Field theoretic atomistics: Learning thermodynamic and variational surrogate to density functional theory

Abstract The Hohenberg-Kohn (HK) theorem—the bedrock of density functional theory (DFT)—establishes a universal map from the external potential to the energy. It also relates the electron density and atomic forces to the variation of the energy with the external potential. But the HK map is rarely utilized in atomistics, where inter-atomic potentials are defined using the molecular or crystal structure rather than the external potential. Beyond the lack of attention to the HK theorem in atomistics, a more severe difficulty is that the HK map applies only to the electronic energy, which is not thermodynamically well-defined, as the electronic energy per atom diverges in the thermodynamic limit. We alleviate this difficulty by deriving an HK-like theorem for the total energy—comprising both electronic energy and nuclear-nuclear repulsion—in terms of auxiliary fields. We refer to this reformulation as the field theoretic atomistics (FTA) map. We machine learn the FTA map using an equivariant architecture. We obtain both forces and electron density from the variation of the FTA map with respect to the auxiliary fields, that are exact relations. Our models attain good accuracy across diverse benchmarks and compete with state-of-the-art machine learned interatomic po-tentials (MLIPs). Notably, through electron density, we predict accurate dipole and quadrupole moments, otherwise nontrivial for interatomic potentials. Our formulation paves the way for a scalable electronic structure surrogate to DFT.