Further

Quick links to online content Annu. Rev. Phys. Chern. 1991.42: 279-309 Copyright © 1991 by Annual Reviews Inc. All rights reserved

COMPUTER SIMULATIONS OF Annu. Rev. Phys. Chem. 1991.42:279-309. Downloaded from www.annualreviews.org by Pennsylvania State University on 05/28/12. For personal use only.

ELEC:TRON-TRANSFER REA� �C'

-'>0 4-C' o

o�

��...,�

A schematic depiction of the potential surfaces for an electron-transfer reaction in a system composed of a fixed donor and

acceptor and a single solvent dipole. The reaction coordinate (x) is represented by the orientation of the dipole. The anharmonic potential energy

surfaces V, and V� are represented by functions of the form dx2± iJ?Jcos (x-n/2) (10). (R(qht) Electron transfer can occur when the librational trajectory of the solvent in the field of the reactants brings the system to a region where plane is

V,(t);

the dotted curve,

Vp(t).

V, and Vp

intersect. The solid curve in the energy-vs-time

The small trajectories breaking away from the main trajectory indicate transitions to Vp.

SIMULATIONS OF ELECTRON TRANSFER REACTIONS

283

veniently obtained by propagating classical molecular-dynamics tra jectories on V" thus obtaining the time-dependent coordinates r(t) for the system and evaluating � Vp,[r(t)]. When a;p/h is much smaller than the time derivative of � Vp., the reaction is non adiabatic and the rate constant is 5. where < >, means an average over many trajectories on Va. The exponent in Equation 4 is a rapidly oscillating function that contributes to the rate constant only when � Vp, O. Thus, kST corresponds to the sum obtained by running a long trajectory and collecting the Landau-Zener transition probability every time V, and Vp intersect (9, 1 0). Although Equation 4 can be derived by a Feynmann path-integral formulation (7, 1 1 ), it reflects a major approximation because the rigorous derivation requires a collection of trajectories at an infinite number of energies (10, I I ). A semiclassical Green Function approach (1 1 ) is more accurate, but is not practical for multidimensional systems. However, the high-temperature limit of Equation 5 can be written in a form that, for harmonic potential surfaces, is identical to the high-temperature limit of the exact quantum mechanical expression for a harmonic system ( 1 0). Such an expression is obtained by rewriting Equations 4 and 5 as

Annu. Rev. Phys. Chem. 1991.42:279-309. Downloaded from www.annualreviews.org by Pennsylvania State University on 05/28/12. For personal use only.

=

k(ST2)

[to < C. p (O)Cp (t), dt J

=

I· t��

=

Icr,p/hI2

yet)

=

__

.

L: L

(ljh)2

exp[(i/h) P + BL .... P + H L .

reaction

This

A direct

simulation

was

carried out by using the density matrix formalism. It includes superexchange. The Franck Condon factors were obtained from a dispersed-polaron analysis similar to that described in Refs. 26 and 27 and werc divided among vibrational modcs spaced 25 cm- I apart. The electronic couplings were

0' 1 2

=

25 and

0' 1 3

=

80 cm

I.

(T 2) relaxation time constants both were set at 1.0 ps.

The longitudinal (T I ) and transverse

Annu. Rev. Phys. Chem. 1991.42:279-309. Downloaded from www.annualreviews.org by Pennsylvania State University on 05/28/12. For personal use only.

SIMULATIONS OF ELECTRON TRANSFER REACTIONS

305

Friesner & Wertheimer ( 1 23) before the Franck-Condon factors for the protein were available. Treutlein et al ( 1 24) have described molecular-dynamics simulations of the Rp. viridis reaction center. These workers did not calculate a rate constant for electron transfer, but focused on the structural reorganization that follows the simulated formation of p+He from P*. They found that the major