Effect of quantum modes in biological electron transfer reactions: A useful approximation for the harmonic model with frequency change and Duchinsky rotation

Although general theory of quantum effects in nonadiabatic electron transfer (ET) reactions based on spin-boson Hamiltonian is well known, its application to problems of biological interest is hampered by the amount of computational work needed to map the details of the real system onto the parameters of the model. In this paper we propose a new formulation of theory of quantum effects which remedies many defects of the usual approach. In the harmonic approximation an exact expression for the rate of electron transfer has long been known that includes effects of frequency change and Duchinsky rotation (mixing) of vibrational modes of donor and acceptor complexes. This expression, however, is not suitable for practical applications due to its complexity. We have developed an exceptionally accurate approximation that is capable of capturing all details of real redox systems typical for biological problems, yet simple enough to be practical. The approximation is based on the well-known Jortner expression for the quantum rate. We describe a method for calculation of the parameters of the Jortner model, average quantum frequency and average excitation number, which are usually treated as adjustable parameters, and in our case are calculated by ab initio quantum chemistry methods. The model is tested against the exact result. We also have tested another useful approximation, which is as good as the first one, however, in a limited region around maximum of ET rate. In this approximation the rate constant has the same form as the semiclassical Marcus expression, except that instead of one reorganization energy λ, it contains two λ’s. We show how these parameters can be calculated for realistic systems. Examples of such calculations are presented for a novel electron transfer between tryptophan and tyrosine, which was discovered recently in photolyase, a DNA repair enzyme, and some other biological systems.