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A. Nitzan, Tel Aviv University
SELECTED TOPICS IN CHEMICAL DYNAMICS IN CONDENSED SYSTEMS
Boulder, Aug 2007
Lecture 2Lecture 2
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Boulder Aug 2007
(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
Chapters 13-15
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Molecular vibrational Molecular vibrational relaxationrelaxation
1
D2
1 2
~ D
c
VRk e
1 /~ DcVRk e
If 1- 2 > D
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Frequency dependent Frequency dependent frictionfriction
consˆ ˆ~ ( ) (0) tantifi tf i T
t
k dte F t F
ˆ ˆ~ ( ) (0)ifi t
f i Tk dte F t F
1
DWIDE BAND APPROXIMATION
MARKOVIAN LIMIT
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Dielectric solvationDielectric solvation
q = + e q = + eq = 0
a b c
C153 / Formamide (295 K)
Wavelength / nm
450 500 550 600
Rel
ativ
e E
mis
sion
Int
ensi
ty
ON O
CF3
2 11 1 2eV
2
(for a molecular charge)
s
q
a
Born solvation energy
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Continuum dielectric theory of Continuum dielectric theory of solvationsolvation
eL D
s
WATER:
D=10 ps L=125 fs
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Electron solvationElectron solvation
Quantum solvation
(1) Increase in the kinetic energy (localization) – seems NOT to affect dynamics
(2) Non-adiabatic solvation (several electronic states involved)
C153 / Formamide (295 K)
Wavelength / nm
450 500 550 600
Rel
ativ
e E
mis
sion
Int
ensi
ty
ON O
CF3
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Activated rate processesActivated rate processes
E B
r e ac t i o nc o o r di nate
KRAMERS THEORY:
Low friction limit
High friction limit
Transition State theory
0 /
2B B
TSTE k Tk e
0 /
2B BB B
TSTE k Tk e k
/0
B BE k TB
B
k J ek T
4k DR
Diffusion controlled
rates
Bk TD
m
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The physics of transition The physics of transition state ratesstate rates
X B
a b
diabatic
X B
1
1
2
Adiabatic
0
( ,TST B f BP xk d P x
v v v v)
Assume:
(1) Equilibrium in the well
(2) Every trajectory on the barrier that goes out makes it
0
( ,TST B abP xk d P
v v v v)
E B
0
B
r e ac t i o nc o o r di nate
THIS IS AN UPPER BOUND ON THE ACTUAL RATE!
Quantum barrier crossing:
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PART B
Electron transfer
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Boulder Aug 2007
(2) Electron transfer processes•Simple models•Marcus theory•The reorganization energy•Adiabatic and non-adiabatic limits•Solvent controlled reactions•Bridge assisted electron transfer•Coherent and incoherent transfer•Electrode processes
Chapter 16
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Theory of Electron TransferTheory of Electron Transfer
Rate – Transition state theoryRate – Transition state theory
Probability to be on barrier (Activation energy)Probability to be on barrier (Activation energy) Transition probabilityTransition probability
Rate – Solvent controlled
NOTE: “solvent controlled” is the term used in this field for the Kramers low friction limit.
0
( ,TST B abP xk d P
v v v v)
Transition rate
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Electron transfer in polar Electron transfer in polar mediamedia
•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
c
q = + e
b
q = + e
a
q = 0
Electronic transition
Nuclear relaxation (solvation)
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q = 1q = 0 q = 0q = 1
Electron transfer
ELECTRONIC ENERGY CONSERVED
Electron transition takes place in unstable nuclear configurations obtained via thermal fluctuations
Nuclear motion
Nuclear motion
q= 0q = 1q = 1q = 0
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Electron transferElectron transfer
E aE A
E b
E
e ne r g y
ab
X a X tr X b
Solvent polarization coordinate
q = 1q = 0 q = 0q = 1
q= 0q = 1q = 1q = 0
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Transition state theoryTransition state theory of of electron transferelectron transfer
Adiabatic and non-adiabatic ET processesE
R
E a(R )
E b(R )
E 1(R )
E 2(R )
R *
tt= 0
V ab
Landau-Zener problem
*
0
( , ) ( )b ak dRR P R R P R
2,
*
2 | |( ) 1 exp a b
b a
R R
VP R
R F
*
2,| |
2Aa b E
NAR R
VKk e
F
Alternatively – solvent control
(For diabatic surfaces (1/2)KR2)
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Solvent controlled electron Solvent controlled electron transfertransfer
Correlation between the fluorescence lifetime and the longitudinal dielectric relaxation time, of 6-N-(4-methylphenylamino-2-naphthalene-sulfon-N,N-dimethylamide) (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))
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Electron transfer – Electron transfer – Marcus theoryMarcus theory
(0) (0) (1) (1)B BA Aq q q q (0) (0) (1) (1)
B BA Aq q q q
D 4
E D 4 P
eP P Pn
1
4e
eP E
4s e
nP E
They have the following characteristics:(1) Pn fluctuates because of thermal motion of solvent nuclei.(2) Pe , as a fast variable, satisfies the equilibrium relationship (3) D = constant (depends on only)Note that the relations E = D-4P; P=Pn + Pe are always satisfied per definition, however D sE. (the latter equality holds only at equilibrium).
We are interested in changes in solvent configuration that take place at constant solute charge distribution
D Es
q = 1q = 0 q = 0q = 1
q= 0q = 1q = 1q = 0
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Electron transfer – Electron transfer – Marcus theoryMarcus theory
0 (0) (0)BAq q
(0) (0) (1) (1)B BA Aq q q q (0) (0) (1) (1)
B BA Aq q q q
D 4
E D 4 P
e nP P P 1
P E4e
e
P E4
s en
D Es
Free energy associated with a nonequilibrium fluctuation of Pn
“reaction coordinate” that characterizes the nuclear polarization
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The Marcus parabolasThe Marcus parabolas
0 1 0( ) Use as a reaction coordinate. It defines the state of the medium that will be in equilibrium with the charge distribution . Marcus calculated the free energy (as function of ) of the solvent when it reaches this state in the systems =0 and =1.
20 0( )W E 2
1 1( ) 1W E
21 1 1 1 1
2 2e s A B AB
qR R R
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Electron transfer: Electron transfer: Activation energyActivation energy
2[( ) ]
4b a
A
E EE
21 1 1 1 1
2 2e s A B AB
qR R R
E aE A
E b
E
e ne r g y
ab
a tr b
20 0( )W E
21 1( ) 1W E
Reorganization energy
Activation energy
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Electron transfer: Effect of Electron transfer: Effect of Driving (=energy gap)Driving (=energy gap)
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Experimental confirmation of the inverted regime
Marcus papers 1955-6
Marcus Nobel Prize: 1992
Miller et al, JACS(1984)
Also seen in proton transfer (Kevin Peters)
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Electron transfer – the Electron transfer – the couplingcoupling
• From Quantum Chemical Calculations
•The Mulliken-Hush formula max 12DA
DA
VeR
• Bridge mediated electron transfer
2 4~
ab
B
E
k Tet abk V e
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Bridge assisted electron Bridge assisted electron transfertransfer
D A
B 1 B 2 B 3
D A
12
3V D 1
V 1 2 V 2 3
V 3 A
1
1 1
1
, 1 , 11
ˆ
1 1
1 1
N
D j Aj
D D AN NA
N
j j j jj
H E D D E j j E A A
V D V D V A N V N A
V j j V j j
, 1 /,j B j j B D AE E V E E
EB
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1
1 1 12 1
21 2 23 2
32
0 0 0
0 0
00
0 0 0
0 0 0
D D D
D
NA
NA A A
N N
E E V c
E E V c
V
V E E V c
V
E E V
V E c
c
E
1 12 1
21 2 23 2
32
1 1,
,
1
1
0 0
00 0
0
0 0
N N N
NN N
D
NA AN
DV cE E V cV E E V c
V
E E V
cV E E V c
1 1 0
0
D D D
A A AN N
E E c V c
E E c V c
ˆ I c uB B BH E
( )ˆc uBB G
1( )ˆ ˆIBB BG E H
D D A A j jjc c c
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Effective donor-acceptor Effective donor-acceptor couplingcoupling
0 0 0 0 0 0 ...G G G VG G VG VG
0
0
D DA
A AD
D A
A D
E c c
V c
E V
E E c
( ) ( )1 111
ˆ ˆ;B BD D D D A A AN NN NAE E V G V E E V G V
( ) ( ) *1 11 1
ˆ ˆ;B BDA D NA AD AN D DAN NV V G V V V G V V
12 23 1,1
/ 1 / 2 / 1 /
1 1 1 1ˆ ...B N NN
D A D A D A N D A N
G V V VE E E E E E E E
(1/ 2) '1
0/
N
NbD NA BDA
B D A B
V V VV V e
V E E
/
2' ln B
D A B
V
b E E
1( )ˆ ˆIBB BG E H
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Marcus expresions for non-Marcus expresions for non-adiabatic ET ratesadiabatic ET rates
2
2 (1
2)1 ( )
|
)2
| ( )
(
2
BD
DA
D
D A AD
N ANA D
V
V
E
GV E
k
E
F
F
2/ 4
( )4
BE k T
B
eE
k T
F
Bridge Green’s Function
Donor-to-Bridge/ Acceptor-to-bridge
Franck-Condon-weighted DOS
Reorganization energy
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Bridge mediated ET rateBridge mediated ET rate
~ ( , )exp( ' )ET AD DAk E T RF
’ (Å-1)=
0.2-0.6 for highly conjugated chains
0.9-1.2 for saturated hydrocarbons
~ 2 for vacuum
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Bridge mediated ET rateBridge mediated ET rateCharge 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.
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Incoherent hoppingIncoherent hopping
........
0 = D
1 2 N
N + 1 = A
k 2 1
k 1 0 = k 0 1 e x p (-E 1 0 ) k N ,N + 1 = k N + 1 ,N e x p (-E 1 0 )
0 1,0 0 0,1 1
1 0,1 2,1 1 1,0 0 1,2 2
1, 1, , 1 1 , 1 1
1 , 1 1 1,
( )
( )N N N N N N N N N N N N
N N N N N N N
P k P k P
P k k P k P k P
P k k P k P k P
P k P k P
constant STEADY STATE SOLUTION
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ET rate from steady state ET rate from steady state hoppinghopping
........
0 = D
1 2 N
N + 1 = A
k
k 1 0 = k 0 1 e x p (-E 1 0 ) k N ,N + 1 = k N + 1 ,N e x p (-E 1 0 )
k k0,1 2,1 1 1,0 0 1,2 2
1, 1, ,
1 1,
1 1
0 ( )
0 ( )N N N N N N N
N N N N
N
k k P k P k P
k k P k P
P k P
0D Ak P
/
1,0
1
1
B BE k T
D A N
N A D
kek k
k kN
k k
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Dependence on Dependence on temperaturetemperature
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 .
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The photosythetic reaction The photosythetic reaction centercenter
Michel - Beyerle et al
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Dependence on bridge Dependence on bridge lengthlength
Ne
11 1up diffk k N
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DNA (Giese et al 2001)DNA (Giese et al 2001)
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ELECTROCHEMISTRYELECTROCHEMISTRY
A
C
R
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DA
, ,( ) 1 ( )b a b ak E E f E
Rate of electron transfer to metal in vacuum
Rate of electron transfer to metal in electrolyte solution
,( ) 1 ( ) b ak dE E f E F E E
2
, , ,2
( ) ( )b a D M M b aE V E
Transition rate to a continuum (Golden Rule)
Donor gives an electron and goes from state a (reduced) to state b (oxidized). Eb,a=Eb- Ea is the energy of the electron given to the metal
2/ 4
( )4
BE k T
B
eE
k T
F
M
EF
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Steady state evaluation Steady state evaluation of ratesof rates
Rate of water flow depends linearly on water height in the cylinder
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
h
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