The Herman-Kluk approximation: Derivation and semiclassical corrections

The Herman-Kluk (HK) approximation for the propagator is derived semiclassically for a multidimensional system as an asymptotic solution of the Schrödinger equation. The propagator is obtained in the form of an expansion in ℏ, in which the lowest-order term is the HK formula. Thus, the result extends the HK approximation to higher orders in ℏ. Examination of the various terms shows that the expansion is a uniform asymptotic series and establishes the HK formula as a uniform semiclassical approximation. Successive terms in the series should allow one to improve the accuracy of the HK approximation for small ℏ in a systematic and purely semiclassical manner, analogous to a higher-order WKB treatment of time-independent wave functions.