The convexity condition of density-functional theory

Area of Science:

• Computational Chemistry
• Quantum Mechanics
• Materials Science

Background:

• Density-functional theory (DFT) is a cornerstone of modern electronic structure calculations.
• A long-standing postulate in DFT is the convexity of total energy with respect to electron count.
• Convexity ensures stability and accurate predictions in theoretical models.

Purpose of the Study:

• To rigorously prove the convexity condition for total energy in finite electronic systems within DFT.
• To establish a stringent constraint on the exact exchange-correlation functional.
• To provide guidance for developing more accurate approximate DFT functionals.

Main Methods:

• Utilized the infinite-separation-limit technique.
• Proved convexity for DFT formulations that are exact for v-representable densities, size-consistent, and translationally invariant.
• Extended the proof to one-body reduced density matrix functional theory.

Main Results:

• The convexity condition (2Ev[N0] ≤ Ev[N0 - 1] + Ev[N0 + 1]) is proven for applicable DFT formulations.
• Demonstrated an analogous result for one-body reduced density matrix functional theory.
• Identified sufficient conditions for convexity in approximate DFT, aiding functional development.

Conclusions:

• Confirms a critical theoretical constraint on the exact exchange-correlation functional in DFT.
• Lifts a standing assumption in the proof of the piecewise linearity condition.
• The findings are crucial for understanding the Kohn-Sham bandgap and derivative discontinuity in DFT.