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

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, wherein interatomic potentials are defined using the molecular or crystal structure rather than the external potential. As a break from this tradition, we present a field theoretic atomistics framework where the external potential assumes the central quantity. We machine learn the HK energy map while satisfying the thermodynamic limit. Further, we obtain both forces and electron density from the variation of the HK energy map, that are exact relations. Our models attain good accuracy across diverse benchmarks and compete with state-of-the-art machine learned interatomic potentials. 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.

💡 Research Summary

This paper introduces a novel “field‑theoretic atomistics” (FTA) framework that places the external potential vₑₓₜ(r) at the core of atomistic modeling, thereby directly exploiting the Hohenberg‑Kohn (HK) theorem. Traditional interatomic potentials (IPs) take atomic positions and species as inputs and output scalar energies and forces, completely ignoring electronic information. In contrast, density‑functional theory (DFT) is built on the one‑to‑one map between the external potential and the ground‑state electron density, but DFT is too costly for large‑scale simulations. The authors bridge this gap by learning the HK energy functional with machine learning while simultaneously enforcing two essential physical constraints: (1) the thermodynamic limit (energy per atom remains bounded as the system size grows) and (2) variational consistency (electron density and forces are exact functional derivatives of the learned energy).

The key methodological step is to replace the raw external potential with two auxiliary fields: a “auxiliary potential” v_aux(r) and an “auxiliary charge density” b_s(r). Each nucleus is surrounded by a compact, non‑overlapping unit charge distribution b_I(r) = −Z_I b_u(|r−R_I|; r_c). The total auxiliary charge is b_s(r)=∑_I b_I(r). The auxiliary potential is defined as v_aux(r)=vₑₓₜ(r)−v_s(r), where v_s(r) is the electrostatic potential generated by b_s. This construction isolates the long‑range electrostatic contributions that scale quadratically with atom number, allowing the total energy E to satisfy the thermodynamic limit.

The authors then formulate a saddle‑point (min‑max) variational problem:

E