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

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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.