Efficient and Equivariant Prediction of Distributed Charges for Accurate Molecular Electrostatics

A machine learning (ML) based, equivariant neural network for constructing distributed charge models (DCMs) of arbitrary resolution─DCM-net─is presented. DCMs efficiently and accurately model the anisotropy of the molecular electrostatic potential (ESP) and go beyond the point charge representation used in conventional molecular mechanics (MM) energy functions. This is particularly relevant for capturing the conformational dependence of the ESP (internal polarization) and chemically relevant features such as lone pairs or σ-holes. Across conformational space, the learned charge positions from DCM-net are stable and continuous. Across the QM9 chemical space, two-charge-per-atom models achieve accuracies comparable to fitted atomic dipoles for previously unseen molecules (0.75 (kcal/mol)/e). Three- and four-charge-per-atom models reach accuracies competitive with atomistic multipole expansions up to quadrupole level (0.55 (kcal/mol)/e). Pronounced improvements of the ESP are found around O- and F-atoms both of which are known to feature strongly anisotropic fields, and for aromatic systems. Across the QM9 reference data set, molecular dipole moments improve by ∼0.1 D compared with fitted monopoles. Transfer learning on dipeptides yields a 0.2 (kcal/mol)/e ESP improvement for unseen samples and a 2-fold MAE reduction for molecular dipole versus fitted monopoles. Overall, DCM-net offers a fast and physically meaningful approach to generating distributed charge models for running pure ML- or mixed ML/MM-based molecular simulations.