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

We present an effective asymptotic Green's function method for solving the time-dependent Schrodinger 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 etal. (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. (C) 2022 Elsevier Inc. All rights reserved.