Kohn-Sham inversion with mathematical guarantees

TL;DR

This work addresses the lack of rigorous guarantees in Kohn–Sham inversion by applying Moreau–Yosida regularization to periodic density functionals. It derives an explicit inversion formula via a proximal density and a duality-mapped potential, where the xc potential is recovered as $v_ ext{xc}^oldsymbol{ε} = rac{1}{oldsymbol{ε}} J( ho^oldsymbol{ε}_ ext{gs}- ho_ ext{gs})$ in the limit $oldsymbol{ε} o0^+$, and provides a provable bound on density-perturbed inversions: $ orm{v_ ext{xc}^oldsymbol{ε}- ilde{v}_ ext{xc}^oldsymbol{ε}}_{oldsymbol{V}} \,\le \\frac{1+Q_oldsymbol{ε}( riangle ho)}{oldsymbol{ε}} orm{ riangle ho}_{oldsymbol{D}}$. The authors implement the scheme for bulk Si, GaAs, and KCl, demonstrating exact inversion and quantitatively validating the error bounds, with a non-expansive proximal mapping playing a central role. This work provides a rigorous framework that can guide error analyses and improve functional development for density-to-potential mappings in KS-DFT, potentially informing embedding methods and the design of better xc functionals.

Abstract

We use an exact Moreau-Yosida regularized formulation to obtain the exchange-correlation potential for periodic systems. We reveal a profound connection between rigorous mathematical principles and efficient numerical implementation, which marks the first computation of a Moreau-Yosida-based inversion for physical systems. We develop a mathematically rigorous inversion algorithm which is demonstrated for representative bulk materials, specifically bulk silicon, gallium arsenide, and potassium chloride. Our inversion algorithm allows the construction of rigorous error bounds that we are able to verify numerically. This unlocks a new pathway to analyze Kohn-Sham inversion methods, which we expect in turn to foster mathematical approaches for developing approximate functionals.

Figures (7)

• Figure 1: An illustration of the Moreau--Yosida (MY) inversion scheme. Given $v_\mathrm{xc}$, a ground-state density $\rho_\mathrm{gs}$ is first computed in a forward, reference calculation. Here, $\Delta\rho$ represent an introduced error, resulting in an inexact reference density $\tilde{\rho}_\mathrm{gs}$. For this $\tilde{\rho}_\mathrm{gs}$, the proximal point $\tilde{\rho}^\varepsilon_\mathrm{gs}$ is found from the MY regularization. An $\varepsilon$-dependent potential is then found by means of the duality mapping $J$, whereupon the potential is obtained by taking $\varepsilon\to 0^+$. The results of the inversion can then be compared with the forward scheme.
• Figure 2: (top) Real-space plot of the reference xc potential along with the potential obtained from inversions for different values of the regularization parameter $\varepsilon$ for bulk silicon. The potential is displayed along a closed path intersecting the high symmetry points Chen_2021FootnotePath. (bottom) The associated pointwise relative error of the potentials obtained from inversions at various $\varepsilon$ compared to the reference xc potential. The crystalline structure and the path is shown in the inset.
• Figure 3: Analogous plots to \ref{['fig:vxc_silicon']} for gallium arsenide (GaAs) along the equivalent path shown in the inset crystalline structure.
• Figure 4: The equivalent plots to \ref{['fig:vxc_silicon', 'fig:vxc_gaas']} for potassium chloride (KCl) displayed along a similar path shown in the inset crystalline structure.
• Figure 5: Convergence of $v_\mathrm{xc}^\varepsilon$ as a function of $\varepsilon$ for various perturbations introduced by basis truncations for bulk silicon. $E_\mathrm{cut}$ determines the cut off in the Fourier basis and $\left\Vert\Delta \rho\right\Vert$ is the corresponding truncation error in the density in $\mathcal{D}$-norm. $E_\mathrm{cut} = 45$ constitutes the (unperturbed) reference calculation.