Machine learning the Hohenberg–Kohn functional: breathing life into a 60-year-old dream

In 1964, Hohenberg and Kohn proved that the ground state of an interacting electron system is completely determined by its electron density. In principle, this result promises a dramatic simplification of electronic structure theory: instead of solving for a many-electron wave function in a 3N-dimensional space, one could obtain the ground state by minimizing an energy functional of the three-dimensional electron density.

In practice, this vision has remained unrealized, because the key ingredient—the kinetic-energy functional of the density—is unknown for real molecular systems. Modern Kohn-Sham density functional theory circumvents this obstacle by introducing auxiliary orbitals, but at the price of cubic scaling with system size.

In this talk I will show that the missing functional can be learned to sufficient accuracy for increasingly complex chemistry using rotation equivariant machine learning models. A key ingredient is the generation of training data by perturbing external potentials, exposing the model to physically meaningful density variations.

These results suggest that machine learning may finally enable a practical realization of the Hohenberg–Kohn program, opening an orbital-free density functional theory route to chemically accurate electronic-structure calculations for large systems, complementing both linear-scaling density functional theory and machine-learned interatomic potentials.