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

TL;DR

This work develops a convex-analysis–based formulation of density-functional theory for periodic systems with a Yukawa electron–electron interaction and employs Moreau–Yosida regularisation to obtain a differentiable, proximally well-posed density–to–potential map that extends to non–$v$-representable densities. The theory is specialised to a one-dimensional periodic Hartree–Fock model, where proximal densities are used to recover local exchange potentials that mimic exact exchange within the regularised framework. A detailed error analysis shows how perturbations of the input density propagate through the proximal map and into the Kohn–Sham potential, with non-expansiveness ensuring controlled sensitivity. Numerically, the method yields converged local exchange potentials for small ε (∼$10^{-8}$–$10^{-6}$) and reveals how model choices, electron count, and Yukawa screening influence convergence, density fidelity, and KS band structure. The results provide a proof-of-principle that exact-exchange effects can be captured via a local potential in a periodic, MY-regularised iKS scheme, paving the way for extending to more accurate correlation treatments.

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.

Figures (20)

• Figure 1: Two examples of real-space lattices. Black dots mark the points in $\mathcal{L}$. Top: $p=d=2$ dimensions, the hatched area is the unit cell and the filled orange area is the $3\times 2$ Born--von Kármán zone. Bottom: $p=1$, $d=2$ dimensions, with a Born--von Kármán zone consisting of three unit cells.
• Figure 2: The Yukawa interaction illustrated for a one-dimensional ($p=d=1$) system with arbitrary lattice constant $a$ and Yukawa parameter $\gamma = 1.5 a$.
• Figure 3: One-dimensional illustration of the effect of the duality map on potentials and densities. The horizontal axis is the position within a unit cell.
• Figure 4: Summary of the forward Hartree--Fock (top two boxes) and the Moreau--Yosida-based inversion procedure used in this work. Note that $\zeta = 1/\varepsilon$ is the inverse regularisation parameter and the model functional parameters are defined in Eq. \ref{['eq:mod_func_alp_xi']}. The output is the proximal point $\rho^{\zeta}$ and the contribution $u^{\zeta}$ to the KS potential.
• Figure 5: An external potential, $v_{\mathrm{ext}}$, and the corresponding Hartree--Fock ground-state density, $\rho_{\mathrm{ref}}$ for a $n=10$ electron system. Note the twin vertical axes.