The Convexity Condition of Density-Functional Theory

TL;DR

This paper proves the convexity of the total energy in density-functional theory under certain conditions, confirming a key assumption and providing criteria for approximate functionals, which impacts understanding of electronic properties.

Contribution

It establishes the convexity condition for DFT and reduced density matrix functional theory, confirming a fundamental assumption and offering criteria for approximate functionals.

Findings

Proves convexity of total energy in DFT for exact, size-consistent, translationally invariant formulations.

Confirms convexity as a fundamental constraint on the exact exchange-correlation functional.

Provides sufficient conditions for convexity in approximate DFT, aiding functional development.

Abstract

It has long been postulated that within density-functional theory (DFT) the total energy of a finite electronic system is convex with respect to electron count, so that 2 E_v[N_0] <= E_v[N_0 - 1] + E_v[N_0 + 1]. Using the infinite-separation-limit technique, this article proves the convexity condition for any formulation of DFT that is (1) exact for all v-representable densities, (2) size-consistent, and (3) translationally invariant. An analogous result is also proven for one-body reduced density matrix functional theory. While there are known DFT formulations in which the ground state is not always accessible, indicating that convexity does not hold in such cases, this proof nonetheless confirms a stringent constraint on the exact exchange-correlation functional. We also provide sufficient conditions for convexity in approximate DFT, which could aid in the development of density-functional approximations. This result lifts a standing assumption in the proof of the piecewise linearity condition with respect to electron count, which has proven central to understanding the Kohn-Sham band-gap and the exchange-correlation derivative discontinuity of DFT.