Interference, fluctuation, and alternation of electron tunneling in protein media. 2. Non-condon theory for the energy gap dependence of electron transfer rate.

Developing the quantum transition rate theory of Prezhdo and Rossky (J. Chem. Phys. 1997, 107, 5863), we produced a new non-Condon theory of the rate of electron transfer (ET) which happens through a protein medium with conformational fluctuation. The new theory is expressed by a convolution form of the power spectrum for the autocorrelation function of the electronic tunneling matrix element T(DA)(t) with quantum correction and the ordinary Franck-Condon factor. The new theory satisfies the detailed balance condition for the forward and backward ET rates. The ET rate formula is divided into two terms of elastic and inelastic tunneling mechanisms on the mathematical basis. The present theory is applied to the ET from Bph(-) to Q(A) in the reaction center of Rhodobacter sphaeroides. Numerical calculations of T(DA)(t) were made by a combined method of molecular dynamics simulations and quantum chemistry calculations. We showed that the normalized autocorrelation function of T(DA)(t) is almost expressed by exponential forms. The calculated energy gap law of the ET rate is nearly Marcus' parabola in most of the normal region and around the maximum region, but it does not decay substantially in the inverted region, which is called the anomalous inverted region. We also showed that the energy gap law at the high uphill energy gap in the normal region is elevated considerably from the Marcus' parabola, which is called the anomalous normal region. Those anomalous energy gap laws are due to the inelastic tunneling mechanism which works actively at the energy gap far from zero. We presented an empirical formula for easily calculating the non-Condon ET rate, which is usable by many researchers. We provided experimental evidence for the anomalous inverted region which was basically reproduced by the present theory. The present theory was extensively compared with the previous non-Condon theories.