
VNA
V12
0
E2  E V23
V32
E N 1  E
0
  c1

  c2
0 

VN 1, N  
E N  E   c N
0
V N , N 1
 E D  E  cD  VD1c1  0
 E A  E  c A  VAN c N  0


 V1 D c D 



0



  




0




V c 

 NA A 

Hˆ B  EI B c B  u
( B)
ˆ
cB  G u

ˆ
Gˆ ( B )  EI B  H
B

1
Effective donor-acceptor coupling
 E D  E  cD  VDA c A  0
 E A  E  c A  VAD cD  0
( B)
( B)
ED  ED  VD1Gˆ 11
VD1 ; E A  E A  VAN Gˆ NN
VNA
VDA
 
Gˆ B
1N

( B)
( B)
*
ˆ
ˆ
 VD1G1 N VNA ; VAD  VAN GN 1 V1D VDA
G  G0  G0VG0  G0VG0VG0  ...
1
 ED / A  E1 
V12
1
 ED / A  E2 
N
VDA
V23 ...
1
1
VN 1, N
 ED / A  EN 1 
 ED / A  E N 

VD1VNA 
VB
 (1/ 2) Nb '


  V0 e
VB   ED / A  EB  
VB
2
 '  ln
b ED / A  EB
Marcus expresions for non-adiabatic
ET rates
k D A 

2
2
| VDA |2 F ( E AD )
VD1VNA
2
2
(B)
G1 N ( E D )
F ( E AD )
Bridge Green’s
Function
Donor-to-Bridge/
Acceptor-to-bridge
   E  / 4 k BT
2
F (E) 
e
4 k BT
Reorganization energy
Franck-Condonweighted DOS
Bridge mediated ET rate
kET ~ F ( E AD , T )exp(  ' RDA )
’ (Å-1)=
0.2-0.6
for highly conjugated chains
0.9-1.2
for saturated hydrocarbons
~2
for vacuum
Bridge mediated ET rate
Charge recombination
lifetimes in the
compounds shown in
the inset in dioxane
solvent. (J. M. Warman
et al, Adv. Chem. Phys.
Vol 106, 1999). The
process starts with a
photoinduced electron
transfer – a charge
separation process. The
lifetimes shown are for
the back electron
transfer (charge
recombination) process.
Incoherent hopping
k21
2
N
........
1
k10=k01exp(-E10)
kN,N+1=kN+1,Nexp(-E10)
0=D
P0   k1,0 P0  k0,1 P1
N+1 = A
constant
P1  ( k0,1  k2,1 ) P1  k1,0 P0  k1,2 P2
STEADY STATE
SOLUTION
PN  ( k N 1, N  k N 1, N ) PN  k N , N 1 PN 1  k N , N 1 PN 1
PN 1   k N , N 1 PN 1  k N 1, N PN
ET rate from steady state hopping
0   ( k0,1  k2,1 ) P1  k1,0 P0  k1,2 P2
k
2
k
k
........
1
k10=k01exp(-E10)
N
kN,N+1=kN+1,Nexp(-E10)
0   ( k N 1, N  k N 1, N ) PN  k N , N 1 PN 1
PN 1  k N 1, N PN  kD A P0
0=D
k D A  k N 1,0 
ke
k
kN  A
N+1 = A
  E B / k BT 

k
k1 D
1 N
Dependence on temperature
The integrated elastic (dotted line) and activated (dashed line)
components of the transmission, and the total transmission
probability (full line) displayed as function of inverse temperature.
Parameters are as in Fig. 3.
The photosythetic reaction center
Michel - Beyerle et al
Dependence on bridge length
e N
1
1
 kup
 kdiff
N 
1
DNA (Giese et al 2001)
Steady state evaluation of rates
Rate of water flow depends linearly
on water height in the cylinder
h
Two ways to get the rate of water
flowing out:
(1) Measure h(t) and get the rate
coefficient from k=(1/h)dh/dt
(1) Keep h constant and measure the
steady state outwards water flux
J. Get the rate from k=J/h
= Steady state rate