Efficient grid-based method in nonequilibrium Green's function calculations. Application to model atoms and molecules

TL;DR

This paper introduces a finite-element discrete variable representation method for nonequilibrium Green's function calculations, enabling efficient and accurate modeling of strongly inhomogeneous quantum systems and complex atomic/molecular states.

Contribution

It presents a novel finite-element approach that reduces computational complexity and allows direct solutions of two-time quantum equations on spatial grids.

Findings

Successfully computed ground states of atoms and molecules in 1D.

Achieved accurate energies, densities, and bond lengths compared to Schrödinger solutions.

Demonstrated efficiency over traditional basis methods.

Abstract

We propose and apply the finite-element discrete variable representation to express the nonequilibrium Green's function for strongly inhomogeneous quantum systems. This method is highly favorable against a general basis approach with regard to numerical complexity, memory resources, and computation time. Its flexibility also allows for an accurate representation of spatially extended hamiltonians, and thus opens the way towards a direct solution of the two-time Schwinger/Keldysh/Kadanoff-Baym equations on spatial grids, including e.g. the description of highly excited states in atoms. As first benchmarks, we compute and characterize, in Hartree-Fock and second Born approximation, the ground states of the He atom, the H$_2$ molecule and the LiH molecule in one spatial dimension. Thereby, the ground-state/binding energies, densities and bond-lengths are compared with the direct solution of the time-dependent Schr\"odinger equation.