Expression of the Electron Transfer Rate

In a first model, these motions are represented by harmonic vibrations, and the functions (Q) and Xbw (Q) are then replaced by products of harmonic oscillator-like wavefunctions. The solutions of Eqs. (9) take this particular form when the T jJ are negligible and when and H b can be expanded in terms of normal coordinates  

From the expressions given for example in Refs. [4,9,29], it can be seen that the nuclear factor, and consequently the electron transfer rate, becomes temperature independent when the temperature is low enough for only the ground level of each oscillator to be populated (nuclear tunneling effect). In the opposite limit where IcgT is greater than all the vibrational quanta hco , the nuclear factor takes an activated form similar to that of Eq. 1 with AG replaced by AU [4,9,29]. The model has been refined to take into account the frequency shifts that may accompany the change of redox state [22]. 

As pointed out in Ref. [4], no entropy variation appears in the description given by the harmonic model, apart from the weak contribution arising from the frequency shifts of the oscillators. The applications of this model are then a priori restricted to redox reactions in which entropic contributions can be neglected. We shall see in Sect. 3 that the current interpretations of most electron transfer processes which take place in bacterial reaction centers are based on this assumption. 

One may wonder whether a purely harmonic model is always realistic in biological systems, since strongly unharmonic motions are expected at room temperature in proteins [30,31,32] and in the solvent. Marcus has demonstrated that it is possible to go beyond the harmonic approximation for the nuclear motions if the temperature is high enough so that they can be treated classically. More specifically, he has examined the situation in which the motions coupled to the electron transfer process include quantum modes, as well as classical modes which describe the reorientations of the medium dipoles. Marcus has shown that the rate expression is then identical to that obtained when these reorientations are represented by harmonic oscillators in the high temperature limit, provided that AU° is replaced by the free energy variation AG [33]. In practice, tractable expressions can be derived only in special cases, and we will summarize below the formulae that are more commonly used in the applications. 

When the electron transfer process is coupled to classical reorientation modes and to only one harmonic oscillator whose energy quantum h( is high enough for only the ground vibrational level to be populated, the expression of the electron transfer rate is given by [4, 9]  

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Expression of the Electron Transfer Rate for a Non-adiabatic Process... 

In this chapter, we focus on the hopping regime and start with a primer on electron-transfer theory in Section 1.1.2. This section will underline the three major parameters that enter the expression of the electron-transfer rate reorganization energy, electronic coupling, and driving force. We then discuss some examples of the impact of chemical structure and packing mode on these parameters. 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. 

With these reservations in mind, we can compare observed and calculated rate constants. Where appropriate, the full Marcus expression, which includes the log f term (Equation 9), has been used in the calculation of the electron transfer rates. 

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... 

Note that assumptions (2) and (3) are about timescales. Denoting by x, and tlz the characteristic times (inverse rates) of the electron transfer reaction, the solvent relaxation, and the Landau-Zener transition, respectively, (the latter is the duration of a single curve-crossing event) we are assuming that the inequalities Tr A Ts tlz hold. The validity of this assumption has to be addressed, but for now let us consider its consequences. When assumptions (1)—(3) are satisfied we can invoke the extended transition-state theory of Section 14.3.5 that leads to an expression for the electron transfer rate coefficient of the form (cf. Eq. 14.32)... 

The electrocatalytic activity of an electrocatalyst can also be described using the same concept of TOP as shown in Eqn (3.1). However, the electrocatalytic reaction involves the electron transfer from the catalyst to the reactant, the reaction rate or TOP is better being expressed as the electron transfer rate at one catalytic active site of the catalyst ... 

ITIES at (1) 50 and (2) 5 mM concentrations of RufCN)/". For other experimental parameters, see Figure 8.4. A cp is expressed in terms of [C104"] according to Equation 8.11. (B) Potential dependence of the effective bimolecular rate constant. (Reprinted with permission from Tsionsky, M., Bard, A.J., and Mirkin, M.V., Scanning electrochemical microscopy 34. Potential dependence of the electron-transfer rate and film formation at the liquid/liquid interface, /. Phys. Chem., 100, 17881-17888, 1996. Copyright 1996 American Chemical Society.)... 

The expression (21) of the electron transfer rate together with the functional behavior of Q2 t) suggests that one may employ the method of steepest descent, at least in the high temperature limit, for an approximate evaluation. This approximation is based on a quadratic expansion of Q2 t) around its minimum at t = 0. The procedure requires one to determine the quantity... 

In our simple model, the expression in A2.4.135 corresponds to the activation energy for a redox process in which only the interaction between the central ion and the ligands in the primary solvation shell is considered, and this only in the fonn of the totally synnnetrical vibration. In reality, the rate of the electron transfer reaction is also infiuenced by the motion of molecules in the outer solvation shell, as well as by other... 

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-... 

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 rate constants of the electron transfers vary with the electrode potential. In particular, in their Arrhenius form, they are expressed 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 ... 

Electron transfer reactions have also been treated from the quantum mechanical point of view in formal analogy to radiationless transitions, considering the weakly interacting states of a supermolecule AB the probability (rate constant) of the electron transfer is given by a golden rule expression of the type17... 

Recently, the electron-transfer kinetics in the DSSC, shown as a schematic diagram in Fig. 10, have been under intensive investigation. Time-resolved laser spectroscopy measurements are used to study one of the most important primary processes—electron injection from dye photosensitizers into the conduction band of semiconductors [30-47]. The electron-transfer rate from the dye photosensitizer into the semiconductor depends on the configuration of the adsorbed dye photosensitizers on the semiconductor surface and the energy gap between the LUMO level of the dye photosensitizers and the conduction-band level of the semiconductor. For example, the rate constant for electron injection, kini, is given by Fermi s golden rule expression ... 

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... 

An expression for the overall rate constant, ket, of the electron transfer process is given in Equation 6.106 for competing diffusional and activated modes of reaction. 

The technique of complex-valued dielectric functions was originally applied to solvation problems by Ovchinnikov and Ovchinnikova [2] in the context of the electron transfer theory. They reformulated in terms of s(k, to) the familiar golden rule rate expression for electron transfer [3], This idea, thoroughly elaborated and extended by Dogonadze, Kuznetsov and their associates [4-7], constitutes a background for subsequent nonlocal solvation theories. 

The manifestation of the dipole-dipole approximation can be seen explicitly in Equation (3.134) as the R 6 dependence of the energy transfer rate. In Equation (3.134) the electronic and nuclear factors are entangled because the dipole-dipole electronic coupling is partitioned between k24>d/(td R6) and the Forster spectral overlap integral, which contains the acceptor dipole strength. Therefore, for the purposes of examining the theory it is useful to write the Fermi Golden Rule expression explicitly,... 

The electron transfer mechanism of azurin, a well known example for this type of proteins, has been systematically studied using the chemical relaxation method and a well defined inorganic outer sphere redox couple. In parallel, the investigations of the reaction with its presumed physiological partner, cytochrome c, were pursued (7). The specificity of the interaction between azurin and cytochrome c P-551 is expressed in higher specific rates and in the control of the electron transfer equilibrium by conformational transitions of both proteins. 

Equation (7) expresses an important distinction between the activation free energy for the overall electrochemical reaction in the absence of a net driving force, AG 0, and the intrinsic barrier for the electron-transfer step, AG, t. The former is most directly related to the experimental standard rate constant, whereas the latter is of more fundamental significance from a theoretical standpoint (vide infra). It is therefore desirable to provide reasonable estimates of wp and ws so that the experimental kinetics can be related directly to the energetics of the electron-transfer step. 

In the classical limit when the thermal energy K T is much higher than the energy ha of the vibrational frequencies that are coupled to the electron transfer reaction, the Franck-Condon factor can be expressed in terms of AG and X and equation 2 converts to the classical Marcus formula for the electron transfer rate ... 

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