Regularized density-potential inversion for periodic systems: Application to exact exchange in one dimension.

Area of Science:

• Computational Chemistry
• Quantum Mechanics
• Materials Science

Background:

• Density Functional Theory (DFT) is a cornerstone of modern computational chemistry and condensed matter physics.
• Accurate calculation of electron-electron interactions, particularly exchange-correlation effects, remains a significant challenge.
• Existing DFT formulations can be sensitive to perturbations, hindering numerical stability and implementation.

Purpose of the Study:

• To develop a robust, convex analysis-based formulation of DFT for periodic systems applicable in any dimension.
• To introduce Moreau-Yosida regularization to enhance the stability and numerical tractability of the Hohenberg-Kohn mapping.
• To provide a proof-of-principle for recovering accurate exchange potentials, paving the way for improved exchange-correlation functional development.

Main Methods:

• Formulation of DFT using convex analysis for periodic systems with Yukawa-type electron-electron interactions.
• Application of Moreau-Yosida regularization to non-interacting density functionals.
• Numerical implementation using a Hartree-Fock approach for one-dimensional systems, focusing on self-consistent field optimization.

Main Results:

• A novel, perturbation-insensitive Hohenberg-Kohn mapping is derived using convex analysis and regularization.
• Demonstration of the practical feasibility of recovering local Kohn-Sham potentials that accurately represent exact exchange.
• Theoretical and numerical error analysis of the regularized inverse Kohn-Sham algorithm, quantifying density perturbation propagation.

Conclusions:

• The proposed convex analysis framework offers a stable and numerically implementable approach to DFT for periodic systems.
• The regularization technique successfully addresses challenges in self-consistent field optimization and potential recovery.
• This work validates a pathway towards more accurate and reliable computation of exchange-correlation potentials in advanced DFT methods.