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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">LCAO-Methode</span></h1>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Die <b>LCAO-Methode</b> (von engl. <i><span lang="en">linear combination of atomic orbitals</span></i> ‚lineare Kombination von Atomorbitalen‘) ist eine <a href="Quantensuperposition" class="mw-redirect" title="Quantensuperposition">Quantensuperposition</a> von <a href="Atomorbital" title="Atomorbital">Atomorbitalen</a> und eine Methode zur Berechnung von <a href="Molek%C3%BClorbital" class="mw-redirect" title="Molekülorbital">Molekülorbitalen</a> in der <a href="Quantenchemie" title="Quantenchemie">Quantenchemie</a>.<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> In der <a href="Quantenmechanik" title="Quantenmechanik">Quantenmechanik</a> werden <a href="Elektronenkonfiguration" title="Elektronenkonfiguration">Elektronenkonfigurationen</a> von <a href="Atom" title="Atom">Atomen</a> als <a href="Wellenfunktion" title="Wellenfunktion">Wellenfunktion</a> beschrieben, in Bezug auf <a href="Wasserstoff" title="Wasserstoff">Wasserstoff</a> in der <a href="Schr%C3%B6dinger-Gleichung" class="mw-redirect" title="Schrödinger-Gleichung">Schrödinger-Gleichung</a>. In einer <a href="Chemische_Reaktion" title="Chemische Reaktion">chemischen Reaktion</a> werden die Orbitalwellenfunktionen modifiziert, d. h. die <a href="Elektronenwolke" class="mw-redirect" title="Elektronenwolke">Elektronenwolke</a> wird je nach den an einer <a href="Chemische_Bindung" title="Chemische Bindung">chemischen Bindung</a> teilnehmenden Atomen verändert.
</p><p>Die LCAO-Methode wurde 1929 durch <a href="John_Lennard-Jones" title="John Lennard-Jones">John Lennard-Jones</a> mit der Beschreibung der Bindung diatomischer <a href="Molek%C3%BCl" title="Molekül">Moleküle</a> der zweiten <a href="Periodensystem" title="Periodensystem">Periode</a> veröffentlicht, wurde aber zuvor bereits durch <a href="Linus_Pauling" title="Linus Pauling">Linus Pauling</a> für <a href="Diwasserstoff-Kation" title="Diwasserstoff-Kation">H<sub>2</sub><sup>+</sup></a> verwendet.<sup id="cite_ref-kutzelnigg_2-0" class="reference"><a href="#cite_note-kutzelnigg-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>
</p>
<div class="mw-heading mw-heading2"><h2 id="Prinzip">Prinzip</h2></div>
<p>Eine Grundannahme ist, dass die Anzahl der Molekülorbitale der Anzahl der Atomorbitale in der linearen Expansion gleicht. Die n Atomorbitale werden zu n Molekülorbitalen kombiniert, die mit einem Index von 1 bis n nummeriert werden und die nicht alle gleich sein müssen. Der Ausdruck der linearen Expansion für das <i>i</i>-te Molekülorbital ist:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \phi _{i}=c_{1i}\chi _{1}+c_{2i}\chi _{2}+c_{3i}\chi _{3}+\cdots +c_{ni}\chi _{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mtext> </mtext>
<msub>
<mi>ϕ<!-- ϕ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mi>i</mi>
</mrow>
</msub>
<msub>
<mi>χ<!-- χ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mi>i</mi>
</mrow>
</msub>
<msub>
<mi>χ<!-- χ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
<mi>i</mi>
</mrow>
</msub>
<msub>
<mi>χ<!-- χ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<mo>+</mo>
<mo>⋯<!-- ⋯ --></mo>
<mo>+</mo>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mi>i</mi>
</mrow>
</msub>
<msub>
<mi>χ<!-- χ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \ \phi _{i}=c_{1i}\chi _{1}+c_{2i}\chi _{2}+c_{3i}\chi _{3}+\cdots +c_{ni}\chi _{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1b79d8c86067fa2567d40b3b365686ef0518ab4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:40.829ex; height:2.509ex;" alt="{\displaystyle \ \phi _{i}=c_{1i}\chi _{1}+c_{2i}\chi _{2}+c_{3i}\chi _{3}+\cdots +c_{ni}\chi _{n}}" loading="lazy"></span></dd></dl>
<p>oder
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \phi _{i}=\sum _{r}c_{ri}\chi _{r}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mtext> </mtext>
<msub>
<mi>ϕ<!-- ϕ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>=</mo>
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>r</mi>
</mrow>
</munder>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>r</mi>
<mi>i</mi>
</mrow>
</msub>
<msub>
<mi>χ<!-- χ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>r</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \ \phi _{i}=\sum _{r}c_{ri}\chi _{r}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f2c26737077e0cb57d0869bae2e82de331e44faa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.583ex; height:5.509ex;" alt="{\displaystyle \ \phi _{i}=\sum _{r}c_{ri}\chi _{r}}" loading="lazy"></span></dd></dl>
<p>bei dem <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \phi _{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mtext> </mtext>
<msub>
<mi>ϕ<!-- ϕ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \ \phi _{i}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8f5800e3f63827cdf9c631c01615213ec6239753.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.766ex; height:2.509ex;" alt="{\displaystyle \ \phi _{i}}" loading="lazy"></span> (<a href="Phi" title="Phi">phi</a>) ein <a href="Molek%C3%BClorbital" class="mw-redirect" title="Molekülorbital">Molekülorbital</a> ist, repräsentiert als <a href="Summe" title="Summe">Summe</a> von n <a href="Atomorbital" title="Atomorbital">Atomorbitalen</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \chi _{r}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mtext> </mtext>
<msub>
<mi>χ<!-- χ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>r</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \ \chi _{r}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d2f63594c9666e033a8f625f238c6869b6d467a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.01ex; height:2.009ex;" alt="{\displaystyle \ \chi _{r}}" loading="lazy"></span> (<a href="Chi" title="Chi">chi</a>), jedes multipliziert mit einem korrespondierenden Koeffizienten <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ c_{ri}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mtext> </mtext>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>r</mi>
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \ c_{ri}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9934d4b61dd13be589622987713a2c84144776b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.129ex; height:2.009ex;" alt="{\displaystyle \ c_{ri}}" loading="lazy"></span>, und <i>r</i> (nummeriert von 1 bis n) das im Term kombinierte Atomorbital darstellt. Die Koeffizienten sind die Anteile des Beitrags der Atomorbitale am Molekülorbital. Die <a href="Hartree-Fock-Methode" title="Hartree-Fock-Methode">Hartree-Fock-Methode</a> wird verwendet, um die Koeffizienten der Expansion zu ermitteln.
</p><p>Die Orbitale werden als lineare Kombinationen der <a href="Basisfunktion" class="mw-redirect" title="Basisfunktion">Basisfunktionen</a> ausgedrückt, welche als Ein-<a href="Elektron" title="Elektron">Elektronen</a>-Funktionen auf den <a href="Atomkern" title="Atomkern">Atomkernen</a> der beteiligten Atome im Molekül zentriert sind. Die verwendeten Atomorbitale sind normalerweise vom <a href="Wasserstoff" title="Wasserstoff">Wasserstoff</a> (z. B. <a href="Slater_Type_Orbitals" title="Slater Type Orbitals">Slater Type Orbitals</a>), da diese analytisch bekannt sind, jedoch können auch andere ausgewählt werden, wie z. B. die Gauss-Orbitale im Standardbasissatz.
</p><p>Durch die Minimierung der Gesamtenergie des Systems wird ein passender Satz an Koeffizienten der linearen Kombination bestimmt. Dieser quantitative Ansatz ist heute als <a href="Hartree-Fock-Methode" title="Hartree-Fock-Methode">Hartree-Fock-Methode</a> bekannt. Seit der Einführung der <a href="Molekulare_Modellierung" title="Molekulare Modellierung">molekularen Modellierung</a> wird die LCAO-Methode jedoch weniger zur Optimierung einer Wellenfunktion als vielmehr zu einer qualitativen Bewertung herangezogen, welche bei der Vorhersage und bei der Erklärung der mit modernen Methoden erhaltenen Ergebnisse hilfreich ist. In diesem Fall werden die Form der Molekülorbitale und ihre jeweiligen <a href="Energie" title="Energie">Energien</a> näherungsweise aus dem Vergleich der Energien der einzelnen Atomorbitale (oder Molekülfragmente) abgeleitet und die Level repulsion oder ähnliches angewendet. Die zur Klarstellung erzeugten <a href="Funktionsgraph" title="Funktionsgraph">Graphen</a> werden als <i>Korrelationsdiagramme</i> (engl. <i><span lang="en">correlation diagrams</span></i>) bezeichnet. Die benötigten Energien der Atomorbitale stammen aus Berechnungen oder können über das <a href="Koopmans-Theorem" title="Koopmans-Theorem">Koopmans-Theorem</a> experimentell bestimmt werden.
</p><p>Der erste Schritt besteht aus der Zuweisung einer <a href="Punktgruppe" title="Punktgruppe">Punktgruppe</a> zu dem Molekül. Ein häufiges Beispiel ist <a href="Wasser" title="Wasser">Wasser</a>, welches eine C<sub>2v</sub> Symmetrie aufweist. Im Folgenden wird die <a href="Darstellungstheorie" title="Darstellungstheorie">reduzible Darstellung</a> der Bindung im Wasser aufgeführt:
</p><p><span typeof="mw:File"></span>
</p><p>Jede Operation in der Punktgruppe wird in Bezug auf das Molekül durchgeführt. Die Anzahl an unveränderten Bindungen ist der Charakter einer Operation. Diese reduzible Darstellung wird in die Summe der irreduziblen Darstellungen zerlegt. Die irreduziblen Darstellungen korrespondieren mit der Symmetrie der beteiligten Orbitale.
</p><p><a href="MO-Theorie" class="mw-redirect" title="MO-Theorie">MO-Diagramme</a> bieten eine einfache qualitative Behandlung der LCAO-Näherung.
</p><p><span typeof="mw:File"></span>
</p><p>Quantitative Theorien sind die <i><a href="H%C3%BCckel-N%C3%A4herung" title="Hückel-Näherung">Hückel-Näherung</a></i>, die <i>erweiterte Hückel-Methode</i> und die <i>Pariser-Parr-Pople-Methode</i>.
</p><p>Betrachtet man ein System mit mehreren Elementen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span> (z. B. Atome), zentriert auf <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {r}}_{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>r</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {r}}_{i}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/de93eb4c8bca39012a94e9809c45d7fd677bf975.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.023ex; height:2.676ex;" alt="{\displaystyle {\vec {r}}_{i}}" loading="lazy"></span>, so stellt man fest, dass <span class="mwe-math-element mwe-math-element-inline"><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 _{i}({\vec {r}}-{\vec {r}}_{i})\rangle }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<msub>
<mi mathvariant="normal">Ψ<!-- Ψ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>r</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>−<!-- − --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>r</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |\Psi _{i}({\vec {r}}-{\vec {r}}_{i})\rangle }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6bb73f3d0f71c4a3321f9f7a0972bd69c9e83759.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.055ex; height:2.843ex;" alt="{\displaystyle |\Psi _{i}({\vec {r}}-{\vec {r}}_{i})\rangle }" loading="lazy"></span>
die Wellenfunktion des Elektrons beschreibt, wenn das Element <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span> isoliert ist.
Die Wellenfunktion <span class="mwe-math-element mwe-math-element-inline"><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 ({\vec {r}})\rangle }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi mathvariant="normal">Ψ<!-- Ψ --></mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>r</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo stretchy="false">)</mo>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |\Psi ({\vec {r}})\rangle }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1e201aefcd0f935d351a8ee5785d5e0a63498d09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.392ex; height:2.843ex;" alt="{\displaystyle |\Psi ({\vec {r}})\rangle }" loading="lazy"></span>, die das Elektron im gesamten System beschreibt, kann durch eine lineare Kombination von Wellenfunktionen <span class="mwe-math-element mwe-math-element-inline"><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 _{i}({\vec {r}}-{\vec {r}}_{i})\rangle }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<msub>
<mi mathvariant="normal">Ψ<!-- Ψ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>r</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>−<!-- − --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>r</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |\Psi _{i}({\vec {r}}-{\vec {r}}_{i})\rangle }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6bb73f3d0f71c4a3321f9f7a0972bd69c9e83759.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.055ex; height:2.843ex;" alt="{\displaystyle |\Psi _{i}({\vec {r}}-{\vec {r}}_{i})\rangle }" loading="lazy"></span> genähert werden:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><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 ({\vec {r}})\rangle \simeq \sum _{i}\alpha _{i}|\Psi _{i}({\vec {r}}-{\vec {r}}_{i})\rangle }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi mathvariant="normal">Ψ<!-- Ψ --></mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>r</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo stretchy="false">)</mo>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
<mo>≃<!-- ≃ --></mo>
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</munder>
<msub>
<mi>α<!-- α --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<msub>
<mi mathvariant="normal">Ψ<!-- Ψ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>r</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>−<!-- − --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>r</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |\Psi ({\vec {r}})\rangle \simeq \sum _{i}\alpha _{i}|\Psi _{i}({\vec {r}}-{\vec {r}}_{i})\rangle }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ba3e08bdd35c23e9e9b29f3af75e469b2307ece.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.575ex; height:5.509ex;" alt="{\displaystyle |\Psi ({\vec {r}})\rangle \simeq \sum _{i}\alpha _{i}|\Psi _{i}({\vec {r}}-{\vec {r}}_{i})\rangle }" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Herleitung">Herleitung</h2></div>
<p>Die Wellenfunktion <span class="mwe-math-element mwe-math-element-inline"><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 _{i}({\vec {r}}-{\vec {r}}_{i})\rangle }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<msub>
<mi mathvariant="normal">Ψ<!-- Ψ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>r</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>−<!-- − --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>r</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |\Psi _{i}({\vec {r}}-{\vec {r}}_{i})\rangle }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6bb73f3d0f71c4a3321f9f7a0972bd69c9e83759.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.055ex; height:2.843ex;" alt="{\displaystyle |\Psi _{i}({\vec {r}}-{\vec {r}}_{i})\rangle }" loading="lazy"></span> beschreibt ein Elektron, wenn das Element <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span> isoliert ist.
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{i}|\Psi _{i}({\vec {r}}-{\vec {r}}_{i})\rangle =\left({-\hbar ^{2} \over {2m_{e}}}\nabla ^{2}+V_{i}({\vec {r}}-{\vec {r}}_{i})\right)|\Psi _{i}({\vec {r}}-{\vec {r}}_{i})\rangle }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<msub>
<mi mathvariant="normal">Ψ<!-- Ψ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>r</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>−<!-- − --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>r</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mo>−<!-- − --></mo>
<msup>
<mi class="MJX-variant">ℏ<!-- ℏ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<msub>
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>e</mi>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
<msup>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>+</mo>
<msub>
<mi>V</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>r</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>−<!-- − --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>r</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<msub>
<mi mathvariant="normal">Ψ<!-- Ψ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>r</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>−<!-- − --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>r</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E_{i}|\Psi _{i}({\vec {r}}-{\vec {r}}_{i})\rangle =\left({-\hbar ^{2} \over {2m_{e}}}\nabla ^{2}+V_{i}({\vec {r}}-{\vec {r}}_{i})\right)|\Psi _{i}({\vec {r}}-{\vec {r}}_{i})\rangle }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47caeab6712c8eaecd8b9387d791c56e3179eb1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:54.45ex; height:6.343ex;" alt="{\displaystyle E_{i}|\Psi _{i}({\vec {r}}-{\vec {r}}_{i})\rangle =\left({-\hbar ^{2} \over {2m_{e}}}\nabla ^{2}+V_{i}({\vec {r}}-{\vec {r}}_{i})\right)|\Psi _{i}({\vec {r}}-{\vec {r}}_{i})\rangle }" loading="lazy"></span></dd></dl>
<p>Unter der Hypothese, dass die Größenordnung
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \Psi _{i}({\vec {r}}-{\vec {r}}_{i})\mid V_{j}({\vec {r}}-{\vec {r}}_{j})\mid \Psi _{i}({\vec {r}}-{\vec {r}}_{i})\rangle =\int _{\Omega }\Psi _{i}^{*}({\vec {r}}-{\vec {r}}_{i})V({\vec {r}}-{\vec {r}}_{j})\Psi _{i}({\vec {r}}-{\vec {r}}_{i})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<msub>
<mi mathvariant="normal">Ψ<!-- Ψ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>r</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>−<!-- − --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>r</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>∣<!-- ∣ --></mo>
<msub>
<mi>V</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>r</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>−<!-- − --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>r</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>∣<!-- ∣ --></mo>
<msub>
<mi mathvariant="normal">Ψ<!-- Ψ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>r</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>−<!-- − --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>r</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
<mo>=</mo>
<msub>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">Ω<!-- Ω --></mi>
</mrow>
</msub>
<msubsup>
<mi mathvariant="normal">Ψ<!-- Ψ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>∗<!-- ∗ --></mo>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>r</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>−<!-- − --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>r</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mi>V</mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>r</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>−<!-- − --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>r</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<msub>
<mi mathvariant="normal">Ψ<!-- Ψ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>r</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mo>−<!-- − --></mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>r</mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \langle \Psi _{i}({\vec {r}}-{\vec {r}}_{i})\mid V_{j}({\vec {r}}-{\vec {r}}_{j})\mid \Psi _{i}({\vec {r}}-{\vec {r}}_{i})\rangle =\int _{\Omega }\Psi _{i}^{*}({\vec {r}}-{\vec {r}}_{i})V({\vec {r}}-{\vec {r}}_{j})\Psi _{i}({\vec {r}}-{\vec {r}}_{i})}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a31ea4586f146638b5c491d4a11ed97cbf4323b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:74.212ex; height:5.676ex;" alt="{\displaystyle \langle \Psi _{i}({\vec {r}}-{\vec {r}}_{i})\mid V_{j}({\vec {r}}-{\vec {r}}_{j})\mid \Psi _{i}({\vec {r}}-{\vec {r}}_{i})\rangle =\int _{\Omega }\Psi _{i}^{*}({\vec {r}}-{\vec {r}}_{i})V({\vec {r}}-{\vec {r}}_{j})\Psi _{i}({\vec {r}}-{\vec {r}}_{i})}" loading="lazy"></span></dd></dl>
<p>nicht signifikativ ist außer für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j=i}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>j</mi>
<mo>=</mo>
<mi>i</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle j=i}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/64b78ea5c581ea5b6cdd76486a96f8af9e0d9e39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:4.886ex; height:2.509ex;" alt="{\displaystyle j=i}" loading="lazy"></span>, ist die potentielle Modifikation durch ein Element <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j\neq i}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>j</mi>
<mo>≠<!-- ≠ --></mo>
<mi>i</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle j\neq i}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/66b7538abe3f69348eb371e48d7b33a8994610e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.027ex; width:4.886ex; height:2.676ex;" alt="{\displaystyle j\neq i}" loading="lazy"></span> für die Wellenfunktion <span class="mwe-math-element mwe-math-element-inline"><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 _{i}(r-R_{i})\rangle }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<msub>
<mi mathvariant="normal">Ψ<!-- Ψ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>r</mi>
<mo>−<!-- − --></mo>
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |\Psi _{i}(r-R_{i})\rangle }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4c92a547cb481610e29c1153521a4de7e6c312e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.421ex; height:2.843ex;" alt="{\displaystyle |\Psi _{i}(r-R_{i})\rangle }" loading="lazy"></span> nicht so wichtig.
</p><p>Jede Lösung der Gleichung des Gesamtsystems
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E|\Psi \rangle =\left(-{\hbar ^{2} \over {2m_{e}}}{\nabla ^{2}}+\sum _{i}V_{i}(r-R_{i})\right)|\Psi \rangle }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>E</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi mathvariant="normal">Ψ<!-- Ψ --></mi>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi class="MJX-variant">ℏ<!-- ℏ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<msub>
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>e</mi>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<msup>
<mi mathvariant="normal">∇<!-- ∇ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</munder>
<msub>
<mi>V</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>r</mi>
<mo>−<!-- − --></mo>
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi mathvariant="normal">Ψ<!-- Ψ --></mi>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle E|\Psi \rangle =\left(-{\hbar ^{2} \over {2m_{e}}}{\nabla ^{2}}+\sum _{i}V_{i}(r-R_{i})\right)|\Psi \rangle }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0134974d5fa0ffae53880bcef65718a271ebb34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:42.497ex; height:7.509ex;" alt="{\displaystyle E|\Psi \rangle =\left(-{\hbar ^{2} \over {2m_{e}}}{\nabla ^{2}}+\sum _{i}V_{i}(r-R_{i})\right)|\Psi \rangle }" loading="lazy"></span></dd></dl>
<p>kann durch eine lineare Kombination der isolierten Wellenfunktionen genähert werden:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><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 \rangle =\sum _{i}\alpha _{i}|\Psi _{i}(r-R_{i})\rangle }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi mathvariant="normal">Ψ<!-- Ψ --></mi>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
<mo>=</mo>
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</munder>
<msub>
<mi>α<!-- α --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<msub>
<mi mathvariant="normal">Ψ<!-- Ψ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>r</mi>
<mo>−<!-- − --></mo>
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |\Psi \rangle =\sum _{i}\alpha _{i}|\Psi _{i}(r-R_{i})\rangle }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/605ba48ce3d2e2ab6e4b9bfc5faaf76ef4088e72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:24.909ex; height:5.509ex;" alt="{\displaystyle |\Psi \rangle =\sum _{i}\alpha _{i}|\Psi _{i}(r-R_{i})\rangle }" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>
<ul><li><a href="Tight-Binding-Methode" title="Tight-Binding-Methode">Tight-Binding-Methode</a></li>
<li><a href="Holstein-Herring-Methode" title="Holstein-Herring-Methode">Holstein-Herring-Methode</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>
<ul><li>LCAO @ chemistry.umeche.maine.edu <a rel="nofollow" class="external text" href="http://chemistry.umeche.maine.edu/Modeling/lcao.html">Link</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>
<ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><a href="James_Huheey" class="mw-redirect" title="James Huheey">Huheey, James</a>. <i>Inorganic Chemistry:Principles of Structure and Reactivity</i>.</span>
</li>
<li id="cite_note-kutzelnigg-2"><span class="mw-cite-backlink"><a href="#cite_ref-kutzelnigg_2-0">↑</a></span> <span class="reference-text"><a href="Werner_Kutzelnigg" title="Werner Kutzelnigg">Werner Kutzelnigg</a>: <i><a href="Friedrich_Hund" title="Friedrich Hund">Friedrich Hund</a> and Chemistry.</i> In: <i>Angewandte Chemie International Edition in English.</i> 35, 1996, S. 572–586,
<a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1002/anie.199605721">10.1002/anie.199605721</a></span>.</span>
</li>
<li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.nobelprize.org/nobel_prizes/chemistry/laureates/1966/mulliken-lecture.pdf">Robert S. Mulliken: Spectroscopy, molecular orbitals, and chemical bonding. Nobel Lecture</a>.</span>
</li>
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