Partitioning dynamic and thermal factors in quantum rate calculations: a coherent state approach

Abstract

At the heart of the search for a quantum transition state theory is the partitioning of dynamic from thermal factors in the quantum rate expression. We explore the possibility of achieving an approximate partitioning by using the coherent state basis. The coherent states provide a tetradic representation of both the dynamic and thermal factors; the degree to which these factors partition is tied to the degree to which one or both of the tetradics is diagonal, and hence phase space localized. We find that for the dynamical factor the off-diagonal contributions are small, except for matrix elements between coherent states positioned anywhere along the stable branch of the classical separatrix. The thermal factor is nearly diagonal at high temperatures, but has significant off-diagonal contributions at low temperatures. As a result, at high temperatures the thermal factor cuts off long range correlation, leading to the classical limit. At low temperatures, there is a subtle interplay of the thermal and dynamical factors, with the long range off-diagonal portions of the thermal factor combining with the long range off-diagonal portions of the dynamical factor. This phase space picture sheds light on the physical assumptions underlying several commonly applied approximations for calculating thermal reaction rates. In particular, by elucidating the subtlety of the contributions to the low temperature rate it becomes clear why a simple, yet accurate, estimate of the rate in this regime is elusive, if not impossible.