A. Nitzan, Tel Aviv University

SELECTED TOPICS IN CHEMICAL DYNAMICS IN CONDENSED SYSTEMS

Boulder, Aug 2007

Lecture 2Lecture 2

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

Molecular vibrational Molecular vibrational relaxationrelaxation

1

D2

1 2

~ D

c

VRk e

1 /~ DcVRk e

If 1- 2 > D

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

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

Continuum dielectric theory of Continuum dielectric theory of solvationsolvation

eL D

s

WATER:

D=10 ps L=125 fs

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

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

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:

PART B

Electron transfer

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

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

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)

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

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

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)

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))

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

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

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

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

Electron transfer: Effect of Electron transfer: Effect of Driving (=energy gap)Driving (=energy gap)

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)

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

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

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

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

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

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

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.

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

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

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 .

The photosythetic reaction The photosythetic reaction centercenter

Michel - Beyerle et al

Dependence on bridge Dependence on bridge lengthlength

Ne

11 1up diffk k N

DNA (Giese et al 2001)DNA (Giese et al 2001)

ELECTROCHEMISTRYELECTROCHEMISTRY

A

C

R

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

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