Electron transfer rate constant expressions

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-... 

Early studies of ET dynamics at externally biased interfaces were based on conventional cyclic voltammetry employing four-electrode potentiostats [62,67 70,79]. The formal pseudo-first-order electron-transfer rate constants [ket(cms )] were measured on the basis of the Nicholson method [99] and convolution potential sweep voltammetry [79,100] in the presence of an excess of one of the reactant species. The constant composition approximation allows expression of the ET rate constant with the same units as in heterogeneous reaction on solid electrodes. However, any comparison with the expression described in Section II.B requires the transformation to bimolecular units, i.e., M cms . Values of of the order of 1-2 x lO cms (0.05 to O.IM cms ) were reported for Fe(CN)g in the aqueous phase and the redox species Lu(PC)2, Sn(PC)2, TCNQ, and RuTPP(Py)2 in DCE [62,70]. Despite the fact that large potential perturbations across the interface introduce interferences in kinetic analysis [101], these early estimations allowed some preliminary comparisons to established ET models in heterogeneous media. 

A general difficulty encountered in kinetic studies of outer-sphere electron-transfer processes concerns the separation of the precursor formation constant (K) and the electron-transfer rate constant (kKT) in the reactions outlined above. In the majority of cases, precursor formation is a diffusion controlled step, followed by rate-determining electron transfer. In the presence of an excess of Red, the rate expression is given by... 

The Landau-Zener expression is calculated in a time-dependent semiclassical manner from the diabatic surfaces (those depicted in Fig. 1) exactly because these surfaces, which describe the failure to react, are the appropriate zeroth order description for the long-range electron transfer case. As can be seen, in the very weak coupling limit (small A) the k l factor and hence the electron transfer rate constant become proportional to the absolute square of A ... 

Early attempts at observing electron transfer in metalloproteins utilized redox-active metal complexes as external partners. The reactions were usually second-order and approaches based on the Marcus expression allowed, for example, conjectures as to the character and accessibility of the metal site. xhe agreement of the observed and calculated rate constants for cytochrome c reactions for example is particularly good, even ignoring work terms. The observations of deviation from second-order kinetics ( saturation kinetics) allowed the dissection of the observed rate constant into the components, namely adduct stability and first-order electron transfer rate constant (see however Sec. 1.6.4). Now it was a little easier to comment on the possible site of attack on the proteins, particularly when a number of modifications of the proteins became available. 

The expression in equation (25) gives the rate constant for electron transfer from a particular vibrational distribution of the reactants to a particular distribution of the products. In order to calculate the total electron transfer rate constant it is necessary to include two additional features. One is all possible transitions from a particular vibrational distribution of the reactants to all of the possible vibrational distributions of the products. In addition, it is necessary to recognize that a statistical distribution exists in solution amongst the various vibrational distributions of the reactants. The total transition rate will include contributions from each distribution of the reactants to all distributions of the products. 

The expression for ket in equation (29) is still not a complete expression for the total electron transfer rate constant. Both the electronic coupling term V and A0 are dependent upon the interreactant separation distance r, and, therefore, so is ktt in equation (29). The dependence of /.0 on r is shown in equation (23) in the dielectric continuum limit. The magnitude of V depends upon the extent of donor-acceptor electronic orbital overlap (equation 17) and the electronic wave-functions fall off exponentially from the centers of the reactants. In order to make comparisons between ktt and experimental values of electron transfer rate constants, it is necessary to include the dependence of ktt on r as discussed in a later section. 

Equation (36), which attempts to include both t and re, has been proposed as a more general expression for et.48 Note that in the limit, when Te -4 t , the expression for the electron transfer rate constant (equation 37) no longer depends on the extent of electronic coupling since vel > vn. In this limit the rate constant for electron transfer for a vibrational distribution near the intersection region is dictated by rates of repopulation of those intramolecular and/or solvent modes which cause the trapping of the exchanging electron. 

A question that arises in consideration of the annihilation pathways is why the reactions between radical ions lead preferentially to the formation of excited state species rather than directly forming products in the ground state. The phenomenon can be explained in the context of electron transfer theory [34-38], Since electron transfer occurs on the Franck-Condon time scale, the reactants have to achieve a structural configuration that is along the path to product formation. The transition state of the electron transfer corresponds to the area of intersection of the reactant and product potential energy surfaces in a multidimensional configuration space. Electron transfer rates are then proportional to the nuclear frequency and probability that a pair of reactants reaches the energy in which they have a common conformation with the products and electron transfer can occur. The electron transfer rate constant can then be expressed as... 

The related expression for the electron-transfer rate constant... 

Improta et studied large molecules with a well-separated electron donor and electron acceptor pair. In this case, the electron-transfer process may lead to a dissociation of the molecule. Improta et al. applied the Marcus theory to obtain an expression for the electron transfer rate constant. Subsequently, the values of the parameters entering this expression were calculated using a density-functional approach for the solute together with the polarizable continuum model for the solvent. They applied their approach for a specific system (for the present purpose, the details are less important) and found the results of Table 24. Here, three different... 

The electronic coupling of the reactant state with the product state, F, is a function of the overlap of the donor and acceptor orbitals. This in turn depends on energetic, spatial, geometric, and symmetry factors. At relatively large donor acceptor separations, it can be assumed that the relevant orbitals decay exponentially with distance. In these cases, the electron transfer rate constant will depend on this separation as per Eq. 2, where Rda is the donor-acceptor separation and y is a constant that expresses the sensitivity of the... 

Other investigations of the effect of solvent on electron transfer rate constants in porphyrin-quinone dyads have been reported [68, 80, 115, 116]. In general, the results are in qualitative agreement with expectations based on Eqs. 1, 3, and 4 or related expressions, but there are usually quantitative deviations due, at least in part, to the specific interactions mentioned above. 

Recent theoretical developments have been directed at incorporating the distance dependence of electron-transfer rate constants into the kinetic equations for diffusion-controlled reactions on the micelle surface, rather than assuming that these are collisional processes [82]. Both forward and back electron-electron transfers have been considered, with fitting of experimental data by numerical integration of appropriate differential equations. A standard Marcus-type expression was used to describe the electron-transfer rate constants. In the particular case of electron transfer from a donor to a single acceptor located initially at 0, one has instead of Eq. 10 [82c]... 

The most commonly employed electron transfer theory was proposed by Marcus and Hush [358, 359]. In their approach, the electron transfer rate constant is expressed as... 

Provided that the intramolecular electron-transfer rate constants can be expressed in the form ... 

To make the effect of the electronic continuum on the reaction dynamics as transparent as possible, consider the discrete two-state couphng problem representative of homogeneous electron transfer. In this problem, an electron is transferred between two localised molecular states. The electron-transfer rate constant ( et) is given by the well-known expression (Marcus and Sutin, 1985)... 

Fig. 9. Working curves for spectroelectrochemical determination of heterogeneous electron transfer rate constants for quasi-reversible reactions. Numerical values correspond to n T) where tj is expressed in millivolts. ...

The vibrational relaxation processes that accompany electron transfer range from some that are very fast (sub-picosecond) to some that are very slow (for some systems and some conditions, days). The rates of vibrational relaxation depend on heat flow from the molecule to the bulk solvent, the amplitude (or number of quanta) for the nuclear motion involved, the molecular and solvent vibrations that couple strongly to the nuclear motion, etc. Some aspects of the relevance of these relaxation times to the analytical expression describing the rate constant have been noted above. Apart from the general structure of the analytical expression, these relaxation times can affect the electron transfer rate constant through their effect on AGda°, Xr and the pre-exponential j/nu factor. ... 

Are the rate constants estimated from the Bloch type treatment consistent with expectations based on theory The -1 mixed valence state of complex (Id) is a Robin-Day class II complex, and thus its electron transfer rate constant can be independently estimated from Marcus theory. The semiclassical expression for the rate constant for intramolecular electron transfer, k, in a symmetric mixed valence complex with no net free energy change is given by... 

The relationship between the standard heterogeneous electron transfer rate constant, k, h> and structural factors is expressed in terms of the Marcus-Hush theory applied to electrode reactions [2] ... 

The following generalized rate expression, derived from Fermi s golden rule (see e.g., [9,11]), is useful for discussing the effects of cofactor structure changes on the electron transfer rate constant (kgj) ... 

On the basis of a nonadiabatic electron-transfer theory, which exposes the homogeneous width of the nuclear factor from low frequency modes (phonons), and hole burning data we conclude that this nonexponentiality is not due to a distribution of values, f, for the relevant adiabatic electronic energy gap(s) 2. Dispersive kinetics from f in the low temperature limit are judged to be unlikely. Nevertheless, the expression (. 2) for the average electron-transfer rate constant suggests that samples which exhibit sufficiently different Fj-values for the P-band should have measurably different values for in... 

Both Marcus27 and Hush28 have addressed electron transfer rates, and have given detailed mathematical developments. Marcus s approach has resulted in an important equation that bears his name. It is an expression for the rate constant of a net electron transfer (ET) expressed in terms of the electron exchange (EE) rate constants of the two partners. The k for ET is designated kAS, and the two k s for EE are kAA and bb- We write the three reactions as follows ... 

---