Complex trajectory method in time-dependent WKB

We present a significant improvement to a time-dependent WKB (TDWKB) formulation developed by Boiron and Lombardi [JCP {\bf108}, 3431 (1998)] in which the TDWKB equations are solved along classical trajectories that propagate in the complex plane. Boiron and Lombardi showed that the method gives very good agreement with the exact quantum mechanical result as long as the wavefunction does not exhibit interference effects such as oscillations and nodes. In this paper we show that this limitation can be overcome by superposing the contributions of crossing trajectories. We also demonstrate that the approximation improves when incorporating higher order terms in the expansion. These improvements could make the TDWKB formulation a competitive alternative to current time-dependent semiclassical methods.