Solvent controlled charge transfer dynamics on diabatic surfaces with different curvatures

We investigate electron transfer reactions on diabatic surfaces with different curvatures within the framework of Zusman theory. First, a generalization of the nonadiabatic Marcus–Levich–Dogonadze rate expression is obtained for the case of different forward and backward reorganization energies. Second, we provide a corresponding generalization of the Zusman rate expression which bridges between nonadiabatic and solvent controlled adiabatic electron transfer. The derived analytical rate expressions compare favorably against the precise numerics. A detailed comparison with previous work of Tang is presented. 2002 Elsevier Science B.V. All rights reserved. The kinetics of electron transfer (ET) processes in condensed media is essentially mediated by solvent effects. In the theories developed by Marcus [1–3], Hush [4], Levich and Dogonadze [5] the possible influence of the solvent dynamics on the transfer rate is neglected. Over 20 years ago, Zusman [6] and Alexandrov [7] introduced a set of equations to incorporate the important solvent relaxation effects. In the Zusman–Alexandrov theory, the limits of nonadiabatic and solvent controlled adiabatic transfer are described in a unified way. In the theoretical analysis, the reaction coordinate is usually assumed to move on parabolic diabatic energy curves of identical curvatures. The curvature is related to the reorganization energy, so that equal curvatures implies that the reorganization energies for the forward and backward reactions are identical. Previous analytical studies and numerical simulations indicate that the values of these two reorganization energies can be different [8–13]. A few years ago, Tang [14] presented an extension of the Zusman–Alexandrov theory to the case of parabolic diabatic energy curves of different curvatures. Tang’s analysis relies on the use of the contact approximation (cf. Eq. (11)). The main objective of our study is to further extend the analysis of Zusman equations to the case of two parabolas with different curvatures beyond the contact approximation and to compare the analytical predictions with the numerical solutions of Zusman equations. In the limit of nonadiabatic electron transfer we present a rate expression which is favorably tested against numerical solutions of 10 July 2002 Chemical Physics Letters 360 (2002) 333–339 www.elsevier.com/locate/cplett * Corresponding author. Fax: +49-821-598-3222. E-mail addresses: [email protected], [email protected] (P. H€anggi). 0009-2614/02/$ see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S0009-2614 (02 )00845-X Zusman equations. We will also obtain a generalization of the Zusman rate expression that allows us to span the nonadiabatic to solvent controlled adiabatic electron transfer regimes in a unified way. Our expression is compared with a result obtained by Tang [14]. The basic elements to describe electron transfer processes are two diabatic electronic energy curves VjðxÞ, j 1⁄4 1; 2, and a generalized one-dimensional reaction coordinate x with effective mass m. The electronic states before and after the charge transfer will be denoted as donor, j1i, and acceptor, j2i, respectively. The reaction coordinate represents a combination of the selected nuclear modes coupled directly to the electronic transfer system [15]. The reaction coordinate is also coupled to the rest of nuclear modes. This coupling introduces friction in the dynamics of the reaction coordinate. In the overdamped limit, Zusman equations provide an appropriate description for the time evolution of the matrix elements qjkðx; tÞ :1⁄4 hj; xjq̂ðtÞjx; ki of the reduced density operator in the electron and reaction coordinate Hilbert space. Zusman equations and their validity conditions have been repeatedly derived and discussed in the literature [16–21]. These equations read in matrix form