Electron transfer

Non-adiabatic electron transfer is ``weak" only at sufficiently large distances where it is limited by interparticle, ``exchange" interaction and described by second order perturbation theory. However, at the closest approach distance it may become so ``strong" that not an interaction but the attainment of the cross-point of energy levels controls the reaction. In the overdamped limit the transfer rate should be proportional to diffusion along the reaction coordinate. Since diffusion is hindered by friction (velocity relaxation) in reactant well the same is valid for the reaction. This result known as "Relaxation Hindrance" of the reaction [1] or "Dynamical Solvent Effect" (in polar liquids) [2] is qualitatively similar to the famous "high friction" limit of Kramers adiabatic theory [3]. If the motion in reactant well is underdamped the transfer rate is also limited by diffusion but in the energy space, from the bottom of the well to an over-barrier region. The energy diffusion as well as corresponding transfer rate estimated in [4] is accelerated by energy (velocity) relaxation. Considered as a function of relaxation (friction) the non-adiabatic transfer rate exhibits the cross-over similar to that obtained by Kramers in adiabatic limit. However, the insensitive to friction ``perturbation theory plateau" that separates the energy diffusion and ``relaxation hindrance" limits are wider and lower than adiabatic TST plateau. The complex relation between adiabatic and non-adiabatic results (at any interaction and friction) was recently established for highly activated process in normal Marcus' region [5]. The same work has been also done for inverted region where strong interaction not saturates but hampers the electron transfer [6].

Between normal and inverted regions where electron transfer is highly activated there is a narrow activationless strip that should be considered separately. It was shown in 1979 that activationless nonadiabatic transfer in overdamped limit proceeds non-exponentially in time [7]. Hence the transfer at corresponding distance may not be characterized by a unimolecular rate constant. As a result the encounter theory of activationless reactions has to be "non-local", taking into account simultaneously both the translational diffusion of the reactants and the system diffusion along the reaction coordinate. The very first approach to such a problem has been proposed in [8]. Then it was shown that dynamical solvent effect (or relaxation hindrance) can never affect the rate of kinetic controlled reactions [9]. The saturation is removed in the fast diffusion limit, because at short residence time in a thin activationless layer the electron transfer becomes weak.

A lot of attempts were made to check the quasi-parabolic free energy dependence of activation energy which is large in both normal and inverted regions but zero in between. This is the famous Marcus ``free energy gap" (FEG) law. The FEG law is inherent to any transfer rate (weak and strong), intermolecular and intramolecular, and in the latter case was confirmed experimentally. On the contrary, the very existence of inverted region in intermolecular reactions was put in doubt when Rehm and Weller [10] discovered a plateau instead of descending branch of parabola in free energy dependence of ionization rate constant. The discrepancy was finally attributed to a modulation of position dependent transfer rate by encounter diffusion: the very fast activationless reactions, at the top of FEG parabola, are limited by diffusion rather than reaction, so their rate should be almost independent of the reaction free energy [11,12,13].

The situation with charge recombination (back electron transfer) is quite the opposite: most of experiments were done in inverted region and the results look as though they fit the FEG law. Unfortunately, the relevant rate constants were obtained within the naive ``exponential model" which predicts the exponential kinetics of ion pair recombination and separation. The model assumes that both ionization and recombination take place at contact. In much better, ``contact" approximation only recombination remains contact while initial separation of ions is considered as a fitting parameter [14]. Subjected to revision in this approximation the exponential model was shown to be wrong: not only the process does not develop exponentially but also diffusion (viscosity) dependence of separation time is in fact qualitatively different [15]. Moreover, at slow diffusion even recombination should be considered as a remote transfer to avoid principle difficulties inherent to contact (Smoluchowskii-like) approximation [16].

When both forward and backward transfers are given by position dependent rates the initial separation of ions remains a single enigma. To get over the uncertainty of initial charge distribution a unified theory was recently developed for geminate recombination followed binary photoionization [17,18]. The latter is well described by encounter theory that can predict the accumulation and the final distribution of ions which are products of primary ionization but reactants in subsequent recombination. This distribution was shown to be never contact even for normal ionization which rate is a sharp exponential function of distance [19]. If ionization is kinetically controlled the distribution reproduces the exponential rate shape but in diffusion control limit it has a maximum shifted to effective reaction radius that is larger than contact. In inverted region the initial distribution is never contact [13].

The shape of initial distribution determines the charge separation quantum yield that was experimentally studied as a function of recombination free energy. We confirmed that this dependence should semi-quantitatively follow the FEG law predictions but only for fast (``kinetically controlled") recombination. For slower diffusion the process is controlled by attainment of the recombination layer from initial ion position. The latter may be either inside the layer (in inverted region) or outside it (in normal one) as shown in Figure [13,20]. Figure of quantum yield The separation quantum yield may be always presented as 1/(1+X) but unlike exponential model X does not fit the parabolic FEG law when geminate recombination is controlled by diffusion. The top of it is qualitatively distorted and the more the slower is diffusion. The FEG law is valid only for X << 1.

The unified theory describes not only the quantum yield but the kinetic of ions separation as well. The latter consists from ascending (charge accumulation) and descending (recombination-separation) branches. In general, two stages can not be separated in time because recombination starts before ionization is finished. The full kinetics of the process may be calculated now for any position dependent rates of back and forward electron transfer and arbitrary diffusion, but in [13,19,20] it has been done mainly for highly polar solvents and for single reaction mode. We plan to study now less polar solvents where situation is different in a few respects and the assistance of high frequency quantum modes [21,22] should be taken into account. On the other hand, the full revision of interpretation given to experimental data has to be undertaken now to obtain reliable information about forward and back electron transfer rates and their free energy dependence.

REFERENCES

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Last Revised: December 18, 1995