The classical theory of Marcus

2 THEORIES OF ELECTRON-TRANSFER REACTIONS 16.2.1 The classical theory of Marcus 

ET reactions between molecular species in cases where no bond-breaking-bond-forming processes occur are usually considered to be very different in nature from atom-transfer reactions. For the latter case, we can use a Bom-Oppenheimer adiabatic potential-energy surface and there is a strong interaction with the attacking atom, which effectively determines the barrier for the reaction. In contrast, for the former processes the interaction between the electron donor and its acceptor is very weak, the nuclear configuration of the products resembles that of the reactants and, in retrospect with a certain naivety, chemists expected that ET would in general be fast processes, and would mainly be diffusion controlled. 

The current view is that the electron-transfer event itself is a fast activationless process the barrier for the reaction stems from the necessity to adjust the orientation of the solvent dipoles around ions and the lengths of some bond in the inner-coordination shells prior to the transfer step. According to this view, which was due largely to Rudolph Marcus [3], for the solvent, and to Noel Hush [4], for the metal-Ugand bond lengths, there are no proper transition states in electron-transfer reactions, because the solvent molecules are not in equilibrium distribution with the charges of the oxidised and reduced species. 

The theoretical formalism proposed to estimate the rates of ET reactions is known as the theory of Marcus (TM) [3,5,6]. However it is relevant to make a distinction between two components of the theory. One component is concerned with the estimation of the intrinsic barrier, AG(0), for homonuclear reactions in terms of molecular parameters of the reactants, which we will call TM-1. The other component of the theory addresses the effect of the reaction energy, AG°, on the reaction rates, presented in Chapter 7 in terms of the quadratic expression of Marcus (eq. (7.6)) this will be called TM-2. It is currently employed to estimate the rates of cross-reactions, when the reaction energies of the heteronuclear reactions are known together with the rates of the corresponding homonuclear reactions. 

In general terms, the theory of Marcus involves a model for ET reactions based on the approximation that the inner-coordination sphere energy is independent of the outer-sphere reorganisation. In its classical formulation, TM provides the rate for a self-exchange reaction such as reaction (16.1) 

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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 the classical theory of Marcus, the rate determining factors involve nuclear reorganization. We write the first-order rate constant, ke, as [37]... 

The approximations in which the classical theory of Marcus is based have been mentioned above. They require a discussion between the effects of having enough interaction between the reactants in the coUisional complexes to achieve adiabaticity, but not enough to decrease the barrier. The near-adiabaticity assumed for ET reactions, may not be applicable to all molecular systems, in particular when spin changes are involved or when the redox-active orbitals of the two reactants cannot approach each other very closely. On the other hand, if the redox partners interact very strongly, the electron coupling will increase to a point where its value reduces the energy barrier appreciably. 

We begin by reviewing the elements of the classical theory of ET, developed independently by Marcus and Hush.4,5 While more sophisticated theoretical treatments of ET exist,6,7 Marcus-Hush theory will largely serve the purposes of... 

The theory of photoinduced electron transfer is based on the classical work of Marcus, Hush, Mulliken, Murrell and many others and it has been extensively reviewed in Chapter 1 of Part 1. The study of isolated, ultra-cold systems provides an opportunity to check some of the basic assumptions of these theories. In particular, one can easily discriminate between different structural isomers of a given system... 

The classical theory of electron transfer developed by Marcus starts with the same kind of hard spheres in a dielectric continuum model that is used to derive the free energy of solvation of an ion. A central role in the theory is played by the reorganisation energy X, which in its simplest definition is given by... 

The theoretical description of the kinetics of electron transfer reactions starts fi om the pioneering work of Marcus [1] in his work the convenient expression for the free energy of activation was defined. However, the pre-exponential factor in the expression for the reaction rate constant was left undetermined in the framework of that classical (activate-complex formalism) and macroscopic theory. The more sophisticated, semiclassical or quantum-mechanical, approaches [37-41] avoid this inadequacy. Typically, they are based on the Franck-Condon principle, i.e., assuming the separation of the electronic and nuclear motions. The Franck-Condon principle... 

The classical theory developed by Marcus and Hush can offer no explanation for this, any more than it can for the occurrence of long-range ET, first observed in an ET protein by Chance and Nishimura (1960) and in a rigid glass by Miller (1975), or the sensitivity of ET rates to the nature of any molecules or molecular groups lying between the donor and acceptor, established by the pioneering work of Taube et al. (1953) and Tanbe and Myers (1954) on the inner-sphere ET reactions of transition-metal complexes. 

The intent here is not to provide a rigorous and comprehensive treatment of the theory, but rather to help researchers understand basic principles, classical models derived from the theory, and the assumptions upon which they are based. This focus is consistent with the goal of this chapter, which is to enable those new to this area to apply the classical forms of the Marcus model to their own science. 

Electron-transfer kinetics in solutions have often been analyzed and interpreted in the framework of the general adiabatic theory of Marcus (43). Although electron-transfer dynamics are not always characterized by a classical rate constant (44), a general formulation of the chemical reaction concerns the rate constant k, which can be expressed as ... 

In this spirit, we will also briefly describe the basis for some of the microscopic kinetic theories of unimolecular reaction rates that have arisen from nonlinear dynamics. Unlike the classical versions of Rice-Ramsperger-Kassel-Marcus (RRKM) theory and transition state theory, these theories explicitly take into account nonstatistical dynamical effects such as barrier recrossing, quasiperiodic trapping (both of which generally slow down the reaction rate), and other interesting effects. The implications for quantum dynamics are currently an active area of investigation. 

Several theories have been developed for the unimolecular reactions. The earliest, proposed by KASSEL, HINSHELWOOD, RICE, and RAMSPERGER /136/, as well as the later theory of SLATER /137/, are based on classical models. The most recent and important theory of MARCUS ans RICE /138/ rests on the semiclassical activated complex theory which makes use of potential energy surfaces. 

A well defined theory of chemical reactions is required before analyzing solvent effects on this special type of solute. The transition state theory has had an enormous influence in the development of modern chemistry [32-37]. Quantum mechanical theories that go beyond the classical statistical mechanics theory of absolute rate have been developed by several authors [36,38,39], However, there are still compelling motivations to formulate an alternate approach to the quantum theory that goes beyond a theory of reaction rates. In this paper, a particular theory of chemical reactions is elaborated. In this theoretical scheme, solvent effects at the thermodynamic and quantum mechanical level can be treated with a fair degree of generality. The theory can be related to modern versions of the Marcus theory of electron transfer [19,40,41] but there is no... 

Before we can enter a discussion of the redox processes involved in the two mechanisms defined above, we need a simple theoretical background which provides relevant insights into the phenomenon of ET. The Marcus theory of outer-sphere ET provides such a framework for the delineation of mechanistic domains, thanks to its origin in a simple model and its classical nature (Marcus, 1964 Marcus and Sutin, 1985 for applications in organic chemistry, see Eberson, 1982b, 1987). 

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