Theory of electron transfer in donor–acceptor pairs

Abstract

The wave functions of donor-acceptor pairs before and after electron transfer are written as a product of the electron-vibrational wave functions of the donor and acceptor with allowance for the change in the number of electrons on these particles by one after transition. In this approximation, the energy of the initial state is represented as a sum of the electron-vibrational levels of the donor and acceptor and that of the final state as a sum of donor cation and acceptor anion levels. Formulas for the electron transfer probability of symmetrical and nonsymmetrical donor-acceptor pairs have been derived that express the dependence of this process on the ionization potential difference of the donor and the electron affinity of the acceptor, on the vibrational frequencies of these particles, and on temperature.

Theory and mechanism of electron transfer at low temperatures

“…In all these cases of the arbitrary choice of electron transfer matrix ele-ments, the detailed balancing principle for direct and inverse electron transfer is violated [l]. Indeed, for an inverse process, yf of the one-electron Hamiltonian Hf [161, Vf21 of the many-electron Hamiltonian Hf [2] and Vo [ l l , 171 do not coincide with the operators V ' , Vi2 and V,, respectively, for the direct process. It follows that (ilv'lf) f (f\Vfli), (ilvf2lp) f (flvf2li) and (polV,lp,) f (p~IVolpo).…”

“…Here E , and ' p , are the electronic energy and wavefunction of the part p of the system for equilibrium positions of nuclei; E;,g=I (Up,), is the interaction energy between the parts of the system, including exchange energy, which is attained through the electron rearrangement operator A ; w,$, nJpk and are the frequencies, quantum numbers, and wavefunctions of oscillators relating to normal vibrations with coordinates Q, , the equilibrium values of which are Q,+. On substituting (2) and (3) into (l), summing over final vibrational states, and statistically averaging with Gibbs oscillator distribution-functions [4], converting to the integral form of the &function and writing (nlpr I nlpl) as (Alp are the displacement of the positions of potential surface minima in the case of electron transfer on each part of the system), introducing the density matrices pp X 9 QL I PI, + itwiP ) and PP (QiP -Alp, Q,p -A, I -itwJp) (PIP = w l p h / W 111 by analogy with the equilibrium density-matrix of a harmonic oscillator [4], and integrating with respect to Q,p and Q,p, we obtain, at qPr = wIpl = wlp, the formula where 3 = IIWJ + AdN2) + A e 3 + 1 A u f i p g = e2(N2 + 1) + eI(NI) + Ae3 + 2 AUfipg. p .…”