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class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Aproximaci%C3%B3_de_Born-Oppenheimer" title="Aproximació de Born-Oppenheimer – Catalaans" lang="ca" hreflang="ca" data-title="Aproximació de Born-Oppenheimer" data-language-autonym="Català" data-language-local-name="Catalaans" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Bornova%E2%80%93Oppenheimerova_aproximace" title="Bornova–Oppenheimerova aproximace – Tsjechisch" lang="cs" hreflang="cs" data-title="Bornova–Oppenheimerova aproximace" data-language-autonym="Čeština" data-language-local-name="Tsjechisch" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Born-Oppenheimer-approksimationen" title="Born-Oppenheimer-approksimationen – Deens" lang="da" hreflang="da" data-title="Born-Oppenheimer-approksimationen" data-language-autonym="Dansk" data-language-local-name="Deens" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Born-Oppenheimer-N%C3%A4herung" title="Born-Oppenheimer-Näherung – Duits" lang="de" hreflang="de" data-title="Born-Oppenheimer-Näherung" data-language-autonym="Deutsch" data-language-local-name="Duits" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Born%E2%80%93Oppenheimer_approximation" title="Born–Oppenheimer approximation – Engels" lang="en" hreflang="en" data-title="Born–Oppenheimer approximation" data-language-autonym="English" data-language-local-name="Engels" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Aproximaci%C3%B3n_de_Born-Oppenheimer" title="Aproximación de Born-Oppenheimer – Spaans" lang="es" hreflang="es" data-title="Aproximación de Born-Oppenheimer" data-language-autonym="Español" data-language-local-name="Spaans" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Born-Oppenheimerren_hurbilketa" title="Born-Oppenheimerren hurbilketa – Baskisch" lang="eu" hreflang="eu" data-title="Born-Oppenheimerren hurbilketa" data-language-autonym="Euskara" data-language-local-name="Baskisch" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D9%82%D8%B1%DB%8C%D8%A8_%D8%A8%D9%88%D8%B1%D9%86%E2%80%93%D8%A7%D9%88%D9%BE%D9%86%D9%87%D8%A7%DB%8C%D9%85%D8%B1" title="تقریب بورن–اوپنهایمر – Perzisch" lang="fa" hreflang="fa" data-title="تقریب بورن–اوپنهایمر" data-language-autonym="فارسی" data-language-local-name="Perzisch" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Approximation_de_Born-Oppenheimer" title="Approximation de Born-Oppenheimer – Frans" lang="fr" hreflang="fr" data-title="Approximation de Born-Oppenheimer" data-language-autonym="Français" data-language-local-name="Frans" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A7%D7%99%D7%A8%D7%95%D7%91_%D7%91%D7%95%D7%A8%D7%9F-%D7%90%D7%95%D7%A4%D7%A0%D7%94%D7%99%D7%99%D7%9E%D7%A8" title="קירוב בורן-אופנהיימר – Hebreeuws" lang="he" hreflang="he" data-title="קירוב בורן-אופנהיימר" data-language-autonym="עברית" data-language-local-name="Hebreeuws" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Born%E2%80%93Oppenheimer-k%C3%B6zel%C3%ADt%C3%A9s" title="Born–Oppenheimer-közelítés – Hongaars" lang="hu" hreflang="hu" data-title="Born–Oppenheimer-közelítés" data-language-autonym="Magyar" data-language-local-name="Hongaars" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Hampiran_Born%E2%80%93Oppenheimer" title="Hampiran Born–Oppenheimer – Indonesisch" lang="id" hreflang="id" data-title="Hampiran Born–Oppenheimer" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesisch" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Approssimazione_di_Born-Oppenheimer" title="Approssimazione di Born-Oppenheimer – Italiaans" lang="it" hreflang="it" data-title="Approssimazione di Born-Oppenheimer" data-language-autonym="Italiano" data-language-local-name="Italiaans" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%9C%E3%83%AB%E3%83%B3%E2%80%93%E3%82%AA%E3%83%83%E3%83%9A%E3%83%B3%E3%83%8F%E3%82%A4%E3%83%9E%E3%83%BC%E8%BF%91%E4%BC%BC" title="ボルン–オッペンハイマー近似 – Japans" lang="ja" hreflang="ja" data-title="ボルン–オッペンハイマー近似" data-language-autonym="日本語" data-language-local-name="Japans" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B3%B4%EB%A5%B8-%EC%98%A4%ED%8E%9C%ED%95%98%EC%9D%B4%EB%A8%B8_%EA%B7%BC%EC%82%AC" title="보른-오펜하이머 근사 – Koreaans" lang="ko" hreflang="ko" data-title="보른-오펜하이머 근사" data-language-autonym="한국어" data-language-local-name="Koreaans" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Przybli%C5%BCenie_Borna-Oppenheimera" title="Przybliżenie Borna-Oppenheimera – Pools" lang="pl" hreflang="pl" data-title="Przybliżenie Borna-Oppenheimera" data-language-autonym="Polski" data-language-local-name="Pools" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Aproxima%C3%A7%C3%A3o_de_Born-Oppenheimer" title="Aproximação de Born-Oppenheimer – Portugees" lang="pt" hreflang="pt" data-title="Aproximação de Born-Oppenheimer" data-language-autonym="Português" data-language-local-name="Portugees" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D1%80%D0%B8%D0%B1%D0%BB%D0%B8%D0%B6%D0%B5%D0%BD%D0%B8%D0%B5_%D0%91%D0%BE%D1%80%D0%BD%D0%B0_%E2%80%94_%D0%9E%D0%BF%D0%BF%D0%B5%D0%BD%D0%B3%D0%B5%D0%B9%D0%BC%D0%B5%D1%80%D0%B0" title="Приближение Борна — Оппенгеймера – Russisch" lang="ru" hreflang="ru" data-title="Приближение Борна — Оппенгеймера" data-language-autonym="Русский" data-language-local-name="Russisch" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Born%E2%80%93Oppenheimer-approximationen" title="Born–Oppenheimer-approximationen – Zweeds" lang="sv" hreflang="sv" data-title="Born–Oppenheimer-approximationen" data-language-autonym="Svenska" data-language-local-name="Zweeds" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Born-Oppenheimer_yakla%C5%9F%C4%B1m%C4%B1" 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</div> <div id="siteSub" class="noprint">Uit Wikipedia, de vrije encyclopedie</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="nl" dir="ltr"><p>De <b>Born-Oppenheimerbenadering</b>, ook wel verkort tot de <b>Born-Oppenheimer</b>, is een vereenvoudigingsmethode in de <a href="/wiki/Molecuulfysica" title="Molecuulfysica">molecuulfysica</a> en de <a href="/wiki/Kwantumchemie" title="Kwantumchemie">kwantumchemie</a>, waarmee de berekening van de <a href="/wiki/Energie" title="Energie">energie</a> en <a href="/wiki/Golffunctie" title="Golffunctie">golffunctie</a> van grotere systemen, zoals <a href="/wiki/Molecuul" title="Molecuul">moleculen</a>, vergemakkelijkt wordt. </p> <div class="mw-heading mw-heading2"><h2 id="Probleem">Probleem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Born-Oppenheimerbenadering&veaction=edit&section=1" title="Bewerk dit kopje: Probleem" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Born-Oppenheimerbenadering&action=edit&section=1" title="De broncode bewerken van de sectie: Probleem"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bij moleculen kan het aantal <a href="/wiki/Vrijheidsgraad" title="Vrijheidsgraad">vrijheidsgraden</a>, manieren waarop (delen van) het molecuul zich kan (kunnen) bewegen, snel oplopen. In het geval van water is bijvoorbeeld sprake van drie <a href="/wiki/Atoomkern" title="Atoomkern">kernen</a> en tien <a href="/wiki/Elektron" title="Elektron">elektronen</a>. Deze hebben, in de tijdsonafhankelijke <a href="/wiki/Schr%C3%B6dingervergelijking" title="Schrödingervergelijking">schrödingervergelijking</a>, allemaal drie vrijheidsgraden (x, y en z), wat neerkomt op 39 variabelen, aangezien deze allemaal onafhankelijk van, maar beïnvloed door elkaar kunnen bewegen. Met zoveel vrijheidsgraden wordt het al snel onmogelijk om de schrödingervergelijking op te lossen, zelfs als men numerieke benaderingen zoekt. Alleen aan het allerkleinste atoom, het <a href="/wiki/Waterstofatoom" title="Waterstofatoom">waterstofatoom</a>, kunnen berekeningen gedaan worden zonder enige benadering te doen en dus met exacte uitkomst. Maar zelfs numerieke benaderingen zijn vrijwel ondoenlijk voor moleculen die gecompliceerder zijn dan het kleinst mogelijke molecuul, het waterstofmolecuul <a href="/wiki/Diwaterstof" title="Diwaterstof">H<sub>2</sub></a>. </p> <div class="mw-heading mw-heading2"><h2 id="Benadering">Benadering</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Born-Oppenheimerbenadering&veaction=edit&section=2" title="Bewerk dit kopje: Benadering" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Born-Oppenheimerbenadering&action=edit&section=2" title="De broncode bewerken van de sectie: Benadering"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/1927" title="1927">1927</a> stelden <a href="/wiki/Max_Born" title="Max Born">Max Born</a> en <a href="/wiki/Robert_Oppenheimer" title="Robert Oppenheimer">Robert Oppenheimer</a> een benadering voor om toch berekeningen te kunnen doen aan grotere systemen: de Born-Oppenheimerbenadering. Hierbij wordt er een onderscheid gemaakt in de coördinaten en de golffunctie van de kernen en van de elektronen. Hierbij is de redenering dat de atoomkernen vele <a href="/wiki/Orde_van_grootte" title="Orde van grootte">malen</a> zwaarder zijn dan de elektronen. Hierdoor is het aannemelijk dat de atoomkernen, vergeleken met de elektronen, praktisch stilstaan (dikwijls wordt hiervoor de <a href="/wiki/Evenwicht" title="Evenwicht">evenwichtstoestand</a> genomen). De Born-Oppenheimerbenadering gaat er dan ook van uit dat binnen een redelijk kort tijdsbestek de kernen stilstaan ten opzichte van de elektronen. Dit komt er uiteindelijk op neer dat bij de Born-Oppenheimer ervan wordt uitgegaan dat de golffunctie te splitsen valt: </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 \Psi _{\text{total}}=\psi _{\text{electronic}}\times \psi _{\text{nuclear}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>total</mtext> </mrow> </msub> <mo>=</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>electronic</mtext> </mrow> </msub> <mo>×</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>nuclear</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi _{\text{total}}=\psi _{\text{electronic}}\times \psi _{\text{nuclear}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b7584ad17431bf6c84e2cc3d53699560b14d572" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.918ex; height:2.509ex;" alt="{\displaystyle \Psi _{\text{total}}=\psi _{\text{electronic}}\times \psi _{\text{nuclear}}}"></span></dd></dl> <p>In de praktijk benadert men dan de kernen als stilstaande puntladingen, die een elektrostatische potentiaal genereren waarin zich de elektronen bevinden. Op deze manier kan men dan het simpelste molecuul, het eenwaardig positief geladen waterstofion H<sub>2</sub><sup>+</sup>, "exact" oplossen. </p><p>In de praktijk zal voor berekeningen nog verdere benadering noodzakelijk zijn, bijvoorbeeld de <a href="/wiki/Hartree-Fock-methode" title="Hartree-Fock-methode">Hartree-Fock-methode</a> en <a href="/wiki/Post-Hartree-Fock" title="Post-Hartree-Fock">post-Hartree-Fockmethoden</a>, of met <a href="/wiki/Dichtheidsfunctionaaltheorie" title="Dichtheidsfunctionaaltheorie">dichtheidsfunctionaaltheorie</a>. </p>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Overgenomen van "<a dir="ltr" href="https://nl.wikipedia.org/w/index.php?title=Born-Oppenheimerbenadering&oldid=55572850">https://nl.wikipedia.org/w/index.php?title=Born-Oppenheimerbenadering&oldid=55572850</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Categorie:Alles" title="Categorie:Alles">Categorieën</a>: <ul><li><a href="/wiki/Categorie:Kwantummechanica" title="Categorie:Kwantummechanica">Kwantummechanica</a></li><li><a href="/wiki/Categorie:Computationele_chemie" title="Categorie:Computationele chemie">Computationele chemie</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Deze pagina is voor het laatst bewerkt op 31 jan 2020 om 15:11.</li> <li id="footer-info-copyright">De tekst is beschikbaar onder de licentie <a rel="nofollow" class="external text" href="//creativecommons.org/licenses/by-sa/4.0/deed.nl">Creative Commons Naamsvermelding/Gelijk delen</a>, er kunnen aanvullende voorwaarden van toepassing zijn. 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