Electron-transfer . nonadiabatic solvent transitions

Electron transfer (ET) is of course accompanied by rearrangement of the solvent as shown by the horizontal displacement in Figure 26. Tradiational theories for ET fall into two cases. In the nonadiabatic case it is assumed that the rate of ET is controlled by the process of crossing from one electronic state (e.g., LE) to the other (e.g., CT) [60,61]. Alternatively in the weakly adiabatic case, it is assumed that the solvent polarization is always in equilibrium with the changing charge distribution. For this latter case transition state theory is applicable [59]. 

The Marcus theory, as described above, is a transition state theory (TST, see Section 14.3) by which the rate of an electron transfer process (in both the adiabatic and nonadiabatic limits) is assumed to be determined by the probability to reach a subset of solvent configurations defined by a certain value of the reaction coordinate. The rate expressions (16.50) for adiabatic, and (16.59) or (16.51) for nonadiabatic electron transfer were obtained by making the TST assumptions that (1) the probability to reach transition state configuration(s) is thermal, and (2) once the reaction coordinate reaches its transition state value, the electron transfer reaction proceeds to completion. Both assumptions rely on the supposition that the overall reaction is slow relative to the thermal relaxation of the nuclear environment. We have seen in Sections 14.4.2 and 14.4.4 that the breakdown of this picture leads to dynamic solvent effects, that in the Markovian limit can be characterized by a friction coefficient y The rate is proportional to y in the low friction, y 0, limit where assumption (1) breaks down, and varies like y when y oo and assumption (2) does. What stands in common to these situations is that in these opposing limits the solvent affects dynamically the reaction rate. Solvent effects in TST appear only through its effect on the free energy surface of the reactant subspace. 

We have presented several approaches to calculate the rate constants of electron transfer occurring in solvent from the weak to strong electronic couplings. In the fast solvent relaxation limit, the approach based on the nonadiabatic transition state theory can be adopted. It is related to the Marcus formula by a prefactor and referred as a modified Marcus formula. When the solvent dynamics begin to play a role, the quantum Kramers-like theory is applicable. For the case where the intramoleeular vibrational motions are much faster than the solvent motion, the extended Sumi-Mareus theory is a better ehoice. As the coherent motion of eleetron is ineorporated, such as in the organic semiconductors, the time-dependent wavepaeket diffusion approach is proposed. Several applications show that the proposed approaches, together with electronic structure calculations for the faetors eontrolling eleetron transfer, can be used to theoretically predict electron transfer rates correctly. 

Fig. 9.1. PES of the adiabatic electron transfer reaction of Eq. (9.4) in polar solvent obtained through superposition of the terms of the initial , and the final Ef states q is the reaction coordinate that includes the solvent reorganization). The process is shown schematically as a change in the polarization of the medium when passing from the equilibrium configuration (qf) of the initial state to the equilibrium configuration (qj) of the state with the transferred electron. The electron transfer occurs in the region q. When V.j. is small, there exists a probability for the reaction trajectory to cross the transition state region without leading to product formation (nonadiabatic reaction)...

In the simplest case, the R mode is characterized by a low frequency and is not dynamically coupled to the fluctuations of the solvent. The system is assumed to maintain an equilibrium distribution along the R coordinate. In this case, ve can exclude the R mode from the dynamical description and consider an equilibrium ensemble of PCET systems with fixed proton donor-acceptor distances. The electrons and transferring proton are assumed to be adiabatic with respect to the R coordinate and solvent coordinates within the reactant and product states. Thus, the reaction is described in terms of nonadiabatic transitions between two sets of intersecting free energy surfaces ( R, and ej, Zp, corresponding to... 

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