![]() Therefore, definitions of probabilities or rate constants related to scattering eigenstates are inapplicable. Reactions in condensed phase are typically described by the potential energy profile along the reaction coordinate of a double-well character. These flux-based approaches naturally connect to quantum and classical transition state theories (beyond the scope of this paper). (The atomic units are used throughout.) The cumulative reaction probability can also be equivalently determined from expressions involving correlation functions of the flux operator eigenstates evolved in time. From the scattering matrix the cumulative reaction probability,, may be computed asįurthermore, determines the thermal reaction rate constant as the Boltzmann average over the collision energy : Its elements, subscripts and label specific internal states of reactants and products, are defined in terms of the energy eigenstates with incoming/outgoing boundary conditions in the reaction coordinate for a potential which vanishes at large reactant-product separation. The ultimate information about a reactive process in gas phase is contained in the scattering matrix defined for a given collision energy. (The latter is not included in the WKB except via the harmonic zero-point energy of the reactant well.) There are two QM effects that should be considered: (i) decay of the wavefunction inside a barrier for a range of energies, treated approximately in the WKB approach, and (ii) quantization of the energy levels due to the bound character of a potential. 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. This was done, for example, in recent studies of the barrier shape effect on the classical and QM contributions to reactivity in enzyme catalysis and in a study of the tunneling control of reactivity in carbenes. 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. A few other examples, where tunneling defines the reaction rate are isomerization of methylhydroxycarbene and reaction of hydroxyl radical with methanol. One example, which motivated many theoretical investigations, is the unusually high experimental kinetic isotope effect ( ) of the transferring hydrogen/deuterium substitution on the reaction rate constant in soybean lipoxygenase-1, attributed to QM tunneling as the dominant reaction mechanism. There is great interest in understanding the role of quantum-mechanical (QM) effects on reactivity at low temperature in complex systems, such as liquids, proteins, at surfaces, and so forth. Applications are givenįor several model systems, including proton transfer in a HO–H–CH 3 model, and the differences between the quantum-mechanical and quasiclassical tunneling probabilities are examined. The proposed definition involvesĮnergy eigenstates of the bound potential and exact quantum-mechanical transmission probability through the barrier region of the corresponding scattering potential. 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. 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. The potential energy profile for such processes is typically represented as a double-well potential along the reaction coordinate. ![]() The quantum-mechanical tunneling is often important in low-energy reactions, which involve motion of light nuclei, occurring in condensed phase.
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