Eliminating wavepacket reflection from grid boundaries using complex coordinate contours

Abstract

It is demonstrated that complex scaling of coordinates near the boundary of a grid for numerical solution of the time-dependent Schrödinger equation analytically prevents reflection of wavepackets from the boundary. The complex coordinates method has been shown to be successful in the computation of a large class of Green's function matrix elements in time-independent collision theory, and the connection is established here with the behavior of wavepackets as functions of complex coordinates. The procedure, which is applicable using finite elements or finite difference, is demonstrated numerically using a complex contour known as the exterior scaling contour and provides an elegant solution to the numerical difficulty of wavepacket reflection from grid boundaries in one or more dimensions.