A. Nitzan, Tel Aviv University SELECTED TOPICS IN CHEMICAL DYNAMICS IN CONDENSED SYSTEMS Boulder, Aug 2007 Lecture 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 relaxation If  1 -  2 >  D

Frequency dependent friction WIDE BAND APPROXIMATION MARKOVIAN LIMIT

Dielectric solvation Born solvation energy

Continuum dielectric theory of solvation WATER:  D =10 ps  L =125 fs

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 KRAMERS THEORY: Low friction limit High friction limit Transition State theory Diffusion controlled rates

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:

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 Transfer Rate – Transition state theory Rate – Transition state theory Probability to be on barrier (Activation energy) Probability to be on barrier (Activation energy) Transition probability Transition probability  Rate – Solvent controlled NOTE: “solvent controlled” is the term used in this field for the Kramers low friction limit. Transition rate

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 (solvation)

Electron transfer ELECTRONIC ENERGY CONSERVED Electron transition takes place in unstable nuclear configurations obtained via thermal fluctuations Nuclear motion

Electron transfer Solvent polarization coordinate

Transition state theory of electron transfer Adiabatic and non-adiabatic ET processes Landau-Zener problem Alternatively – solvent control ( For diabatic surfaces (1/2)KR 2 )

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 They have the following characteristics: (1) P n fluctuates because of thermal motion of solvent nuclei. (2) P e, as a fast variable, satisfies the equilibrium relationship (3) D = constant (depends on  only) Note that the relations E = D-4  P; P=P n + P e are always satisfied per definition, however D   s E. (the latter equality holds only at equilibrium). We are interested in changes in solvent configuration that take place at constant solute charge distribution 

Electron transfer – Marcus theory  Free energy associated with a nonequilibrium fluctuation of P n “reaction coordinate” that characterizes the nuclear polarization

The Marcus parabolas 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.

Electron transfer: Activation energy Reorganization energy Activation energy

Electron transfer: Effect of Driving (=energy gap)

Experimental confirmation of the inverted regime Marcus papers Marcus Nobel Prize: 1992 Miller et al, JACS(1984) Also seen in proton transfer (Kevin Peters)

Electron transfer – the coupling From Quantum Chemical Calculations The Mulliken-Hush formula Bridge mediated electron transfer

Bridge assisted electron transfer EBEB

Effective donor-acceptor coupling

Marcus expresions for non-adiabatic ET rates Bridge Green’s Function Donor-to-Bridge/ Acceptor-to-bridge Franck-Condon- weighted DOS Reorganization energy

Bridge mediated ET rate  ’ ( Å -1 ) = for highly conjugated chains 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 constant STEADY STATE SOLUTION

ET rate from steady state hopping

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

DNA (Giese et al 2001)

ELECTROCHEMISTRY

D A Rate of electron transfer to metal in vacuum Rate of electron transfer to metal in electrolyte solution Transition rate to a continuum (Golden Rule) Donor gives an electron and goes from state a (reduced) to state b (oxidized). E b,a =E b - E a is the energy of the electron given to the metal M EFEF

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: (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