Coarse-grained V-representability

TL;DR

This paper proves that all strictly positive coarse-grained densities in an interacting electron system are ensemble V-representable with a unique potential, advancing the understanding of density functional theory's foundational problems.

Contribution

It establishes the coarse-grained ensemble V-representability for all strictly positive densities and proves the differentiability of the Lieb functional in this context.

Findings

Every positive coarse-grained density is V-representable.

There exists a unique potential for each coarse-grained density.

Lieb functional is Gateaux differentiable for confined systems.

Abstract

The unsolved problem of determining which densities are ground state densities of an interacting electron system in some external potential is important to the foundations of density functional theory. A coarse-grained version of this ensemble V-representability problem is shown to be thoroughly tractable. Averaging the density of an interacting electron system over the cells of a regular partition of space produces a coarse-grained density. It is proved that every strictly positive coarse-grained density is coarse-grained ensemble V-representable: there is a unique potential, constant over each cell of the partition, which has a ground state with the prescribed coarse-grained density. For a system confined to a box, the (coarse-grained) Lieb [Intl. J. Quantum Chem. 24, 243 (1983)] functional is also shown to be Gateaux differentiable. All results extend to open systems.