Derivation of a true (t->0+) quantum transition-state theory. Part I: Uniqueness and equivalence to ring-polymer molecular dynamics transition-state-theory

TL;DR

This paper introduces a new quantum transition-state theory based on a generalized Kubo transform, establishing its equivalence to ring-polymer molecular dynamics TST and demonstrating its positive-definite quantum statistics at all temperatures.

Contribution

It derives a unique quantum TST that is equivalent to RPMD-TST, ensuring positive-definite quantum statistics and extending the classical TST framework to quantum regimes.

Findings

The new quantum TST is identical to RPMD-TST.

It provides positive-definite quantum statistics at all temperatures.

RPMD-TST approximates the exact quantum rate when recrossing is negligible.

Abstract

Surprisingly, there exists a quantum flux-side time-correlation function which has a non-zero short-time (t->0+) limit, and thus yields a rigorous quantum generalization of classical transition-state theory (TST). In this Part I of two articles, we introduce the new time-correlation function, and derive its short-time limit. The new ingredient is a generalized Kubo transform which allows the flux and side dividing surfaces to be the same function of path-integral space. Choosing this common dividing surface to be a single point gives a short-time limit which is identical to an expression introduced on heuristic grounds by Wigner in 1932, but which does not give positive-definite quantum statistics, causing it to fail while still in the shallow-tunnelling regime. Choosing the dividing surface to be invariant to imaginary-time translation gives, uniquely, a short-time limit that gives the correct positive- definite quantum statistics at all temperatures, and which is identical to ring-polymer molecular dynamics (RPMD) TST. We find that the RPMD-TST rate is not a strict upper bound to the exact quantum rate, but a good approximation to one if real-time coherence effects are small. Part II will show that the RPMD-TST rate is equal to the exact quantum rate in the absence of recrossing.