Exact form of the universal energy-density functional in DFT

Determine the exact form of the Hohenberg–Kohn universal energy-density functional F[n] in density functional theory, which maps a ground-state density n(r) to the universal contribution to the ground-state energy. In the noninteracting single-particle case this functional reduces to the kinetic-energy functional, while for interacting electron systems it must include correlation effects beyond the Hartree term.

Background

Density functional theory relies on the existence of a universal energy functional F[n] that, together with the external potential term, yields the ground-state energy via E[n] = F[n] + ∫dr n(r) V(r). While the Hohenberg–Kohn theorems guarantee this mapping, the paper emphasizes that the exact form of F[n] is not known, which is a foundational limitation for both orbital-free and Kohn–Sham approaches.

The authors motivate machine-learning strategies (including their VAE-based approach) as data-driven surrogates for this unknown functional, especially in regimes where traditional approximations fail, such as systems with strong electron correlations. Despite the progress reported, the fundamental question of the exact functional form remains unresolved.