Sumi-Marcus theory

The theoretical ket values were nearly two orders smaller than those experimentally obtained. This may be attributed to disregarding the internal vibration and/or the nuclear tunneling effect. An estimation of the former effect by Sumi-Marcus theory predicts one-two orders increase in the rate with conserving the relative difference in the rates in different solvents. 

The most important achievement of this work is the observation of greatly reduced intramolecular ET rates in the presence of chemically inert counterions. The dynamic counter-ion control of the ET rate could be observed because the studied electron transfer reaction was veiy weakly exothermic. Therefore, even in the case of the large organic cations and relatively weak electrostatic interactions, the condition Xionic > AG was satisfied, and the evolution along the slow coordinate of the Sumi-Marcus theory, that is, the diffusion of the cation, could become the rate-determining step. 

One should keep in mind that the rate constant increases monotonically with coupling only if the relaxation time of the fast mode is infinitely small. If it is finite, as it should be, the rate versus couphng dependence starts as the Golden Rule prescribes, then continues according to the Sumi-Marcus theory, but finally saturates at a certain value determined by the relaxation time of the fast mode. This effect is illustrated in Figure 9.18(h) for the case where both fast and slow modes relax exponentially. In this case, the problem is reduced to solution of a system of 2D diffusion equations along parabolic potential surfaces corresponding to two coordinates and... 

In order for the reaction to take place with the mechanism in the Grote-Hynes theory as well as in the Kramers theory, the reactant must surmount over the transition-state barrier only by diffusional Brownian motions regulated by solvent fluctuations. In the two-step mechanism of the Sumi-Marcus model, on the other hand, surmounting over the transition-state barrier is accomplished as a result of sequential two steps. That is, the barrier is climbed first by diffusional Brownian motions only up to intermediate heights, from which much faster intramolecular vibrational motions take the reactant to the transition state located at the top of the barrier. 

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. 

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. 

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... 

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