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Using methods based on the constrained random phase approximation (cRPA) and moment RPA (mRPA), we construct static effective Hamiltonians that include screening through the environment. We derive expressions for the energy contribution from the environment and for the effective one- and two-body terms, taking into account double-counting corrections. cRPA requires additional consideration due to its frequency dependence, while mRPA provides a static Hamiltonian by construction. For the ground state energy of benzene and bond dissociation curves, we discuss the differences and similarities between the different flavors of RPA-based screening. We show that downfolding using cRPA describes both dynamical and strong correlation well, while mRPA and cRPA restricted to screening the particle-hole matrix elements can fail to describe bond dissociation due to a dominating dynamical correlation term. In the static limit, these two methods are shown to be almost indistinguishable.<\/jats:p>","DOI":"10.1021\/acs.jctc.6c01128","type":"journal-article","created":{"date-parts":[[2026,9,8]],"date-time":"2026-09-08T11:02:19Z","timestamp":1788865339000},"page":"9468-9482","source":"Crossref","is-referenced-by-count":0,"title":["Static Effective\nHamiltonians for Molecular Systems\nthrough RPA-Based Downfolding"],"prefix":"10.1021","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0009-0000-3866-9357","authenticated-orcid":true,"given":"Erik","family":"Verzijl","sequence":"first","affiliation":[{"name":"Vrije Universiteit , , De Boelelaan 1105 , ,","place":["Amsterdam, The Netherlands, 1081 HV"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0957-4081","authenticated-orcid":true,"given":"Arno","family":"Fo\u0308rster","sequence":"additional","affiliation":[{"name":"Vrije Universiteit , , De Boelelaan 1105 , ,","place":["Amsterdam, The Netherlands, 1081 HV"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"316","published-online":{"date-parts":[[2026,9,8]]},"reference":[{"key":"2026092209063927500_cit1","doi-asserted-by":"publisher","first-page":"1185","DOI":"10.1021\/acs.jctc.3c01190","article-title":"Distributed Implementation\nof Full Configuration Interaction for One Trillion Determinants","volume":"20","author":"Gao","year":"2024","journal-title":"J. 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