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

Area of Science:

• Chemical Physics
• Quantum Mechanics
• Theoretical Chemistry

Background:

• Classical transition state theory (TST) defines reaction rates as a t → 0+ limit of flux-side time correlation functions.
• Quantum mechanics' noncommutativity of population and flux measurements prevents direct extension of TST to the quantum regime.
• Quantum TST (QTST) is broadly defined as any quantum rate theory matching TST in the classical limit, with no unique QTST universally accepted.

Purpose of the Study:

• To question and clarify assumptions in a recent proposal for a unique QTST by Hele and Althorpe (HA).
• To investigate the relationship between HA's proposed QTST, ring polymer molecular dynamics TST (RPMD-TST), and established quantum rate theories.
• To formulate an alternative quantum rate expression based on linear response theory and real-time dynamics of imaginary-time path integrals.

Main Methods:

• Analysis of the time correlation function used by HA, assessing its relation to kinetic rate constants via linear response theory.
• Theoretical examination of a key step in HA's proof to verify its reliance on exact quantum mechanical identities.
• Formulation of a new quantum rate expression using linear response theory and a formalism for real-time dynamics of imaginary-time path integrals.

Main Results:

• The time correlation function used by HA is not directly related to the kinetic rate constant through linear response theory.
• A key step in HA's proof was found to be questionable, leading to a corrected t → 0+ limit for their QTST.
• The corrected limit of HA's QTST aligns with the path integral quantum transition state theory rate expression using a centroid dividing surface, not RPMD-TST.
• The newly formulated quantum rate expression's t → 0+ limit vanishes in the exact quantum limit.

Conclusions:

• The proposed QTST by Hele and Althorpe is not equivalent to RPMD-TST and requires correction.
• The established path integral quantum transition state theory with a centroid dividing surface provides a more accurate quantum rate expression in this context.
• A novel quantum rate expression derived from linear response theory and path integral dynamics shows a vanishing rate in the exact quantum limit, highlighting fundamental differences in quantum rate theories.