Electron transfer frequency factors

Nuclear, electronic and frequency factors in electron transfer reactions. N. Sutin, Acc. Chem. Res., 1982,15, 275-282 (78). 

The expressions for ket in equations (29) and (30) are the products of two factors (1) an exponential term which, assuming a common force constant, gives the fractional population of reactants at temperature T with vibrational distributions at the intersection regions for each of the trapping vibrations, and (2) a pre-exponential term ve, the electron transfer frequency ve is defined in equation (31). 

The book Electrons in Chemical Reactions by L. Salem places electron transfer in a broad context ranging from basic wave mechanics to ion-solvent interactions. Sutin has reviewed theories of electron transfer with emphasis on nuclear, electronic, and frequency factors and Wong and Schatz have given a useful unified account of the dynamics of mixed-valence complexes using the vibronic coupling model. ... 

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]. 

In semiclassical ET theory, three parameters govern the reaction rates the electronic couphng between the donor and acceptor (%) the free-energy change for the reaction (AG°) and a parameter (X.) related to the extent of inner-shell and solvent nuclear reorganization accompanying the ET reaction [29]. Additionally, when intrinsic ET barriers are small, the dynamics of nuclear motion can limit ET rates through the frequency factor v. These parameters describe the rate of electron transfer between a donor and acceptor held at a fixed distance and orientation (Eq. 1),... 

Distance The affects of electron donor-acceptor distance on reaction rate arises because electron transfer, like any reaction, requires the wavefunctions of the reactants to mix (i.e. orbital overlap must occur). Unlike atom transfer, the relatively weak overlap which can occur at long distances (> 10 A) may still be sufficient to allow reaction at significant rates. On the basis of work with both proteins and models, it is now generally accepted that donor-acceptor electronic coupling, and thus electron transfer rates, decrease exponentially with distance kji Ve, exp . FCF where v i is the frequency of the mode which promotes reaction (previously estimated between 10 -10 s )FCF is a Franck Condon Factor explained below, and p is empirically estimated to range from 0.8-1.2 with a value of p 0.9 A most common for proteins. 

An improved and direct correlation between the experimental rate constant and [obtained using Eq. (49)] is observed if v = /zd is used instead of v = 1/Tt, the solvent-dependent tunneling factor is utilized, and only AG (het) of Eq. (8) is used in Eq. (49) (see triangles in Fig. 18). Furthermore, the inverse of the longitudinal solvent relaxation time Xi is not necessarily the relevant one to use as the frequency factor v (see empty circles in Fig. 18). Similar conclusions were reached by Barbara and Jerzeba for the electron transfer reaction in homogeneous solutions. Barbara and Jerzeba measured the electron transfer time... 

Electron transfer from a donor to an acceptor represents the transition of this particle from one discrete electron state to another. For this transition to become possible it is necessary to change the coordinates of the atomic nuclei which determine the energy of the discrete states of the electron. For this reason, the frequency factor in eqn. (1), as will be shown below, characterizes the motion of the nuclei rather than that of the electron. Therefore, there are no reasons to consider its value to be of the order of 1016 s 1. It will be shown in further discussion that the frequency factor depends on many characteristics of a donor, an acceptor, and a medium, and its value can vary over a very wide range, reaching as high a value as 1020s. ... 

Fig. 7. The frequency factor for electron transfer reactions of the biphenyl anion as a function of exothermicity [79J. The points are the frequency factors for various acceptors calculated by using eqn. (35) of Chap. 4 on the assumption that av = 0.83 A the line was calculated using eqn. (40) of Chap. 3.

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