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Moreover, we derive a lower bound for the effective rank in the form \u221dNAO2\/log27NAO, where NAO is the number of atomic orbitals in the molecule, under mild conditions imposed on the decomposition threshold \u025b. As a result, while a subquadratic growth of the CPD rank is not excluded, a linear relationship between the rank and NAO cannot hold universally. The implications of these findings for the use of the canonical polyadic format to represent electron repulsion integrals in quantum chemistry are analyzed.<\/jats:p>","DOI":"10.1063\/5.0343714","type":"journal-article","created":{"date-parts":[[2026,8,13]],"date-time":"2026-08-13T11:20:52Z","timestamp":1786620052000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":["On the effective rank of canonical polyadic decomposition of electron repulsion integrals"],"prefix":"10.1063","volume":"165","author":[{"ORCID":"https:\/\/orcid.org\/0009-0004-6589-2705","authenticated-orcid":false,"given":"Aleksandra","family":"Oszmian","sequence":"first","affiliation":[{"name":"University of Warsaw, Faculty of Chemistry , Pasteura 1, 02-093 Warsaw,","place":["Poland"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7928-4450","authenticated-orcid":false,"given":"Micha\u0142","family":"Lesiuk","sequence":"additional","affiliation":[{"name":"University of Warsaw, Faculty of Chemistry , Pasteura 1, 02-093 Warsaw,","place":["Poland"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"317","published-online":{"date-parts":[[2026,8,13]]},"reference":[{"key":"2026081306413960400_c1","doi-asserted-by":"publisher","first-page":"4496","DOI":"10.1063\/1.1679012","volume":"58","year":"1973","journal-title":"J.\u00a0Chem. 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