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The formalism is based on dividing the full orbital space into orbitals localized to the molecule and orbitals localized to the environment. This enables a separation of the full electronic Hamiltonian into terms referencing only molecule, environment, or interaction terms. The interaction terms are divided into particle-conserving interactions and particle-non-conserving (particle-breaking) interactions. The particle-conserving interactions are dominant and may be included using standard embedding schemes. The particle-breaking terms are responsible for inducing fractional charging, and we show that the local orbital space approach provides a convenient framework for different types of perturbative treatments. In the local orbital basis, we generate a basis of many-electron states for the composite system, in which a specific molecular charge may label each state. This basis is used to construct a projection operator acting on the Liouville\u2013von Neumann equation for the composite system to yield an equation for the reduced density matrix for the molecule. The diagonal elements of the reduced density matrix represent populations of different molecular charge states and determine the fractional charging. The projected Liouville\u2013von Neumann equation is the starting point for two perturbative treatments: damped response theory and Redfield theory. The damped response framework introduces energy broadening of electronic states. Phenomenological broadening is also introduced into the Redfield equation. We illustrate the presented formalism by considering benzene physisorbed on a finite graphene sheet as a toy model.<\/jats:p>","DOI":"10.1063\/5.0251855","type":"journal-article","created":{"date-parts":[[2025,3,7]],"date-time":"2025-03-07T05:23:28Z","timestamp":1741325008000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":5,"title":["Fractional charging of electronically open molecules: An explicit projection operator approach"],"prefix":"10.1063","volume":"162","author":[{"ORCID":"https:\/\/orcid.org\/0009-0000-7797-336X","authenticated-orcid":false,"given":"Bendik St\u00f8a","family":"Sannes","sequence":"first","affiliation":[{"name":"Department of Chemistry, Norwegian University of Science and Technology 1 , NO-7491 Trondheim,","place":["Norway"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8571-2872","authenticated-orcid":false,"given":"Jacob","family":"Pedersen","sequence":"additional","affiliation":[{"name":"Department of Chemistry, Norwegian University of Science and Technology 1 , NO-7491 Trondheim,","place":["Norway"]},{"name":"Department of Chemistry, Technical University of Denmark 2 , DK-2800 Kongens Lyngby,","place":["Denmark"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1239-7776","authenticated-orcid":false,"given":"Ida-Marie","family":"H\u00f8yvik","sequence":"additional","affiliation":[{"name":"Department of Chemistry, Norwegian University of Science and Technology 1 , NO-7491 Trondheim,","place":["Norway"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"317","published-online":{"date-parts":[[2025,3,7]]},"reference":[{"key":"2025030710232355900_c1","doi-asserted-by":"publisher","first-page":"204120","DOI":"10.1063\/1.4767981","article-title":"Calculation of electronic excitations using wave-function in wave-function frozen-density embedding","volume":"137","year":"2012","journal-title":"J.\u00a0Chem. 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