ARTICLES Variation of the Resonant Transfer Rate When Passing from Nonadiabatic to Adiabatic Electron Transfer V. Gladkikh and A. I. Burshtein* Chemical Physics Department, Weizmann Institute of Science, RehoVot 76100, Israel I. Rips Department of Sciences, Holon Academic Institute of Technology, Holon 58102, Israel ReceiVed: December 14, 2004; In Final Form: April 11, 2005 Two competing theories are used for bridging the gap between the nonadiabatic and the deeply adiabatic electron transfer between symmetric parabolic wells. For the high friction limit, a simple analytic interpolation is proposed as a reasonable alternative to them, well-fitted to the results of numerical simulations. It provides a continuous description of the electron transfer rate in the whole range of variation of the nonadiabatic coupling between the diabatic states. For lower friction, the original theories are used for the same goal. With an increase in coupling, the cusped barrier transforms into the parabolic one. Correspondingly, the pre-exponent of the Arrhenius transfer rate first increases with coupling, then levels off approaching the “dynamic solvent effect” plateau but finally reduces reaching the limit of the adiabatic Kramers theory for the parabolic barrier. These changes proceeding with a reduction in the particle separation affect significantly the spatial dependence of the total transfer rate. When approaching the contact distance, the exact rate becomes smaller than in the theory of dynamical solvent effects and much smaller than predicted by perturbation theory (golden rule), conventionally used in photochemistry and electrochemistry. I. Introduction The electron transfer rate is a fundamental property used in the theories of intramolecular and intermolecular reactions in dense media. 1-4 At high temperatures, the system motion is adiabatic everywhere except at the crossing point of the intersecting energy levels where the electron tunneling occurs. For electron exchange reactions, the potential surface consists of the two symmetric diabatic energy levels, which are com- monly assumed to be parabolic (Figure 1). The free energy gap for electron transfer in both directions is zero, and the transfer rate is given by the conventional Arrhenius equation: Here, U is the energetic height of the crossing point, 2V is the nonadiabatic splitting of the energy levels 1 and 2 at this point, λ is the reorganization energy of transfer, and k B ) 1. The preexponential factor, k, depends on the nonadiabatic coupling and the dynamic of motion along the reaction coordinate. The evaluation of this factor constitutes a complex problem that cannot be solved universally within a single theory. A number of theories have to be used to cover the whole domain of k(V,γ) where γ is a friction along the reaction coordinate. This two-dimensional domain was used in a few works 5-7 to indicate the results of different theories and their mutual borders as shown in Figure 2, taken from ref 7. This figure establishes all of the results and their regions of applicability but does not provide bridging between them. Particularly, the variation of the prefactor k with the nonadiabatic coupling V at a fixed dissipation strength γ (in the vertical cross-section of the domain from bottom to top) is due to the monotonic increase of the coupling, with reduction of the inter-reactant separation (up to their closest approach at r ) σ). Passing this way at high friction, one starts from the nonadiabatic perturbation theory subregion, where transfer is limited by tunneling, crosses the intermediate subregion of the dynamical solvent effect (DSE), but finishes in the adiabatic subregion where the reaction is controlled by W ) ke -(U-V)/T , U ) λ/4 (1.1) Figure 1. Energetic scheme of resonant electron transfer. V(r) ) V 0 e -(r-σ)/L (1.2) 4983 J. Phys. Chem. A 2005, 109, 4983-4988 10.1021/jp044311y CCC: $30.25 © 2005 American Chemical Society Published on Web 05/18/2005