Marcus theory semiclassical equation

The classical (or semiclassical) equation for the rate constant of e.t. in the Marcus-Hush theory is fundamentally an Arrhenius-Eyring transition state equation, which leads to two quite different temperature effects. The preexponential factor implies only the usual square-root dependence related to the activation entropy so that the major temperature effect resides in the exponential term. The quadratic relationship of the activation energy and the reaction free energy then leads to the prediction that the influence of the temperature on the rate constant should go through a minimum when AG is zero, and then should increase as AG° becomes either more negative, or more positive (Fig. 12). In a quantitative formulation, the derivative dk/dT is expected to follow a bell-shaped function [83]. 

An alternative approach to polaron transport in organic solids is in terms of electron transfer (ET). The process can be viewed as a special case of the non-radiative decay of an electronic state. The derivation of the theory is developed in various books or review papers [13-15]. The parameter of importance here is the transition probability per unit time (or transition rate) kif between an initial and a final state. The rate is estimated within the Franck-Condon approximation. In the high-temperature regime ( cOif < kT) the Franck-Condon-weighted density (FCWD) reduces to a standard Arrhenius equation, so the rate takes its semiclassical Marcus theory expression [16] ... 

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