Integrating Charge Equilibration with Equivariant Machine-Learning Interatomic Potentials

Machine-learning interatomic potentials (MLIPs) based on local atomic environments have achieved remarkable accuracy and efficiency, yet they often struggle in systems where long-range electrostatics, charge transfer, and nonlocal electronic effects play a decisive role. In this work, we augment the equivariant Multi-Atomic Cluster Expansion (MACE) potential with a charge equilibration (QEq) framework, enabling self-consistent, environment-dependent charge redistribution within a high-accuracy MLIP. We assess the capabilities and limitations of this approach through two representative applications: charged oxygen vacancies in wurtzite ZnO and a transferable water potential trained solely on molecular cluster data. For ZnO defects, the model accurately reproduces charge state-dependent relaxations and migration pathways in small to medium-sized supercells, demonstrating that ML-enhanced QEq can capture complex defect physics. However, further increasing system size reveals intrinsic limitations of the quadratic QEq formalism, manifesting as spurious charge delocalization and a collapse of distinct charge states. In the water case, we show that initializing long-range models from pretrained short-range representations substantially improves data efficiency and can help transfer from gas-phase water cluster structures to bulk liquid. Together, these results highlight both the promise and the fundamental constraints of QEq-based MLIPs, and emphasize the importance of physically informed architectures and pretrained representations for extending ML potentials to systems governed by long-range electrostatics and nonlocal charge response.