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Here, the CI-NEB and the TI methods are used to find a transition state along the reaction path and evaluate the free energy path, respectively. Our algorithm is applied to the Walden inversion reaction of the hydronium ions (H3O+). The numerical results show that the computational effort by our algorithm is significantly reduced compared to that of the previously proposed algorithm combining PIHMC without losing accuracy. We also demonstrate the importance of temperature and isotope effects on the addition reaction of hydrogen isocyanide to formaldehyde. In this reaction, the nuclear quantum effect causes the structural change at the TS and decreases the energy barrier.<\/jats:p>","DOI":"10.1063\/5.0225208","type":"journal-article","created":{"date-parts":[[2024,11,1]],"date-time":"2024-11-01T11:04:40Z","timestamp":1730459080000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":2,"title":["Efficient quantum mechanical minimum free energy path calculation by combining path integral hybrid Monte Carlo and climbing image nudged elastic band methods, and its application to the addition reaction of hydrogen isocyanide to formaldehyde"],"prefix":"10.1063","volume":"161","author":[{"ORCID":"https:\/\/orcid.org\/0009-0005-2268-3214","authenticated-orcid":false,"given":"Aiko","family":"Io","sequence":"first","affiliation":[{"name":"Quantum Chemistry Division, Yokohama City University 1 , Seto 22-2, Kanazawa-Ku, Yokohama 236-0027, Kanagawa,","place":["Japan"]},{"name":"Analysis Research Department, Chemical Research Laboratories, Nissan Chemical Corporation 2 , 2-10-1, Tsuboi-Nishi, Funabashi, Chiba 274-8507,","place":["Japan"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4080-5036","authenticated-orcid":false,"given":"Kimichi","family":"Suzuki","sequence":"additional","affiliation":[{"name":"Institute for Chemical Reaction Design and Discovery (WPI-ICReDD), Hokkaido University 3 , Sapporo 001-0021,","place":["Japan"]},{"name":"Department of Chemistry, Faculty of Science, Hokkaido University 4 , Sapporo 060-0810,","place":["Japan"]},{"name":"JST, ERATO Maeda Artificial Intelligence for Chemical Reaction Design and Discovery Project 5 , Sapporo 060-0810,","place":["Japan"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5767-7942","authenticated-orcid":false,"given":"Makito","family":"Takagi","sequence":"additional","affiliation":[{"name":"Quantum Chemistry Division, Yokohama City University 1 , Seto 22-2, Kanazawa-Ku, Yokohama 236-0027, Kanagawa,","place":["Japan"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5489-5714","authenticated-orcid":false,"given":"Masanori","family":"Tachikawa","sequence":"additional","affiliation":[{"name":"Quantum Chemistry Division, Yokohama City University 1 , Seto 22-2, Kanazawa-Ku, Yokohama 236-0027, Kanagawa,","place":["Japan"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"317","published-online":{"date-parts":[[2024,11,1]]},"reference":[{"issue":"5301","key":"2024110111043048800_c1","doi-asserted-by":"publisher","first-page":"817","DOI":"10.1126\/science.275.5301.817","article-title":"On the quantum nature of the shared proton in hydrogen bonds","volume":"275","year":"1997","journal-title":"Science"},{"issue":"6720","key":"2024110111043048800_c2","doi-asserted-by":"publisher","first-page":"601","DOI":"10.1038\/17579","article-title":"The nature of the hydrated excess proton in water","volume":"397","year":"1999","journal-title":"Nature"},{"issue":"6892","key":"2024110111043048800_c3","doi-asserted-by":"publisher","first-page":"925","DOI":"10.1038\/nature00797","article-title":"The nature and transport mechanism of hydrated hydroxide ions in aqueous solution","volume":"417","year":"2002","journal-title":"Nature"},{"issue":"21","key":"2024110111043048800_c4","doi-asserted-by":"publisher","first-page":"4233","DOI":"10.1021\/acs.jpclett.5b01899","article-title":"Stability of complex biomolecular structures: van der Waals, hydrogen bond cooperativity, and nuclear quantum effects","volume":"6","year":"2015","journal-title":"J.\u00a0Phys. 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