Sumi and Marcus

It remains now to solve Eq. (2.3). Here, there are various approaches, depending on the conditions. When a non-steady-state solution is required, one can introduce the decoupling approximation of Sumi and Marcus, if there is the difference in time scales mentioned earlier. Or one can integrate Eq. (2.3) numerically. For the steady-state approximation either Eq. (2.3) can again be solved numerically or some additional analytical approximation can be introduced. For example, one introduced elsewhere [44] is to consider the case that most of the reacting systems cross the transition state in some narrow window (X, X i jA), narrow compared with the X region of the reactant [e.g., the interval (O,Xc) in Fig. 2]. In that case the k(X) can be replaced by a delta function, fc(Xi)A5(X-Xi). Equation (2.3) is then readily integrated and the point X is obtained as the X that maximizes the rate expression. The A is obtained from the width of the distribution of rates in that system [44]. 

In the remaining part of Section II.A we review the formal relationship of C(t) to fundamental quantities in the statistical mechanical description of solvation. The derivation we review is adopted from the work of Van der Zwan and Hynes. A useful result of the derivation is that a physical basis for the solvent coordinate in Figure 1 is established [54], The reader is referred to papers by Bagchi et al. [53], and Sumi and Marcus [54] for related treatments. 

Most polaron models consider only electron transfer steps parallel and antiparallel to the applied field. Van der Auweraer et al. (1994) derived an expression for the mobility that takes into account isotropic hopping in three dimensions. The treatment is based on the Marcus theoiy (Marcus, 1964, 1968, 1984 Kester et al., 1974 Jortner, 1976 Sumi and Marcus, 1986 Jortner and Bixon, 1988) and assumes that energetic and positional disorder can be neglected. 

Recently much attention has been aroused on solution reactions whose rates decrease as the viscosity Tj of solvents increases. These reactions cannot be rationalized in the framework of the transition state theory. To describe them, two currents of theories have been developed by extending the Kramers theory. One was initiated by Grote and Hynes, while the other by Sumi and Marcus. Recent data on thermal Z/E isomerization of substituted azobenzenes and A/ -benzyU-deneanilines confirms the applicability of the latter for 77 variation over 10 times under pressure. 

Forty years after Kramers seminal paper on the effect of solvent dynamics on chemical reaction rates (Kramers, 1940), Zusman (1980) was the first to consider the effect of solvent dynamics on ET reactions, and later treatments have been provided by Friedman and Newton (1982), Calef and Wolynes (1983a, 1983b), Sumi and Marcus (1986), Marcus and Sumi (1986), Onuchic et al. (1986), Rips and Jortner (1987), Jortner and Bixon (1987) and Bixon and Jortner (1993). The response of a solvent to a change in local electric field can be characterised by a relaxation time, r. For a polar solvent, % is the longitudinal or constant charge solvent dielectric relaxation time given by, where is the usual constant field dielectric relaxation time... 

The recent theoretical approaches include a theory of barrierless electronic relaxation which draws on the model of nonradiative excited state decay, and a general treatment of the effect of solvent dielectric relaxation based on the theory of optical line shapes, as well as treatments based on classical and quantum rate theories. Equation(5) does not hold for all solvents and, more generally, may be frequency-dependent. Papers by Hynes, Rips and Jortner, Sumi and Marcus, and Warshel and Hwang " contain good overviews of the theoretical developments. 

For organic molecules in solution, both the molecular vibrational motions and the solvent relaxation play important roles in ET. Usually, the vibrational motions are much faster than the solvent polarization motion because the vibrational motions have much higher frequencies. Thus, the thermal equilibrium may be maintained for those vibrational modes during the course of ET. In this case, one may separately consider the intramolecular vibrational motions and solvent motions. Along the directions of the high-frequency modes, ET can still be described by NA-TST, but the solvent motion has to be treated with the dynamic equations. Sumi and Marcus proposed a method to describe such a kind of ET, where the solvent motion satisfies the dififusion reaction equation and intramolecular ET is incorporated with a sink function in the diffusion equation. In the original SM theory, the sink function is obtained from the... 

Sumi H. and Marcus R. A. (1986), Dielectric relaxation and intramolecular electron transfers and Dynamical effects in electron transfer reactions , J. Chem. Phys. 84, 4272-4276 and 4894-4914. 

Figure C3.2.12. Experimentally observed electron transfer time in psec (squares) and theoretical electron transfer times (survival times, Tau a and Tau b) predicted by an extended Sumi-Marcus model. For fast solvents tire survival times are a strong Emction of tire characteristic solvent relaxation dynamics. For slower solvents tire electron transfer occurs tlirough tire motion of intramolecular degrees of freedom. From [451.

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