Regularised density-potential inversion for periodic systems: application to exact exchange in one dimension

TL;DR

This paper develops a convex analysis-based formulation of density-functional theory for periodic systems, introducing regularisation techniques to improve numerical stability and demonstrate the recovery of exact exchange potentials in one-dimensional models.

Contribution

It presents a novel regularisation approach for the density-to-potential map in density-functional theory, enabling stable numerical inversion and application to exact exchange calculations.

Findings

Regularised inverse Kohn--Sham algorithm is feasible for 1D systems.

The method accurately recovers local Kohn--Sham potentials for exact exchange.

Error propagation in the regularised scheme is quantitatively analyzed.

Abstract

A detailed convex analysis-based formulation of density-functional theory for periodic systems in arbitrary dimensions is presented. The electron-electron interaction is taken to be of Yukawa type, harmonising with underlying function spaces for densities and wave functions. Moreau--Yosida regularisation of the underlying non-interacting density functionals is then considered, allowing us to recast the Hohenberg--Kohn mapping in a form that is insensitive to perturbations (non-expansiveness) and lends itself to numerical implementation. The general theory is exemplified with a numerical Hartree--Fock implementation for one-dimensional systems. We discuss in particular the challenge of self-consistent field optimisation in calculations related to the regularised noninteracting Hohenberg--Kohn map. The implementation is used to demonstrate that it is practically feasible to recover local Kohn--Sham potentials reproducing the effects of exact exchange within this scheme, which provides a proof-of-principle for recovering the exchange-correlation potential at more accurate levels of theory. Error analysis is performed for the regularised inverse Kohn--Sham algorithm by quantifying, both theoretically and numerically, how perturbations of the input ground-state density propagate through the regularised density-to-potential map.