Solvent dynamical effects in electron transfer: Numerical predictions of molecularity effects using the mean spherical approximation

The influence of solvent molecularity upon the adiabatic barrier-crossing frequency νn and barrier height ΔG*, for electron–exchange reactions involving only solvent reorganization is examined numerically on the basis of a mean spherical approximation (MSA) treatment of the reaction coordinate time-correlation function, Δ(t) [Refs. 3(c) and 7(c)] for Debye solvents. The calculated ΔG* values for a spatially isolated redox couple fall increasingly below the corresponding dielectric continuum quantity ΔG*con as the ratio of the solvent to reactant radii (rsol /rre ) increases. For the experimentally common circumstance (rsol /rre )≲2 and for zero and infinite-frequency dielectric constants ε0 and ε∞ over the ‘‘typical’’ ranges 20–100 and 1.75–2.5, respectively, the calculated ΔG* values are up to ∼30% below ΔG*con , the deviations tending to be larger for smaller ε0 and ε∞ values. Two kinetic models are utilized to extract νn values from Δ(t): Hynes’ approach derived from the generalized Langevin equation (GLE), and that based on the Smoluchowski diffusion equation. The νn values derived using both models decrease progressively below the corresponding dielectric continuum frequency factor νconn as (rsol /rre ) increases, the deviations between νn and νconn being greater for smaller ε∞ and/or larger ε0 . The magnitude of these deviations, however, is noticeably smaller for the Hynes GLE than the diffusion model. Thus for (rsol /rre )≲2 with the above dielectric parameters, νn is calculated to be up to ∼two and five fold smaller than νconn on the basis of the Hynes GLE and diffusion model, respectively. These differences can be understood in terms of the relative influences upon νn of the slower relaxation components arising from short-range solvation, responsible for the deviations from νconn , in comparison with the faster dynamics associated with more distant solvent molecules. Slightly larger deviations from the continuum predictions are obtained using the GLE approach in the presence of barrier-top curvature. At least for (rsol/rre)<2, the MSA-prescribed rate constants also do not deviate greatly from the corresponding continuum prediction, the corresponding decreases in νn and ΔG* being partly compensatory.