<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE article PUBLIC "-//NLM//DTD JATS (Z39.96) Journal Publishing DTD v1.3 20210610//EN" "JATS-journalpublishing1-3.dtd">
<article article-type="research-article" dtd-version="1.3" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xml:lang="en"><front><journal-meta><journal-id journal-id-type="publisher-id">najo</journal-id><journal-title-group><journal-title xml:lang="en">Nanosystems: Physics, Chemistry, Mathematics</journal-title><trans-title-group xml:lang="ru"><trans-title>Наносистемы: физика, химия, математика</trans-title></trans-title-group></journal-title-group><issn pub-type="ppub">2220-8054</issn><issn pub-type="epub">2305-7971</issn><publisher><publisher-name>Университет ИТМО</publisher-name></publisher></journal-meta><article-meta><article-id custom-type="elpub" pub-id-type="custom">najo-1496</article-id><article-categories><subj-group subj-group-type="heading"><subject>Research Article</subject></subj-group><subj-group subj-group-type="section-heading" xml:lang="en"><subject>CHEMISTRY AND MATERIALS SCIENCE</subject></subj-group><subj-group subj-group-type="section-heading" xml:lang="ru"><subject>ХИМИЯ И НАУКА О МАТЕРИАЛАХ</subject></subj-group></article-categories><title-group><article-title>Bridging accuracy and efficiency: assessing universal ML potentials for niobium-oxygen clusters</article-title><trans-title-group xml:lang="ru"><trans-title></trans-title></trans-title-group></title-group><contrib-group><contrib contrib-type="author" corresp="yes"><contrib-id contrib-id-type="orcid">https://orcid.org/0000-0001-8312-7071</contrib-id><name-alternatives><name name-style="western" xml:lang="en"><surname>Popov</surname><given-names>Ilya</given-names></name></name-alternatives><email xlink:type="simple">popov@yandex.ru</email><xref ref-type="aff" rid="aff-1"/></contrib><contrib contrib-type="author" corresp="yes"><contrib-id contrib-id-type="orcid">https://orcid.org/0000-0003-1656-732X</contrib-id><name-alternatives><name name-style="western" xml:lang="en"><surname>Valeeva</surname><given-names>Albina</given-names></name></name-alternatives><email xlink:type="simple">anibla_v@mail.ru</email><xref ref-type="aff" rid="aff-2"/></contrib><contrib contrib-type="author" corresp="yes"><contrib-id contrib-id-type="orcid">https://orcid.org/0000-0001-6195-7971</contrib-id><name-alternatives><name name-style="western" xml:lang="en"><surname>Enyashin</surname><given-names>Andrey</given-names></name></name-alternatives><email xlink:type="simple">enyashin@ihim.uran.ru</email><xref ref-type="aff" rid="aff-2"/></contrib></contrib-group><aff-alternatives id="aff-1"><aff xml:lang="ru"><institution>Институт химии твердого тела УрО РАН</institution></aff><aff xml:lang="en"><institution>Institute of Solid State Chemistry, Ural Branch of Russian Academy of Sciences</institution></aff></aff-alternatives><aff xml:lang="en" id="aff-2"><institution>Institute of Solid State Chemistry, Ural Branch of Russian Academy of Sciences</institution><country>Russian Federation</country></aff><pub-date pub-type="collection"><year>2025</year></pub-date><pub-date pub-type="epub"><day>05</day><month>02</month><year>2026</year></pub-date><volume>16</volume><issue>5</issue><elocation-id>1496</elocation-id><permissions><copyright-statement>Copyright &amp;#x00A9; Popov I., Valeeva A., Enyashin A., 2025</copyright-statement><copyright-year>2025</copyright-year><copyright-holder xml:lang="ru">Popov I., Valeeva A., Enyashin A.</copyright-holder><copyright-holder xml:lang="en">Popov I., Valeeva A., Enyashin A.</copyright-holder><license xml:lang="ru" license-type="creative-commons-attribution" xlink:href="https://creativecommons.org/licenses/by/4.0/" xlink:type="simple"><license-p>Данная работа распространяется под лицензией Creative Commons Attribution 4.0.</license-p></license><license xml:lang="en" license-type="creative-commons-attribution" xlink:href="https://creativecommons.org/licenses/by/4.0/" xlink:type="simple"><license-p>This work is licensed under a Creative Commons Attribution 4.0 License.</license-p></license></permissions><self-uri xlink:href="https://nanojournal.ifmo.ru/jour/article/view/1496">https://nanojournal.ifmo.ru/jour/article/view/1496</self-uri><abstract><p>Machine Learning Interatomic Potentials (MLIPs) promise to combine the accuracy of DFT with the speed of classical force fields. However, their reliability for complex, multi-component systems requires rigorous validation. Here, we perform a targeted evaluation of three leading universal MLIPs using niobium oxide clusters (NbnOm, n≤6, m≤6) as a challenging test case. The Nb‑O system is very well suited for this task due to its complex electronic interactions, manifested in existence of the bulk phase with 25% vacancy-ordered lattice and, at nanoscale, by a diverse range of non-stoichiometric clusters. We employ a dataset of global minima structures identified via DFT-based evolutionary search as a strict reference. A comparative analysis is then performed by executing evolutionary searches with the MLIPs. By directly comparing predicted structures, energies, and relative stability, we provide a comprehensive assessment of the accuracy and limitations of current universal potentials for modeling complex nanoscale oxides.</p></abstract><kwd-group xml:lang="en"><kwd>DFT</kwd><kwd>Machine learning potential</kwd><kwd>MLIP</kwd><kwd>Evolutionary algorithm</kwd><kwd>Niobium oxide</kwd><kwd>NbO</kwd><kwd>Nanoparticle</kwd><kwd>Cluster</kwd><kwd>USPEX.</kwd></kwd-group><funding-group><funding-statement xml:lang="en">Russian Science Foundation (project 19-73-20012-П, https://rscf.ru/en/project/19-73-20012/)</funding-statement></funding-group></article-meta></front><back><ref-list><title>References</title><ref id="cit1"><label>1</label><citation-alternatives><mixed-citation xml:lang="ru">Wan K., He J., Shi X. Construction of High Accuracy Machine Learning Interatomic Potential for Surface/Interface of Nanomaterials – A Review. Adv. Mater., 2024, 36(22), 2305758. (https://doi.org/10.1002/adma.202305758).</mixed-citation><mixed-citation xml:lang="en">Wan K., He J., Shi X. Construction of High Accuracy Machine Learning Interatomic Potential for Surface/Interface of Nanomaterials – A Review. Adv. Mater., 2024, 36(22), 2305758. (https://doi.org/10.1002/adma.202305758).</mixed-citation></citation-alternatives></ref><ref id="cit2"><label>2</label><citation-alternatives><mixed-citation xml:lang="ru">Wang G., Wang C., Zhang X., Li Z., Zhou J., Sun Z. Machine learning interatomic potential: Bridge the gap between small-scale models and realistic device-scale simulations. iScience, 2024, 27(5), 109673. (https://doi.org/10.1016/j.isci.2024.109673).</mixed-citation><mixed-citation xml:lang="en">Wang G., Wang C., Zhang X., Li Z., Zhou J., Sun Z. Machine learning interatomic potential: Bridge the gap between small-scale models and realistic device-scale simulations. iScience, 2024, 27(5), 109673. (https://doi.org/10.1016/j.isci.2024.109673).</mixed-citation></citation-alternatives></ref><ref id="cit3"><label>3</label><citation-alternatives><mixed-citation xml:lang="ru">Mortazavi B., Zhuang X., Rabczuk T., Shapeev A.V. Atomistic modeling of the mechanical properties: the rise of machine learning interatomic potentials. Mater. Horiz., 2023, 10, 1956–1968. (https://doi.org/10.1039/D3MH00125C).</mixed-citation><mixed-citation xml:lang="en">Mortazavi B., Zhuang X., Rabczuk T., Shapeev A.V. Atomistic modeling of the mechanical properties: the rise of machine learning interatomic potentials. Mater. Horiz., 2023, 10, 1956–1968. (https://doi.org/10.1039/D3MH00125C).</mixed-citation></citation-alternatives></ref><ref id="cit4"><label>4</label><citation-alternatives><mixed-citation xml:lang="ru">Barroso-Luque L., Shuaibi M., Fu X., Wood B.M., Dzamba M., Gao M., Rizvi A., Zitnick C.L., Ulissi Z.W. Open Materials 2024 (OMat24) Inorganic Materials Dataset and Models. arXiv, 2024, [cond-mat.mtrl-sci], arXiv:2410.12771. (https://doi.org/10.48550/arXiv.2410.12771).</mixed-citation><mixed-citation xml:lang="en">Barroso-Luque L., Shuaibi M., Fu X., Wood B.M., Dzamba M., Gao M., Rizvi A., Zitnick C.L., Ulissi Z.W. Open Materials 2024 (OMat24) Inorganic Materials Dataset and Models. arXiv, 2024, [cond-mat.mtrl-sci], arXiv:2410.12771. (https://doi.org/10.48550/arXiv.2410.12771).</mixed-citation></citation-alternatives></ref><ref id="cit5"><label>5</label><citation-alternatives><mixed-citation xml:lang="ru">Ramlaoui A., Siron M., Djafar I., Musielewicz J., Rossello A., Schmidt V., Duval A. LeMat-Traj: A Scalable and Unified Dataset of Materials Trajectories for Atomistic Modeling. arXiv, 2025, [cs.LG], arXiv:2508.20875. (https://doi.org/10.48550/arXiv.2508.20875).</mixed-citation><mixed-citation xml:lang="en">Ramlaoui A., Siron M., Djafar I., Musielewicz J., Rossello A., Schmidt V., Duval A. LeMat-Traj: A Scalable and Unified Dataset of Materials Trajectories for Atomistic Modeling. arXiv, 2025, [cs.LG], arXiv:2508.20875. (https://doi.org/10.48550/arXiv.2508.20875).</mixed-citation></citation-alternatives></ref><ref id="cit6"><label>6</label><citation-alternatives><mixed-citation xml:lang="ru">Jain A., Ong S.P., Hautier G., Chen W., Richards W.D., Dacek S., Cholia S., Gunter D., Skinner D., Ceder G., Persson K.A. Commentary: The Materials Project: A materials genome approach to accelerating materials innovation. APL Materials, 2013, 1, 011002. (https://doi.org/10.1063/1.4812323).</mixed-citation><mixed-citation xml:lang="en">Jain A., Ong S.P., Hautier G., Chen W., Richards W.D., Dacek S., Cholia S., Gunter D., Skinner D., Ceder G., Persson K.A. Commentary: The Materials Project: A materials genome approach to accelerating materials innovation. APL Materials, 2013, 1, 011002. (https://doi.org/10.1063/1.4812323).</mixed-citation></citation-alternatives></ref><ref id="cit7"><label>7</label><citation-alternatives><mixed-citation xml:lang="ru">Schmidt J., Cerqueira T.F.T., Romero A.H., Loew A., Jäger F., Wang H.-C., Botti S., Marques M.A.L. Improving machine-learning models in materials science through large datasets. Materials Today Physics, 2024, 48, 101560. (https://doi.org/10.1016/j.mtphys.2024.101560).</mixed-citation><mixed-citation xml:lang="en">Schmidt J., Cerqueira T.F.T., Romero A.H., Loew A., Jäger F., Wang H.-C., Botti S., Marques M.A.L. Improving machine-learning models in materials science through large datasets. Materials Today Physics, 2024, 48, 101560. (https://doi.org/10.1016/j.mtphys.2024.101560).</mixed-citation></citation-alternatives></ref><ref id="cit8"><label>8</label><citation-alternatives><mixed-citation xml:lang="ru">Saal J.E., Kirklin S., Aykol M., Meredig B., Wolverton C. Materials Design and Discovery with High-Throughput Density Functional Theory: The Open Quantum Materials Database (OQMD). Jom., 2013, 65(11), 1501–1509. (https://doi.org/10.1007/s11837-013-0755-4).</mixed-citation><mixed-citation xml:lang="en">Saal J.E., Kirklin S., Aykol M., Meredig B., Wolverton C. Materials Design and Discovery with High-Throughput Density Functional Theory: The Open Quantum Materials Database (OQMD). Jom., 2013, 65(11), 1501–1509. (https://doi.org/10.1007/s11837-013-0755-4).</mixed-citation></citation-alternatives></ref><ref id="cit9"><label>9</label><citation-alternatives><mixed-citation xml:lang="ru">Riebesell J., Goodall R.E.A., Benner P., Chiang Y., Deng B., Ceder G., Asta M., Lee A.A., Jain A., Persson K.A. A framework to evaluate machine learning crystal stability predictions. Nat. Mach. Intell., 2025, 7, 836–847. (https://doi.org/10.1038/s42256-025-01055-1).</mixed-citation><mixed-citation xml:lang="en">Riebesell J., Goodall R.E.A., Benner P., Chiang Y., Deng B., Ceder G., Asta M., Lee A.A., Jain A., Persson K.A. A framework to evaluate machine learning crystal stability predictions. Nat. Mach. Intell., 2025, 7, 836–847. (https://doi.org/10.1038/s42256-025-01055-1).</mixed-citation></citation-alternatives></ref><ref id="cit10"><label>10</label><citation-alternatives><mixed-citation xml:lang="ru">Okatz A.M., Keesom P.H. Specific heat and magnetization of the superconducting monoxides: NbO and TiO. Phys. Rev. B, 1975, 12, 4917. (https://doi.org/10.1103/PhysRevB.12.4917).</mixed-citation><mixed-citation xml:lang="en">Okatz A.M., Keesom P.H. Specific heat and magnetization of the superconducting monoxides: NbO and TiO. Phys. Rev. B, 1975, 12, 4917. (https://doi.org/10.1103/PhysRevB.12.4917).</mixed-citation></citation-alternatives></ref><ref id="cit11"><label>11</label><citation-alternatives><mixed-citation xml:lang="ru">Bowman A.L., Wallace T.C., Yarnell J.L., Wenzel R.G. The crystal structure of niobium monoxide. Acta Crystallogr., 1966, 21, 843. (https://doi.org/10.1107/S0365110X66004043).</mixed-citation><mixed-citation xml:lang="en">Bowman A.L., Wallace T.C., Yarnell J.L., Wenzel R.G. The crystal structure of niobium monoxide. Acta Crystallogr., 1966, 21, 843. (https://doi.org/10.1107/S0365110X66004043).</mixed-citation></citation-alternatives></ref><ref id="cit12"><label>12</label><citation-alternatives><mixed-citation xml:lang="ru">Kurmaev E.Z., Moewes A., Bureev O.G., Nekrasov I.A., Cherkashenko V.M., Korotin M.A., Ederer D.L. Electronic structure of niobium oxides. J. Alloys Compd., 2002, 347, 213–218. (https://doi.org/10.1016/S0925-8388(02)00765-X).</mixed-citation><mixed-citation xml:lang="en">Kurmaev E.Z., Moewes A., Bureev O.G., Nekrasov I.A., Cherkashenko V.M., Korotin M.A., Ederer D.L. Electronic structure of niobium oxides. J. Alloys Compd., 2002, 347, 213–218. (https://doi.org/10.1016/S0925-8388(02)00765-X).</mixed-citation></citation-alternatives></ref><ref id="cit13"><label>13</label><citation-alternatives><mixed-citation xml:lang="ru">Hu Z., Qian G., Li S., Yang L., Chen X., Weng M., Tan W., Pan F. Discovery of space aromaticity in transition–metal monoxide crystal Nb3O3 enabled by octahedral Nb6 structural units. Sci. Bull., 2019, 65, 367–372. (https://doi.org/10.1016/j.scib.2019.10.026),</mixed-citation><mixed-citation xml:lang="en">Hu Z., Qian G., Li S., Yang L., Chen X., Weng M., Tan W., Pan F. Discovery of space aromaticity in transition–metal monoxide crystal Nb3O3 enabled by octahedral Nb6 structural units. Sci. Bull., 2019, 65, 367–372. (https://doi.org/10.1016/j.scib.2019.10.026),</mixed-citation></citation-alternatives></ref><ref id="cit14"><label>14</label><citation-alternatives><mixed-citation xml:lang="ru">Miura A., Takei T., Kumada N., Wada S., Magome E., Moriyoshi C., Kuroiwa Y. Bonding Preference of Carbon, Nitrogen, and Oxygen in Niobium-Based Rock-Salt Structures. Inorg. Chem., 2013, 52, 9699. (https://doi.org/10.1021/ic400830b).</mixed-citation><mixed-citation xml:lang="en">Miura A., Takei T., Kumada N., Wada S., Magome E., Moriyoshi C., Kuroiwa Y. Bonding Preference of Carbon, Nitrogen, and Oxygen in Niobium-Based Rock-Salt Structures. Inorg. Chem., 2013, 52, 9699. (https://doi.org/10.1021/ic400830b).</mixed-citation></citation-alternatives></ref><ref id="cit15"><label>15</label><citation-alternatives><mixed-citation xml:lang="ru">Music D., Schmidt P., Mráz S. Adsorption of film-forming species on NbO and NbO2 surfaces. J. Vac. Sci. Technol. A., 2017, 35, 061512. (https://doi.org/10.1116/1.4995492).</mixed-citation><mixed-citation xml:lang="en">Music D., Schmidt P., Mráz S. Adsorption of film-forming species on NbO and NbO2 surfaces. J. Vac. Sci. Technol. A., 2017, 35, 061512. (https://doi.org/10.1116/1.4995492).</mixed-citation></citation-alternatives></ref><ref id="cit16"><label>16</label><citation-alternatives><mixed-citation xml:lang="ru">Fielicke A., Meijer G., Helden G. Infrared Spectroscopy of Niobium Oxide Cluster Cations in a Molecular Beam: Identifying the Cluster Structures. J. Am. Chem. Soc., 2003, 125, 3659–3667. (https://doi.org/10.1021/ja0288946).</mixed-citation><mixed-citation xml:lang="en">Fielicke A., Meijer G., Helden G. Infrared Spectroscopy of Niobium Oxide Cluster Cations in a Molecular Beam: Identifying the Cluster Structures. J. Am. Chem. Soc., 2003, 125, 3659–3667. (https://doi.org/10.1021/ja0288946).</mixed-citation></citation-alternatives></ref><ref id="cit17"><label>17</label><citation-alternatives><mixed-citation xml:lang="ru">Deng H.T., Kerns K.P., Castleman A.W. Formation, Structures, and Reactivities of Niobium Oxide Cluster Ions. J. Phys. Chem., 1996, 100, 13386–13392. (https://doi.org/10.1021/jp953100d).</mixed-citation><mixed-citation xml:lang="en">Deng H.T., Kerns K.P., Castleman A.W. Formation, Structures, and Reactivities of Niobium Oxide Cluster Ions. J. Phys. Chem., 1996, 100, 13386–13392. (https://doi.org/10.1021/jp953100d).</mixed-citation></citation-alternatives></ref><ref id="cit18"><label>18</label><citation-alternatives><mixed-citation xml:lang="ru">Popov I.S., Valeeva A.A., Enyashin A.N. Identifying stable Nb-O clusters using evolutionary algorithm and DFT: A foundation for machine learning potentials. Chem. Phys., 2025, 590, 112533. (https://doi.org/10.1016/j.chemphys.2024.112533).</mixed-citation><mixed-citation xml:lang="en">Popov I.S., Valeeva A.A., Enyashin A.N. Identifying stable Nb-O clusters using evolutionary algorithm and DFT: A foundation for machine learning potentials. Chem. Phys., 2025, 590, 112533. (https://doi.org/10.1016/j.chemphys.2024.112533).</mixed-citation></citation-alternatives></ref><ref id="cit19"><label>19</label><citation-alternatives><mixed-citation xml:lang="ru">Lyakhov A.O., Oganov A.R., Stokes H., Zhu Q. New developments in evolutionary structure prediction algorithm USPEX. Comp. Phys. Comm., 2013, 184, 1172–1182. (https://doi.org/10.1016/j.cpc.2012.12.009).</mixed-citation><mixed-citation xml:lang="en">Lyakhov A.O., Oganov A.R., Stokes H., Zhu Q. New developments in evolutionary structure prediction algorithm USPEX. Comp. Phys. Comm., 2013, 184, 1172–1182. (https://doi.org/10.1016/j.cpc.2012.12.009).</mixed-citation></citation-alternatives></ref><ref id="cit20"><label>20</label><citation-alternatives><mixed-citation xml:lang="ru">Lepeshkin S.V., Baturin V.S., Uspenskii Y.A., Oganov A.R. Method for simultaneous prediction of atomic structure of nanoclusters in a wide area of compositions. J. Phys. Chem. Lett., 2019, 10, 102–106. (https://doi.org/10.1021/acs.jpclett.8b03510).</mixed-citation><mixed-citation xml:lang="en">Lepeshkin S.V., Baturin V.S., Uspenskii Y.A., Oganov A.R. Method for simultaneous prediction of atomic structure of nanoclusters in a wide area of compositions. J. Phys. Chem. Lett., 2019, 10, 102–106. (https://doi.org/10.1021/acs.jpclett.8b03510).</mixed-citation></citation-alternatives></ref><ref id="cit21"><label>21</label><citation-alternatives><mixed-citation xml:lang="ru">Heydariyan S., Nouri M.R., Alaei M., Allahyari Z., Niehaus T.A. New candidates for the global minimum of medium-sized silicon clusters: A hybrid DFTB/DFT genetic algorithm applied to Sin, n = 8-80. J. Chem. Phys., 2018, 149, 074313. (https://doi.org/10.1063/1.5037159).</mixed-citation><mixed-citation xml:lang="en">Heydariyan S., Nouri M.R., Alaei M., Allahyari Z., Niehaus T.A. New candidates for the global minimum of medium-sized silicon clusters: A hybrid DFTB/DFT genetic algorithm applied to Sin, n = 8-80. J. Chem. Phys., 2018, 149, 074313. (https://doi.org/10.1063/1.5037159).</mixed-citation></citation-alternatives></ref><ref id="cit22"><label>22</label><citation-alternatives><mixed-citation xml:lang="ru">Olvera-Neria O., García-Cruz R., Gonzalez-Torres J., García-Cruz L.M., Castillo-Sánchez J.L., Poulain E. Strongly Bound Frenkel Excitons on TiO2 Nanoparticles: An Evolutionary and DFT Approach. Int. J. Photoenergy, 2024, 2024, 4014216. (https://doi.org/10.1155/2024/4014216).</mixed-citation><mixed-citation xml:lang="en">Olvera-Neria O., García-Cruz R., Gonzalez-Torres J., García-Cruz L.M., Castillo-Sánchez J.L., Poulain E. Strongly Bound Frenkel Excitons on TiO2 Nanoparticles: An Evolutionary and DFT Approach. Int. J. Photoenergy, 2024, 2024, 4014216. (https://doi.org/10.1155/2024/4014216).</mixed-citation></citation-alternatives></ref><ref id="cit23"><label>23</label><citation-alternatives><mixed-citation xml:lang="ru">Yu X., Oganov A.R., Zhu Q., Qi F., Qian G. The stability and unexpected chemistry of oxide clusters. Phys. Chem. Chem. Phys., 2018, 20, 30437. (https://doi.org/10.1039/c8cp03519a).</mixed-citation><mixed-citation xml:lang="en">Yu X., Oganov A.R., Zhu Q., Qi F., Qian G. The stability and unexpected chemistry of oxide clusters. Phys. Chem. Chem. Phys., 2018, 20, 30437. (https://doi.org/10.1039/c8cp03519a).</mixed-citation></citation-alternatives></ref><ref id="cit24"><label>24</label><citation-alternatives><mixed-citation xml:lang="ru">Mahdavifar Z. Prediction of unexpected BnPn structures: promising materials for non-linear optical devices and photocatalytic activities. Nanoscale Adv., 2021, 3, 2846. (https://doi.org/10.1039/d0na01040e).</mixed-citation><mixed-citation xml:lang="en">Mahdavifar Z. Prediction of unexpected BnPn structures: promising materials for non-linear optical devices and photocatalytic activities. Nanoscale Adv., 2021, 3, 2846. (https://doi.org/10.1039/d0na01040e).</mixed-citation></citation-alternatives></ref><ref id="cit25"><label>25</label><citation-alternatives><mixed-citation xml:lang="ru">Sandu M.P., Kovtunov M.A., Baturin V.S., Oganov A.R., Kurzina I.A. Influence of the Pd:Bi ratio on Pd–Bi/Al2O3 catalysts: structure, surface and activity in glucose oxidation. Phys. Chem. Chem. Phys., 2021, 23, 14889. (https://doi.org/10.1039/d1cp01305j).</mixed-citation><mixed-citation xml:lang="en">Sandu M.P., Kovtunov M.A., Baturin V.S., Oganov A.R., Kurzina I.A. Influence of the Pd:Bi ratio on Pd–Bi/Al2O3 catalysts: structure, surface and activity in glucose oxidation. Phys. Chem. Chem. Phys., 2021, 23, 14889. (https://doi.org/10.1039/d1cp01305j).</mixed-citation></citation-alternatives></ref><ref id="cit26"><label>26</label><citation-alternatives><mixed-citation xml:lang="ru">Zhou T., Ma L., Chen H. Electronic structure and stability of Al6CMn (M = Li, Na, K; n = 2, 4, 6) clusters. Comput. Theor. Chem., 2020, 1178, 112780, (https://doi.org/10.1016/j.comptc.2020.112780).</mixed-citation><mixed-citation xml:lang="en">Zhou T., Ma L., Chen H. Electronic structure and stability of Al6CMn (M = Li, Na, K; n = 2, 4, 6) clusters. Comput. Theor. Chem., 2020, 1178, 112780, (https://doi.org/10.1016/j.comptc.2020.112780).</mixed-citation></citation-alternatives></ref><ref id="cit27"><label>27</label><citation-alternatives><mixed-citation xml:lang="ru">Steshin I.S., Panteleev S.V., Petukhov I.V., Ignatov S.K. Parametrization of Gaussian approximation potential for the global optimization of magnesium clusters Mg N (N≤ 100).</mixed-citation><mixed-citation xml:lang="en">Steshin I.S., Panteleev S.V., Petukhov I.V., Ignatov S.K. Parametrization of Gaussian approximation potential for the global optimization of magnesium clusters Mg N (N≤ 100).</mixed-citation></citation-alternatives></ref><ref id="cit28"><label>28</label><citation-alternatives><mixed-citation xml:lang="ru">Phys. Chem. Chem. Phys., 2025, 27, 18960–18977. (https://doi.org/10.1039/D5CP02189H).</mixed-citation><mixed-citation xml:lang="en">Phys. Chem. Chem. Phys., 2025, 27, 18960–18977. (https://doi.org/10.1039/D5CP02189H).</mixed-citation></citation-alternatives></ref><ref id="cit29"><label>29</label><citation-alternatives><mixed-citation xml:lang="ru">Kresse G., Furthmüller J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B, 1996, 54, 11169. (https://doi.org/10.1103/PhysRevB.54.11169).</mixed-citation><mixed-citation xml:lang="en">Kresse G., Furthmüller J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B, 1996, 54, 11169. (https://doi.org/10.1103/PhysRevB.54.11169).</mixed-citation></citation-alternatives></ref><ref id="cit30"><label>30</label><citation-alternatives><mixed-citation xml:lang="ru">Fu X., Wood B.M., Barroso-Luque L., Levine D.S., Gao M., Dzamba M., Zitnick C.L. Learning Smooth and Expressive Interatomic Potentials for Physical Property Prediction. arXiv, 2025, [physics.comp-ph], arXiv:2502.12147. (https://doi.org/10.48550/arXiv.2502.12147).</mixed-citation><mixed-citation xml:lang="en">Fu X., Wood B.M., Barroso-Luque L., Levine D.S., Gao M., Dzamba M., Zitnick C.L. Learning Smooth and Expressive Interatomic Potentials for Physical Property Prediction. arXiv, 2025, [physics.comp-ph], arXiv:2502.12147. (https://doi.org/10.48550/arXiv.2502.12147).</mixed-citation></citation-alternatives></ref><ref id="cit31"><label>31</label><citation-alternatives><mixed-citation xml:lang="ru">Rhodes B., Vandenhaute S., Šimkus V., Gin J., Godwin J., Duignan T., Neumann M. Orb-v3: atomistic simulation at scale. arXiv, 2025, [cond-mat.mtrl-sci], arXiv:2504.06231.  (https://doi.org/10.48550/arXiv.2504.06231).</mixed-citation><mixed-citation xml:lang="en">Rhodes B., Vandenhaute S., Šimkus V., Gin J., Godwin J., Duignan T., Neumann M. Orb-v3: atomistic simulation at scale. arXiv, 2025, [cond-mat.mtrl-sci], arXiv:2504.06231.  (https://doi.org/10.48550/arXiv.2504.06231).</mixed-citation></citation-alternatives></ref><ref id="cit32"><label>32</label><citation-alternatives><mixed-citation xml:lang="ru">Kim J., Kim J., Kim J., Lee J., Park Y., Kang Y., Han S. Data-efficient multi-fidelity training for high-fidelity machine learning interatomic potentials. arXiv, 2025, [cond-mat.mtrl-sci], arXiv:2504.06231. (https://doi.org/10.48550/arXiv.2409.07947).</mixed-citation><mixed-citation xml:lang="en">Kim J., Kim J., Kim J., Lee J., Park Y., Kang Y., Han S. Data-efficient multi-fidelity training for high-fidelity machine learning interatomic potentials. arXiv, 2025, [cond-mat.mtrl-sci], arXiv:2504.06231. (https://doi.org/10.48550/arXiv.2409.07947).</mixed-citation></citation-alternatives></ref><ref id="cit33"><label>33</label><citation-alternatives><mixed-citation xml:lang="ru">Wood B.M., Dzamba M., Fu X., Gao M., Shuaibi M., Barroso-Luque L., Abdelmaqsoud K., Gharakhanyan V., Kitchin J.R., Levine D.S., Michel K., Sriram A., Cohen T., Das A., Rizvi A., Sahoo S.J., Ulissi Z.W., Zitnick C.L. UMA: A Family of Universal Models for Atoms. arXiv, 2025, [cs.LG], arXiv:2506.23971. (https://doi.org/10.48550/arXiv.2506.23971).</mixed-citation><mixed-citation xml:lang="en">Wood B.M., Dzamba M., Fu X., Gao M., Shuaibi M., Barroso-Luque L., Abdelmaqsoud K., Gharakhanyan V., Kitchin J.R., Levine D.S., Michel K., Sriram A., Cohen T., Das A., Rizvi A., Sahoo S.J., Ulissi Z.W., Zitnick C.L. UMA: A Family of Universal Models for Atoms. arXiv, 2025, [cs.LG], arXiv:2506.23971. (https://doi.org/10.48550/arXiv.2506.23971).</mixed-citation></citation-alternatives></ref><ref id="cit34"><label>34</label><citation-alternatives><mixed-citation xml:lang="ru">Larsen A.H., Mortensen J.J., Blomqvist J., Castelli I.E., Christensen R., Dułak M., Friis J., Groves M.N., Hammer B., Hargus C., Hermes E.D., Jennings P.C., Jensen P.B., Kermode J., Kitchin J.R., Kolsbjerg E.L., Kubal J., Kaasbjerg K., Lysgaard S., Maronsson J.B., Maxson T., Olsen T., Pastewka L., Peterson A., Rostgaard C., Schiøtz J., Schütt O., Strange M., Thygesen K.S., Vegge T., Vilhelmsen L., Walter M., Zeng Z., Jacobsen K.W. The atomic simulation environment – a Python library for working with atoms. J. Phys.: Condens. Matter, 2017, 29, 273002. (https://doi.org/10.1088/1361-648X/aa680e).</mixed-citation><mixed-citation xml:lang="en">Larsen A.H., Mortensen J.J., Blomqvist J., Castelli I.E., Christensen R., Dułak M., Friis J., Groves M.N., Hammer B., Hargus C., Hermes E.D., Jennings P.C., Jensen P.B., Kermode J., Kitchin J.R., Kolsbjerg E.L., Kubal J., Kaasbjerg K., Lysgaard S., Maronsson J.B., Maxson T., Olsen T., Pastewka L., Peterson A., Rostgaard C., Schiøtz J., Schütt O., Strange M., Thygesen K.S., Vegge T., Vilhelmsen L., Walter M., Zeng Z., Jacobsen K.W. The atomic simulation environment – a Python library for working with atoms. J. Phys.: Condens. Matter, 2017, 29, 273002. (https://doi.org/10.1088/1361-648X/aa680e).</mixed-citation></citation-alternatives></ref><ref id="cit35"><label>35</label><citation-alternatives><mixed-citation xml:lang="ru">Thompson A.P., Aktulga H.M., Berger R., Bolintineanu D.S., Brown W.M., Crozier P. S., Veld P.J., Kohlmeyer A., Moore S.G., Nguyen T.D., Shan R., Stevens M.J., Tranchida J., Trott C., Plimpton S.J. LAMMPS - a flexible simulation tool for particle-based materials modeling at the atomic, meso, and continuum scales. Comp. Phys. Comm., 2022, 271, 10817. (https://doi.org/10.1016/j. cpc.2021.108171).</mixed-citation><mixed-citation xml:lang="en">Thompson A.P., Aktulga H.M., Berger R., Bolintineanu D.S., Brown W.M., Crozier P. S., Veld P.J., Kohlmeyer A., Moore S.G., Nguyen T.D., Shan R., Stevens M.J., Tranchida J., Trott C., Plimpton S.J. LAMMPS - a flexible simulation tool for particle-based materials modeling at the atomic, meso, and continuum scales. Comp. Phys. Comm., 2022, 271, 10817. (https://doi.org/10.1016/j. cpc.2021.108171).</mixed-citation></citation-alternatives></ref><ref id="cit36"><label>36</label><citation-alternatives><mixed-citation xml:lang="ru">Stukowski A. Visualization and analysis of atomistic simulation data with OVITO - the Open Visualization Tool. Modelling Simul. Mater. Sci. Eng., 2010, 18, 015012. (https://doi.org/10.1088/0965-0393/18/1/015012).</mixed-citation><mixed-citation xml:lang="en">Stukowski A. Visualization and analysis of atomistic simulation data with OVITO - the Open Visualization Tool. Modelling Simul. Mater. Sci. Eng., 2010, 18, 015012. (https://doi.org/10.1088/0965-0393/18/1/015012).</mixed-citation></citation-alternatives></ref><ref id="cit37"><label>37</label><citation-alternatives><mixed-citation xml:lang="ru">Hunter J.D. Matplotlib: A 2D Graphics Environment. Computing in Science &amp; Engineering, 2007, 9(3), 90–95. (https://doi.org/10.1109/MCSE.2007.55).</mixed-citation><mixed-citation xml:lang="en">Hunter J.D. Matplotlib: A 2D Graphics Environment. Computing in Science &amp; Engineering, 2007, 9(3), 90–95. (https://doi.org/10.1109/MCSE.2007.55).</mixed-citation></citation-alternatives></ref></ref-list><fn-group><fn fn-type="conflict"><p>The authors declare that there are no conflicts of interest present.</p></fn></fn-group></back></article>
