# Non-uniqueness of quantum transition state theory and general dividing surfaces in the path integral space

## TL;DR

This paper analyzes the non-uniqueness of quantum transition state theory (QTST) and shows that recent formulations like HA-QTST are limited to specific dividing surfaces, revealing ambiguities in their general applicability.

## Contribution

The work provides a detailed theoretical analysis of HA-QTST, demonstrating its limitations and clarifying its relation to RPMD rate theory for various path integral dividing surfaces.

## Key findings

- HA-QTST matches RPMD rate theory for quadratic low-frequency modes
- Different results arise for higher frequency imaginary time paths
- HA-QTST does not generally derive RPMD-TST, indicating ambiguity

## Abstract

Despite the fact that quantum mechanical principles do not allow the establishment of an exact quantum analogue of the classical transition state theory (TST), the development of a quantum TST (QTST) with a proper dynamical justification, while recovering the TST in the classical limit, has been a long standing theoretical challenge in chemical physics. One of the most recent efforts of this kind was put forth by Hele and Althorpe (HA) [ J. Chem. Phys. 138 , 084108 (2013)], which can be specified for any cyclically invariant dividing surface defined in the space of the imaginary time path integral. The present work revisits the issue of the non-uniqueness of QTST and provides a detailed theoretical analysis of HA-QTST for a general class of such path integral dividing surfaces. While we confirm that HA-QTST reproduces the result based on the ring polymer molecular dynamics (RPMD) rate theory for dividing surfaces containing only a quadratic form of low frequency Fourier modes, we find that it produces different results for those containing higher frequency imaginary time paths which accommodate greater quantum fluctuations. This result confirms the assessment made in our previous work [J. Chem. Phys. 144, 084110 (2016)] that HA-QTST does not provide a derivation of RPMD-TST in general, and points to a new ambiguity of HA-QTST with respect to its justification for general cyclically invariant dividing surfaces defined in the space of imaginary time path integrals. Our analysis also offers new insights into similar path integral based QTST approaches.