Electron transfer reaction dynamics in non-Debye solvents

Abstract

The dynamics of electron transfer in a non-Debye solvent is described by multidimensional Markovian reaction-diffusion equation. To highlight differences with existing approaches in the simplest possible context, the irreversible outer-sphere reaction in a solvent with a biexponential energy-gap autocorrelation function, Δ(t), is studied in detail. In a Debye solvent, Δ(t)= exp (−t/τ L ) and the rate can be rigorously expressed as an explicit functional of exp (−t/τ L ). It has been suggested that the exact rate in a non-Debye solvent can be found by replacing exp (−t/τ L ) with the appropriate (nonexponential) Δ(t). For a “biexponential” solvent, our approach is based on an anisotropicdiffusionequation for motion on a harmonic surface in the presence of a two-dimensional delta function sink. Three approximations, which reduce the solution of this equation to effective one-dimensional ones, are considered and compared with exact Brownian dynamics simulation results. The crudest approximation replaces the non-Debye solvent with an effective Debye one with τ eff −1 =(−dΔ/dt) t=0 . The second is obtained by invoking the Wilemski–Fixman-type closure approximation for the equivalent two-dimensional integral equation. This approximation turns out to be identical to the above mentioned “substitution” procedure. When the relaxation times of the two exponentials are sufficiently different, it is shown how the two-dimensional problem can be reduced to a one-dimensional one with a nonlocal sink function. This anisotropicrelaxation time approximation is in excellent agreement with simulations when the relaxation times differ by at least a factor of three and the activation energy is greater than k B T. Finally, it is indicated how the influence of intramolecular vibrational modes (i.e., nonlocal sink functions) can be treated within the framework of this formalism.