SELF-CONSISTENT COULOMB INTERACTIONS FOR MACHINE LEARNING INTERATOMIC POTENTIALS

Aug 4, 2026

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A general mathematical framework is developed for incorporating long-range electrostatic (Coulomb) effects into transferable machine learning interatomic potentials. The approach decomposes total energy into local atom-centered contributions while enabling charge equilibration, preserving locality to maintain transferability across atomistic systems. Partial explanations are provided for the effectiveness of existing ML potentials that use equilibration. A practical fitting scheme is also presented for energy and electron density of water clusters to complement the theory with implementable results.

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Abstract

Introduction

arXiv :2406 . 10915v1 [physics .comp-ph] 16 Jun 2024 SELF-CONSISTENT COULOMB INTERACTIONS FOR MACHINE LEARNING INTERATOMIC POTENTIALS JACK THOMAS, WILL BALDWIN, GABOR CSANYI, AND CHRISTOPH ORTNER Abstract. A ubiquitous approach to obtain transferable machine learning-based models of potential energy surfaces for atomistic systems is to decompose the total energy into a sum of local atom-centred contributions. However, in many systems non-negligible long-range electrostatic effects must be taken into account as well. We introduce a general mathematical framework to study how such long-range effects can be included ina way that (i) allows charge equilibration and (ii) retains the locality of the learnable atom-centred contributions to ensure transferability. Our results give partial explanations for the success of existing machine learned potentials that include equilibriation and provide perspectives how to design such schemes in a systematic way. To complement the rigorous theoretical results, we describe a practical scheme for fitting the energy and electron density of water clusters. 1. Introduction Electronic structure models are widely used to predict optical, magnetic, and mechanical properties of materials and molecules. Today, ab initio methods, such as density functional theory (DFT) [20, 27 , 30 , 42], are too computationally expensive for large-scale simulations (but are still a popular choice for the simulation of systems up to a few hundred atoms, for example), whereas empirical force fields remain useful for large system sizes and longtimescales. The introduction of machine-learning (ML) methodology into this field offers the prospect of bridging the gap between ab initio and empirical models in order to derive models with ab initio accuracy but at a fraction of the computational cost, enabling a systematic extension of the predictive first principles approach beyond the electronic structure length-scale to which it has been limited up until recently [6–8 , 12 , 17 , 39] . This is often motivated by invoking the nearsightedness principle of electronic matter (NEM) [45] . NEM concerns an electron density which is the ground state of an external potential v, given a fixed chemical potential. The statement is that if v is changed in some region Ω, then the response of the electron density at point x decays towards zero as x moves away from Ω . NEM therefore suggests that local machine learning models are effective so long as a change in geometry does not induce changes in potential at some distant point. This is not the case when, for instance, a reorientation of a polar molecule leads to a change in external potential even at distant points. In order to account for electrostatic effects, a long-range pairwise term can be added to the standard machine learning energy contribution that maps local geometry to energy [5] . However, changes in [jack.thomas@universite-paris-saclay.fr.](jack.thomas@universite-paris-saclay.fr. Laboratoire)[ Laboratoire](jack.thomas@universite-paris-saclay.fr. Laboratoire) de Math´ematiques d’Orsay, Universit´e Paris– Saclay, CNRS, 91405, Orsay, France. [wjb48@cam.ac.uk](wjb48@cam.ac.uk), [gc121@cam.ac.uk](gc121@cam.ac.uk. Engineering Laboratory)[. Engineering Laboratory](gc121@cam.ac.uk. Engineering Laboratory), University of Cambridge, Trumpington Street, Cambridge, CB2 1PZ, United Kingdom [ortner@math.ubc.ca](ortner@math.ubc.ca. Department)[. Department](ortner@math.ubc.ca. Department) of Mathematics, University of British Columbia, Vancouver, Canada. 2020 Mathematics Subject Classification: 65E05; 74E15; 81V45; 81V70 . Keywords and phrases: Coulomb interactions; machine learning; electronic structure; tight binding; interatomic potentials; locality; body-order expansion. Date: Tuesday 18th June, 2024 . SELF-CONSISTENT COULOMB INTERACTIONS 2 the chemical environment may induce changes in the charge distribution at long-range (even if the local geometry is unchang

FAQs

Why do standard local machine learning energy models need additional electrostatics?

Many systems exhibit long-range electrostatic effects where charge distributions change at distant points even if the local geometry is unchanged, so purely local mappings can miss important contributions.

What is the key goal of the proposed framework?

To include long-range effects in a way that allows charge equilibration while retaining the locality of learnable atom-centered contributions for transferability.

How is the theory complemented beyond rigorous results?

A practical scheme is provided for fitting both the energy and electron density of water clusters, demonstrating how the framework can be implemented.

Aug 4, 2026

Upload by Miles