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Perspective on Moreau-Yosida Regularization in Density-Functional Theory

Markus Penz, Michael F. Herbst, Trygve Helgaker, Andre Laestadius

TL;DR

The paper addresses the lack of differentiability and robust inversion in density-functional theory by introducing Moreau–Yosida (MY) regularization as a unifying framework. It presents three equivalent MY routes—on the universal density functional, on the total energy functional, and through density–potential mixing—yielding differentiable functionals and well-posed, convergent Kohn–Sham iterations. By connecting density–potential inversion to proximal-point methods and Dyson-like resolvent forms, the work links established schemes such as ZMP to a rigorous convex-analytic foundation, and extends these ideas to periodic systems and mean-field theories with auto-regularization. The results pave the way for mathematically rigorous, computationally viable extensions of DFT, with enhanced inversion capabilities, clearer physical interpretation via space topology, and potential integration with quantum-electrodynamical formalisms and beyond.

Abstract

Within density-functional theory, Moreau-Yosida regularization enables both a reformulation of the theory and a mathematically well-defined definition of the Kohn-Sham approach. It is further employed in density-potential inversion schemes and, through the choice of topology for the density and potential space, can be directly linked to classical field theories. This perspective collects various appearances of the regularization technique within density-functional theory alongside possibilities for their future development.

Perspective on Moreau-Yosida Regularization in Density-Functional Theory

TL;DR

The paper addresses the lack of differentiability and robust inversion in density-functional theory by introducing Moreau–Yosida (MY) regularization as a unifying framework. It presents three equivalent MY routes—on the universal density functional, on the total energy functional, and through density–potential mixing—yielding differentiable functionals and well-posed, convergent Kohn–Sham iterations. By connecting density–potential inversion to proximal-point methods and Dyson-like resolvent forms, the work links established schemes such as ZMP to a rigorous convex-analytic foundation, and extends these ideas to periodic systems and mean-field theories with auto-regularization. The results pave the way for mathematically rigorous, computationally viable extensions of DFT, with enhanced inversion capabilities, clearer physical interpretation via space topology, and potential integration with quantum-electrodynamical formalisms and beyond.

Abstract

Within density-functional theory, Moreau-Yosida regularization enables both a reformulation of the theory and a mathematically well-defined definition of the Kohn-Sham approach. It is further employed in density-potential inversion schemes and, through the choice of topology for the density and potential space, can be directly linked to classical field theories. This perspective collects various appearances of the regularization technique within density-functional theory alongside possibilities for their future development.

Paper Structure

This paper contains 24 sections, 117 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Functional analysis preliminaries
  4. Convex analysis preliminaries
  5. Moreau--Yosida regularization
  6. Formal density-functional theory
  7. Kohn--Sham method
  8. Moreau--Yosida regularization in density-functional theory
  9. MY based on the universal density functional
  10. MY based on the total energy functional
  11. MY based on density--potential mixing
  12. Density--potential inversion
  13. Proximal-point iteration and relation to ZMP
  14. Moreau--Yosida in inverse Kohn--Sham
  15. Resolvent form and Dyson-like equation
  16. ...and 9 more sections

Figures (3)

  • Figure 1: Example of a pure-state functional $\tilde{F}^\lambda(x)$, the convex $F^\lambda(x)$, and its MY regularization $F_\varepsilon^\lambda(x)$. The unregularized functionals jump to $+\infty$ for all $x$ outside their effective domain $D$, but the regularized functional remains finite everywhere.
  • Figure 2: Illustration of one iteration step in the regularized KS scheme.
  • Figure 3: Convergence of $\rho$ and $v_{\mathrm{xc},\varepsilon}$ as a function of $\varepsilon$ for the bulk silicon test case and a basis truncation of $E_\text{cut} = 30$, leading to an error $\Delta \rho = 2.7 \times 10^{-5}$. We refer the reader to the original reference Herbst2025 for details of the computational setup.