Calculation of the Quantum-Mechanical Tunneling in Bound Potentials

The quantum-mechanical tunneling is often important in low-energy reactions, which involve motion of light nuclei, occurring in condensed phase. The potential energy profile for such processes is typically represented as a double-well potential along the reaction coordinate. In a potential of this type defining reaction probabilities, rigorously formulated only for unbound potentials in terms of the scattering states with incoming/outgoing scattering boundary conditions, becomes ambiguous. Based on the analysis of a rectangular double-well potential, a modified expression for the reaction probabilities and rate constants suitable for arbitrary double- (or multiple-) well potentials is developed with the goal of quantifying tunneling. The proposed definition involves energy eigenstates of the bound potential and exact quantum-mechanical transmission probability through the barrier region of the corresponding scattering potential. Applications are given for several model systems, including proton transfer in a HO–H–CH3 model, and the differences between the quantum-mechanical and quasiclassical tunneling probabilities are examined. 1. Introduction There is great interest in understanding the role of quantum-mechanical (QM) effects on reactivity at low temperature in complex systems, such as liquids [1], proteins [2], at surfaces [3], and so forth. One example, which motivated many theoretical investigations [4], is the unusually high experimental kinetic isotope effect ( ) of the transferring hydrogen/deuterium substitution on the reaction rate constant in soybean lipoxygenase-1 [5], attributed to QM tunneling as the dominant reaction mechanism. A few other examples, where tunneling defines the reaction rate are isomerization of methylhydroxycarbene [6] and reaction of hydroxyl radical with methanol [7]. To understand the role of tunneling for a specific system a typical approach is to construct an electronic potential energy profile of a double-well character, along one-dimensional reaction coordinate, and to compute tunneling probabilities using the quasiclassical Wentzel-Kramers-Brillouin (WKB) approximation. This was done, for example, in recent studies of the barrier shape effect on the classical and QM contributions to reactivity in enzyme catalysis [8–10] and in a study of the tunneling control of reactivity in carbenes [6]. Our goal here is to define and analyze QM reaction probabilities and thermal reaction rate constants in the bound potentials in a more rigorous way than the WKB theory. There are two QM effects that should be considered: (i)