Electron transfer theories

Electron transfer theory predicts that the rate of an electron transfer reaction will vary predictably with temperature (T), AG, and donor-acceptor distance (r) according to the relationships given in Eq. 4 and 5 (Marcus and Sutin, 1985), where h is Planckis constant, R is 

The factor P is related to the nature of the intervening medium between redox centers. The experimentally-derived value of r will be dependent upon the value of P which is used in the analysis. Values which are 

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). 

The Marcus model of intermolecular ET refers to reaction (17) and is detailed in Fig. 1. 

The two reacting species, D (after donor, the reductant) and A (after acceptor, the oxidant) are approximated as two spheres of charges Zx and Z2 and radii rx and r2, D being symbolized by the larger sphere in which the arrow represents the electron to be moved. The two spheres first have to diffuse 

The reorganization energy of a self-exchange reaction is denoted A(0) (from the fact that AG° = 0) and is an important quantity in the Marcus theory, where it can be shown that the activation free energy of a self-exchange reaction, AG(0), is equal to X.(0)/4. It is also possible to measure rate constants of self-exchange processes experimentally and thus get access to (0) via this relationship. 

It is important to remember that the Marcus model refers to a weakly electronically coupled model, as embodied in the term outer-sphere ET. Thus it must be assumed that the electronic overlap between the two reactants is so small that no quantum-chemical effects ensue, yet that there must be enough overlap for the transmission coefficient k of the Eyring equation to be equal to 1 (the reaction must be adiabatic). Usually, this minimum overlap requirement is put at a fairly low level, around 0.1 kcal mol-1, which causes no problems for most reactions involving at least one organic species. 

When the quenching reactions are rapid, the observed rate constant must be corrected for the effect of diffusion in order to obtain the activation controlled rate constant. The correction is made by applying 

Since both reactions are in the normal energy region (i.e., the minima in the potential curves are on different sides of the intersection region), the rate constants k2i and ki2 are given by 

From Eqs. (5.26) and (5.27) it follows that if AG23 is positive for the quenching step, the back electron transfer to the excited state is thermodynamically favored. When kz2 (ks4 + k3o), then from Eq. (5.24) a plot of RTln kq should have a slope of -1. In the other limit when fe2 ( 34 + feo), the slope should be -1/2. These predictions are found to be borne out in practice for a number of chemical systems that undergo quenching reactions with Ru(bpy)3 .  

In Table 5.11 are shown the quenching rate constants for a series of cyanide complexes. Up to the values found for the fastest quenchers in this table, a plot of kq against shows a good straight line correlation. Such plots should be taken with caution since the quenchers have different charge types, and will differ in their inherent barriers to electron transfer because of their different self-exchange rates. Nevertheless, even for systems such as these, it is still possible to test whether an outer-sphere electron transfer mechanism is operable by determining whether the data fit to the Marcus theory. 

This equation relates the rate constant for the reaction ( 12) between donor A and acceptor B in Eq. (5.28) to the self-exchange rate constants for the reactionsA/A and B/B  

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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... 

RB Yelle, T Ichiye. Solvation free energy reaction curves for electron transfer Theory and simulation. J Phys Chem B 101 4127-4135, 1997. 

Bertini I, Luchinat C, Scozzafava A (1982) Carbonic Anhydrase An Insight into the Zinc Binding Site and into the Active Cavity Through Metal Substitution. 48 45-91 Bertrand P (1991) Application of Electron Transfer Theories to Biological Systems. 75 1-48 Bill E, see Trautwein AX (1991) 78 1-96 Bino A, see Ardon M (1987) 65 1-28 Blanchard M, see Linares C (1977) 33 179-207 Blasse G, see Powell RC (1980) 42 43-96... 

EPR studies on electron transfer systems where neighboring centers are coupled by spin-spin interactions can yield useful data for analyzing the electron transfer kinetics. In the framework of the Condon approximation, the electron transfer rate constant predicted by electron transfer theories can be expressed as the product of an electronic factor Tab by a nuclear factor that depends explicitly on temperature (258). On the one hand, since iron-sulfur clusters are spatially extended redox centers, the electronic factor strongly depends on how the various sites of the cluster are affected by the variation in the electronic structure between the oxidized and reduced forms. Theoret-... 

The nonadiabatic transition state theory given in the Section II.C, namely, Eq. (17), can be applied to the electron-transfer problem [28]. Since the electron transfer theory should be formulated in the free energy space, we introduce the... 

The electron transfer discussed above corresponds to the so-called normal case in which the NT type of nonadiabatic transition plays the essential role. There is another important case called inverted case, in which the LZ type of nonadiabatic transition plays a role. Since the ZN theory can describe this type of transition also, the corresponding electron-transfer theory can be formulated [114]. On the other hand, the realistic electron transfer occurs in solution and... 

Bertrand, P. Apphcation of Electron Transfer Theories to Biological Systems. Vol. 75, pp. 1-48. 

In this chapter, we wiU review electrochemical electron transfer theory on metal electrodes, starting from the theories of Marcus [1956] and Hush [1958] and ending with the catalysis of bond-breaking reactions. On this route, we will explore the relation to ion transfer reactions, and also cover the earlier models for noncatalytic bond breaking. Obviously, this will be a tour de force, and many interesting side-issues win be left unexplored. However, we hope that the unifying view that we present, based on a framework of model Hamiltonians, will clarify the various aspects of this most important class of electrochemical reactions. 

In the electron transfer theories discussed so far, the metal has been treated as a structureless donor or acceptor of electrons—its electronic structure has not been considered. Mathematically, this view is expressed in the wide band approximation, in which A is considered as independent of the electronic energy e. For the. sp-metals, which near the Fermi level have just a wide, stmctureless band composed of. s- and p-states, this approximation is justified. However, these metals are generally bad catalysts for example, the hydrogen oxidation reaction proceeds very slowly on all. sp-metals, but rapidly on transition metals such as platinum and palladium [Trasatti, 1977]. Therefore, a theory of electrocatalysis must abandon the wide band approximation, and take account of the details of the electronic structure of the metal near the Fermi level [Santos and Schmickler, 2007a, b, c Santos and Schmickler, 2006]. 

Anderson AB, Cai Y, Sidik RA, Kang DB. 2005. Advancements in the local reaction center electron transfer theory and the transition state structure in the first step of oxygen reduction over platinum. J Electroanal Chem 580 17-22. 

We have established an important principle in electron transfer theory that is not present in conventional one-dimensional models. The reaction coordinate is always localizing and corresponds to coordinate Aj. The coordinate X2 corresponds to the direction in which the matrix element between ground and excited states is switched on. If this coordinate has zero length then the branching space becomes one dimensional and an adiabatic reaction path does not exist. We now consider two examples. 

Marcus, R. A., Chemical and electrochemical electron-transfer theory, Ann. Rev. Phys. Chem., 15, 155 (1964). 

In addition, the determination of metal-ligand bond distances in solution and their oxidation state dependence is critical to the application of electron transfer theories since such changes can contribute significantly to the energy of activation through the so-called inner-sphere reorganizational energy term. 

Kavamos GJ, Turro NJ (1986) Photosensitization by reversible electron transfer theories, experimental evidence, and examples. Chem Rev 86 401 -49... 

The first attempt to describe the dynamics of dissociative electron transfer started with the derivation from existing thermochemical data of the standard potential for the dissociative electron transfer reaction, rx r.+x-,12 14 with application of the Butler-Volmer law for electrochemical reactions12 and of the Marcus quadratic equation for a series of homogeneous reactions.1314 Application of the Marcus-Hush model to dissociative electron transfers had little basis in electron transfer theory (the same is true for applications to proton transfer or SN2 reactions). Thus, there was no real justification for the application of the Marcus equation and the contribution of bond breaking to the intrinsic barrier was not established. 

However, a more accurate comparison between the experimental reaction kinetics and the predictions of the dissociative electron transfer theory revealed that the agreement is good when steric hindrance is maximal (tertiary carbon acceptors) and that the reaction is increasingly faster than predicted as steric hindrance decreases.31 These results were interpreted as indicating an increase... 

Many of the features of electron transfer reactions involving excited states can be understood based on electron transfer theory. 

Bolton, J. R. and Archer, M. D. (1991) Basic electron-transfer theory,in Bolton, J. R., Mataga, N. and McLendon, G.(eds.), Electron Transfer in Inorganic, Organic and Biological Systems, American Chemical Society, 7-23. 

In the case of stepwise processes, the cleavage of the primary radical intermediate (often an ion radical) may be viewed in a number of cases as an intramolecular dissociative electron transfer. An extension of the dissociative electron transfer theory gives access to the dynamics of the cleavage of a primary radical into a secondary radical and a charged or neutral leaving group. The theory applies to the reverse reaction (i.e., the coupling of a radical with a nucleophile), which is the key step of the vast family of... 

Coming back to aromatic anion radicals, a more accurate comparison between the experimental reaction kinetics and the predictions of the dissociative electron transfer theory revealed that the agreement is good when steric hindrance is maximal (tertiary carbon acceptors) and that the reaction is faster and faster than predicted as steric hindrance decreases, as discussed in detail in Section 3.2.2 (see, particularly, Figure 3.1). These results were interpreted as indicating an increase in the ET character of the reaction as steric hindrance increases. Similar conclusions were drawn from the temperature dependence of the kinetics, showing that the entropy of activation increases with steric hindrance, paralleling the increase in the ET character of the reaction. 

We next consider the expression for k in the classical formalism. According to the Franck-Condon principle, internuclear distances and nuclear velocities do not change during the actual electron transfer. This requirement is incorporated into the classical electron-transfer theories by postulating that the electron transfer occurs at the intersection of two potential energy surfaces, one for the reactants... 

Electron transfer theories in mixed-valence and related systems have been summarized elsewhere ((5) and references therein). Conventionally, the electron transfer rate is calculated perturb tionally using the Fermi golden rule assuming that the electronic perturbation (e) is small. The most detailed... 

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