[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-123407-en":3,"doc-seo-123407-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},123407,13056703019662,"Evangeline","https://ap-avatar.wpscdn.com/avatar/be000253a8e92610077?_k=1778726343310543188",8,"Research & Report","Self-consistent Coulomb interactions for machine learning interatomic potentials","A ubiquitous approach to build transferable machine learning models of potential energy surfaces decomposes total energy into local atom-centred contributions. Many systems, however, require non-negligible long-range electrostatic effects. A general mathematical framework is introduced to incorporate long-range terms while enabling charge equilibration and preserving locality of the learnable contributions. The results partially explain existing equilibrating machine-learned potentials and outline systematic design principles, complemented by a practical fitting scheme for water-cluster energies and electron density.","Nonlinearity 38 (2025) 095024 (38pp) [https://doi.org/10.1088/1361-6544/ae0402](https://doi.org/10.1088/1361-6544/ae0402)  \nSelf-consistent Coulomb interactions for machine learning interatomic potentials  \nJack Thomas 1 􀁂, Will Baldwin2, Gabor Csanyi2  \nand Christoph Ortner3, ∗ 􀁂  \n1 School of Mathematics, University of Minnesota Twin Cities, Minneapolis, MN 55455, United States of America  \n2 Engineering Laboratory, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, United Kingdom  \n3 Department of Mathematics, University of British Columbia, Vancouver, Canada  \nE-mail: [ortner@math.ubc.ca](ortner@math.ubc.ca), [thom9218@umn.edu](thom9218@umn.edu), [wjb48@cam.ac.uk and](wjb48@cam.ac.uk and)  \n[gc121@cam.ac.uk](gc121@cam.ac.uk)  \nReceived 18 June 2024; revised 7 August 2025 Accepted for publication 5 September 2025 Published 17 September 2025  \nRecommended by Dr Helen Davis  \nAbstract  \nA 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 in a 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 equilibration and provide perspectiveshow 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.  \n∗ Author to whom any correspondence should be addressed.  \nOriginal Content from this work may be used under the terms of the Creative Commons Attribution  \n4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.  \n© 2025 The Author(s) . Published by IOP Publishing Ltd and the London Mathematical Society. 1  \nKeywords: Coulomb interactions, machine learning, electronic structure, tight binding, interatomic potentials, locality, body-order expansion  \nMathematics Subject Classification numbers: 65E05, 74E15, 81V45, 81V70  \n1. Introduction  \nElectronic 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 long timescales. 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] .  \nThis 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 asa 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 potent","cbCaiiBblE1E3Bj4","https://ap.wps.com/l/cbCaiiBblE1E3Bj4","pdf",364956,1,38,"English","en",105,"# Introduction\n## Electronic structure and modeling trade-offs\n## Nearesightedness principle and limitations\n## Incorporating long-range electrostatics and electronic information\n## Decomposition into local many-body contributions","[{\"question\":\"Why are long-range electrostatic effects important in machine learning interatomic potentials?\",\"answer\":\"Many systems exhibit electrostatic influences that do not decay quickly enough to be captured solely by local atom-centred contributions, so long-range effects must be included to maintain accuracy.\"},{\"question\":\"What does the proposed framework enable regarding charge information?\",\"answer\":\"It provides a mathematical scheme that allows charge equilibration while retaining the locality of the learnable atom-centred contributions, supporting transferability.\"},{\"question\":\"How is the approach supported beyond theory?\",\"answer\":\"A practical fitting scheme is described for training on the energy and electron density of water clusters, complementing the rigorous theoretical results.\"}]","Self-consistent Coulomb interactions for machine learning interatomic potentials | 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are long-range electrostatic effects important in machine learning interatomic potentials?","Question",{"text":75,"@type":76},"Many systems exhibit electrostatic influences that do not decay quickly enough to be captured solely by local atom-centred contributions, so long-range effects must be included to maintain accuracy.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What does the proposed framework enable regarding charge information?",{"text":80,"@type":76},"It provides a mathematical scheme that allows charge equilibration while retaining the locality of the learnable atom-centred contributions, supporting transferability.",{"name":82,"@type":73,"acceptedAnswer":83},"How is the approach supported beyond theory?",{"text":84,"@type":76},"A practical fitting scheme is described for training on the energy and electron density of water clusters, complementing the rigorous theoretical 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