Activationless dynamic solvent effect Serguei Feskov a , Vladislav Gladkikh b , Anatoly I. Burshtein b, * a Department of Physics, Volgograd State University, University Avenue, 100, Volgograd 400062, Russia b Weizmann Institute of Science, Rehovot 76100, Israel Received 15 June 2007; in final form 30 August 2007 Available online 6 September 2007 Abstract The dynamic solvent effect (DSE) is studied numerically at any free energy of electron transfer. In the activationless case the exact result known for an irreversible transfer is confirmed but the commonly accepted approximate results for reversible transfer were corrected. The general rate dependence on electron coupling is shown to fit the Zusman interpolation between the perturbation theory and DSE, not only in the normal and inverted region but in the activationless case as well. DSE is responsible for the saturation of the electron transfer at short distances where the electron coupling becomes strong. Ó 2007 Published by Elsevier B.V. 1. Introduction The electron transfer between the molecules separated by distance r is mainly determined by the electron coupling between the donor and acceptor parabolic terms, V. Since the tunnelling length L is short compared to the contact distance r, the coupling V ¼ V 0 e 2ðrrÞ=L ; ð1:1Þ sharply decreases with inter-particle separation. At rela- tively long r it is rather small and the perturbation theory quadratic in coupling leads to the famous Marcus rate of electron transfer [1]: W PT ðrÞ¼ V 2  h ffiffiffi p p ffiffiffiffiffiffi kT p exp  ðDG þ kÞ 2 4kT  ! ; ð1:2Þ where DG is the free energy and k is the reorganization energy of electron transfer (k B = 1). At shorter distances the coupling becomes too large and the weak non-adiabatic transfer gives way to a strong one [2]. The latter known as the dynamical solvent effect (DSE) can be also considered as adiabatic passage over a cusped barrier separating the wells. The transition from the weak to strong non-adiabatic transfer is given by the general Zusman interpolation derived for highly activated transfer [3,4]: W Z ¼ W PT 1 þ W PT =W DSE ¼ W PT at V ! 0; W DSE at V !1:  ð1:3Þ The DSE rate W DSE ¼ 1 s exp  ðDG þ kÞ 2 4kT  ! ð1:4Þ is inverse in s which is the time of reaching the crossing point by diffusional motion along the reaction coordinate. The latter is proportional to s L which is the longitudinal relaxation time of polar solvent assisting the transfer. For highly activated transfer in either the normal (DG < k) or inverted Marcus region (DG > k) there is the following relationship between these times established by Zusman [3]: 1 s ¼ 1 s L ffiffiffiffiffiffiffiffiffiffiffi 4pkT p jDG þ kjjDG  kj jDG þ kjþjDG  kj : ð1:5Þ The resonant transfer (DG = 0) is really highly activated when its activation energy k/4  T. According to the Zusman formula (1.5) its rate is 1 s 0 ¼ 1 4s L ffiffiffiffiffiffi k pT r : ð1:6Þ 0009-2614/$ - see front matter Ó 2007 Published by Elsevier B.V. doi:10.1016/j.cplett.2007.08.090 * Corresponding author. Fax: +972 89344123. E-mail address: cfbursh@wisemail.weizmann.ac.il (A.I. Burshtein). www.elsevier.com/locate/cplett Chemical Physics Letters 447 (2007) 162–167