Can quantum transition state theory be defined as an exact t=0+ limit?

TL;DR

This paper critically examines the possibility of defining an exact quantum transition state theory as a t=0+ limit, challenging recent claims and clarifying the relationship between quantum flux correlation functions and measurable rates.

Contribution

It clarifies the assumptions behind recent QTST proposals, corrects a key step in their derivation, and proposes an alternative path integral quantum rate expression.

Findings

The HA flux-side correlation function is not directly related to the quantum rate via linear response.

A key step in HA's proof does not rely on an exact quantum identity.

The proposed t=0+ limit of HA's QTST differs from RPMD-TST and aligns with centroid-based path integral TST.

Abstract

The definition of the classical transition state theory (TST) as a t= 0+ limit of the flux-side time correlation function relies on the assumption that simultaneous measurement of population and flux is a well defined physical process. However, the noncommutativity of the two measurements in quantum mechanics makes the extension of such a concept to the quantum regime impossible. For this reason, quantum TST (QTST) has been generally accepted as any kind of quantum rate theory reproducing the TST in the classical limit, and there has been a broad consensus that no unique QTST retaining all the properties of TST can be defined. Hele and Althorpe (HA) [J. Chem. Phys. 138, 084108 (2013)] recently suggested that a true QTST can be defined as the exact t=0+ limit of a certain kind of quantum flux-side time correlation function and that it is equivalent to the ring polymer molecular dynamics (RPMD) TST. This work seeks to question and clarify certain assumptions underlying these suggestions and their implications. First, the time correlation function used by HA as a starting expression is not related to the kinetic rate constant by virtue of linear response theory, which is the first important step in relating a t=0+ limit to a physically measurable rate. Second, a theoretical analysis calls into question a key step in HA's proof which appears not to rely on an exact quantum mechanical identity. The correction of this makes the true t=0+ limit of HA's QTST different from the RPMD-TST rate expression, but rather equal to the well-known path integral quantum transition state theory rate expression for the case of centroid dividing surface. An alternative path integral quantum rate expression is then formulated based on the linear response theory. It is shown that the t=0+ limit of the new rate expression vanishes in the exact quantum limit.