On the anomaly of the tunneling matrix element in long-range electron transfer

The electron tunneling matrix element of long-range electron transfer has usually been calculated by the two-state model. This model is useful when couplings between a donor and mediator and between an acceptor and mediator are weak. In the resent paper, we make correction for this electron tunneling matrix element on the basis of the diabatic picture so that the element can be used when the couplings are not necessarily weak. This correction is due to the normalization factors for the electronic wave functions of the initial and final states. We prove that these normalization factors are equivalent to the "dilution factors" derived by Katz and Stuchebrukhov (J. Chem. Phys. 1998, 109, 4960) on the basis of the adiabatic picture by adopting the truncation of Taylor expansion. We find that the normalized tunneling matrix element for the electron transfer in protein has a dip when the tunneling energy approaches the energy of the localized state whose group is close to either the donor or acceptor. This dip can be eliminated by choosing a combined system of the molecular group corresponding to the localized state with the original donor or acceptor as a new donor or acceptor. Making use of this anomaly, we can determine the proper size of the donor or acceptor in any long-range electron-transfer reaction.