Reaction coordinate Marcus theory

Figure 1. Schematic illustration of the diabatic (solid) and adiabatic (long-dashed) electronic free energy curves as functions of the solvent coordinate z<. for a single electron transfer reaction. The Marcus theory quantities AG° and A are indicated.

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... 

It is often useful to compare the reactivity of one compound with that of similar compounds. What we would like to do is to find out how a reaction coordinate (and in particular the transition state) changes when one reactant molecule is replaced by a similar molecule. Marcus theory is a method for doing this. ... 

Rates of addition to carbonyls (or expulsion to regenerate a carbonyl) can be estimated by appropriate forms of Marcus Theory. " These reactions are often subject to general acid/base catalysis, so that it is commonly necessary to use Multidimensional Marcus Theory (MMT) - to allow for the variable importance of different proton transfer modes. This approach treats a concerted reaction as the result of several orthogonal processes, each of which has its own reaction coordinate and its own intrinsic barrier independent of the other coordinates. If an intrinsic barrier for the simple addition process is available then this is a satisfactory procedure. Intrinsic barriers are generally insensitive to the reactivity of the species, although for very reactive carbonyl compounds one finds that the intrinsic barrier becomes variable. ... 

In summary, to apply the Marcus theory of electron transfer, it is necessary to see if the temperature dependence of the electron transfer rate constant can be described by a function of the Arrhenius form. When this is valid, one can then determine the activation energy AEa only under this condition can we use AEa to determine if the parabolic dependence on AG/ is valid and if the reaction coordinate is defined. 

One might ask how the Marcus case of crossing in the harmonic region can arise. In a sense, that is the surprising situation. How can significant reaction occur without reaching the anharmonic part of the potential To think of this, it is probably wise to remember that Marcus theory was first applied to electron transfer of the outer-sphere variety. Albery answers by pointing out that a reaction coordinate which is constructed from the intersection of parabolic surfaces for several... 

Fig. 1 Comparison of Marcus theory of outer sphere electron transfer (a) with the Saveant theory (b) of concerted dissociative electron transfer. The reaction coordinate is a solvent parameter. The reaction coordinate, r, is the A—B bond length.

R. A. Marcus It certainly is a good point that transition state theory, and hence RRKM, provides an upper bound to the reactive flux (apart from nuclear tunneling) as Wigner has noted. Steve Klippenstein [1] in recent papers has explored the question of the best reaction coordinate, e.g., in the case of a unimolecular reaction ABC — AB + C, where A, B, C can be any combination of atoms and groups, whether the BC distance is the best choice for defining the transition state, or the distance between C and the center of mass of AB, or some other combination. The best combination is the one which yields the minimum flux. In recent articles Steve Klippenstein has provided a method of determining the best (in coordinate space) transition state [1]. 

R. A. Marcus My interests in variational microcanonical transition state theory with J conservation goes back to a J. Chem. Phys. 1965 paper [1], and perhaps I could make a few comments. First, using a variational treatment we showed with Steve Klippenstein a few years ago that the transition-state switching mentioned by Prof. Lorquet poses no major problem The calculations sometimes reveal two, instead of one, bottlenecks (transition states, position of minimum entropy along the reaction coordinate) [2], and then one can use a method described by Miller and partly anticipated by Wigner and Hirschfelder to calculate the net dux. 

Marcus attempted to calculate the minimum energy reaction coordinate or reaction trajectory needed for electron transfer to occur. The reaction coordinate includes contributions from all of the trapping vibrations of the system including the solvent and is not simply the normal coordinate illustrated in Figure 1. In general, the reaction coordinate is a complex function of the coordinates of the series of normal modes that are involved in electron trapping. In this approach to the theory of electron transfer the rate constant for outer-sphere electron transfer is given by equation (18). 

The RRKM theory is a ubiquitous tool for studying dissociation or isomerization rates of molecules as a function of their vibrational energy. Still highly active in the theoretical field, Marcus has tackled such issues as the semiclassical theory for inelastic and reactive collisions, devising reaction coordinates, new tunneling paths, and exploring solvent dynamics effects on unim-olecular reactions in clusters. 

In a previous paper [22], it has been shown that the reaction coordinate is well defined only in the Marcus theory, that is in the classical regime. In this case, the reaction coordinate is along the minimum crossing point in the multi-dimensional potential surfaces (i.e., multi-mode case). In the quantum regime, the reaction coordinate is not well defined especially in the case of the non-relaxed ET like Wm. 

Figure 4.2 illustrates the parabolic free-energy surfaces as a function of reaction coordinate. In particular, three different kinetic regimes are shown in accordance with the classical Marcus theory. The reorganization energy, X, represents the change in free energy upon transformation of the equilibrium conformation of the reactants to the equilibrium conformation of the products when no electron is... 

Perhaps the point to emphasise in discussing theories of translational energy release is that the quasiequilibrium theory (QET) neither predicts nor seeks to describe energy release [576, 720], Neither does the Rice— Ramspergei Kassel—Marcus (RRKM) theory, which for the purposes of this discussion is equivalent to QET. Additional assumptions are necessary before QET can provide a basis for prediction of energy release (see Sect. 8.1.1) and the nature of these assumptions is as fundamental as the assumption of energy randomisation (ergodic hypothesis) or that of separability of the transition state reaction coordinate (Sect. 2.1). The only exception arises, in a sense by definition, with the case of the loose transition state [Sect. 8.1.1(a)]. 

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