Semiclassical initial value representation for the Boltzmann operator in thermal rate constants

The thermal rate constant for a chemical reaction, $\mathrm{k(T),}$ can be expressed as the long time limit of the flux-side correlation $C_{\mathrm{fs}}\mathrm{(t)=}\mathrm{tr}{\mathrm{[e}}^{\text{−βĤ/2}}{\mathrm{F̂e}}^{\text{−βĤ/2}}{e}^{\mathrm{iĤt/\hslash }}{\mathrm{ĥe}}^{\mathrm{-iĤt/\hslash }}\mathrm{].}$ Previous work has focused on semiclassical (SC) approximations [implemented via an initial value representation (IVR)] for the time evolution operators $\exp \mathrm{(\pm iĤt/\hslash )}$ in the correlation function, and this paper shows how an SC-IVR can also be used to approximate the Boltzmann operators $\exp \text{(−βĤ/2).}$ Test calculations show that over a wide temperature range little error is introduced in the rate constant by this SC approximation for the Boltzmann operator.