Transition-state dynamics in complex quantum systems

TL;DR

This paper develops a quantum model to analyze reaction dynamics in complex systems with internal barriers, questioning the validity of transition-state theory assumptions and revealing conditions under which the theory applies or fails.

Contribution

The study introduces a Gaussian Orthogonal Ensemble-based model that derives transition-state theory from quantum principles and examines its assumptions in complex systems.

Findings

Transition-state formula can be derived under certain approximations.

Reaction rates are insensitive to decay widths across a broad parameter range.

Transmission factor T is rarely close to unity without a strong collective component.

Abstract

A model is proposed for studying the reaction dynamics in complex quantum systems in which the complete mixing of states is hindered by an internal barrier. Such systems are often treated by the transition-state theory, also known in chemistry as RRKM theory, but the validity of the theory is questionable when there is no identifiable coordinate associated with the barrier. The model consists of two Gaussian Orthogonal Ensembles (GOE) of internal levels coupled to each other and to the wave functions in the entrance and decay channels. We find that the transition-state formula can be derived from the model under some easily justifiable approximations. In particular, the assumption in transition-state theory that the reaction rates are insensitive to the decay widths of the internal states on the far side of the barrier is fulfilled for broad range of Hamiltonian parameters. More doubtful is the common assumption that the transmission factor $T$ across the barrier is unity or can be modeled by a one-dimensional Hamiltonian giving $T$ close to unity above the barrier. This is not the case in the model; we find that the transmission factor only approaches one under special conditions that are not likely to be fulfilled without a strong collective component in the Hamiltonian.