Long-range electrostatics for machine learning interatomic potentials is easier than we thought

TL;DR

The paper tackles the difficulty of incorporating long-range electrostatics into machine-learning interatomic potentials (MLIPs). It introduces Latent Ewald Summation (LES), which adds a physically grounded Coulomb energy term using environment-dependent latent charges learned from standard energy/force data rather than DFT charges. The authors establish two design principles and show that LES improves energy/force accuracy across architectures, yields interpretable charges, and enables prediction of electrical response properties like dipoles and Born effective charges with low data requirements and modest overhead. LES is shown to be universal across short-range MLIPs and extensible with dipole/BEC fine-tuning, Qeq, and other extensions, offering a practical path to accurate, scalable simulations of interfaces, polar/ionic materials, and biomolecules. The work positions long-range electrostatics as simpler and more broadly applicable in MLIPs than previously believed, while outlining limitations and future directions such as interfacial systems and foundation models.

Abstract

The lack of long-range electrostatics is a key limitation of modern machine learning interatomic potentials (MLIPs), hindering reliable applications to interfaces, charge-transfer reactions, polar and ionic materials, and biomolecules. In this Perspective, we distill two design principles behind the Latent Ewald Summation (LES) framework, which can capture long-range interactions, charges, and electrical response just by learning from standard energy and force training data: (i) use a Coulomb functional form with environment-dependent charges to capture electrostatic interactions, and (ii) avoid explicit training on ambiguous density functional theory (DFT) partial charges. When both principles are satisfied, substantial flexibility remains: essentially any short-range MLIP can be augmented; charge equilibration schemes can be added when desired; dipoles and Born effective charges can be inferred or finetuned; and charge/spin-state embeddings or tensorial targets can be further incorporated. We also discuss current limitations and open challenges. Together, these minimal, physics-guided design rules suggest that incorporating long-range electrostatics into MLIPs is simpler and perhaps more broadly applicable than is commonly assumed.

Figures (2)

• Figure 1: A map of representative strategies for learning long-range interactions. Black solid arrows indicate an extension or variant of a method. Black dashed arrows show conceptual connections. Red arrows indicate the learning of physical quantities. The blue shaded region indicates methods that use additional DFT training labels, the green denotes methods with charge equilibration, the purple marks the methods using high-dimensional long-range representations, the yellow represents the classical force fields with fixed charges, and the red shows the methods that can learn electrical response properties. Abbreviations: LES, latent Ewald summation Cheng2025Latent; HDNNP, high-dimensional neural network potential Morawietz2012neuralko2021fourth; BAMBOO, ByteDance AI molecular simulation booster gong_predictive_2025; AIMNET, atoms-in-molecules neural network potential Anstine2025; Ewald MP, Ewald message passing; LODE, long-distance equivariant grisafi2019incorporating; RANGE, relaying attention nodes for global encoding caruso2025Extending; DPLR, deep potential long-range zhang2022deep; SCFNN, self-consistent field neural network gao2022self; Qeq, charge equilibration scheme proposed in Ref. rappe1991chargeGhasemi2015Interatomic; BEC, Born effective charge tensors; Equivariant features, direct prediction of the tensorial quantities grisafi2018symmetryschienbein2023SpectroscopyGrisafi2019Transferable; Differentiable learning, learn corresponding response properties by taking derivatives falletta2025unifiedSchmiedmayer2024. $U$ is electric enthalpy, $U^0$ is the energy in the absence of the electric field, $P$ is polarization, and $\mathcal{E}$ is a uniform electric field falletta2025unified.
• Figure 2: Benchmark results of short-range and LES-augmented MLIPs for bulk RPBE-D3 water. a: The test force root mean square errors (RMSEs) for baseline MLIPs (hollow bars), and the corresponding LES-augmented models (solid bars) for different architectures (MACE batatia2022mace, NequIP batzner20223, CHGNet deng2023chgnet, CACE cheng2024cartesian, and Allegro musaelian2023learning). The cutoff $r$, the number of layers $n_l$, the order of irreducible representations (rotation order) $\ell$, and body order $\nu$ are indicated for each MLIP. Results are reproduced from Ref. Kim2025Universalb. b: Parity plots comparing the diagonal components of the BEC tensor ($Z^{*}_{\alpha\alpha}$) of CACE LES with RPBE-D3 reference values for 100 bulk water configurations Schmiedmayer2024; insets show the off-diagonal components ($Z^{*}_{\alpha\beta}$ with $\alpha\neq\beta$). E+F denotes a CACE LES model trained on energy and forces, E+F+BEC is reproduced from Ref. zhong2025machine and was trained on a smaller set of 100 configurations with BEC labels, and E+F+Qeq incorporates a charge equilibration (Qeq) scheme rappe1991chargeGhasemi2015Interatomic. The test RMSE values for forces in meV/Å are 21.0 (E+F), 25.3 (E+F+BEC), and 21.1 (E+F+Qeq). All three models use the same CACE settings ($r=4.5$ Å, $n_l=1$, $\nu=3$). c: The test force RMSEs and BEC $R^2$ values of MACE LES models ($r=4.5$ Å, $n_l=2$, $\ell=1$) as a function of training set size. d: Computational performance benchmarks of MD simulations of bulk liquid water for varying system sizes ($N$) using LES-augmented MLIPs in single-precision (float32) with ASE implementations HjorthLarsen2017atomic performed on an NVIDIA L40S GPU (48 GB memory). The left panel shows the timing of MD simulations using the MACE models ($r=4.5$ Å, $n_l=2$) with different $\ell$, both without (empty symbols and dashed lines) and with LES (filled symbols and solid lines). The right panel shows the timing of the CACE models ($r=4.5$ Å, $n_l=1$, $\nu=3$), without LES (blue line), with LES (red line), and with Qeq scheme implementation (orange line). A reference line for $N^3$ scaling is included to illustrate the performance. The CACE LES model with the Qeq scheme shows cubic scaling with respect to the number of atoms. e: The test RMSEs for energy and force of CACE LES models ($r=4.5$ Å, $n_l=1$, $\nu=2$) with varying charge output dimensions. f: Infrared (IR) absorption spectra of bulk liquid water obtained with the E+F, E+F+BEC, and E+F+Qeq CACE LES models. The experimental IR spectrum Bertie1996Infrared is included for comparison. E+F and E+F+BEC results are reproduced from Ref. zhong2025machine.