The pair density method

Kinetic equations for the electron tunneling reactions in the case of non-pair distributions of reagents have been obtained [8-11]. Two methods have been used in the literature to obtain these equations. Both of them have been used earlier to describe the kinetics of energy transfer processes. These are the method of pair density and that of conditional concentrations. It has been shown [15] that these two different methods are, in fact, equivalent and lead to identical results. The detailed description of the pair density method can be found in refs. 3,5,7,13,16, and 33 and that of the method of conditional concentrations in refs. 5,8,15, and 17. 

At the present time, the precise analytical solution of the problem is obtained only when the concentration of one of the reagents (for clarity let it be the acceptor) exceeds substantially that of the other. In this case each donor particle can be considered independently of other donor particles further, it can be assumed that the number of acceptor particles does not depend on time. Let the acceptors be located around the donor in a certain fixed manner. Then the probability that by the time t the given donor has not reacted is evidently 

The probability P(t) is connected with the instantaneous concentration of donors n(t) and with their initial concentration n(0) by the relationship P(t) = n(t)/n(0). Differentiating this ratio with respect to time, we find that n(t) obeys the equation 

If the concentration of the donor considerably exceeds that of the acceptor, n IV, then the kinetics of acceptor decay is, apparently, described by the equation 

Comparison of eqns. (3), (18), and (20) shows that both for the pair and non-pair reagent distributions the derivation of kinetic equations for the electron tunneling reactions is reduced, as a matter of fact, to the calculation of the value 

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When the concentrations of reagents have comparable values, it is necessary to pay attention to the correlation effect in the decay of different donors, i.e. to consider the fact that the spatial distribution of acceptors near the chosen donor can change as a result of the decay of the acceptors in the reactions with other donors neighbouring the chosen one. The rigorous derivation of kinetic equations with the consideration of such a correlation is, as far as we know, unavailable. The approximate description of the kinetics of a biomolecular electron tunneling reaction at n(t) = N t) can be given in terms of the pair density method with the help of eqn. (19) in which, however, N is not a constant quantity but depends on time in the same way as n(t), i.e. 

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