Damped motion in a two-dimensional double well I. Proton tunneling in a hydrogen bond

A model is presented to decribe proton transfer between two sites of a hydrogen bond forming part of a larger molecular complex in a thermal environment. The tunneling motion of the proton is assumed to couple strongly to the end atom vibration of the hydrogen bond via an interaction of the small polaron type. The end atom vibration interacts with the remaining vibrations of the molecular complex and the surrounding medium. These degrees of freedom are treated as an ensemble of harmonic oscillators forming a heat bath and therefore giving rise to a random force acting on, and a damping of, the end atom vibration. Via this vibration the tunneling motion of the proton is damped, too, thus describing an effective transfer from one site to another. The rate coefficient for the transfer reaction is calculated according to Kubo's theor, as an integral over the time correlation function of the reactive flux. Both the end atom vibration and the heat both have an influence on the reaction rate, which therefore depends on both types of coupling constants. The rate constant for systems like a hydrogen bond between two oxygen atoms is not directly related to the tunneling frequency, but may show resonance features when the oscillator frequency is comparable to the level splitting of the two lowest lying proton states.