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

TL;DR

This work introduces field theoretic atomistics (FTA), a thermodynamically and variationally consistent surrogate for DFT that reformulates the HK map in terms of auxiliary fields $v_{\text{aux}}$ and $b_{\text{s}}$, enabling direct prediction of electron density and electrostatic quantities alongside total energy and forces. By learning the saddle-point energy $\widetilde{E}[v_{\text{aux}}, b_{\text{s}}, N_e]$ with an ACE+NN architecture and enforcing field-loss terms, FTA achieves competitive accuracy on molecular benchmarks (aspirin, 3BPA) and delivers accurate dipole and quadrupole moments through the predicted densities. The framework unifies electronic-structure information with atomistic potentials, supports external-field coupling, and retains variational relations linking density, potential, and energy, offering a scalable electronic-structure surrogate suitable for large-scale simulations and future extensions (nonlocal descriptors, grand-canonical open systems).

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, 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.

Figures (8)

• Figure 1: Schematic of field theoretic atomistics (FTA) approach. The original Hohenberg-Kohn (HK) map pertains to the electronic energy, which does not satisfy the thermodynamic limit. We reformulate the HK map for the total energy, in terms of auxiliary fields ($v_{\text{\footnotesize{aux}}}(\boldsymbol{\textbf{r}})$, $b_{\text{\footnotesize{s}}}(\boldsymbol{\textbf{r}})$), to ensure the thermodynamic limit. The variation of the energy with the auxiliary fields provide the ground-state density and electrostatic potential. We machine-learn the FTA energy map using a combination of atomic cluster expansion Drautz2019 and neural networks.
• Figure 2: Learning curves of force and dipole moment for aspirin with training samples. For force, it shows the mean absolute error in meV/Å . For dipole moment, it shows the mean absolute error divided by the mean of the absolute test set values (dimensionless quantity).
• Figure 3: Comparison of the reference and predicted $\Delta\rho_g$ for a test aspirin configuration. The yellow and blue iso-surfaces represent $\Delta\rho_g$ values of $0.0095$ and $-0.0095$, respectively, which corresponds to 10% of the maximum value of the reference $\Delta\rho_g$. Brown, white and red colored atoms denote C, H, and O species respectively.
• Figure 4: Comparison of the potential energy surface of 3BPA for the $\beta=120^{\circ}$ dihedral slice.
• Figure 5: Comparison of the dipole moment magnitude ($\abs{\boldsymbol{d}^{\Delta \rho_g}}$) of 3BPA for the $\beta=120^{\circ}$ dihedral slice.