Exploring a spectral filtering approach to electronic structure calculations

Abstract

ABSTRACT We make an initial exploration of a spectral filtering approach to numerically approximating solutions to the time-independent Schrödinger equation for a model system of two spinless fermions with harmonic interactions. Fourier transformation of the time evolution of an arbitrary antisymmetrized wave packet results in resonance at frequencies proportional to eigenenergies of the system Hamiltonian, while the Fourier coefficient approximates the corresponding eigenfunction. Spectral filtering is explored using the two-particle model through both direct numerical integration of the Schrödinger equation and a semiclassical parametrisation. In the former, a discrete position basis describes the time-evolution operator, which is then repeatedly applied to an antisymmetrized wave packet to obtain the packet's time evolution. In the semiclassical parametrisation, equations of motion for the parameters are obtained from the Dirac-Frenkel-McLachlan functional. Integration of the parameter equations of motion describes the packet's time evolution. The semiclassical approach requires less memory and accurately obtains the eigenspectrum. In a step towards application of spectral filtering to realistic systems, a possible ansatz for use with Coulomb potentials is explored, using a simple model Hamiltonian.