Abstract
We explore the relation between the quantum and semiclassical instanton approximations for the reaction rate constant. From the quantum instanton expression, we analyze the contributions to the rate constant in terms of minimum-action paths and find that two such paths dominate the expression. For symmetric barriers, these two paths join together to describe the semiclassical instanton periodic orbit. However, for asymmetric barriers, one of the two paths takes an unphysically low energy and dominates the expression, leading to order-of-magnitude errors in the rate predictions. Nevertheless, semiclassical instanton theory remains accurate. We conclude that semiclassical instanton theory can be obtained directly from the semiclassical limit of the quantum instanton only for symmetric systems. We suggest a modification of the quantum instanton approach which avoids sampling the spurious path and thus has a stronger connection to semiclassical instanton theory, giving numerically accurate predictions even for very asymmetric systems in the low temperature limit.
Highlights
The inclusion of nuclear quantum effects in molecular dynamics calculations is a challenging task
In order to validate our semiclassical analysis of the quantum instanton and cumulant expansion, it is useful to compute the numerical values of the various terms making up these approximations and to compare the values obtained from quantum mechanics and from the semiclassical approximation
All these approximations are an order of magnitude larger than the exact rate constant, showing that the source of the error is in the quantum instanton (QI) or 2OCE rate formulas themselves. (The system and temperature were chosen to demonstrate this order of magnitude error.) This justifies our use of a semiclassical evaluation of the relevant quantities for analyzing the QI results
Summary
The inclusion of nuclear quantum effects in molecular dynamics calculations is a challenging task. Semiclassical instanton (SCI) theory provides one of the simplest formulations of a QTST for multidimensional tunneling.14,15 It can be derived rigorously by taking asymptotic (h → 0) approximations to the flux-flux correlation function and is known in a number of formulations, including one obtained from the. Known as the quantum instanton (QI), was developed as an attempt to correct the quantitative deficiencies of the semiclassical approximation This method goes beyond the harmonic approximation employed by SCI theory and samples paths using efficient path-integral Monte Carlo or molecular dynamics methods.. Ring-polymer transition-state theory has been rederived from a generalized flux-side correlation function, which yields a good approximation to the rate at zero time.36,37 This provides an extension of the centroid-based QTST of Voth, Chandler, and Miller, a method which does not dominantly sample the instanton and fails for asymmetric barriers..
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