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

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.