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The reference interaction site model (RISM) is one of the solvation theories used to study solution systems and has shown good performance, especially in the polar solvent systems. Although the analytical first derivative based on the RISM coupled with quantum methods (RISM-SCF) has already been derived, the analytical second derivative has not been proposed yet. Therefore, in this study, the analytical second derivative was derived using RISM-SCF explicitly including constrained spatial electron density distribution (RISM-SCF-cSED). The performance of this method was validated with the Hessian calculations of formaldehyde and para-nitroaniline in solution, and the results demonstrated that the method accurately calculated frequency values at a small computational cost.<\/jats:p>","DOI":"10.1063\/5.0067248","type":"journal-article","created":{"date-parts":[[2021,11,10]],"date-time":"2021-11-10T14:05:41Z","timestamp":1636553141000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":8,"title":["Analytical second derivatives of the free energy in solution by the reference interaction site model self-consistent field explicitly including constrained spatial electron density distribution"],"prefix":"10.1063","volume":"155","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7574-0965","authenticated-orcid":false,"given":"Daisuke","family":"Yokogawa","sequence":"first","affiliation":[{"name":"Graduate School of Arts and Sciences, The University of Tokyo , 3-8-1 Komaba, Meguro-ku, Tokyo 153-8902, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4717-7159","authenticated-orcid":false,"given":"Kayo","family":"Suda","sequence":"additional","affiliation":[{"name":"Graduate School of Arts and Sciences, The University of Tokyo , 3-8-1 Komaba, Meguro-ku, Tokyo 153-8902, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"317","published-online":{"date-parts":[[2021,11,23]]},"reference":[{"key":"2023063015530770400_c1","doi-asserted-by":"publisher","first-page":"225","DOI":"10.1002\/qua.560160825","article-title":"Derivative studies in Hartree-Fock and M\u00f8ller-Plesset theories","volume":"16","year":"1979","journal-title":"Int. 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