Marcus formula system

Having the quanmm chemical estimate of the system reorganization energy values one can further estimate the electron transfer activation energies E according to the Marcus formula given in [8] ... 

One may wonder whether a purely harmonic model is always realistic in biological systems, since strongly unharmonic motions are expected at room temperature in proteins [30,31,32] and in the solvent. Marcus has demonstrated that it is possible to go beyond the harmonic approximation for the nuclear motions if the temperature is high enough so that they can be treated classically. More specifically, he has examined the situation in which the motions coupled to the electron transfer process include quantum modes, as well as classical modes which describe the reorientations of the medium dipoles. Marcus has shown that the rate expression is then identical to that obtained when these reorientations are represented by harmonic oscillators in the high temperature limit, provided that AU° is replaced by the free energy variation AG [33]. In practice, tractable expressions can be derived only in special cases, and we will summarize below the formulae that are more commonly used in the applications. 

With the model of Equation (19) and with a reasonable estimate of the free energies A(fn and AG°3 we can start to evaluate the apparent activation barrier. Before doing so, we must clarify several points (i) A Marcus type relationship and the corresponding LFERs are only valid for a two-state system (1 —>2), i.e., for a reaction with a single step. However, we have a three-state process that involves a two-step mechanism (1 ->2->3). Fitting such a system to a Marcus type formula can lead to nonphysical parameters (e.g., too small of a value for X). (ii) In order to use the HAW approach in a three-state system (or in a four-state system) we must consider the elementary rate constants and then consider the preequilibrium concentrations. 

In eqn (12.19), the effects of nonadiabatic transition including the nuclear tunneling are properly taken into account by rj and naturally the main task is to evaluate the thermally averaged transition probability P fl, < ), which has to be evaluated using the Monte Carlo technique for multi-dimensional systems. It is easily shown that the Marcus-Hush formula in adiabatic and nonadiabatic limits can be recovered from eqn (12.19) and eqn (12.20) within the high-temperature approximation. 

ET rate via electronic coupling for a multi-dimensional system in the Marcus inverted regime, (a) pEa—6.7, (b) pSa = 10.0, and (c) pEa = 20.0. Ea represents the minimum energy on the seam surface. Solid line present result dashed line the results predicted from the LZ formula dotted line results from perturbation theory. 

Calculations of A<, from the electrostatic continuum model and from quantum mechanical models have been reviewed and compared/ The distinction between electrostatic displacement D and field E is emphasized. Values of A are compared for different physical models of the reacting molecules, e.g., conducting spheres (the model usually considered in previous literature) and cavities of various dimensions. In the electrostatic model a formula for A has been given, which applies to any system which has a symmetrical binuclear structure, and from which Marcus two-sphere " and Cannon s ellipsoidal " models can be deduced as special cases. 

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