Nonadiabatic electron transfer rate constant

In general, for the types of linked donor-acceptor systems to be discussed in this review, electron transfer is assumed to occur in the nonadiabatic regime. That is, the mixing between the electronic state of the donor and acceptor before electron transfer occurs and the corresponding state after electron transfer is weak (< kBT) [9], The actual electron transfer event is assumed to be fast compared to the time scale of nuclear motions. Marcus has proposed [11, 15] that the electron transfer rate constant ke, is given by Eq. 1. 

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... 

The electron transfer rates in biological systems differ from those between small transition metal complexes in solution because the electron transfer is generally long-range, often greater than 10 A [1]. For long-range transfer (the nonadiabatic limit), the rate constant is... 

Let us summarize the results obtained. The theory is restricted to nonadiabatic electron-transfer reactions. If only classical modes are reorganized during the transition, the rate constant for the oxidation is ... 

When the appropriate equilibrium constants are not extremely small, direct electron (or hole) transfer to bridge units may occur and the electron (or hole) may then hop from moiety-to-moiety (randomly) along the bridge. When this happens, the electron-transfer rate will decrease only slowly with bridge length. This behavior has been treated in terms of standard nonadiabatic electron-transfer theory. ... 

This is the most direct experimental manifestation of the existence of an electronic interaction. It can occur spontaneously in mixed-valence complexes, but also in bimetallic systems after a photochemical excitation (photoinduced electron transfer). The general theory considers electron transfer as a special case of radiationless transition, with a perturbative treatment based on Fermi s Golden Rule [42]. In the nonadiabatic case, the rate constant can be written as [43] ... 

Maximal Rate Constant. Ic ay, and Maximal Electron-Exchange Matrix Element. Vjpax Non-Bridaed ir-Svstems. The maximal rate constant of nonadiabatic electron transfer in the medium temperature regime is given by (43)... 

The electron-transfer reactions that occur within and between proteins typically involve prosthetic groups separated by distances that are often greater than 10 A. When we consider these distant electron transfers, an explicit expression for the electronic factor is required. In the nonadiabatic limit, the rate constant for reaction between a donor and acceptor held at fixed distance and orientation is ... 

The research groups of Lewis and Wasielewski estimated the rate constants of the charge separation (kcs) and recombination (kcR) between the nucleobase, which acts as the electron donor, and the electron acceptor at the loop position of the DNA hairpins, and also investigated the free energy dependence of the electron transfer rate. " It was found that the single-step electron transfer in DNA mediated by nucleobases can be described by the Marcus theory (4) developed for nonadiabatic electron transfer system. 

We have presented several approaches to calculate the rate constants of electron transfer occurring in solvent from the weak to strong electronic couplings. In the fast solvent relaxation limit, the approach based on the nonadiabatic transition state theory can be adopted. It is related to the Marcus formula by a prefactor and referred as a modified Marcus formula. When the solvent dynamics begin to play a role, the quantum Kramers-like theory is applicable. For the case where the intramoleeular vibrational motions are much faster than the solvent motion, the extended Sumi-Mareus theory is a better ehoice. As the coherent motion of eleetron is ineorporated, such as in the organic semiconductors, the time-dependent wavepaeket diffusion approach is proposed. Several applications show that the proposed approaches, together with electronic structure calculations for the faetors eontrolling eleetron transfer, can be used to theoretically predict electron transfer rates correctly. 

For a nonadiabatic electron transfer to or from an electrode, denoting by e an energy state of the electrode, the first-order rate constant rate is given by [186,187]... 

The ZN formulas can also be utihzed to formulate a theory for the direct evaluation of thermal rate constant of electronically nonadiabatic chemical reactions based on the idea of transition state theory [27]. This formulation can be further utilized to formulate a theory of electron transfer and an improvement of the celebrated Marcus formula can be done [28]. 

The total Hamiltonian is the sum of the two terms H = H + //osc- The way in which the rate constant is obtained from this Hamiltonian depends on whether the reaction is adiabatic or nonadiabatic, concepts that are explained in Fig. 2.2, which shows a simplified, one-dimensional potential energy surface for the reaction. In the absence of an electronic interaction between the reactant and the metal (i.e., all Vk = 0), there are two parabolic surfaces one for the initial state labeled A, and one for the final state B. In the presence of an electronic interaction, the two surfaces split at their intersection point. When a thermal fluctuation takes the system to the intersection, electron transfer can occur in this case, the system follows the path... 

A recently proposed semiclassical model, in which an electronic transmission coefficient and a nuclear tunneling factor are introduced as corrections to the classical activated-complex expression, is described. The nuclear tunneling corrections are shown to be important only at low temperatures or when the electron transfer is very exothermic. By contrast, corrections for nonadiabaticity may be significant for most outer-sphere reactions of metal complexes. The rate constants for the Fe(H20)6 +-Fe(H20)6 +> Ru(NH3)62+-Ru(NH3)63+ and Ru(bpy)32+-Ru(bpy)33+ electron exchange reactions predicted by the semiclassical model are in very good agreement with the observed values. The implications of the model for optically-induced electron transfer in mixed-valence systems are noted. 

Temperature and pressure effects on rate constants for [Fe(phen)3] +/[Fe(phen)3] + electron transfer in water and in acetonitrile have yielded activation parameters AF was discussed in relation to possible nonadiabaticity and solvation contributions. Solvation effects on AF° for [Fe(diimine)3] " " " " half-cells, related diimine/cyanide ternary systems (diimine = phen, bipy), and also [Fe(CN)6] and Fe aq/Fe aq, have been assessed. Initial state-transition state analyses for base hydrolysis and for peroxodisulfate oxidation for [Fe(diimine)3] +, [Fe(tsb)2] ", [Fe(cage)] " " in DMSO-water mixtures suggest that base hydrolysis is generally controlled by hydroxide (de)hydration, but that in peroxodisulfate oxidation solvation changes for both reactants are significant in determining the overall reactivity pattern. ... 

The effect of temperature on the photoinduced electron transfer from [Ru(bpy)3]2+ to methyl viologen solubilized in cellophane has been investigated 98 K The first-order rate constant which depends exponentially on the distance between the reactants shows a non-Arrhenius type of behavior in the temperature interval from 77 to 294 K. This phenomenon, previously found to be of great importance in biological systems, is quantitatively interpreted in terms of a nonadiabatic multiphonon non-radiative process. 

In the high-resolution ESR (326 GHz) study of the biradical state Qa - Qb - in the Rb. Spheroids, RC determines the exchange integral in the biradical (Jo = 109 s 1) (Calvo et al., 2001). Because the rate constant of electron transfer from Qa to Qb is essentially less (kET 104 s 1) (Feher et al., 1992 Xu et al, 2000) than expected for an nonadiabatic activationless ET and the kET values considerably deviate from the dependence of the supperexchange attenuation parameter (yET) on the distance between donor and acceptor centers in RCs (Fig. XXX), we can conclude that the ET is adiabatic and requires thermal activation. 

The dependence of the logarithm of the rate constants of electron transfer (log kEr) between the donor (D) and acceptor (A) centers on the D-A distances is similar to the correspondent dependence for the superexchange attenuation coefficient (log Yet) (Fig.2.6). Therefore, we can conclude that, similar to primary events in RCs from bacteria, the primary fast electron transfers take place as nonadiabatic and conformationally nonequilibrium processes. 

The Fermi Golden rule describes the first-order rate constant for the electron transfer process, according to equation (11), where the summation is over all the vibrational substates of the initial state i, weighted according to their probability Pi, times the square of the electron transfer matrix element in brackets. The delta function ensures conservation of energy, in that only initial and final states of the same energy contribute to the observed rate. This treatment assumes a weak coupling between D and A, also known as the nonadiabatic limit. 

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