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### Lecture 2

A. Nitzan, Tel Aviv University

ELECTRON TRANSFER AND TRANSMISSION IN MOLECULES AND MOLECULAR JUNCTIONS

AEC, 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

Dielectric solvation

Born solvation energy

Electron solvation

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

Diffusion controlled rates

KRAMERS THEORY:

Low friction limit

High friction limit

Transition State theory

The physics of transition state rates

Assume:

(1) Equilibrium in the well

(2) Every trajectory on the barrier that goes out makes it

THIS IS AN UPPER BOUND ON THE ACTUAL RATE!

Quantum barrier crossing:

Electron transfer

- (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

Coming March 2006

Chapter 16

Theory of Electron Transfer

- Activation energy
- Transition probability
- Rate – Transition state theory

Transition rate

- 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

Electronic transition

Nuclear relaxation

Nuclear motion

Nuclear motion

Electron transition takes place in unstable nuclear configurations obtained via thermal fluctuations

Electron transfer

Solvent polarization coordinate

Transition state theory of electron transfer

Alternatively – solvent control

Adiabatic and non-adiabatic ET processes

Landau-Zener problem

(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,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 – Marcus theory

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.

(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).

Electron transfer – Marcus theory

Free energy associated with a nonequilibrium fluctuation of Pn

q

“reaction coordinate” that characterizes the nuclear polarization

The Marcus parabolas

Use q as a reaction coordinate. It defines the state of the medium that will be in equilibrium with the charge distribution rq. 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.

Experimental confirmation of the inverted regime

Marcus papers 1955-6

Miller et al, JACS(1984)

Marcus Nobel Prize: 1992

Electron transfer – the coupling

- From Quantum Chemical Calculations
- The Mulliken-Hush formula

- Bridge mediated electron transfer

Donor-to-Bridge/ Acceptor-to-bridge

Bridge Green’s Function

Franck-Condon-weighted DOS

Reorganization energy

Marcus expresions for non-adiabatic ET ratesBridge mediated ET rate

b’ (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.

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

Steady state evaluation of rates

- Rate of water flow depends linearly on water height in the cylinder
- Two ways to get the rate of water flowing out:
- Measure h(t) and get the rate coefficient from k=(1/h)dh/dt
- 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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