Exponential operator for the Schrödinger equation in stretched coordinates

We present a new scheme for solving the time-dependent Schrödinger equation in analytically stretched coordinates. The algorithm is based on splitting the time evolution operator e−iε(T+V) and evaluating the momentum operation (e−iεT) in a designed reciprocal space. Compared with the conventional finite difference method, the proposed algorithm is stable and the accuracy is greatly improved with a truncation error is proportional to o(ΔN+2), where Δ is the grid spacing and N stands for the grid points in each directions. The total computing effort is roughly same with the finite difference scheme. It is ideal to investigate localized states with confining potential. We have derived the explicit expressions of the momentum operator in the second-order and the 4th-order time evolution operator. To test the proposed algorithm, we employ it in the electronic calculation of 3-dimensional GaN/Al0.2Ga0.8N quantum dot structure.