{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,17]],"date-time":"2026-04-17T12:40:20Z","timestamp":1776429620263,"version":"3.51.2"},"reference-count":29,"publisher":"AIP Publishing","issue":"15","license":[{"start":{"date-parts":[[2026,4,16]],"date-time":"2026-04-16T00:00:00Z","timestamp":1776297600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100005416","name":"Norges Forskningsr\u00e5d","doi-asserted-by":"publisher","award":["262695"],"award-info":[{"award-number":["262695"]}],"id":[{"id":"10.13039\/501100005416","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100005416","name":"Norges Forskningsr\u00e5d","doi-asserted-by":"publisher","award":["315822"],"award-info":[{"award-number":["315822"]}],"id":[{"id":"10.13039\/501100005416","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Sigma2","award":["NN14654K"],"award-info":[{"award-number":["NN14654K"]}]},{"DOI":"10.13039\/501100005357","name":"Agent\u00fara na Podporu V\u00fdskumu a V\u00fdvoja","doi-asserted-by":"publisher","award":["APVV-22-0488"],"award-info":[{"award-number":["APVV-22-0488"]}],"id":[{"id":"10.13039\/501100005357","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100006109","name":"Vedeck\u00e1 Grantov\u00e1 Agent\u00fara M\u0160VVa\u0160 SR a SAV","doi-asserted-by":"publisher","award":["1\/0670\/24"],"award-info":[{"award-number":["1\/0670\/24"]}],"id":[{"id":"10.13039\/501100006109","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100031478","name":"NextGenerationEU","doi-asserted-by":"publisher","award":["09I05-03-V02-00034"],"award-info":[{"award-number":["09I05-03-V02-00034"]}],"id":[{"id":"10.13039\/100031478","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100031478","name":"NextGenerationEU","doi-asserted-by":"publisher","award":["09I01-03-V04-00096"],"award-info":[{"award-number":["09I01-03-V04-00096"]}],"id":[{"id":"10.13039\/100031478","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["pubs.aip.org"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2026,4,21]]},"abstract":"<jats:p>We present an algorithm for efficient evaluation of Boys functions F0,\u2026,Fkmax tailored to modern computing architectures, in particular graphical processing units, where maximum throughput is high and data movement is costly. The method combines rational minimax approximations with upward and downward recurrence relations. The non-negative real axis is partitioned into three regions, [0, \u221e\u27e9 = A \u222a B \u222a C, where regions A and B are treated using rational minimax approximations and region C by an asymptotic approximation. This formulation avoids lookup tables and irregular memory access, making it well-suited for hardware with high maximum throughput and low latency. The rational minimax coefficients are generated using the rational Remez algorithm. For a target maximum absolute error of \u025btol = 5 \u00d7 10\u221214, the corresponding approximation regions and coefficients for Boys functions F0, \u2026, F32 are provided in Appendix\u00a0D.<\/jats:p>","DOI":"10.1063\/5.0316073","type":"journal-article","created":{"date-parts":[[2026,4,17]],"date-time":"2026-04-17T11:42:48Z","timestamp":1776426168000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":["Efficient Boys function evaluation using minimax approximation"],"prefix":"10.1063","volume":"164","author":[{"ORCID":"https:\/\/orcid.org\/0009-0005-6192-2687","authenticated-orcid":false,"given":"Rasmus","family":"Vikhamar-Sandberg","sequence":"first","affiliation":[{"name":"Department of Physical and Theoretical Chemistry, Faculty of Natural Sciences, Comenius University 1 , SK-84215 Bratislava,","place":["Slovakia"]},{"name":"Hylleraas Center for Quantum Molecular Sciences, Department of Chemistry, UiT The Arctic University of Norway 2 , N-9037 Troms\u00f8,","place":["Norway"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0776-4137","authenticated-orcid":false,"given":"Michal","family":"Repisky","sequence":"additional","affiliation":[{"name":"Department of Physical and Theoretical Chemistry, Faculty of Natural Sciences, Comenius University 1 , SK-84215 Bratislava,","place":["Slovakia"]},{"name":"Hylleraas Center for Quantum Molecular Sciences, Department of Chemistry, UiT The Arctic University of Norway 2 , N-9037 Troms\u00f8,","place":["Norway"]}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"317","published-online":{"date-parts":[[2026,4,17]]},"reference":[{"issue":"1063","key":"2026041707424024300_c1","doi-asserted-by":"publisher","first-page":"542","DOI":"10.1098\/rspa.1950.0036","article-title":"Electronic wave functions\u2014I. A general method of calculation for the stationary states of any molecular system","volume":"200","year":"1950","journal-title":"Proc. R. Soc. London, Ser. A"},{"key":"2026041707424024300_c2","doi-asserted-by":"publisher","first-page":"69","DOI":"10.1103\/revmodphys.23.69","article-title":"New developments in molecular orbital theory","volume":"23","year":"1951","journal-title":"Rev. Mod. Phys."},{"key":"2026041707424024300_c3","doi-asserted-by":"publisher","first-page":"541","DOI":"10.1098\/rspa.1951.0048","article-title":"The molecular orbital theory of chemical valency VIII. A method of calculating ionization potentials","volume":"205","year":"1951","journal-title":"Proc. R. Soc. London, Ser. A"},{"key":"2026041707424024300_c4","doi-asserted-by":"publisher","first-page":"163","DOI":"10.1016\/0301-0104(91)80126-3","article-title":"New variations in two-electron integral evaluation in the context of direct SCF procedures","volume":"150","year":"1991","journal-title":"Chem. Phys."},{"key":"2026041707424024300_c5","doi-asserted-by":"publisher","first-page":"290","DOI":"10.1002\/wcms.78","article-title":"Multi-electron integrals","volume":"2","year":"2011","journal-title":"Wiley Interdiscip. Rev.: Comput. Mol. Sci."},{"key":"2026041707424024300_c6","volume-title":"Molecular Electronic-Structure Theory","year":"2013"},{"key":"2026041707424024300_c7","first-page":"1","article-title":"The Gaussian function in calculations of statistical mechanics and quantum mechanics","volume":"2","year":"1963","journal-title":"Methods Comput. Phys."},{"key":"2026041707424024300_c8","doi-asserted-by":"publisher","first-page":"218","DOI":"10.1016\/0021-9991(78)90092-x","article-title":"One- and two-electron integrals over Cartesian Gaussian functions","volume":"26","year":"1978","journal-title":"J.\u00a0Comput. Phys."},{"key":"2026041707424024300_c9","doi-asserted-by":"publisher","first-page":"3963","DOI":"10.1063\/1.450106","article-title":"Efficient recursive computation of molecular integrals over Cartesian Gaussian functions","volume":"84","year":"1986","journal-title":"J.\u00a0Chem. Phys."},{"key":"2026041707424024300_c10","doi-asserted-by":"publisher","first-page":"745","DOI":"10.1002\/qua.560400604","article-title":"Two-electron repulsion integrals over Gaussian s functions","volume":"40","year":"1991","journal-title":"Int. J. Quantum Chem."},{"key":"2026041707424024300_c11","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1002\/(sici)1097-461x(1996)59:3&lt;209::aid-qua4&gt;3.0.co;2-1","article-title":"ACE algorithm for the rapid evaluation of the electron-repulsion integral over Gaussian-type orbitals","volume":"59","year":"1996","journal-title":"Int. J Quantum Chem."},{"key":"2026041707424024300_c12","doi-asserted-by":"publisher","first-page":"1390","DOI":"10.1002\/jcc.23935","article-title":"A rigorous and optimized strategy for the evaluation of the Boys function kernel in molecular electronic structure theory","volume":"36","year":"2015","journal-title":"J.\u00a0Comput. Chem."},{"key":"2026041707424024300_c13","doi-asserted-by":"publisher","first-page":"2022","DOI":"10.1007\/s10910-016-0668-x","article-title":"Boys function evaluation on graphical processing units","volume":"54","year":"2016","journal-title":"J.\u00a0Math. Chem."},{"key":"2026041707424024300_c14","volume-title":"Approximation Theory and Approximation Practice, Extended Edition","year":"2019"},{"key":"2026041707424024300_c15","doi-asserted-by":"publisher","first-page":"1965","DOI":"10.1063\/1.1675126","article-title":"Approximations for the functions Fm(z) occurring in molecular calculations with a Gaussian basis","volume":"54","year":"1971","journal-title":"J.\u00a0Chem. Phys."},{"key":"2026041707424024300_c16","doi-asserted-by":"publisher","first-page":"024104","DOI":"10.1063\/5.0275063","article-title":"Global approximation to the Boys functions for vectorized computation","volume":"163","year":"2025","journal-title":"J.\u00a0Chem. Phys."},{"key":"2026041707424024300_c17","article-title":"The remez method"},{"key":"2026041707424024300_c18","unstructured":"M.\n            Clark\n           and A.Kennedy, Algremez (2005); https:\/\/github.com\/mikeaclark\/AlgRemez."},{"key":"2026041707424024300_c19"},{"key":"2026041707424024300_c20"},{"key":"2026041707424024300_c21","doi-asserted-by":"publisher","first-page":"365","DOI":"10.1007\/s11075-020-01042-0","article-title":"An algorithm for best rational approximation based on barycentric rational interpolation","volume":"88","year":"2021","journal-title":"Numer. Algorithms"},{"key":"2026041707424024300_c22","doi-asserted-by":"publisher","first-page":"2427","DOI":"10.1137\/17M1132409","article-title":"Rational minimax approximation via adaptive barycentric representations","volume":"40","year":"2018","journal-title":"SIAM J. Sci. Comput."},{"key":"2026041707424024300_c23","volume-title":"Approximation Theory and Numerical Methods","year":"1980"},{"key":"2026041707424024300_c24","volume-title":"Algorithms in Real Algebraic Geometry","year":"2006"},{"key":"2026041707424024300_c25","volume-title":"Nonlinear Approximation Theory","year":"1986"},{"key":"2026041707424024300_c26","volume-title":"Practical Optimization Algorithms and Engineering Applications","year":"2007"},{"key":"2026041707424024300_c27","volume-title":"LAPACK Users\u2019 Guide","year":"1999","edition":"3"},{"key":"2026041707424024300_c28","doi-asserted-by":"publisher","first-page":"174117","DOI":"10.1063\/5.0062444","article-title":"A fast algorithm for computing the Boys function","volume":"155","year":"2021","journal-title":"J.\u00a0Chem. Phys."},{"key":"2026041707424024300_c29","doi-asserted-by":"publisher","first-page":"e8328","DOI":"10.1002\/cpe.8328","article-title":"GPU acceleration of the Boys function evaluation in computational quantum chemistry","volume":"37","year":"2024","journal-title":"Concurr. Comput.: Pract. Exper."}],"container-title":["The Journal of Chemical Physics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/pubs.aip.org\/aip\/jcp\/article-pdf\/doi\/10.1063\/5.0316073\/20977763\/154111_1_5.0316073.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/pubs.aip.org\/aip\/jcp\/article-pdf\/doi\/10.1063\/5.0316073\/20977763\/154111_1_5.0316073.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,17]],"date-time":"2026-04-17T11:42:52Z","timestamp":1776426172000},"score":1,"resource":{"primary":{"URL":"https:\/\/pubs.aip.org\/jcp\/article\/164\/15\/154111\/3387281\/Efficient-Boys-function-evaluation-using-minimax"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,4,17]]},"references-count":29,"journal-issue":{"issue":"15","published-print":{"date-parts":[[2026,4,21]]}},"URL":"https:\/\/doi.org\/10.1063\/5.0316073","relation":{},"ISSN":["0021-9606","1089-7690"],"issn-type":[{"value":"0021-9606","type":"print"},{"value":"1089-7690","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2026,4,21]]},"published":{"date-parts":[[2026,4,17]]},"article-number":"154111"}}