Outer-Sphere Electron-Transfer Theory

The outer-sphere theory has been developed using an electrostatic approach to calculate the energy necessary to bring reactants together, to reorganize the solvent around the transition state and to prepare the metal centers for election transfer. 

In North America the theory is associated with the name of Marcus and referred to as the Marcus theory. However, Hush in Australia and Levitch and Dogonadze in the U.S.S.R. have made original contributions. Marcus started from the transition-state theory for ionic reactions. Hush from solid-state electron-transfer theory and Levitch from a consideration of reactions at electrodes. All arrived at essentially the same result, although using different terminologies. More recently, Tachiya and co-workers have developed a model based on the electrostatic interaction of the reactants with a polar solvent which also reduces to the same result under certain conditions. The version of the theory developed by Marcus has remained predominant for kinetic studies because it is framed in more familiar terminology and yields relationships that appear simple to test by experiment. 

The Franck-Condon principle is fundamental to the theory. This principle states that electron movement is much faster than nuclear motion thus, intemuclear distances do not change during the instant of electron transfer. Therefore, it is assumed that on approaching the transition state, the bond lengths of the reactants will adjust to approach those of the products. 

The electron transfer is assumed to be an adiabatic process in the Ehrenfest sense so that the transmission coefficient, k, in the transition-state theory expression (Section 1.6.2), Eq. (6.7), is equal to one. 

This implies that there is enough interaction between reactants in the transition state to make the probability of electron transfer equal to one, although normal bonding forces are assumed to be much weaker than electrostatic ones. There are occasions when the interaction is thought to be so weak that k 1 and the effect of nonadiabaticity on the reaction rate is sometimes used as a rationale for differences between observed and predicted rate constants. 

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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... 

How might one measure the kinetic parameters for the self-exchange k 10 M s ) of (T -arene)2CrV(Ti -tu ene)2Cr in solution What would be the advantages of such a system for testing outer-sphere electron transfer theories How might the rate constant vary with the optical and static dielectric constants for various solvents ... 

R. A. Marcus Outer-sphere electron transfer theory... 

The triplet state also is a strong photoreductant in its reactions with alkylated pyridinium acceptors in acetonitrile solution. As predicted from outer-sphere electron transfer theory, a plot of / rin (Aq) versus (A /A) is linear for quenchers with values of (AVA) that are less than, or approximately equal to, the value of for the IrJ/Irz couple. The plotted data for this fit range from... 

Figure 2.1 (Plate 2.1) shows a classification of the processes that we consider they aU involve interaction of the reactants both with the solvent and with the metal electrode. In simple outer sphere electron transfer, the reactant is separated from the electrode by at least one layer of solvent hence, the interaction with the metal is comparatively weak. This is the realm of the classical theories of Marcus [1956], Hush [1958], Levich [1970], and German and Dogonadze [1974]. Outer sphere transfer can also involve the breaking of a bond (Fig. 2. lb), although the reactant is not in direct contact with the metal. In inner sphere processes (Fig. 2. Ic, d) the reactant is in contact with the electrode depending on the electronic structure of the system, the electronic interaction can be weak or strong. Naturally, catalysis involves a strong...

In order to account for the foregoing kinetic behavior, we rely on the Marcus theory for outer-sphere electron transfer to provide the quantitative basis for establishing the free energy relationship (8), i.e.,... 

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. 

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. 

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. 

Another important aspect of the Marcus theory has also been systematically investigated with organic molecules, namely the quadratic, or at least the non-linear, character of the activation-driving force relationship for outer sphere electron transfer. In other words, does the transfer coefficient (symmetry factor) vary with the driving force, i.e. with the electrode... 

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.

The theoretical results obtained for outer-sphere electron transfer based on self-exchange reactions provide the essential background for discussing the interplay between theory and experiment in a variety of electron transfer processes. The next topic considered is outer-sphere electron transfer for net reactions where AG O and application of the Marcus cross reaction equation for correlating experimental data. A consideration of reactions for which AG is highly favorable leads to some peculiar features and the concept of electron transfer in the inverted region and, also, excited state decay. 

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). 

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