# Ultrafast processes in imaginary time

November 13, 2019**Semiclassical analysis of the quantum instanton approximation**

The inclusion of nuclear quantum effects in molecular dynamics calculations is a challenging task. One of the main difficulties in calculating exact quantum dynamics is the need to solve the time-dependent Schrödinger equation with many degrees of freedom. For atomistic descriptions of most molecular systems, the dynamics are (currently) computationally impossible to solve exactly due to the exponentially increasing size of the basis sets required for these calculations. For this reason, approximate quantum dynamics methods based on path integrals are increasingly favored.

Here, the groups of Jeremy Richardson and Jiri Vanicek explored the relation between the quantum and semiclassical instanton approximations for the reaction rate constant. From the quantum instanton expression, the analyzed 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. The authors concluded that semiclassical instanton theory can be obtained directly from the semiclassical limit of the quantum instanton only for symmetric systems, and 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.

Here, the groups of Jeremy Richardson and Jiri Vanicek explored the relation between the quantum and semiclassical instanton approximations for the reaction rate constant. From the quantum instanton expression, the analyzed 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. The authors concluded that semiclassical instanton theory can be obtained directly from the semiclassical limit of the quantum instanton only for symmetric systems, and 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.

**Reference**: Vaillant, C.L., Thapa, M.J., Vaníček, J., and Richardson, J.O. (2019). Semiclassical analysis of the quantum instanton approximation.

*J Chem Phys 151*, 144111 (doi.org/10.1063/1.5123800)

<<