Semiclassical canonical rate theory

The exact quantum rate may be represented as a phase space trace of a product of two operators: the symmetrized thermal flux operator and a projection operator onto the product space. A semiclassical analysis of the phase space representation of these two operators is presented and used to explain recent results found for a quantum thermodynamic rate theory. For low temperatures, the central object that is responsible for the oscillatory nature of the flux operator is a periodic orbit on the upside down potential surface whose period is $2ħ{/k}_{B}T.\mathrm{}$ The semiclassical analysis of the flux distribution explains why a variation of the dividing surface leads to improved thermodynamic rate estimates in asymmetric systems. The semiclassical limit (stationary phase limit) of the projection operator is shown to be identical to the classical projection operator. A semiclassical rate theory is obtained using the product of the semiclassical flux distribution and either the parabolic barrier or the classical projection operator and compared with the exact rate and approximate quantum thermodynamic estimates.