Regularized 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, harmonizing with underlying function spaces for densities and wave functions. Moreau-Yosida regularization 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 optimization in calculations related to the regularized non-interacting 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 regularized inverse Kohn-Sham algorithm by quantifying, both theoretically and numerically, how perturbations of the input ground-state density propagate through the regularized density-to-potential map.