Theory of Outer-Sphere Electron Transfer Reactions

Chemical and electrochemical reactions in condensed phases are generally quite complex processes only outer-sphere electron-transfer reactions are sufficiently simple that we have reached a fair understanding of them in terms of microscopic concepts. In this chapter we give a simple derivation of a semiclassical theory of outer-sphere electron-transfer reactions, which was first systematically developed by Marcus [1] and Hush [2] in a series of papers. A more advanced treatment will be presented in Chapter 19. 

C. Mechanism and theory of outer sphere electron transfer reactions... 

C. Mechanism and Theory of Outer Sphere Electron Transfer Reactions... 

In contrast to cerium, outer-sphere electron transfer appears to be the dominant reaction route for europium redox reactions with both organic species and transition metal complexes. Interpretation of europium redox reactions in terms of the Marcus theory of outer-sphere electron transfer reactions is limited by the enduring controversy over the self-exchange rate for the Eu(II)/Eu(III) couple. Since the selfexchange rate for the Eu(II)/Eu(IlI) couple has so far proven inaccessible to direct... 

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. 

In the case of stepwise electron-transfer bond-breaking processes, the kinetics of the electron transfer can be analysed according to the Marcus-Hush theory of outer sphere electron transfer. This is a first reason why we will start by recalling the bases and main outcomes of this theory. It will also serve as a starting point for attempting to analyse inner sphere processes. Alkyl and aryl halides will serve as the main experimental examples because they are common reactants in substitution reactions and because, at the same time, a large body of rate data, both electrochemical and chemical, are available. A few additional experimental examples will also be discussed. 

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.

In contrast to the experimentally based work discussed above, in the most recent comprehensive theoretical discussion [21d], Bixon and Jortner state that the question of whether non-adiabatic or adiabatic algorithms describe electron-transfer reactions was settled in the 1960s, and that the majority of outer-sphere electron-transfer reactions are non-adiabatic. This is certainly true for the reactions that occur in the Marcus inverted region in which these authors are interested, but we think the question of whether reactions in the normal region are best treated by adiabatic theory that includes an electronic transmission coefficient or by non-adiabatic equations remains to be established. 

Simulations of outer sphere electron transfer reactions [197, 217-220, 223] show that the Marcus theory is valid over a wide range. Deviations from the linear response regime were found. 

Outer-sphere electron transfers can be treated in a more general way than inner-sphere processes, where specific chemistry and interactions are important. For this reason, the theory of outer-sphere electron transfer is much more highly developed, and the discussion that follows pertains to these kinds of reactions. However, in practical applications, such as in fuel cells and batteries, the more complicated inner-sphere reactions are important. A theory of these requires consideration of specific adsorption effects, as described in Chapter 13, as well as many of the factors important in heterogeneous catalytic reactions (56). 

In our group, we have developed a theoretical framework that can be applied to both kinds of reactions. It is based on a model Hamiltonian incorporating concepts from theories of outer sphere electron transfer [49-51], Anderson-Newns theory [52, 53], and our own ideas. The model as was developed in the 1990s [54, 55] and at that time, it was applied to various processes such as metal deposition/dissolution, anion adsorption, and outer-sphere electron transfer. However, this was at a time when DFT was not widely available, and several important system parameters had to be estimated, so that the applications had a qualitative character nevertheless, they provided a basic understanding of these processes at the molecular level. 

The details of the Marcus theory have been described in several reviews " and in books by Reynolds and Lumry and Cannon. The following discussion will simply outline the features of the theory and give the physical factors that are predicted to be important in determining the rates of outer-sphere electron-transfer reactions. 

Electron-transfer reactions have been attracting interest of theoretical chemists for many years. An important contribution to the theory of outer-sphere electron transfer is that of R. A. Marcus. ... 

A powerful application of outer-sphere electron transfer theory relates the ET rate between D and A to the rates of self exchange for the individual species. Self-exchange rates correspond to electron transfer in D/D (/cjj) and A/A (/c22)- These rates are related through the cross-relation to the D/A electron transfer reaction by the expression... 

However, a closer inspection of the experimental data reveals several differences. For ion-transfer reactions the transfer coefficient a can take on any value between zero and one, and varies with temperature in many cases. For outer-sphere electron-transfer reactions the transfer coefficient is always close to 1/2, and is independent of temperature. The behavior of electron-transfer reactions could be explained by the theory presented in Chapter 6, but this theory - at least in the form we have presented it - does not apply to ion transfer. It can, in fact, be extended into a model that encompasses both types of reactions [7], though not proton-transfer reactions, which are special because of the strong interaction of the proton with water and because of its small mass. 

Overall, the outer-sphere electron-transfer reactions of transition metal complexes reactions are consistent with the expectations of the semiclassical Marcus-Hush theory. h22,25,32,43,57,7i,75 78 agreement... 

A quasi-linear correlation of log/c or AG with the ionization potentials of the electron donors as observed in the FeL3 - + reactions is predicted by Marcus theory for outer-sphere electron transfers. Accordingly, the free-energy dependence of AG can be satisfactorily simulated with the Marcus equation (Eq. 90), taking a (constant) value of A = 41 kcal mol as reorganization energy for all tetraalkyltin compounds (see Figure 19A) [32]. 

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