Long-range electrostatics in atomistic machine learning: a physical perspective

TL;DR

The paper analyzes how to faithfully incorporate long-range electrostatics into atomistic machine learning by contrasting local and nonlocal paradigms, and by examining finite-field coupling to system polarization. It categorizes methods into explicit and implicit local-charge models, and nonlocal approaches including self-consistent, nonlocal descriptors, and nonlocal architectures, highlighting the physical principles and constraints that govern each. Key contributions include formal discussions of charge partitioning via Hirshfeld/GDMA multipoles, Wannier-center polarization, explicit self-consistent charge equilibration (QEq) frameworks, LODE-based nonlocal descriptors, attention- and Fourier-based nonlocal architectures, and unified energy-based models that respect Berry-phase polarization and acoustic sum rules. The work discusses practical implications for electrochemical interfaces and ionic transport, emphasizing how long-range electrostatics shape screening, interfacial dynamics, and charge transport in realistic materials systems. Overall, the paper provides a physics-centered roadmap for selecting and combining ML strategies to capture permanent and induced electrostatics in complex, electrified environments with potential impact on battery science and catalysis.

Abstract

The inclusion of long-range electrostatics in atomistic machine learning (ML) is receiving increasing attention for achieving quantum-mechanical accuracy in predicting a wide range of molecular and material properties. However, there is still no general prescription on how long-range physical effects should be incorporated into the model while preserving well-established locality principles underlying most transferable ML representations. Here, we provide a physical perspective on the problem, by discussing how distinct contributions to the system's electrostatics can be captured through the adoption of different learning paradigms. Specifically, we discern between local charge models, which rely either on explicit charge-density decompositions or implicit auxiliary variables, and models where a notion of nonlocality is deliberately introduced, either via self-consistent procedures or by using nonlocal descriptors and learning architectures. We further address the related aspect of incorporating finite-field effects through the coupling with the system's polarization, relevant for the application of an external electric bias. We conclude by discussing the implications for the simulation of electrochemical interfaces, where long-range electrostatics are essential to capture the interplay between charge redistribution, interfacial dynamics, and ionic screening, and for ionic transport phenomena, which, although less explored, appear far less sensitive to their inclusion.

Figures (1)

• Figure 1: Representation of the various modeling paradigms for including long-range electrostatic interactions in atomistic machine learning. Left: families of ML models that predict charges based on local-environment structural information: (i) quantum-mechanical moments derived from an atomic partitioning of the charge density are learned explicitly, (ii) atomic charges are treated as auxiliary variables and implicitly inferred upon learning electronic energies and/or global dipoles, (iii) an implicit representation of the polarization vector of periodic systems is adopted through learning of Wannier centers, or atomic dipoles treated as auxiliary variables. Right: families of ML models that include nonlocal structural information at inference: (iv) atomic charges are self-consistently optimized through a charge-equilibration procedure coupled with ML predictions of atomic electronegativities, (v) nonlocal representations of the atomic structure are used as input features of the ML model, (vi) nonlocal operations are included as an integral part of the learning architecture.