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data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p><b>Restricted open-shell Hartree–Fock</b> (<b>ROHF</b>) is a variant of <a href="/wiki/Hartree%E2%80%93Fock_method" title="Hartree–Fock method">Hartree–Fock method</a> for <a href="/wiki/Open_shell" class="mw-redirect" title="Open shell">open shell</a> molecules. It uses doubly occupied <a href="/wiki/Molecular_orbitals" class="mw-redirect" title="Molecular orbitals">molecular orbitals</a> as far as possible and then singly occupied orbitals for the unpaired electrons. This is the simple picture for open shell molecules but it is difficult to implement. The foundations of the ROHF method were first formulated by <a href="/wiki/Clemens_C._J._Roothaan" title="Clemens C. J. Roothaan">Clemens C. J. Roothaan</a> in a celebrated paper <sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> and then extended by various authors, see e.g.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> for in-depth discussions. </p><p>As with restricted Hartree–Fock theory for closed shell molecules, it leads to <a href="/wiki/Roothaan_equations" title="Roothaan equations">Roothaan equations</a> written in the form of a <a href="/wiki/Generalized_eigenvalue_problem" class="mw-redirect" title="Generalized eigenvalue problem">generalized eigenvalue problem</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} \mathbf {C} =\mathbf {S} \mathbf {C} \mathbf {\epsilon } }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ϵ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} \mathbf {C} =\mathbf {S} \mathbf {C} \mathbf {\epsilon } }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/551f223d044c3d876cb303f19cbef3a4998280c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.073ex; height:2.176ex;" alt="{\displaystyle \mathbf {F} \mathbf {C} =\mathbf {S} \mathbf {C} \mathbf {\epsilon } }"></span></dd></dl> <p>Where F is the so-called <a href="/wiki/Fock_matrix" title="Fock matrix">Fock matrix</a> (which is a function of C), C is a matrix of coefficients, S is the <a href="/wiki/Overlap_matrix" class="mw-redirect" title="Overlap matrix">overlap matrix</a> of the basis functions, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3837cad72483d97bcdde49c85d3b7b859fb3fd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.944ex; height:1.676ex;" alt="{\displaystyle \epsilon }"></span> is the (diagonal, by convention) matrix of orbital energies. Unlike restricted Hartree–Fock theory for closed shell molecules, the form of the Fock matrix is not unique. Different so-called canonicalisations can be used leading to different orbitals and different orbital energies, but the same total wavefunction, total energy, and other observables. </p><p>In contrast to <a href="/wiki/Unrestricted_Hartree%E2%80%93Fock" title="Unrestricted Hartree–Fock">unrestricted Hartree–Fock</a> (UHF), the ROHF wave function is a satisfactory eigenfunction of the total spin operator - <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {S} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {S} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ab6412080b7fa158001cf0446d05934aafddcf3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.54ex; height:2.676ex;" alt="{\displaystyle \mathbf {S} ^{2}}"></span> (i.e. no <a href="/wiki/Spin_contamination" title="Spin contamination">Spin contamination</a>). </p><p>Developing <a href="/wiki/Post-Hartree%E2%80%93Fock" class="mw-redirect" title="Post-Hartree–Fock">post-Hartree–Fock</a> methods based on a ROHF wave function is inherently more difficult than using a UHF wave function, due to the lack of a unique set of molecular orbitals.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> However, different choices of reference orbitals have shown to provide similar results,<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> and thus many different <a href="/wiki/Post-Hartree%E2%80%93Fock" class="mw-redirect" title="Post-Hartree–Fock">post-Hartree–Fock</a> methods have been implemented in a variety of electronic structure packages. Many (but not all) of these post-Hartree–Fock methods are completely <a href="/wiki/Invariant_(mathematics)" title="Invariant (mathematics)">invariant</a> with respect to orbital choice (assuming that no orbitals are "frozen" and thus not correlated).<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> The ZAPT2 version of <a href="/wiki/M%C3%B8ller%E2%80%93Plesset_perturbation_theory" title="Møller–Plesset perturbation theory">Møller–Plesset perturbation theory</a> specifies the choice of orbitals.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Restricted_open-shell_Hartree%E2%80%93Fock&action=edit&section=1" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFRoothaan1960" class="citation journal cs1">Roothaan, C. C. J. (1960). "Self-consistent field theory for open shells of electronic systems". <i>Reviews of Modern Physics</i>. <b>32</b> (2): 179–185. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1960RvMP...32..179R">1960RvMP...32..179R</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FRevModPhys.32.179">10.1103/RevModPhys.32.179</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Reviews+of+Modern+Physics&rft.atitle=Self-consistent+field+theory+for+open+shells+of+electronic+systems&rft.volume=32&rft.issue=2&rft.pages=179-185&rft.date=1960&rft_id=info%3Adoi%2F10.1103%2FRevModPhys.32.179&rft_id=info%3Abibcode%2F1960RvMP...32..179R&rft.aulast=Roothaan&rft.aufirst=C.+C.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARestricted+open-shell+Hartree%E2%80%93Fock" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCarbóRiera1978" class="citation book cs1">Carbó, R.; Riera, J. 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(1992). <i>Methods of Molecular Quantum Mechanics</i> (2nd ed.). Academic Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-470-01187-4" title="Special:BookSources/978-0-470-01187-4"><bdi>978-0-470-01187-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Methods+of+Molecular+Quantum+Mechanics&rft.edition=2nd&rft.pub=Academic+Press&rft.date=1992&rft.isbn=978-0-470-01187-4&rft.aulast=McWeeny&rft.aufirst=R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARestricted+open-shell+Hartree%E2%80%93Fock" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPlakhutin2002" class="citation book cs1">Plakhutin, B. N. (2002). Sen, K. D. 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"On the Ordering of Orbital Energies in High-Spin ROHF†". <i>The Journal of Physical Chemistry A</i>. <b>114</b> (33): 8772–8777. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2010JPCA..114.8772G">2010JPCA..114.8772G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1021%2Fjp101758y">10.1021/jp101758y</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/20443582">20443582</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Journal+of+Physical+Chemistry+A&rft.atitle=On+the+Ordering+of+Orbital+Energies+in+High-Spin+ROHF%E2%80%A0&rft.volume=114&rft.issue=33&rft.pages=8772-8777&rft.date=2010&rft_id=info%3Apmid%2F20443582&rft_id=info%3Adoi%2F10.1021%2Fjp101758y&rft_id=info%3Abibcode%2F2010JPCA..114.8772G&rft.aulast=Glaesemann&rft.aufirst=Kurt+R.&rft.au=Schmidt%2C+Michael+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARestricted+open-shell+Hartree%E2%80%93Fock" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJensen2007" class="citation book cs1">Jensen, F. (2007). <i>Introduction to Computational Chemistry</i> (2nd ed.). Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-98425-2" title="Special:BookSources/978-0-471-98425-2"><bdi>978-0-471-98425-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Computational+Chemistry&rft.edition=2nd&rft.pub=Wiley&rft.date=2007&rft.isbn=978-0-471-98425-2&rft.aulast=Jensen&rft.aufirst=F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARestricted+open-shell+Hartree%E2%80%93Fock" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCrawfordSchaeferLee1996" class="citation journal cs1">Crawford, T. Daniel; Schaefer, Henry F.; Lee, Timothy J. (1996). <a rel="nofollow" class="external text" href="https://zenodo.org/record/1232940">"On the energy invariance of open-shell perturbation theory with respect to unitary transformations of molecular orbitals"</a>. <i>The Journal of Chemical Physics</i>. <b>105</b> (3): 1060. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1996JChPh.105.1060C">1996JChPh.105.1060C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.471951">10.1063/1.471951</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Journal+of+Chemical+Physics&rft.atitle=On+the+energy+invariance+of+open-shell+perturbation+theory+with+respect+to+unitary+transformations+of+molecular+orbitals&rft.volume=105&rft.issue=3&rft.pages=1060&rft.date=1996&rft_id=info%3Adoi%2F10.1063%2F1.471951&rft_id=info%3Abibcode%2F1996JChPh.105.1060C&rft.aulast=Crawford&rft.aufirst=T.+Daniel&rft.au=Schaefer%2C+Henry+F.&rft.au=Lee%2C+Timothy+J.&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F1232940&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARestricted+open-shell+Hartree%E2%80%93Fock" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWheelerAllenSchaefer_Hf2008" class="citation journal cs1">Wheeler, S. E; Allen, W. D; Schaefer Hf, 3rd (2008). 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