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In particular, through this procedure, one can establish the local counterpart of the chemical potential which, when integrated over all the space, leads to the global quantity and the local counterpart of the hardness that not only provides a function free of ambiguities, but also generates through its integration over all the space the well-defined value of the global quantity given by the difference of the vertical first ionization potential and electron affinity. In the present work, the non-local counterpart of these local reactivity descriptors is derived making use of the Fukui kernel descriptor previously developed by us. Then, the local and non-local chemical potential and hardness, thus obtained, are applied to study site and bond reactivities of several systems, to rationalize the behavior of kinetic and thermodynamic properties, through the chemical information that these indexes provide.<\/jats:p>\n                  <\/jats:sec>\n                  <jats:sec>\n                    <jats:title>Methods<\/jats:title>\n                    <jats:p>The electronic structure calculations required to evaluate the reactivity indexes analyzed in this work were done with the PBE0 exchange\u2013correlation energy functional. The geometry optimization was done in all cases in a modified version of the NWChem program, while the Hirshfeld population analysis was done in a modified version of the demon2k program. For the electrophilic addition of hydrogen halides (HX) to several substituted ethenes and the hydration reaction of aldehydes and ketones, the 6-311G** basis set was used, while for the bond enthalpies of chemical reactions where there is a homolytic bond break and the trans influence in which the lability of the leaving ligand is modified by the ligand opposite to it, the Def2-TZVP was used.<\/jats:p>\n                  <\/jats:sec>","DOI":"10.1007\/s00894-025-06311-0","type":"journal-article","created":{"date-parts":[[2025,2,18]],"date-time":"2025-02-18T06:26:07Z","timestamp":1739859967000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Local and non-local chemical potential and hardness: a grand canonical ensemble approach"],"prefix":"10.1007","volume":"31","author":[{"given":"Paulino","family":"Zer\u00f3n","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Maurizio A.","family":"Pantoja-Hern\u00e1ndez","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Marco","family":"Franco-P\u00e9rez","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jos\u00e9 L.","family":"G\u00e1zquez","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,2,18]]},"reference":[{"key":"6311_CR1","doi-asserted-by":"crossref","first-page":"154103","DOI":"10.1063\/1.4932539","volume":"143","author":"M Franco-P\u00e9rez","year":"2015","unstructured":"Franco-P\u00e9rez M, G\u00e1zquez JL, Ayers PW, Vela A (2015) Revisiting the definition of the electronic chemical potential, chemical hardness, and softness at finite temperatures. 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