# Atomic Cluster Expansion

Linear atomic cluster expansion (ACE) potentials are misspecified when trained on deterministic density-functional labels, so Bayesian posterior variances understate their errors. A delete-one-configuration CR3 jackknife covariance supplies the shape and one conformal quantile from held-out data supplies the scale, bringing per-atom force regions close to nominal coverage on held-out silicon and propagating the same parameter distribution through implicit differentiation to 21 silicon properties; of 20 scored properties, 17 fall inside 95% intervals for the smaller model and 15 for the larger, whose intervals are about 4 times tighter.

Linear atomic cluster expansion (ACE) potentials are misspecified when trained on deterministic density-functional labels, so Bayesian posterior variances understate their errors. A delete-one-configuration CR3 jackknife covariance supplies the shape and one conformal quantile from held-out data supplies the scale, bringing per-atom force regions close to nominal coverage on held-out silicon and propagating the same parameter distribution through implicit differentiation to 21 silicon properties; of 20 scored properties, 17 fall inside 95% intervals for the smaller model and 15 for the larger, whose intervals are about 4 times tighter.

Latent-charge machine-learning potentials learn atomic charges as hidden variables from energies and forces, but their long-range electrostatic term had no exact periodic form, leaving net charge mixed with self-energy. Writing that term as a smeared Epstein–Hurwitz lattice sum separates the net-charge artefact from the local self-energy and yields a sphere-average theorem for cells of any shape, plus forces and virial. Projecting the latent charges onto fixed total charge gives projected latent Ewald summation (pLES); on charged NaCl clusters it recovers the total charge an unconstrained model misses by up to 1.8 and cuts a short-range model's test errors from 1.44 to 0.42 meV/atom and from 37 to 13 meV/Å.

Latent-charge machine-learning potentials learn atomic charges as hidden variables from energies and forces, but their long-range electrostatic term had no exact periodic form, leaving net charge mixed with self-energy. Writing that term as a smeared Epstein–Hurwitz lattice sum separates the net-charge artefact from the local self-energy and yields a sphere-average theorem for cells of any shape, plus forces and virial. Projecting the latent charges onto fixed total charge gives projected latent Ewald summation (pLES); on charged NaCl clusters it recovers the total charge an unconstrained model misses by up to 1.8 and cuts a short-range model's test errors from 1.44 to 0.42 meV/atom and from 37 to 13 meV/Å.