Marcus theory assumptions

This section contains a brief review of the molecular version of Marcus theory, as developed by Warshel [81]. The free energy surface for an electron transfer reaction is shown schematically in Eigure 1, where R represents the reactants and A, P represents the products D and A , and the reaction coordinate X is the degree of polarization of the solvent. The subscript o for R and P denotes the equilibrium values of R and P, while P is the Eranck-Condon state on the P-surface. The activation free energy, AG, can be calculated from Marcus theory by Eq. (4). This relation is based on the assumption that the free energy is a parabolic function of the polarization coordinate. Eor self-exchange transfer reactions, we need only X to calculate AG, because AG° = 0. Moreover, we can write... 

Comparison of equations (2.11) and (2.15) reveals q and r to be kikilk i and A 2//r i, respectively. This enables k to be calculated from qjr. In its simplest forms the structure of the reactive intermediate can be viewed as V(OH)Cr " (when n is 1) or as VOCr (when n is 2). Similar species which have been characterized or implied kinetically are CrOCr (ref. 33), Np02Cr (ref. 37), U02Cr (ref. 31), VOV " (ref. 34), U0Pu02 + (ref. 41), Pu02pe + (ref. 42) and FeOFe + (ref. 38). Predictions on the rate of the V(III)- -Cr(lI) system, based upon Marcus theory", have been made by Dulz and Sutin on the assumption that an outer-sphere process applies. The value arrived at by these authors is 60 times lower than the experimental value. 

The basic assumption of the Marcus theory of electron transfer is that only a weak interaction between the reactants is needed in order for a simple electron-transfer process to occur. Marcus theory considers... 

Marcus theory is based on certain assumptions that will be discussed later. The main goal of computer simulations of electron transfer is to check some of these assumptions and to provide additional microscopic insight into the mechanism of electron transfer and the microscopic factors that influence the rate of transfer. We discuss these issues in the following section for the simple case of outer-sphere electron transfer reactions. 

Q vibration is not directly coupled to the bath of harmonic oscillators. This assumption is similar to the approach employed by Silbey and Suarez who used a tunneling splitting that depends on the oscillating transfer distance Q in their spin-boson Hamiltonian. Borgis and Hynes, too, have made this assumption in the context of Marcus theory. 

One difficulty that arises within the Marcus formulation is that the intrinsic barrier term is ideally treated as a constant. Earlier applications of Marcus theory were based on this assumption (Cohen and Marcus, 1968 Kreevoy and Konasewich, 1970 Kreevoy and Oh, 1973 Albery et al., 1972). However, as regards methyl transfer, this assumption is clearly invalid. 

The above model has been further explored to account for reaction efficiencies in terms of a scheme where nucleophilicities and leaving group abilities can be rationalized by a structure-reactivity pattern. Pellerite and Brau-man (1980, 1983) have proposed that the central energy barrier for an exothermic reaction (see Fig. 3) can be analysed in terms of a thermodynamic driving force, due to the exothermicity of the reaction, and an intrinsic energy barrier. The separation between these two components has been carried out by extending to SN2 reactions the theory developed by Marcus for electron transfer reactions in solutions (Marcus, 1964). While the validity of the Marcus theory to atom and group transfer is open to criticism, the basic assumption of the proposed model is that the intrinsic barrier of reaction (38)... 

Entry no. 11 of Table 15 illustrates many of the difficulties involved in judging the feasibility of a slow electron-transfer step, in this case further complicated by the assumption that it takes place within a pre-formed charge-transfer complex (99). Taking for granted that Marcus theory can be applied to such a... 

Marcus Theory is also able to account for the anomalous values of a outside the range 0 < a < 1 which have been found for proton transfer involving nitroparaffins (Sect. 4.2). In this case the assumptions leading to (119) are not valid. Explanations for the anomalous exponents based on assumed variations in X [81, 200(c)] and WR [80] with AG° along a series of bases have been proposed. 

Such corrections of the original Marcus theory may still be insufficient to describe adequately the activation energy of electron-transfer reactions. However, the discrepancies between the experimental and ealculated rate constant using Eq. (24) may also result from the assumption that the frequency factor developed for the collision in the gas phase should also be valid in condensed media. 

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. 

Many reactions exhibit effects of thermodynamics on reaction rates. Embodied in the Bell-Evans-Polanyi principle and extended and modified by many critical chemists in a variety of interesting ways, the idea can be expressed quantitatively in its simplest form as the Marcus theory (15-18). Murdoch (19) showed some time ago how the Marcus equation can be derived from simple concepts based on the Hammond-Leffler postulate (20-22). Further, in this context, the equation is expected to be applicable to a wide range of reactions rather than only the electron-transfer processes for which it was originally developed and is generally used. Other more elaborate theories may be more correct (for instance, in terms of the physical aspects of the assumptions involving continuity). For the present, our discussion is in terms of Marcus theory, in part because of its simplicity and clear presentation of concepts and in part because our data are not sufficiently reliable to choose anything else. We do have sufficient data to show that Marcus theory cannot explain all of the results, but we view these deviations as fairly minor. 

A further assumption usually made in applying Marcus theory is that the intrinsic barrier for a cross-reaction is the mean of those for the two identity reactions. AE0 can therefore be calculated for an identity reaction if its value is known for a particular cross-reaction and the other identity reaction. 

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