An asymptotic Green's function method for time-dependent Schrödinger equations with application to Kohn-Sham equations

Abstract

We present an effective asymptotic Green's function method for solving the time-dependent Schrödinger equation with scalar and vector potentials, and apply it to the Kohn-Sham equations for electronic structure calculation within the time-dependent density functional theory, where the perfectly matched layer approach is incorporated such that the computation can be performed in a bounded domain of physical interest. The method, which extends the approach in Leung et al. (2014) [40], combines the Huygens' principle or Feynman's path integral for propagating the wavefunction and the semi-classical approximations for approximating the retarded Green's function. Once the analytic approximations for the phase and amplitudes of the asymptotic retarded Green's function are obtained through short-time Taylor series expansions, a short-time propagator for the wavefunction is derived, and the resulting integral can be evaluated by fast Fourier transform after appropriate lowrank approximations. Numerical experiments are presented for demonstration.