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Lecture 2

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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

MARKOVIAN LIMIT

WIDE BAND APPROXIMATION

Born solvation energy

WATER:

tD=10 ps tL=125 fs

Quantum solvation

(1) Increase in the kinetic energy (localization) – seems NOT to affect dynamics

(2) Non-adiabatic solvation (several electronic states involved)

Diffusion controlled rates

KRAMERS THEORY:

Low friction limit

High friction limit

Transition State theory

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:

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
- Coherent and incoherent transfer
- Electrode processes

Coming March 2006

Chapter 16

- 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 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

Electron transfer

Nuclear motion

Nuclear motion

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

Solvent polarization coordinate

Alternatively – solvent control

Adiabatic and non-adiabatic ET processes

Landau-Zener problem

(For diabatic surfaces (1/2)KR2)

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))

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).

Free energy associated with a nonequilibrium fluctuation of Pn

q

“reaction coordinate” that characterizes the nuclear polarization

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.

Reorganization energy

Activation energy

Experimental confirmation of the inverted regime

Marcus papers 1955-6

Miller et al, JACS(1984)

Marcus Nobel Prize: 1992

- From Quantum Chemical Calculations
- The Mulliken-Hush formula

- Bridge mediated electron transfer

EB

Donor-to-Bridge/ Acceptor-to-bridge

Bridge Green’s Function

Franck-Condon-weighted DOS

Reorganization energy

b’ (Å-1)=

0.2-0.6 for highly conjugated chains

0.9-1.2 for saturated hydrocarbons

~ 2 for vacuum

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.

constant

STEADY STATE SOLUTION

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.

Michel - Beyerle et al

- 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