How the Piecewise-Linearity Requirement for the Density Affects Quantities in the Kohn-Sham System

Area of Science:

• Computational Quantum Chemistry
• Materials Science
• Condensed Matter Physics

Background:

• Kohn-Sham density functional theory (KS-DFT) is a widely used method for electronic structure calculations.
• KS-DFT relies on an auxiliary system of non-interacting electrons to model the density of a real, interacting system.
• The exact density in KS-DFT exhibits piecewise-linearity with respect to the number of electrons (N).

Purpose of the Study:

• To investigate how the piecewise-linearity of the exact interacting density is manifested in the Kohn-Sham system.
• To explore the implications of piecewise-linearity for KS quantities, particularly the total electron density, KS subdensities, and highest occupied (HOMO) orbital density.
• To analyze common approximations for the HOMO, including frozen and linear regimes, in light of piecewise-linearity.

Main Methods:

• Formulation of KS quantities using a two-point Taylor expansion in the number of electrons (N).
• Derivation of analytical results based on the piecewise-linearity requirement.
• Numerical investigation employing various exchange-correlation approximations to validate analytical findings.

Main Results:

• The study establishes restrictions on KS expansion coefficients imposed by the piecewise-linearity of the exact density.
• Analytical insights into the behavior of total electron density, KS subdensities, and HOMO orbital density are presented.
• Numerical results confirm the analytical predictions across different exchange-correlation approximations.

Conclusions:

• Understanding and enforcing piecewise-linearity in KS-DFT is crucial for accurate calculations.
• The findings provide a theoretical framework to address and mitigate density-driven errors in KS-DFT, especially for open systems and ensembles.
• This work contributes to the development of more robust and reliable DFT methods.