Machine learning of electronic structure and atomistic properties from the external potential.

Electronic structure calculations remain a major bottleneck in atomistic simulations and, not surprisingly, have attracted significant attention in machine learning (ML). Most existing approaches learn a direct map from molecular geometries, typically represented as graphs or encoded local environments, to molecular properties or use ML as a surrogate for electronic structure theory by targeting quantities, such as Fock or density matrices expressed in an atomic orbital (AO) basis. Inspired by the Hohenberg-Kohn theorem, in this work, we propose an operator-centric framework in which the external (nuclear) potential, expressed in an AO basis, serves as the model input. From this operator, we construct hierarchical, body-ordered representations of atomic configurations that closely mirror the principles underlying several popular atom-centered descriptors. At the same time, the matrix-valued nature of the external potential provides a natural connection to equivariant message-passing neural networks. In particular, we show that successive products of the external potential provide a scalable route to equivariant message passing and enable an efficient description of nonlocal effects. We demonstrate that this approach can be used to model molecular properties, such as energies and dipole moments, from the external potential or to learn effective operator-to-operator maps, including mappings to the Fock matrix from which multiple molecular observables can be simultaneously derived.