Atomorbital – Wikipedia

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border-width: 1px; clear: left; margin-bottom:1em; margin-top:1em; padding: 0.25em; overflow: hidden; word-break: break-word; word-wrap: break-word;"><div class="noviewer noresize" style="display: table-cell; padding-bottom: 0.2em; padding-left: 0.25em; padding-right: 1em; padding-top: 0.2em; vertical-align: middle;" aria-hidden="true" role="presentation"><span typeof="mw:File"><a href="/wiki/Wikipedia:Belege" title="Belege"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Qsicon_Quelle.svg/24px-Qsicon_Quelle.svg.png" decoding="async" width="24" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Qsicon_Quelle.svg/36px-Qsicon_Quelle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Qsicon_Quelle.svg/48px-Qsicon_Quelle.svg.png 2x" data-file-width="24" data-file-height="24" /></a></span></div> <div style="display: table-cell; vertical-align: middle; width: 100%;"> <div> Dieser Artikel oder nachfolgende Abschnitt ist nicht hinreichend mit <a href="/wiki/Wikipedia:Belege" title="Wikipedia:Belege">Belegen</a> (beispielsweise <a href="/wiki/Hilfe:Einzelnachweise" title="Hilfe:Einzelnachweise">Einzelnachweisen</a>) ausgestattet. Angaben ohne ausreichenden Beleg könnten demnächst entfernt werden. Bitte hilf Wikipedia, indem du die Angaben recherchierst und <span style="white-space:nowrap">gute Belege einfügst.</span><br /> <span class="editoronly" style="display:none;"></span></div> </div></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Datei:AOs-3D-dots.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/AOs-3D-dots.png/400px-AOs-3D-dots.png" decoding="async" width="400" height="221" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/AOs-3D-dots.png/600px-AOs-3D-dots.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9b/AOs-3D-dots.png/800px-AOs-3D-dots.png 2x" data-file-width="2000" data-file-height="1106" /></a><figcaption>Darstellung unterschiedlicher Orbitale der ersten und zweiten <a href="/wiki/Elektronenschale" class="mw-redirect" title="Elektronenschale">Elektronenschale</a>.<br />Obere Reihe: Darstellung der Wahrscheinlichkeitsdichten <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 ({\vec {r}})|^{2}}"> <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</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> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\Psi ({\vec {r}})|^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76d4a6aca0b254b2d287d0d6fd5946856b6254f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.188ex; height:3.343ex;" alt="{\displaystyle |\Psi ({\vec {r}})|^{2}}"></span> der Orbitale als Punktwolken.<br />Untere Reihe: Darstellung von <a href="/wiki/Isofl%C3%A4che" title="Isofläche">Isoflächen</a> von <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 ({\vec {r}})|^{2}}"> <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</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> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\Psi ({\vec {r}})|^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76d4a6aca0b254b2d287d0d6fd5946856b6254f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.188ex; height:3.343ex;" alt="{\displaystyle |\Psi ({\vec {r}})|^{2}}"></span>. Die Isofläche ist jeweils so gewählt, dass sich das Elektron innerhalb des von der Isofläche umschlossenen Volumens mit 90 % Wahrscheinlichkeit aufhält.</figcaption></figure> <p>Ein <b>Atomorbital</b> ist in den <a href="/wiki/Quantenmechanik" title="Quantenmechanik">quantenmechanischen</a> Modellen der <a href="/wiki/Atom" title="Atom">Atome</a> die räumliche <a href="/wiki/Wellenfunktion" title="Wellenfunktion">Wellenfunktion</a> eines einzelnen <a href="/wiki/Elektron" title="Elektron">Elektrons</a> in einem <a href="/wiki/Quantenmechanik#Stationäre_Zustände" title="Quantenmechanik">quantenmechanischen Zustand</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>, meist in einem stationären Zustand. Sein <a href="/wiki/Formelzeichen" title="Formelzeichen">Formelzeichen</a> ist meist <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 \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> (kleines <a href="/wiki/Phi" title="Phi">Phi</a>) oder <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span> (kleines <a href="/wiki/Psi_(Buchstabe)" title="Psi (Buchstabe)">Psi</a>). Das <a href="/wiki/Betragsfunktion" title="Betragsfunktion">Betrags</a>quadrat <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 ({\vec {r}})|^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>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> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi ({\vec {r}})|^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f0223d0e4b6c5eaf1357117b8d140ef2b4f2439" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.893ex; height:3.343ex;" alt="{\displaystyle |\psi ({\vec {r}})|^{2}}"></span> als <a href="/wiki/Dichtefunktion" title="Dichtefunktion">Dichtefunktion</a> wird interpretiert als die räumliche Verteilung der <a href="/wiki/Aufenthaltswahrscheinlichkeit" title="Aufenthaltswahrscheinlichkeit">Aufenthaltswahrscheinlichkeit</a>, mit der das Elektron am Ort <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 {\vec {r}}=(x,y,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {r}}=(x,y,z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fe5622ace035bf6747042a78d531deacf8d81a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.772ex; height:2.843ex;" alt="{\displaystyle {\vec {r}}=(x,y,z)}"></span> gefunden werden kann (<a href="/wiki/Bornsche_Wahrscheinlichkeitsinterpretation" title="Bornsche Wahrscheinlichkeitsinterpretation">Bornsche Wahrscheinlichkeitsinterpretation</a> der Quantenmechanik). Zusammen mit der Angabe, ob der <a href="/wiki/Spin" title="Spin">Spin</a> zu einer festen Achse oder zum <a href="/wiki/Bahndrehimpuls" class="mw-redirect" title="Bahndrehimpuls">Bahndrehimpuls</a> des Elektrons parallel oder antiparallel ausgerichtet ist, beschreibt ein Orbital den Elektronenzustand vollständig. </p><p>In den älteren <a href="/wiki/Liste_der_Atommodelle" title="Liste der Atommodelle">Atommodellen</a> nach Niels Bohr (<a href="/wiki/Bohrsches_Atommodell" title="Bohrsches Atommodell">Bohrsches Atommodell</a>, 1913) und Arnold Sommerfeld (<a href="/wiki/Bohr-Sommerfeldsches_Atommodell" class="mw-redirect" title="Bohr-Sommerfeldsches Atommodell">Bohr-Sommerfeldsches Atommodell</a>, 1916) beschreibt ein Orbital eine genaue, durch die <a href="/wiki/Quantisierung_(Physik)" title="Quantisierung (Physik)">Quantisierungsregeln</a> ausgewählte Elektronenbahn. Diese Vorstellung wurde in der Quantenmechanik zugunsten einer diffusen Verteilung der Aufenthaltswahrscheinlichkeit des Elektrons aufgegeben. Das quantenmechanische Atomorbital erstreckt sich für gebundene Elektronen vom <a href="/wiki/Atomkern" title="Atomkern">Atomkern</a> im Zentrum nach außen bis ins Unendliche, wobei die Aufenthaltswahrscheinlichkeit außerhalb weniger 0,1 nm typischerweise sehr klein ist und für größeren Abstand <a href="/wiki/Asymptotisch" class="mw-redirect" title="Asymptotisch">asymptotisch</a> weiter gegen null geht. Der wahrscheinlichste Abstand vom Atomkern ist für das innerste Orbital gleich dem Radius der 1. bohrschen Kreisbahn. </p><p>Anschaulich stellt man ein Orbital gewöhnlich durch die Oberfläche des kleinstmöglichen Volumens dar, in dessen Inneren sich das Elektron mit großer Wahrscheinlichkeit aufhält. Man erhält damit Körper, die ungefähr der Größe und Form der Atome entsprechen, wie sie sich in chemischen <a href="/wiki/Molek%C3%BCl" title="Molekül">Molekülen</a>, <a href="/wiki/Kondensierte_Materie" title="Kondensierte Materie">kondensierter Materie</a> und der <a href="/wiki/Kinetische_Gastheorie" title="Kinetische Gastheorie">kinetischen Gastheorie</a> bemerkbar machen. </p><p>Die gebräuchlichsten Atomorbitale sind die, die sich für das einzige Elektron des Wasserstoffatoms als Lösungen der <a href="/wiki/Schr%C3%B6dingergleichung" title="Schrödingergleichung">Schrödingergleichung</a> des <a href="/wiki/Wasserstoffproblem" class="mw-redirect" title="Wasserstoffproblem">Wasserstoffproblems</a> ergeben und 1926 erstmals veröffentlicht wurden. Sie haben verschiedene Formen, die mit <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 _{nlm_{l}}({\vec {r}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>l</mi> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> </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 stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{nlm_{l}}({\vec {r}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72a980efecc15a9b67c353987f5855c31291e24c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.259ex; height:3.009ex;" alt="{\displaystyle \psi _{nlm_{l}}({\vec {r}})}"></span> bezeichnet werden, wobei der untere Index aus der Haupt<a href="/wiki/Quantenzahl" title="Quantenzahl">quantenzahl</a> <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 n,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/397bfafc701afdf14c2743278a097f6f2957eabb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.042ex; height:2.009ex;" alt="{\displaystyle n,}"></span> der Bahndrehimpulsquantenzahl <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 l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}"></span> und der magnetischen Quantenzahl <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 m_{l}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{l}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3f945c408d692391284a629617fe0b301776222" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.763ex; height:2.009ex;" alt="{\displaystyle m_{l}}"></span> besteht. </p><p>Im <b>Orbitalmodell</b> für Atome mit mehreren Elektronen nimmt man an, dass die Elektronen sich unter Berücksichtigung des <a href="/wiki/Pauli-Prinzip" title="Pauli-Prinzip">Pauli-Prinzips</a> auf die Orbitale verteilen. Ein solcher Zustand heißt <a href="/wiki/Elektronenkonfiguration" title="Elektronenkonfiguration">Elektronenkonfiguration</a> und stellt oft eine brauchbare <a href="/wiki/Approximation#Funktionen" title="Approximation">Näherung</a> für die Struktur der <a href="/wiki/Atomh%C3%BClle" title="Atomhülle">Atomhülle</a> dar, obwohl diese durch zusätzliche Elektronenkorrelationen noch komplizierter ist. </p><p>Zur Beschreibung von Elektronen in Molekülen werden <a href="/wiki/Molek%C3%BClorbital" class="mw-redirect" title="Molekülorbital">Molekülorbitale</a> als <a href="/wiki/Linearkombination" title="Linearkombination">Linearkombination</a> von Atomorbitalen gebildet. Elektronen in Festkörpern werden durch Orbitale beschrieben, die die Form von <a href="/wiki/Blochwellenfunktion" class="mw-redirect" title="Blochwellenfunktion">Blochwellenfunktionen</a> haben. </p><p>In diesem Artikel wird nur auf gebundene Elektronen in Atomen eingegangen. Eine Vereinfachung des Orbitalmodells ist das <a href="/wiki/Schalenmodell_(Atomphysik)" title="Schalenmodell (Atomphysik)">Schalenmodell</a>. </p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="de" dir="ltr"><h2 id="mw-toc-heading">Inhaltsverzeichnis</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Darstellung"><span class="tocnumber">1</span> <span class="toctext">Darstellung</span></a></li> <li class="toclevel-1 tocsection-2"><a href="#Klassifikation"><span class="tocnumber">2</span> <span class="toctext">Klassifikation</span></a> <ul> <li class="toclevel-2 tocsection-3"><a href="#Hauptquantenzahl_n:_Schale"><span class="tocnumber">2.1</span> <span class="toctext">Hauptquantenzahl <i>n</i>: Schale</span></a></li> <li class="toclevel-2 tocsection-4"><a href="#Neben-_oder_Bahndrehimpuls-Quantenzahl_l"><span class="tocnumber">2.2</span> <span class="toctext">Neben- oder Bahndrehimpuls-Quantenzahl <i>l</i></span></a> <ul> <li class="toclevel-3 tocsection-5"><a href="#Form"><span class="tocnumber">2.2.1</span> <span class="toctext">Form</span></a></li> <li class="toclevel-3 tocsection-6"><a href="#Unterschale"><span class="tocnumber">2.2.2</span> <span class="toctext">Unterschale</span></a></li> </ul> </li> <li class="toclevel-2 tocsection-7"><a href="#Magnetquantenzahl_ml:_Neigung_des_Drehimpulsvektors"><span class="tocnumber">2.3</span> <span class="toctext">Magnetquantenzahl <i>m<sub>l</sub></i>: Neigung des Drehimpulsvektors</span></a></li> <li class="toclevel-2 tocsection-8"><a href="#Magnetische_Spinquantenzahl_ms"><span class="tocnumber">2.4</span> <span class="toctext">Magnetische Spinquantenzahl <i>m<sub>s</sub></i></span></a></li> <li class="toclevel-2 tocsection-9"><a href="#Gesamtdrehimpuls_j_und_magnetische_Quantenzahl_mj"><span class="tocnumber">2.5</span> <span class="toctext">Gesamtdrehimpuls <i>j</i> und magnetische Quantenzahl <i>m<sub>j</sub></i></span></a></li> </ul> </li> <li class="toclevel-1 tocsection-10"><a href="#Quantentheorie"><span class="tocnumber">3</span> <span class="toctext">Quantentheorie</span></a></li> <li class="toclevel-1 tocsection-11"><a href="#Natürliches_Orbital"><span class="tocnumber">4</span> <span class="toctext">Natürliches Orbital</span></a></li> <li class="toclevel-1 tocsection-12"><a href="#Zeitabhängigkeit"><span class="tocnumber">5</span> <span class="toctext">Zeitabhängigkeit</span></a></li> <li class="toclevel-1 tocsection-13"><a href="#Hybridisierung"><span class="tocnumber">6</span> <span class="toctext">Hybridisierung</span></a></li> <li class="toclevel-1 tocsection-14"><a href="#Mehr-Elektronen-Wellenfunktionen"><span class="tocnumber">7</span> <span class="toctext">Mehr-Elektronen-Wellenfunktionen</span></a></li> <li class="toclevel-1 tocsection-15"><a href="#Literatur"><span class="tocnumber">8</span> <span class="toctext">Literatur</span></a></li> <li class="toclevel-1 tocsection-16"><a href="#Weblinks"><span class="tocnumber">9</span> <span class="toctext">Weblinks</span></a></li> <li class="toclevel-1 tocsection-17"><a href="#Einzelnachweise"><span class="tocnumber">10</span> <span class="toctext">Einzelnachweise</span></a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Darstellung">Darstellung</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atomorbital&veaction=edit&section=1" title="Abschnitt bearbeiten: Darstellung" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Atomorbital&action=edit&section=1" title="Quellcode des Abschnitts bearbeiten: Darstellung"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Orbital_s1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Orbital_s1.png/220px-Orbital_s1.png" decoding="async" width="220" height="272" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Orbital_s1.png/330px-Orbital_s1.png 1.5x, //upload.wikimedia.org/wikipedia/commons/0/00/Orbital_s1.png 2x" data-file-width="439" data-file-height="543" /></a><figcaption>Darstellung der Wahrscheinlichkeitsdichte des 1s-Orbitals mithilfe einer (sehr feinen) Punktwolke</figcaption></figure> <p>Da die Wellenfunktion <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5471531a3fe80741a839bc98d49fae862a6439a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Psi }"></span> von drei Variablen abhängt und im Allgemeinen komplexe Werte hat, ist eine vollständige grafische Darstellung in einer Abbildung nicht möglich. Häufig zeigen Bilder von Orbitalen eine Darstellung der Wahrscheinlichkeitsdichte <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 ({\vec {r}})|^{2}}"> <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</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> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\Psi ({\vec {r}})|^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76d4a6aca0b254b2d287d0d6fd5946856b6254f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.188ex; height:3.343ex;" alt="{\displaystyle |\Psi ({\vec {r}})|^{2}}"></span>. Dabei wird die Wahrscheinlichkeitsdichte z. B. als <a href="/wiki/Punktwolke" title="Punktwolke">Punktwolke</a> visualisiert: Viele dicht liegende Punkte deuten große Wahrscheinlichkeitsdichte an, während in Gebieten geringer Wahrscheinlichkeitsdichte wenige Punkte eingezeichnet werden. Da die Wahrscheinlichkeitsdichte sich im Prinzip ins Unendliche erstreckt, lässt sich keine äußere Begrenzung des Orbitals angeben. Stattdessen kann man <a href="/wiki/Isofl%C3%A4che" title="Isofläche">Isoflächen</a> gleicher Wahrscheinlichkeitsdichte zeichnen, die durch </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 {\text{const}}=|\Psi ({\vec {r}})|^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>const</mtext> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">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> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{const}}=|\Psi ({\vec {r}})|^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ade98f184a38d4a0243756255b9b73e897a25550" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.595ex; height:3.343ex;" alt="{\displaystyle {\text{const}}=|\Psi ({\vec {r}})|^{2}}"></span></dd></dl> <p>definiert sind. Häufig wird die Konstante so gewählt, dass die Wahrscheinlichkeit, das Elektron in dem von der Isofläche umschlossenen Raum zu finden, 90 % beträgt. Durch Abtasten verschiedener Winkel <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 \theta ,\phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>,</mo> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta ,\phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/460f850fadee91c2f107605ae351dc04dc4dc544" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.51ex; height:2.509ex;" alt="{\displaystyle \theta ,\phi }"></span> erfährt man etwas über die Form der Isofläche und somit etwas über die „Form des Orbitals“. Wie vom Wasserstoffatom bekannt ist, haben die Eigenfunktionen <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 ({\vec {r}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi ({\vec {r}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/860ed8dcca03bb55dda0fe92c92a079038be797b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.841ex; height:2.843ex;" alt="{\displaystyle \Psi ({\vec {r}})}"></span> der <a href="/wiki/Station%C3%A4re_Schr%C3%B6dingergleichung" class="mw-redirect" title="Stationäre Schrödingergleichung">stationären Schrödingergleichung</a> <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 H\Psi ({\vec {r}})=E\Psi ({\vec {r}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mi mathvariant="normal">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>=</mo> <mi>E</mi> <mi mathvariant="normal">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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\Psi ({\vec {r}})=E\Psi ({\vec {r}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b829d48bd68d1a95aaf4f4bc54682e8e8b0a5411" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.619ex; height:2.843ex;" alt="{\displaystyle H\Psi ({\vec {r}})=E\Psi ({\vec {r}})}"></span> einen Radialanteil <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 R(r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R(r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7b97817d8756302ef44f910ec5e76346e8d4f4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.622ex; height:2.843ex;" alt="{\displaystyle R(r)}"></span> und einen Winkelanteil <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 Y_{l}^{m}(\theta ,\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{l}^{m}(\theta ,\phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea58b8719f9025fa0d8eaf6f8be7adc95762f280" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.894ex; height:3.009ex;" alt="{\displaystyle Y_{l}^{m}(\theta ,\phi )}"></span>: </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 ({\vec {r}})=R(r)Y_{l}^{m}(\theta ,\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">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>=</mo> <mi>R</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <msubsup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi ({\vec {r}})=R(r)Y_{l}^{m}(\theta ,\phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c84d62af5cef8435c26c2b3289a55311dac04499" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.455ex; height:3.009ex;" alt="{\displaystyle \Psi ({\vec {r}})=R(r)Y_{l}^{m}(\theta ,\phi )}"></span></dd></dl> <p>Da die Winkelabhängigkeit durch eine universelle Kugelfächenfunktion <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 Y_{l}^{m}(\theta ,\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{l}^{m}(\theta ,\phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea58b8719f9025fa0d8eaf6f8be7adc95762f280" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.894ex; height:3.009ex;" alt="{\displaystyle Y_{l}^{m}(\theta ,\phi )}"></span> gegeben ist, steckt die jeweils spezifische Information im Radialanteil <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 R(r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R(r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7b97817d8756302ef44f910ec5e76346e8d4f4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.622ex; height:2.843ex;" alt="{\displaystyle R(r)}"></span>, der als reellwertige Funktion einer reellen Variablen grafisch dargestellt werden kann. </p><p>Nicht selten wird bei der Darstellung einer Isofläche von <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 ({\vec {r}})|^{2}}"> <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</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> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\Psi ({\vec {r}})|^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76d4a6aca0b254b2d287d0d6fd5946856b6254f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.188ex; height:3.343ex;" alt="{\displaystyle |\Psi ({\vec {r}})|^{2}}"></span> die Fläche entsprechend dem <a href="/wiki/Komplexe_Zahl#Darstellung_von_komplexen_Zahlen_in_der_komplexen_Zahlenebene" title="Komplexe Zahl">komplexen Argument</a> von <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 ({\vec {r}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi ({\vec {r}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/860ed8dcca03bb55dda0fe92c92a079038be797b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.841ex; height:2.843ex;" alt="{\displaystyle \Psi ({\vec {r}})}"></span> koloriert (wie in dem Bild des p-Orbitals). </p><p>Eine einfache Art der schematischen Darstellung der Besetzung von Atomorbitalen ist die <a href="/wiki/Pauling-Schreibweise" title="Pauling-Schreibweise">Pauling-Schreibweise</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Klassifikation">Klassifikation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atomorbital&veaction=edit&section=2" title="Abschnitt bearbeiten: Klassifikation" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Atomorbital&action=edit&section=2" title="Quellcode des Abschnitts bearbeiten: Klassifikation"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Atomorbitale können durch drei Quantenzahlen <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 n,l,m_{l}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>,</mo> <mi>l</mi> <mo>,</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n,l,m_{l}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/329258dfe3841b3bfe476dfe537c73b6b030e926" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.919ex; height:2.509ex;" alt="{\displaystyle n,l,m_{l}}"></span> festgelegt werden und bieten dann Platz für zwei Elektronen mit entgegengesetztem <a href="/wiki/Spin" title="Spin">Spin</a>. Alternativ können Atomorbitale durch vier <a href="/wiki/Quantenzahl" title="Quantenzahl">Quantenzahlen</a> <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 n,l,j,m_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>,</mo> <mi>l</mi> <mo>,</mo> <mi>j</mi> <mo>,</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n,l,j,m_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ad015e5da650b3bb3e9236699b2626eecfe6bf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.098ex; height:2.843ex;" alt="{\displaystyle n,l,j,m_{j}}"></span> festgelegt werden und bieten dann Platz für nur jeweils ein Elektron. </p> <div class="mw-heading mw-heading3"><h3 id="Hauptquantenzahl_n:_Schale">Hauptquantenzahl <i>n</i>: Schale</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atomorbital&veaction=edit&section=3" title="Abschnitt bearbeiten: Hauptquantenzahl n: Schale" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Atomorbital&action=edit&section=3" title="Quellcode des Abschnitts bearbeiten: Hauptquantenzahl n: Schale"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Die Hauptquantenzahl <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 n=1,2,3\dotsc }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=1,2,3\dotsc }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/463c54aa580aa88df1b3ac5e9afbba8ff498dabe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.159ex; height:2.509ex;" alt="{\displaystyle n=1,2,3\dotsc }"></span> bezeichnet die <a href="/wiki/Schalenmodell_(Atomphysik)" title="Schalenmodell (Atomphysik)">Schale</a> (Bezeichnung auch K-Schale, L-Schale, M-Schale …), zu der das Orbital gehört. Im bohrschen Atommodell gibt <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> das <a href="/wiki/Energieniveau" title="Energieniveau">Energieniveau</a> an, beginnend mit dem tiefsten, dem <a href="/wiki/Grundzustand" title="Grundzustand">Grundzustand</a> <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 n=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04d74ade48a04cf5d7a4d8a0f0a94a0bf6050973" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.302ex; height:2.176ex;" alt="{\displaystyle n=1.}"></span> </p><p>Als ungefähre Regel gilt: Je größer <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, desto geringer die <a href="/wiki/Bindungsenergie" title="Bindungsenergie">Bindungsenergie</a> des Elektrons und damit desto größer die Wahrscheinlichkeit, das Elektron weiter entfernt vom Atomkern zu finden. Das gilt auch für Atome mit mehreren Elektronen. Bei Wechselwirkungen zwischen Atomen, die sich nahe kommen, (wie <a href="/wiki/Sto%C3%9F_(Physik)" title="Stoß (Physik)">Stößen</a> von Gasmolekülen, Raumerfüllung in kondensierter Materie, <a href="/wiki/Chemische_Bindung" title="Chemische Bindung">chemischen Bindungen</a>) spielen deshalb die Elektronen mit der größten Hauptquantenzahl die wichtigste Rolle (die Elektronen der <a href="/wiki/Valenzschale" title="Valenzschale">Valenzschale</a>). </p><p>Die Anzahl der <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 (nlm_{l})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mi>l</mi> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (nlm_{l})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18b1829d392d4d6a83daa15853e48ac5a906fb3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.66ex; height:2.843ex;" alt="{\displaystyle (nlm_{l})}"></span>-Orbitale in einer Schale ergibt sich zu <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 n^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4846c73ba44c6ffdc37db7268c4f0d161b88dbe1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.096ex; height:2.676ex;" alt="{\displaystyle n^{2}.}"></span> Unter Berücksichtigung des <a href="/wiki/Pauli-Prinzip" title="Pauli-Prinzip">Pauli-Prinzips</a> kann die Schale mit maximal <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 2\cdot n^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>⋅</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\cdot n^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/867cc403af590c4a97bbcc93f11d9bf54a208112" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.291ex; height:2.676ex;" alt="{\displaystyle 2\cdot n^{2}}"></span> Elektronen besetzt werden, dann ist sie <i>abgeschlossen.</i> Die entsprechenden Atome gehören zu den <a href="/wiki/Edelgas" class="mw-redirect" title="Edelgas">Edelgasen</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Neben-_oder_Bahndrehimpuls-Quantenzahl_l">Neben- oder Bahndrehimpuls-Quantenzahl <i>l</i></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atomorbital&veaction=edit&section=4" title="Abschnitt bearbeiten: Neben- oder Bahndrehimpuls-Quantenzahl l" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Atomorbital&action=edit&section=4" title="Quellcode des Abschnitts bearbeiten: Neben- oder Bahndrehimpuls-Quantenzahl l"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Form">Form</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atomorbital&veaction=edit&section=5" title="Abschnitt bearbeiten: Form" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Atomorbital&action=edit&section=5" title="Quellcode des Abschnitts bearbeiten: Form"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Die Neben- oder Bahndrehimpulsquantenzahl <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 l=0,1,2\dotsc ,(n-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>…</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l=0,1,2\dotsc ,(n-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d781b69111784a2499c2df968606ce3e611965e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.085ex; height:2.843ex;" alt="{\displaystyle l=0,1,2\dotsc ,(n-1)}"></span> innerhalb einer Schale beschreibt den Betrag <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 |{\vec {l}}|=\hbar \cdot {\sqrt {l(l+1)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>l</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mi class="MJX-variant">ℏ</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>l</mi> <mo stretchy="false">(</mo> <mi>l</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\vec {l}}|=\hbar \cdot {\sqrt {l(l+1)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b93c65e73315efc7ecb9b251c8566b00095bda7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.188ex; height:4.843ex;" alt="{\displaystyle |{\vec {l}}|=\hbar \cdot {\sqrt {l(l+1)}}}"></span> des <a href="/wiki/Bahndrehimpuls" class="mw-redirect" title="Bahndrehimpuls">Bahndrehimpulses</a> des Elektrons. Mit der Quantenzahl <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 m_{l}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{l}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3f945c408d692391284a629617fe0b301776222" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.763ex; height:2.009ex;" alt="{\displaystyle m_{l}}"></span> zusammen wird damit die winkelabhängige „Form“ des Orbitals festgelegt. Sie ist für alle Hauptquantenzahlen (beachte <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 n>l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/473353e72ce0efcd4ffffbf91cf4d993792c5771" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.186ex; height:2.176ex;" alt="{\displaystyle n>l}"></span>) dieselbe. </p><p>Statt der Ziffern <i>0, 1, 2 …</i> wird die Nebenquantenzahl in der Literatur meist durch die Buchstaben <i>s, p, d, f …</i> bezeichnet, abgeleitet von den ursprünglich gebrauchten Bezeichnungen <i>„sharp, principal, diffus, fundamental“</i> für die korrespondierenden <a href="/wiki/Spektrallinien" class="mw-redirect" title="Spektrallinien">Spektrallinien</a>; diese konkrete Bedeutung ist seit langem unwesentlich geworden: </p> <table class="wikitable" style="background:#FFFFFF"> <tbody><tr> <th><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 p_{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dbd3c1a6173a7974e0095301da94447c5f67657" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.261ex; height:2.009ex;" alt="{\displaystyle p_{z}}"></span>-Orbital </th> <th><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 d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span>-Orbitale </th> <th><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 4p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f020e39692665a10be8cca01d62d74df15862e36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.332ex; height:2.509ex;" alt="{\displaystyle 4p}"></span>-Orbital </th></tr> <tr> <td><figure class="mw-halign-center" typeof="mw:File"><a href="/wiki/Datei:Pz_orbital.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Pz_orbital.png/120px-Pz_orbital.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Pz_orbital.png/180px-Pz_orbital.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Pz_orbital.png/240px-Pz_orbital.png 2x" data-file-width="1000" data-file-height="1000" /></a><figcaption></figcaption></figure> </td> <td><figure class="mw-halign-center" typeof="mw:File"><a href="/wiki/Datei:Orbitalesd.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Orbitalesd.JPG/300px-Orbitalesd.JPG" decoding="async" width="300" height="323" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/4/47/Orbitalesd.JPG 1.5x" data-file-width="350" data-file-height="377" /></a><figcaption></figcaption></figure> </td> <td><figure class="mw-halign-center" typeof="mw:File"><a href="/wiki/Datei:Atomic-orbital-cloud_n4_px.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a1/Atomic-orbital-cloud_n4_px.png/200px-Atomic-orbital-cloud_n4_px.png" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a1/Atomic-orbital-cloud_n4_px.png/300px-Atomic-orbital-cloud_n4_px.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a1/Atomic-orbital-cloud_n4_px.png/400px-Atomic-orbital-cloud_n4_px.png 2x" data-file-width="1024" data-file-height="1024" /></a><figcaption></figcaption></figure> </td></tr> <tr> <td>Vereinfachte Form eines p-Orbitals <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 (l=1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>l</mi> <mo>=</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (l=1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/429da5e6b9ed06ae2304ee5ecf19b0785323a0d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.764ex; height:2.843ex;" alt="{\displaystyle (l=1)}"></span>.<br />Die Färbung steht für das Vorzeichen der Wellenfunktion. Dargestellt ist eine Isofläche von <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 ({\vec {r}})|^{2}}"> <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</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> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\Psi ({\vec {r}})|^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76d4a6aca0b254b2d287d0d6fd5946856b6254f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.188ex; height:3.343ex;" alt="{\displaystyle |\Psi ({\vec {r}})|^{2}}"></span>. </td> <td>Vereinfachte Formen der verschiedenen d-Orbitale (jeweils <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 l=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/568d606c605ed04ee4beb2bc2d3bed232e0b07f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.954ex; height:2.176ex;" alt="{\displaystyle l=2}"></span>). Für die jeweiligen Orbitale ist eine Isofläche der Wahrscheinlichkeitsdichte <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 ({\vec {r}})|^{2}}"> <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</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> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\Psi ({\vec {r}})|^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76d4a6aca0b254b2d287d0d6fd5946856b6254f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.188ex; height:3.343ex;" alt="{\displaystyle |\Psi ({\vec {r}})|^{2}}"></span> dargestellt. </td> <td>Form eines 4p-Orbitals <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 (l=1,\,m_{x}=0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>l</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (l=1,\,m_{x}=0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15a00f7bd8869cf26ce48fe9b681da2f59f51b69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.658ex; height:2.843ex;" alt="{\displaystyle (l=1,\,m_{x}=0)}"></span>.<br />Die Färbung steht für das Vorzeichen der Wellenfunktion. </td></tr></tbody></table> <table class="wikitable centered" style="text-align:center"> <tbody><tr> <th>Name</th> <th>ehemalige Bedeutung</th> <th>Nebenquantenzahl</th> <th>Form</th> <th>Anzahl <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 2l+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>l</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2l+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad6cc29a87d7b25ec1f294612477d5a38f59c09a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.859ex; height:2.343ex;" alt="{\displaystyle 2l+1}"></span> </th></tr> <tr> <td>s-Orbital</td> <td><i><b>s</b>harp</i></td> <td><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 \,l=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>l</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,l=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7545b1fba1c8729ad64200ba5778099c8bf75ee1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.341ex; height:2.176ex;" alt="{\displaystyle \,l=0}"></span></td> <td><a href="/wiki/Kugelsymmetrisch" class="mw-redirect" title="Kugelsymmetrisch">kugelsymmetrisch</a></td> <td><span style="visibility:hidden;">0</span>1 </td></tr> <tr> <td>p-Orbital</td> <td><i><b>p</b>rincipal</i></td> <td><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 \,l=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>l</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,l=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/907c456f4d1dd83f469889da6e97f9bcc46e5b0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.341ex; height:2.176ex;" alt="{\displaystyle \,l=1}"></span></td> <td><a href="/wiki/Hantel" title="Hantel">hantel</a>förmig</td> <td><span style="visibility:hidden;">000</span>3<style data-mw-deduplicate="TemplateStyles:r247957335">.mw-parser-output .fussnoten-marke{font-size:0.75rem;font-style:normal;font-variant:normal;font-weight:normal;unicode-bidi:isolate;white-space:nowrap}.mw-parser-output .fussnoten-marke.reference,.mw-parser-output span.fussnoten-inhalt{padding-left:0.1rem}.mw-parser-output .fussnoten-marke.reference~.fussnoten-marke.reference,.mw-parser-output span.fussnoten-inhalt~span.fussnoten-inhalt{padding-left:0.15rem}.mw-parser-output .fussnoten-block{margin-bottom:0.1rem}.mw-parser-output div.fussnoten-inhalt{display:inline-block;padding-left:0.8rem;text-indent:-0.8rem}.mw-parser-output div.fussnoten-inhalt p,.mw-parser-output div.fussnoten-inhalt dl,.mw-parser-output div.fussnoten-inhalt ol,.mw-parser-output div.fussnoten-inhalt ul{text-indent:0}.mw-parser-output div.fussnoten-inhalt.fussnoten-floatfix{display:block}.mw-parser-output .fussnoten-box{margin-top:0.5rem;padding-left:0.8rem}.mw-parser-output .fussnoten-box,.mw-parser-output div.fussnoten-inhalt{font-size:94%}.mw-parser-output .fussnoten-box div.fussnoten-inhalt,.mw-parser-output span.fussnoten-inhalt,.mw-parser-output .fussnoten-inhalt .reference-text{font-size:inherit}.mw-parser-output .fussnoten-inhalt .reference-text{display:inline}.mw-parser-output .fussnoten-linie{display:inline-block;position:relative;top:-1em;border-top:solid 1px #808080;width:8rem}.mw-parser-output .fussnoten-linie+p,.mw-parser-output .fussnoten-linie+dl,.mw-parser-output .fussnoten-linie+ol,.mw-parser-output .fussnoten-linie+ul,.mw-parser-output .fussnoten-linie+link+div{margin-top:-1em}.mw-parser-output .fussnoten-marke.reference:target,.mw-parser-output .fussnoten-inhalt:target{background-color:#eaf3ff;box-shadow:0 0 0 0.25em #eaf3ff}.mw-parser-output .fussnoten-marke.reference:target,.mw-parser-output .fussnoten-inhalt:target .fussnoten-marke{font-weight:bold}</style> <span class="reference"><sup class="fussnoten-marke" data-annotationpair-m="A2">A2</sup></span> </td></tr> <tr> <td>d-Orbital</td> <td><i><b>d</b>iffuse</i></td> <td><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 \,l=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>l</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,l=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2842947a412c2e98b7ea974f8d6c5dce0bbbfbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.341ex; height:2.176ex;" alt="{\displaystyle \,l=2}"></span></td> <td>gekreuzte Doppelhantel</td> <td><span style="visibility:hidden;">0</span>5 </td></tr> <tr> <td>f-Orbital</td> <td><i><b>f</b>undamental</i></td> <td><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 \,l=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>l</mi> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,l=3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58e7528ca2252a86826f5f8e1bc00df24217d483" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.341ex; height:2.176ex;" alt="{\displaystyle \,l=3}"></span></td> <td><a href="/wiki/Rosette_(Ornamentik)" class="mw-redirect" title="Rosette (Ornamentik)">rosetten</a>förmig</td> <td><span style="visibility:hidden;">0</span>7 </td></tr> <tr> <td>g-Orbital<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r247957335"> <span class="reference"><sup class="fussnoten-marke" data-annotationpair-m="A1">A1</sup></span></td> <td>(alphabetische Fortsetzung)</td> <td><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 \,l=4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>l</mi> <mo>=</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,l=4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47f7e389527c8b2fe40bce2ad33c6f7c596dcddc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.341ex; height:2.176ex;" alt="{\displaystyle \,l=4}"></span></td> <td>rosettenförmig</td> <td><span style="visibility:hidden;">0</span>9 </td></tr> <tr> <td>h-Orbital<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r247957335"> <span class="reference"><sup class="fussnoten-marke" data-annotationpair-m="A1">A1</sup></span></td> <td>(alphabetische Fortsetzung)</td> <td><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 \,l=5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>l</mi> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,l=5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29c617cac1ee5c0c5079858e99c5bc1456795de1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.341ex; height:2.176ex;" alt="{\displaystyle \,l=5}"></span></td> <td>rosettenförmig</td> <td>11 </td></tr></tbody></table> <p><b>Anmerkungen:</b> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r247957335"><div class="fussnoten-box"> <div class="fussnoten-linie" aria-hidden="true" role="presentation"></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r247957335"><div class="fussnoten-block"><div class="fussnoten-inhalt references"><sup class="fussnoten-marke mw-cite-backlink" data-annotationpair-a="A1">A1</sup> <div class="reference-text">Kann als <a href="/wiki/Angeregter_Zustand" title="Angeregter Zustand">angeregter Zustand</a> vorkommen. Für den <a href="/wiki/Grundzustand" title="Grundzustand">Grundzustand</a> wird es theoretisch erst für Atome ab der <a href="/wiki/Unbiunium" title="Unbiunium">Ordnungszahl 121</a> erwartet.</div></div></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r247957335"><div class="fussnoten-block"><div class="fussnoten-inhalt references"><sup class="fussnoten-marke mw-cite-backlink" data-annotationpair-a="A2">A2</sup> <div class="reference-text">Entsprechend den drei Raumachsen.</div></div></div> </div> <p>Die Orbitale charakterisieren streng genommen nur die stationären <a href="/wiki/Materiewelle" title="Materiewelle">Elektronen-Wellen</a> in Systemen mit nur einem Elektron (wie z. B. <a href="/wiki/Wasserstoff" title="Wasserstoff">Wasserstoffatom</a> H, <a href="/wiki/Helium" title="Helium">Heliumion</a> He<sup>+</sup>, <a href="/wiki/Lithium" title="Lithium">Lithiumion</a> Li<sup>2+</sup> usw.). Da die Form der Orbitale auch in Mehrelektronensystemen in etwa erhalten bleibt, reicht ihre Kenntnis aus, um viele qualitative Fragen zur chemischen Bindung und zum Aufbau von Stoffen zu beantworten. </p><p>Dabei ist zu beachten, dass die in der Literatur dargestellten Orbitale zuweilen <i>nicht</i> die <a href="/wiki/Eigenzustand" title="Eigenzustand">Eigenzustände</a> zur magnetischen Quantenzahl <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 m_{l}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{l}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3f945c408d692391284a629617fe0b301776222" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.763ex; height:2.009ex;" alt="{\displaystyle m_{l}}"></span> der z-Komponente des <a href="/wiki/Drehimpulsoperator" class="mw-redirect" title="Drehimpulsoperator">Drehimpulsoperators</a> <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 {\hat {l}}_{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>l</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {l}}_{z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80858792edd88b1eba41017b1e83bee5050c776d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.358ex; height:3.176ex;" alt="{\displaystyle {\hat {l}}_{z}}"></span> sind. Z. B. wird von den p-Orbitalen nur der eine Eigenzustand für den <a href="/wiki/Eigenwert" class="mw-redirect" title="Eigenwert">Eigenwert</a> <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 m_{l}{\mathord {=}}0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo>=</mo> </mrow> </mrow> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{l}{\mathord {=}}0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e47c5d8f721e0a2bf174d753b59b932428f5965f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.733ex; height:2.509ex;" alt="{\displaystyle m_{l}{\mathord {=}}0}"></span> dargestellt und als p<sub>z</sub> bezeichnet. Die mit p<sub>x</sub> und p<sub>y</sub> bezeichneten Orbitale sind jedoch <i>nicht</i> die entsprechenden Eigenzustände für <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 m_{l}=\pm 1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo>=</mo> <mo>±</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{l}=\pm 1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/932805f89d48f991f117b1bd9cac3069d4500bb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.479ex; height:2.509ex;" alt="{\displaystyle m_{l}=\pm 1,}"></span> sondern sind deren <a href="/wiki/Superposition_(Physik)" title="Superposition (Physik)">Superpositionen</a>. Sie sind Eigenzustände zu den Operatoren <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 {\hat {l}}_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>l</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {l}}_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0251ecf7a5f39cb2811596e6d787fc7ab7d7c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.528ex; height:3.176ex;" alt="{\displaystyle {\hat {l}}_{x}}"></span> bzw. <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 {\hat {l}}_{y},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>l</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {l}}_{y},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c10cfddd30b3ea38ff323b1bda33995e9916cad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.052ex; height:3.509ex;" alt="{\displaystyle {\hat {l}}_{y},}"></span> jeweils zu <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 m_{x,y}{\mathord {=}}0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo>=</mo> </mrow> </mrow> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{x,y}{\mathord {=}}0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94782dbb163df22786db63542d8bb6db0e8ce92d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.105ex; height:2.843ex;" alt="{\displaystyle m_{x,y}{\mathord {=}}0,}"></span> die aber nicht mit <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 {\hat {l}}_{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>l</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {l}}_{z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80858792edd88b1eba41017b1e83bee5050c776d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.358ex; height:3.176ex;" alt="{\displaystyle {\hat {l}}_{z}}"></span> kommutieren. Für die Schlussfolgerungen ist das kein Problem, solange die entsprechenden Wellenfunktionen <a href="/wiki/Orthogonal" class="mw-redirect" title="Orthogonal">orthogonal</a> sind. </p> <div class="mw-heading mw-heading4"><h4 id="Unterschale">Unterschale</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atomorbital&veaction=edit&section=6" title="Abschnitt bearbeiten: Unterschale" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Atomorbital&action=edit&section=6" title="Quellcode des Abschnitts bearbeiten: Unterschale"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Je größer <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 l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}"></span>, desto größer ist bei festem <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> die mittlere Entfernung des Elektrons vom Atomkern: </p> <ul><li>Bei <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 l=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66485a3e3da13d226eb36a131bf1fc7e16403a5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.954ex; height:2.176ex;" alt="{\displaystyle l=0}"></span> ist das Orbital kugelförmig und hat auch bei <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 r=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/894a83e863728b4ee2e12f3a999a09f5f2bf1c89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.176ex;" alt="{\displaystyle r=0}"></span>, also im Kern, eine nichtverschwindende Aufenthaltswahrscheinlichkeit.</li> <li>Der Maximalwert <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 l=n-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> <mo>=</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l=n-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72a1d9783b86f15f6fa8adbe80986bb4fd756859" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.189ex; height:2.343ex;" alt="{\displaystyle l=n-1}"></span> entspricht der bohrschen Kreisbahn, hier konzentriert sich die Aufenthaltswahrscheinlichkeit bei dem im bohrschen Modell berechneten Radius.</li></ul> <p>Da bei Atomen mit mehreren Elektronen die <a href="/wiki/Inneres_Elektron" title="Inneres Elektron">inneren Elektronen</a> die anziehende <a href="/wiki/Kernladung" title="Kernladung">Kernladung</a> <a href="/wiki/Abschirmung_(Atomphysik)" title="Abschirmung (Atomphysik)">abschirmen</a>, verringert sich die Bindungsenergie der äußeren Elektronen. Da die mittleren Kernabstände von der Nebenquantenzahl abhängen, ergeben sich zum gleichen <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> je nach Nebenquantenzahl verschiedene Energieniveaus innerhalb derselben Schale. Diese werden auch als <i>Unterschalen</i> der <i>Hauptschale</i> (zu festem <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>) bezeichnet. </p><p>Die Anzahl der Unterschalen je Schale ist gleich der Hauptquantenzahl <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>: </p> <ul><li>Für <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 n=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9ec7e1edc2e6d98f5aec2a39ae5f1c99d1e1425" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=1}"></span> gibt es nur die 1s-Schale.</li> <li>Für <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 n=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a02c8bd752d2cc859747ca1f3a508281bdbc3b34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=2}"></span> gibt es zu <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 l=0,1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l=0,1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69e1d060a5123b7115fd972c0ac874e67bf79faf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.151ex; height:2.509ex;" alt="{\displaystyle l=0,1}"></span> die 2s- und die 2p-Schale.</li> <li>Für <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 n=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c5a5a42ced00df920fad4ab2d4acdb960a4105b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=3}"></span> sind drei Unterschalen <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 l=0,1,2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l=0,1,2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a42e766a1314d070c8efec00a3f006e65a5a6c53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.347ex; height:2.509ex;" alt="{\displaystyle l=0,1,2}"></span> möglich, die mit 3s, 3p, 3d bezeichnet werden.</li></ul> <p>Pro Unterschale gibt es <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 2l+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>l</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2l+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad6cc29a87d7b25ec1f294612477d5a38f59c09a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.859ex; height:2.343ex;" alt="{\displaystyle 2l+1}"></span> Orbitale (jeweils mit anderer Magnetquantenzahl <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 m_{l}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{l}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3f945c408d692391284a629617fe0b301776222" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.763ex; height:2.009ex;" alt="{\displaystyle m_{l}}"></span>, s. folgenden Abschnitt), was auf insgesamt <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 n^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac9810bbdafe4a6a8061338db0f74e25b7952620" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.449ex; height:2.676ex;" alt="{\displaystyle n^{2}}"></span> Orbitale pro Schale führt. </p> <div class="mw-heading mw-heading3"><h3 id="Magnetquantenzahl_ml:_Neigung_des_Drehimpulsvektors">Magnetquantenzahl <i>m<sub>l</sub></i>: Neigung des Drehimpulsvektors</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atomorbital&veaction=edit&section=7" title="Abschnitt bearbeiten: Magnetquantenzahl ml: Neigung des Drehimpulsvektors" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Atomorbital&action=edit&section=7" title="Quellcode des Abschnitts bearbeiten: Magnetquantenzahl ml: Neigung des Drehimpulsvektors"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Die Magnetquantenzahl </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 m_{l}=-l,-(l-1),\dotsc ,0,\dotsc ,(l-1),l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>l</mi> <mo>,</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>l</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mi>l</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{l}=-l,-(l-1),\dotsc ,0,\dotsc ,(l-1),l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47c44a10eff718391face41d51bf546721074300" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.462ex; height:2.843ex;" alt="{\displaystyle m_{l}=-l,-(l-1),\dotsc ,0,\dotsc ,(l-1),l}"></span></dd></dl> <p>gibt die z-Komponente <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 m_{l}\hbar }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mi class="MJX-variant">ℏ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{l}\hbar }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1c735c6c1af10c308f8e9753d69feb5726bab1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.069ex; height:2.509ex;" alt="{\displaystyle m_{l}\hbar }"></span> des Bahndrehimpulsvektors gegenüber einer (frei gewählten) z-Achse an. Das entspricht anschaulich einem Neigungswinkel </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 \cos \vartheta ={\frac {m_{l}}{\sqrt {l(l+1)}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>ϑ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <msqrt> <mi>l</mi> <mo stretchy="false">(</mo> <mi>l</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \vartheta ={\frac {m_{l}}{\sqrt {l(l+1)}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3225c70a4a4c56a0e3409d5943cc96d53dd1f7ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:18.976ex; height:6.009ex;" alt="{\displaystyle \cos \vartheta ={\frac {m_{l}}{\sqrt {l(l+1)}}}.}"></span></dd></dl> <ul><li>Bei <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 m_{l}=+l\Leftrightarrow \cos \vartheta ={\text{max}}\Leftrightarrow \vartheta \approx 0^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo>=</mo> <mo>+</mo> <mi>l</mi> <mo stretchy="false">⇔</mo> <mi>cos</mi> <mo>⁡</mo> <mi>ϑ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>max</mtext> </mrow> <mo stretchy="false">⇔</mo> <mi>ϑ</mi> <mo>≈</mo> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{l}=+l\Leftrightarrow \cos \vartheta ={\text{max}}\Leftrightarrow \vartheta \approx 0^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e18c7308e80f298653768d0af48dbb2d72f7e2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:34.576ex; height:2.676ex;" alt="{\displaystyle m_{l}=+l\Leftrightarrow \cos \vartheta ={\text{max}}\Leftrightarrow \vartheta \approx 0^{\circ }}"></span> liegt der Bahndrehimpuls (etwa) <a href="/wiki/Parallelit%C3%A4t_(Vektorrechnung)" class="mw-redirect" title="Parallelität (Vektorrechnung)">parallel</a> zur Achse,</li> <li>bei <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 m_{l}=-l\Leftrightarrow \cos \vartheta ={\text{min}}\Leftrightarrow \vartheta \approx 180^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>l</mi> <mo stretchy="false">⇔</mo> <mi>cos</mi> <mo>⁡</mo> <mi>ϑ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>min</mtext> </mrow> <mo stretchy="false">⇔</mo> <mi>ϑ</mi> <mo>≈</mo> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{l}=-l\Leftrightarrow \cos \vartheta ={\text{min}}\Leftrightarrow \vartheta \approx 180^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af1a59b54532cc25b9379496d4a291b928d10de0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:36.451ex; height:2.676ex;" alt="{\displaystyle m_{l}=-l\Leftrightarrow \cos \vartheta ={\text{min}}\Leftrightarrow \vartheta \approx 180^{\circ }}"></span> (etwa) <a href="/wiki/Antiparallelit%C3%A4t_(Vektorrechnung)" class="mw-redirect" title="Antiparallelität (Vektorrechnung)">antiparallel</a>.</li></ul> <p>Dass bei gegebenem <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 l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}"></span> genau <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 2l+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>l</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2l+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad6cc29a87d7b25ec1f294612477d5a38f59c09a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.859ex; height:2.343ex;" alt="{\displaystyle 2l+1}"></span> verschiedene Werte möglich sind, wird als <a href="/wiki/Richtungsquantelung" title="Richtungsquantelung">Richtungsquantelung</a> bezeichnet. </p><p>Wenn kein äußeres Feld anliegt, haben die <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 2l+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>l</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2l+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad6cc29a87d7b25ec1f294612477d5a38f59c09a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.859ex; height:2.343ex;" alt="{\displaystyle 2l+1}"></span> einzelnen Orbitale einer Unterschale gleiche Energie. Dagegen spaltet im <a href="/wiki/Magnetismus" title="Magnetismus">Magnetfeld</a> die Energie innerhalb der Unterschale in <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 2l+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>l</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2l+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad6cc29a87d7b25ec1f294612477d5a38f59c09a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.859ex; height:2.343ex;" alt="{\displaystyle 2l+1}"></span> äquidistante Werte auf (<a href="/wiki/Zeeman-Effekt" title="Zeeman-Effekt">Zeeman-Effekt</a>), d. h., jedes einzelne Orbital entspricht dann einem separaten Energieniveau. </p> <div class="mw-heading mw-heading3"><h3 id="Magnetische_Spinquantenzahl_ms">Magnetische Spinquantenzahl <i>m<sub>s</sub></i></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atomorbital&veaction=edit&section=8" title="Abschnitt bearbeiten: Magnetische Spinquantenzahl ms" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Atomorbital&action=edit&section=8" title="Quellcode des Abschnitts bearbeiten: Magnetische Spinquantenzahl ms"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bei den leichteren Atomen braucht man den <a href="/wiki/Elektronenspin" title="Elektronenspin">Elektronenspin</a> nur in der Form zu berücksichtigen, dass jedes Orbital <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 _{nlm_{l}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>l</mi> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{nlm_{l}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bd4aa6ba745a7323654a3b4957daa1a107ef11f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.227ex; height:2.843ex;" alt="{\displaystyle \psi _{nlm_{l}}}"></span> von genau einem <a href="/wiki/Elektronenpaar" title="Elektronenpaar">Elektronenpaar</a> besetzt werden kann, dessen zwei Elektronen nach dem Pauli-Prinzip entgegengesetzte magnetische Spinquantenzahlen aufweisen (<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 m_{s}=\pm {\tfrac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{s}=\pm {\tfrac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6bbed7f7b212caf8dcf5cbc6ca8aa4b7e52de0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.608ex; height:3.509ex;" alt="{\displaystyle m_{s}=\pm {\tfrac {1}{2}}}"></span>). </p> <div class="mw-heading mw-heading3"><h3 id="Gesamtdrehimpuls_j_und_magnetische_Quantenzahl_mj">Gesamtdrehimpuls <i>j</i> und magnetische Quantenzahl <i>m<sub>j</sub></i></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atomorbital&veaction=edit&section=9" title="Abschnitt bearbeiten: Gesamtdrehimpuls j und magnetische Quantenzahl mj" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Atomorbital&action=edit&section=9" title="Quellcode des Abschnitts bearbeiten: Gesamtdrehimpuls j und magnetische Quantenzahl mj"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Zu den schweren Atomen hin wird die <a href="/wiki/Spin-Bahn-Wechselwirkung" class="mw-redirect" title="Spin-Bahn-Wechselwirkung">Spin-Bahn-Wechselwirkung</a> stärker. Sie bewirkt die Aufspaltung der Energie einer Unterschale mit bestimmten <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 n>1,l>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>1</mn> <mo>,</mo> <mi>l</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>1,l>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30f61e6fd7d9652a0b8ab55325307814414bbc2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.644ex; height:2.509ex;" alt="{\displaystyle n>1,l>0}"></span> in zwei Unterschalen, je nach Wert des Gesamtdrehimpulses <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 j=l\pm {\tfrac {1}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>=</mo> <mi>l</mi> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j=l\pm {\tfrac {1}{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d038616d32270b8fb73a7a59c3a0510563b41526" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; margin-left: -0.027ex; width:9.922ex; height:3.509ex;" alt="{\displaystyle j=l\pm {\tfrac {1}{2}}.}"></span> Die magnetische Quantenzahl <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 m_{j}=-j,-(j-1),\dotsc ,+j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>j</mi> <mo>,</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>j</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mo>+</mo> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{j}=-j,-(j-1),\dotsc ,+j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20552ca184c9f5d8a6c5dd43d9e1f765d3b9e87a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.371ex; height:3.009ex;" alt="{\displaystyle m_{j}=-j,-(j-1),\dotsc ,+j}"></span> durchläuft <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 2j+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2j+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0a258cc33d54429124aedc154d9b3a968c4d99b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.123ex; height:2.509ex;" alt="{\displaystyle 2j+1}"></span> Werte. Jedes dieser Orbitale kann von einem Elektron besetzt werden, sodass die Gesamtzahl der Plätze gleich bleibt. In der Bezeichnung wird der Wert für <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 j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}"></span> als unterer Index an das Symbol für <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 nl}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle nl}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d32f8e8b4d4b18151191de7b1809a9d552c33d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.088ex; height:2.176ex;" alt="{\displaystyle nl}"></span> angefügt, z. B. <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 2p_{3/2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2p_{3/2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e22f4ce20c0df58cfbbe981eeab085d78620376a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.677ex; height:3.009ex;" alt="{\displaystyle 2p_{3/2}.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Quantentheorie">Quantentheorie</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atomorbital&veaction=edit&section=10" title="Abschnitt bearbeiten: Quantentheorie" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Atomorbital&action=edit&section=10" title="Quellcode des Abschnitts bearbeiten: Quantentheorie"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="hauptartikel" role="navigation"><span class="hauptartikel-pfeil" title="siehe" aria-hidden="true" role="presentation">→ </span><i><span class="hauptartikel-text">Hauptartikel</span>: <a href="/wiki/Wasserstoffproblem" class="mw-redirect" title="Wasserstoffproblem">Wasserstoffproblem</a></i></div> <p>Aus der nichtrelativistischen Quantentheorie ergeben sich die Orbitale wie folgt: Die Wechselwirkung zwischen Elektron und Atomkern wird durch das <a href="/wiki/Coulombsches_Gesetz" title="Coulombsches Gesetz">Coulombpotential</a> beschrieben, der Atomkern als fix angenommen. Der <a href="/wiki/Hamiltonoperator" title="Hamiltonoperator">Hamiltonoperator</a> für das <a href="/wiki/Ein-Elektron-System" title="Ein-Elektron-System">Ein-Elektron-System</a> ist </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 {\hat {H}}={\frac {{\hat {p}}^{2}}{2m}}+V(r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {H}}={\frac {{\hat {p}}^{2}}{2m}}+V(r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab540cf943cc57a3c6367579ac2d46eb2e822454" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.687ex; height:5.843ex;" alt="{\displaystyle {\hat {H}}={\frac {{\hat {p}}^{2}}{2m}}+V(r)}"></span></dd></dl> <p>mit dem Potential </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 V(r)={\frac {Ze}{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Z</mi> <mi>e</mi> </mrow> <mi>r</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(r)={\frac {Ze}{r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a8d0261a8f01dc1197ce2500b01a0223fc633e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.344ex; height:5.176ex;" alt="{\displaystyle V(r)={\frac {Ze}{r}}}"></span>.</dd></dl> <p>Da der Hamiltonoperator mit dem Drehimpulsoperator kommutiert, bilden <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 {\hat {H}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {H}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ec15924944b3f54b9aa0f4d6a902e6adbf0fae6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.71ex; height:3.176ex;" alt="{\displaystyle {\hat {H}},}"></span> <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 {\hat {l}}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>l</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {l}}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/841cd5221bab32d3f9ecb4537efc30e255f5381b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.41ex; height:3.343ex;" alt="{\displaystyle {\hat {l}}^{2}}"></span> und <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 {\hat {l}}_{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>l</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {l}}_{z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80858792edd88b1eba41017b1e83bee5050c776d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.358ex; height:3.176ex;" alt="{\displaystyle {\hat {l}}_{z}}"></span> ein <a href="/wiki/Vollst%C3%A4ndiger_Satz_kommutierender_Observablen" title="Vollständiger Satz kommutierender Observablen">vollständiges System kommutierender Observablen</a>. Zu diesen drei Operatoren gibt es also gemeinsame Eigenzustände, die durch die drei zugehörigen Quantenzahlen <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 n,l,m_{l}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>,</mo> <mi>l</mi> <mo>,</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n,l,m_{l}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/329258dfe3841b3bfe476dfe537c73b6b030e926" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.919ex; height:2.509ex;" alt="{\displaystyle n,l,m_{l}}"></span> bestimmt sind. </p><p>Die Schrödingergleichung </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 {\hat {H}}\cdot \psi _{n,l,m_{l}}(r,\vartheta ,\phi )=E_{n,l,m_{l}}\cdot \psi _{n,l,m_{l}}(r,\vartheta ,\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>l</mi> <mo>,</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>ϑ</mi> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>l</mi> <mo>,</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> </mrow> </msub> <mo>⋅</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>l</mi> <mo>,</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>ϑ</mi> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {H}}\cdot \psi _{n,l,m_{l}}(r,\vartheta ,\phi )=E_{n,l,m_{l}}\cdot \psi _{n,l,m_{l}}(r,\vartheta ,\phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bb9ee5fdfc45e819f68235d24b9811c6c06dad1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:42.517ex; height:3.509ex;" alt="{\displaystyle {\hat {H}}\cdot \psi _{n,l,m_{l}}(r,\vartheta ,\phi )=E_{n,l,m_{l}}\cdot \psi _{n,l,m_{l}}(r,\vartheta ,\phi )}"></span></dd></dl> <p>lässt sich in einen radius- und einen winkelabhängigen Teil zerlegen. Die Eigenfunktionen <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 _{n,l,m_{l}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>l</mi> <mo>,</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{n,l,m_{l}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1df9bca2492fe6ffeacf827e082aeb14a4160b7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.142ex; height:2.843ex;" alt="{\displaystyle \psi _{n,l,m_{l}}}"></span> sind das Produkt aus einer <a href="/wiki/Kugelfl%C3%A4chenfunktion" class="mw-redirect" title="Kugelflächenfunktion">Kugelflächenfunktion</a> <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 Y_{lm_{l}}(\vartheta ,\varphi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ϑ</mi> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{lm_{l}}(\vartheta ,\varphi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/456bf3150740a52b22170fe4361a631def3f2c33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.815ex; height:3.009ex;" alt="{\displaystyle Y_{lm_{l}}(\vartheta ,\varphi )}"></span> (Eigenfunktion des Drehimpulsoperators) und einer radialen Funktion <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 \Phi _{nl}(r)\colon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>l</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{nl}(r)\colon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c703ff1dfc73c9b9a4e05b798f8c4d14b8bf5daa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.892ex; height:2.843ex;" alt="{\displaystyle \Phi _{nl}(r)\colon }"></span> </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 _{n,l,m_{l}}(r,\vartheta ,\phi )=Y_{lm_{l}}(\vartheta ,\varphi )\cdot \Phi _{nl}(r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>l</mi> <mo>,</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>ϑ</mi> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ϑ</mi> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>l</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{n,l,m_{l}}(r,\vartheta ,\phi )=Y_{lm_{l}}(\vartheta ,\varphi )\cdot \Phi _{nl}(r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80f77e55babf5557c7aed10f831ba13825beca95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.664ex; height:3.009ex;" alt="{\displaystyle \psi _{n,l,m_{l}}(r,\vartheta ,\phi )=Y_{lm_{l}}(\vartheta ,\varphi )\cdot \Phi _{nl}(r)}"></span></dd></dl> <p>Diese sind bis <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 n{\mathord {=}}3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo>=</mo> </mrow> </mrow> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n{\mathord {=}}3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40c02c3f2e0259e24676c328fcfd303ede3dabda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.365ex; height:2.176ex;" alt="{\displaystyle n{\mathord {=}}3}"></span> in der folgenden Tabelle normiert dargestellt. Dabei bezeichnen <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 a_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/693ad9f934775838bd72406b41ada4a59785d7ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{0}}"></span> den <a href="/wiki/Bohrscher_Radius" title="Bohrscher Radius">Bohrschen Radius</a> und <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 Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}"></span> die Kernladungszahl. </p><p>Die in der folgenden Tabelle dargestellten Orbitale sind alle um die z-Achse ausgerichtet, weil es sich um Eigenfunktionen des <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 {\hat {l}}_{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>l</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {l}}_{z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80858792edd88b1eba41017b1e83bee5050c776d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.358ex; height:3.176ex;" alt="{\displaystyle {\hat {l}}_{z}}"></span>-Operators handelt. Für Ausrichtung eines Orbitals mit gegebenem Bahndrehimpuls <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 l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}"></span> in eine beliebige andere Richtung muss man Linearkombinationen der Wellenfunktionen zu den verschiedenen <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 m_{l}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{l}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3f945c408d692391284a629617fe0b301776222" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.763ex; height:2.009ex;" alt="{\displaystyle m_{l}}"></span> bilden. Die grafische Darstellung zeigt ein Volumen, auf dessen Oberfläche die Aufenthaltswahrscheinlichkeitsdichte <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 ({\vec {r}})|^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>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> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi ({\vec {r}})|^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f0223d0e4b6c5eaf1357117b8d140ef2b4f2439" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.893ex; height:3.343ex;" alt="{\displaystyle |\psi ({\vec {r}})|^{2}}"></span> konstant ist. Die Farben kodieren die komplexe Phase der Wellenfunktion. </p> <table class="wikitable"> <caption>Komplexe Wellenfunktionen in <a href="/wiki/Wasserstoff%C3%A4hnliches_Ion" class="mw-redirect" title="Wasserstoffähnliches Ion">wasserstoffähnlichen Atomen</a> </caption> <tbody><tr> <th>Orbital </th> <th colspan="4">Wellenfunktion des Orbitals </th> <th>Form des Orbitals <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 ({\vec {r}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi ({\vec {r}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d084550a5205083592573704216ae699e4cf7013" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.546ex; height:2.843ex;" alt="{\displaystyle \psi ({\vec {r}})}"></span><br />(nicht maßstäblich) </th></tr> <tr> <th></th> <th><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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span></th> <th><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 l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}"></span></th> <th><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 m_{l}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{l}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3f945c408d692391284a629617fe0b301776222" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.763ex; height:2.009ex;" alt="{\displaystyle m_{l}}"></span></th> <th><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 _{n,l,m_{l}}(r,\theta ,\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>l</mi> <mo>,</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>θ</mi> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{n,l,m_{l}}(r,\theta ,\phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b734b23eccf1e98e24f94cb196be94772c638514" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.543ex; height:3.009ex;" alt="{\displaystyle \psi _{n,l,m_{l}}(r,\theta ,\phi )}"></span></th> <th> </th></tr> <tr> <td>1s</td> <td>1</td> <td>0</td> <td><span style="visibility:hidden;">0</span>0</td> <td><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 {\frac {1}{\sqrt {\pi }}}\left({\frac {Z}{a_{0}}}\right)^{\frac {3}{2}}e^{-\textstyle {\frac {Zr}{a_{0}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mi>π</mi> </msqrt> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Z</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Z</mi> <mi>r</mi> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {\pi }}}\left({\frac {Z}{a_{0}}}\right)^{\frac {3}{2}}e^{-\textstyle {\frac {Zr}{a_{0}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/380a75ac0d2f723bab2684c4521515aa18d17237" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:17.741ex; height:7.676ex;" alt="{\displaystyle {\frac {1}{\sqrt {\pi }}}\left({\frac {Z}{a_{0}}}\right)^{\frac {3}{2}}e^{-\textstyle {\frac {Zr}{a_{0}}}}}"></span> </td> <td style="text-align:center"><span typeof="mw:File"><a href="/wiki/Datei:Hydrogen_eigenstate_n1_l0_m0_wedgecut.png" class="mw-file-description" title="1s-Orbital"><img alt="1s-Orbital" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/15/Hydrogen_eigenstate_n1_l0_m0_wedgecut.png/45px-Hydrogen_eigenstate_n1_l0_m0_wedgecut.png" decoding="async" width="45" height="45" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/15/Hydrogen_eigenstate_n1_l0_m0_wedgecut.png/68px-Hydrogen_eigenstate_n1_l0_m0_wedgecut.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/15/Hydrogen_eigenstate_n1_l0_m0_wedgecut.png/90px-Hydrogen_eigenstate_n1_l0_m0_wedgecut.png 2x" data-file-width="2560" data-file-height="2560" /></a></span> </td></tr> <tr> <td>2s</td> <td>2</td> <td>0</td> <td><span style="visibility:hidden;">0</span>0</td> <td><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 {\frac {1}{4{\sqrt {2\pi }}}}\left({\frac {Z}{a_{0}}}\right)^{\frac {3}{2}}\left(2-{\frac {Zr}{a_{0}}}\right)e^{-\textstyle {\frac {Zr}{2a_{0}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Z</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Z</mi> <mi>r</mi> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Z</mi> <mi>r</mi> </mrow> <mrow> <mn>2</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{4{\sqrt {2\pi }}}}\left({\frac {Z}{a_{0}}}\right)^{\frac {3}{2}}\left(2-{\frac {Zr}{a_{0}}}\right)e^{-\textstyle {\frac {Zr}{2a_{0}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8f6adee5a9d2a4ab318369bd1cb8e4d541d4947" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:32.423ex; height:7.676ex;" alt="{\displaystyle {\frac {1}{4{\sqrt {2\pi }}}}\left({\frac {Z}{a_{0}}}\right)^{\frac {3}{2}}\left(2-{\frac {Zr}{a_{0}}}\right)e^{-\textstyle {\frac {Zr}{2a_{0}}}}}"></span> </td> <td style="text-align:center"><span typeof="mw:File"><a href="/wiki/Datei:Hydrogen_eigenstate_n2_l0_m0_wedgecut.png" class="mw-file-description" title="2s-Orbital"><img alt="2s-Orbital" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Hydrogen_eigenstate_n2_l0_m0_wedgecut.png/75px-Hydrogen_eigenstate_n2_l0_m0_wedgecut.png" decoding="async" width="75" height="75" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Hydrogen_eigenstate_n2_l0_m0_wedgecut.png/113px-Hydrogen_eigenstate_n2_l0_m0_wedgecut.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/71/Hydrogen_eigenstate_n2_l0_m0_wedgecut.png/150px-Hydrogen_eigenstate_n2_l0_m0_wedgecut.png 2x" data-file-width="2560" data-file-height="2560" /></a></span> </td></tr> <tr> <td>2p<sub>0</sub></td> <td>2</td> <td>1</td> <td><span style="visibility:hidden;">0</span>0</td> <td><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 {\frac {1}{4{\sqrt {2\pi }}}}\left({\frac {Z}{a_{0}}}\right)^{\frac {3}{2}}{\frac {Zr}{a_{0}}}e^{-\textstyle {\frac {Zr}{2a_{0}}}}\cos \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Z</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Z</mi> <mi>r</mi> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Z</mi> <mi>r</mi> </mrow> <mrow> <mn>2</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> </msup> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{4{\sqrt {2\pi }}}}\left({\frac {Z}{a_{0}}}\right)^{\frac {3}{2}}{\frac {Zr}{a_{0}}}e^{-\textstyle {\frac {Zr}{2a_{0}}}}\cos \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4154f14f0d19df467c2e1043abb3d081739d2a78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:29.201ex; height:7.676ex;" alt="{\displaystyle {\frac {1}{4{\sqrt {2\pi }}}}\left({\frac {Z}{a_{0}}}\right)^{\frac {3}{2}}{\frac {Zr}{a_{0}}}e^{-\textstyle {\frac {Zr}{2a_{0}}}}\cos \theta }"></span> </td> <td style="text-align:center"><span typeof="mw:File"><a href="/wiki/Datei:Hydrogen_eigenstate_n2_l1_m0.png" class="mw-file-description" title="2p0-Orbital"><img alt="2p0-Orbital" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Hydrogen_eigenstate_n2_l1_m0.png/75px-Hydrogen_eigenstate_n2_l1_m0.png" decoding="async" width="75" height="75" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Hydrogen_eigenstate_n2_l1_m0.png/113px-Hydrogen_eigenstate_n2_l1_m0.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Hydrogen_eigenstate_n2_l1_m0.png/150px-Hydrogen_eigenstate_n2_l1_m0.png 2x" data-file-width="2560" data-file-height="2560" /></a></span> </td></tr> <tr> <td>2p<sub>−1/+1</sub></td> <td>2</td> <td>1</td> <td>±1</td> <td><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 {\frac {1}{8{\sqrt {\pi }}}}\left({\frac {Z}{a_{0}}}\right)^{\frac {3}{2}}{\frac {Zr}{a_{0}}}e^{-\textstyle {\frac {Zr}{2a_{0}}}}\sin \theta e^{\pm i\phi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>8</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>π</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Z</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Z</mi> <mi>r</mi> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Z</mi> <mi>r</mi> </mrow> <mrow> <mn>2</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> </msup> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> <mi>i</mi> <mi>ϕ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{8{\sqrt {\pi }}}}\left({\frac {Z}{a_{0}}}\right)^{\frac {3}{2}}{\frac {Zr}{a_{0}}}e^{-\textstyle {\frac {Zr}{2a_{0}}}}\sin \theta e^{\pm i\phi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b71f8c70d145fe5d4458e66c03abc7301f3f2936" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:31.924ex; height:7.676ex;" alt="{\displaystyle {\frac {1}{8{\sqrt {\pi }}}}\left({\frac {Z}{a_{0}}}\right)^{\frac {3}{2}}{\frac {Zr}{a_{0}}}e^{-\textstyle {\frac {Zr}{2a_{0}}}}\sin \theta e^{\pm i\phi }}"></span> </td> <td style="text-align:center"><span typeof="mw:File"><a href="/wiki/Datei:Hydrogen_eigenstate_n2_l1_m-1.png" class="mw-file-description" title="2p−1-Orbital"><img alt="2p−1-Orbital" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/Hydrogen_eigenstate_n2_l1_m-1.png/75px-Hydrogen_eigenstate_n2_l1_m-1.png" decoding="async" width="75" height="75" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/Hydrogen_eigenstate_n2_l1_m-1.png/113px-Hydrogen_eigenstate_n2_l1_m-1.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/07/Hydrogen_eigenstate_n2_l1_m-1.png/150px-Hydrogen_eigenstate_n2_l1_m-1.png 2x" data-file-width="2560" data-file-height="2560" /></a></span> <span typeof="mw:File"><a href="/wiki/Datei:Hydrogen_eigenstate_n2_l1_m1.png" class="mw-file-description" title="2p1-Orbital"><img alt="2p1-Orbital" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Hydrogen_eigenstate_n2_l1_m1.png/75px-Hydrogen_eigenstate_n2_l1_m1.png" decoding="async" width="75" height="75" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Hydrogen_eigenstate_n2_l1_m1.png/113px-Hydrogen_eigenstate_n2_l1_m1.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Hydrogen_eigenstate_n2_l1_m1.png/150px-Hydrogen_eigenstate_n2_l1_m1.png 2x" data-file-width="2560" data-file-height="2560" /></a></span> </td></tr> <tr> <td>3s</td> <td>3</td> <td>0</td> <td><span style="visibility:hidden;">0</span>0</td> <td><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 {\frac {1}{81{\sqrt {3\pi }}}}\left({\frac {Z}{a_{0}}}\right)^{\frac {3}{2}}\left(27-18{\frac {Zr}{a_{0}}}+2{\frac {Z^{2}r^{2}}{a_{0}^{2}}}\right)e^{-\textstyle {\frac {Zr}{3a_{0}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>81</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> <mi>π</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Z</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mn>27</mn> <mo>−</mo> <mn>18</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Z</mi> <mi>r</mi> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Z</mi> <mi>r</mi> </mrow> <mrow> <mn>3</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{81{\sqrt {3\pi }}}}\left({\frac {Z}{a_{0}}}\right)^{\frac {3}{2}}\left(27-18{\frac {Zr}{a_{0}}}+2{\frac {Z^{2}r^{2}}{a_{0}^{2}}}\right)e^{-\textstyle {\frac {Zr}{3a_{0}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbc1e919726c418a56f0ff670cce5d794a0bed7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:47.037ex; height:8.009ex;" alt="{\displaystyle {\frac {1}{81{\sqrt {3\pi }}}}\left({\frac {Z}{a_{0}}}\right)^{\frac {3}{2}}\left(27-18{\frac {Zr}{a_{0}}}+2{\frac {Z^{2}r^{2}}{a_{0}^{2}}}\right)e^{-\textstyle {\frac {Zr}{3a_{0}}}}}"></span> </td> <td style="text-align:center"><span typeof="mw:File"><a href="/wiki/Datei:Hydrogen_eigenstate_n3_l0_m0_wedgecut.png" class="mw-file-description" title="3s-Orbital"><img alt="3s-Orbital" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Hydrogen_eigenstate_n3_l0_m0_wedgecut.png/105px-Hydrogen_eigenstate_n3_l0_m0_wedgecut.png" decoding="async" width="105" height="105" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Hydrogen_eigenstate_n3_l0_m0_wedgecut.png/158px-Hydrogen_eigenstate_n3_l0_m0_wedgecut.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/85/Hydrogen_eigenstate_n3_l0_m0_wedgecut.png/210px-Hydrogen_eigenstate_n3_l0_m0_wedgecut.png 2x" data-file-width="2560" data-file-height="2560" /></a></span> </td></tr> <tr> <td>3p<sub>0</sub></td> <td>3</td> <td>1</td> <td><span style="visibility:hidden;">0</span>0</td> <td><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 {\frac {\sqrt {2}}{81{\sqrt {\pi }}}}\left({\frac {Z}{a_{0}}}\right)^{\frac {3}{2}}\left(6-{\frac {Zr}{a_{0}}}\right){\frac {Zr}{a_{0}}}e^{-\textstyle {\frac {Zr}{3a_{0}}}}\cos \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> </msqrt> <mrow> <mn>81</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>π</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Z</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mn>6</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Z</mi> <mi>r</mi> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Z</mi> <mi>r</mi> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Z</mi> <mi>r</mi> </mrow> <mrow> <mn>3</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> </msup> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sqrt {2}}{81{\sqrt {\pi }}}}\left({\frac {Z}{a_{0}}}\right)^{\frac {3}{2}}\left(6-{\frac {Zr}{a_{0}}}\right){\frac {Zr}{a_{0}}}e^{-\textstyle {\frac {Zr}{3a_{0}}}}\cos \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04935f23ffb25bb4670462a406d95e997d38a01b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:40.964ex; height:7.676ex;" alt="{\displaystyle {\frac {\sqrt {2}}{81{\sqrt {\pi }}}}\left({\frac {Z}{a_{0}}}\right)^{\frac {3}{2}}\left(6-{\frac {Zr}{a_{0}}}\right){\frac {Zr}{a_{0}}}e^{-\textstyle {\frac {Zr}{3a_{0}}}}\cos \theta }"></span> </td> <td style="text-align:center"><span typeof="mw:File"><a href="/wiki/Datei:Hydrogen_eigenstate_n3_l1_m0.png" class="mw-file-description" title="3p0-Orbital"><img alt="3p0-Orbital" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Hydrogen_eigenstate_n3_l1_m0.png/105px-Hydrogen_eigenstate_n3_l1_m0.png" decoding="async" width="105" height="105" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Hydrogen_eigenstate_n3_l1_m0.png/158px-Hydrogen_eigenstate_n3_l1_m0.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/21/Hydrogen_eigenstate_n3_l1_m0.png/210px-Hydrogen_eigenstate_n3_l1_m0.png 2x" data-file-width="2560" data-file-height="2560" /></a></span> </td></tr> <tr> <td>3p<sub>−1/+1</sub></td> <td>3</td> <td>1</td> <td>±1</td> <td><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 {\frac {1}{81{\sqrt {\pi }}}}\left({\frac {Z}{a_{0}}}\right)^{\frac {3}{2}}\left(6-{\frac {Zr}{a_{0}}}\right){\frac {Zr}{a_{0}}}e^{-\textstyle {\frac {Zr}{3a_{0}}}}\sin \theta e^{\pm i\phi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>81</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>π</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Z</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mn>6</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Z</mi> <mi>r</mi> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Z</mi> <mi>r</mi> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Z</mi> <mi>r</mi> </mrow> <mrow> <mn>3</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> </msup> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> <mi>i</mi> <mi>ϕ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{81{\sqrt {\pi }}}}\left({\frac {Z}{a_{0}}}\right)^{\frac {3}{2}}\left(6-{\frac {Zr}{a_{0}}}\right){\frac {Zr}{a_{0}}}e^{-\textstyle {\frac {Zr}{3a_{0}}}}\sin \theta e^{\pm i\phi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6989ec903fe1c484050668d3d0833660459bfe12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:44.85ex; height:7.676ex;" alt="{\displaystyle {\frac {1}{81{\sqrt {\pi }}}}\left({\frac {Z}{a_{0}}}\right)^{\frac {3}{2}}\left(6-{\frac {Zr}{a_{0}}}\right){\frac {Zr}{a_{0}}}e^{-\textstyle {\frac {Zr}{3a_{0}}}}\sin \theta e^{\pm i\phi }}"></span> </td> <td style="text-align:center"><span typeof="mw:File"><a href="/wiki/Datei:Hydrogen_eigenstate_n3_l1_m-1.png" class="mw-file-description" title="3p−1-Orbital"><img alt="3p−1-Orbital" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Hydrogen_eigenstate_n3_l1_m-1.png/105px-Hydrogen_eigenstate_n3_l1_m-1.png" decoding="async" width="105" height="105" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Hydrogen_eigenstate_n3_l1_m-1.png/158px-Hydrogen_eigenstate_n3_l1_m-1.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Hydrogen_eigenstate_n3_l1_m-1.png/210px-Hydrogen_eigenstate_n3_l1_m-1.png 2x" data-file-width="2560" data-file-height="2560" /></a></span> <span typeof="mw:File"><a href="/wiki/Datei:Hydrogen_eigenstate_n3_l1_m1.png" class="mw-file-description" title="3p1-Orbital"><img alt="3p1-Orbital" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Hydrogen_eigenstate_n3_l1_m1.png/105px-Hydrogen_eigenstate_n3_l1_m1.png" decoding="async" width="105" height="105" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Hydrogen_eigenstate_n3_l1_m1.png/158px-Hydrogen_eigenstate_n3_l1_m1.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/00/Hydrogen_eigenstate_n3_l1_m1.png/210px-Hydrogen_eigenstate_n3_l1_m1.png 2x" data-file-width="2560" data-file-height="2560" /></a></span> </td></tr> <tr> <td>3d<sub>0</sub></td> <td>3</td> <td>2</td> <td><span style="visibility:hidden;">0</span>0</td> <td><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 {\frac {1}{81{\sqrt {6\pi }}}}\left({\frac {Z}{a_{0}}}\right)^{\frac {3}{2}}{\frac {Z^{2}r^{2}}{a_{0}^{2}}}e^{-\textstyle {\frac {Zr}{3a_{0}}}}(3\cos ^{2}\theta -1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>81</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6</mn> <mi>π</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Z</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Z</mi> <mi>r</mi> </mrow> <mrow> <mn>3</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> </msup> <mo stretchy="false">(</mo> <mn>3</mn> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{81{\sqrt {6\pi }}}}\left({\frac {Z}{a_{0}}}\right)^{\frac {3}{2}}{\frac {Z^{2}r^{2}}{a_{0}^{2}}}e^{-\textstyle {\frac {Zr}{3a_{0}}}}(3\cos ^{2}\theta -1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be5c42268ccc20c5863448fbed90630d4fd9ccc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:40.528ex; height:7.843ex;" alt="{\displaystyle {\frac {1}{81{\sqrt {6\pi }}}}\left({\frac {Z}{a_{0}}}\right)^{\frac {3}{2}}{\frac {Z^{2}r^{2}}{a_{0}^{2}}}e^{-\textstyle {\frac {Zr}{3a_{0}}}}(3\cos ^{2}\theta -1)}"></span> </td> <td style="text-align:center"><span typeof="mw:File"><a href="/wiki/Datei:Hydrogen_eigenstate_n3_l2_m0.png" class="mw-file-description" title="3d0-Orbital"><img alt="3d0-Orbital" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Hydrogen_eigenstate_n3_l2_m0.png/105px-Hydrogen_eigenstate_n3_l2_m0.png" decoding="async" width="105" height="105" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Hydrogen_eigenstate_n3_l2_m0.png/158px-Hydrogen_eigenstate_n3_l2_m0.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Hydrogen_eigenstate_n3_l2_m0.png/210px-Hydrogen_eigenstate_n3_l2_m0.png 2x" data-file-width="2560" data-file-height="2560" /></a></span> </td></tr> <tr> <td>3d<sub>−1/+1</sub></td> <td>3</td> <td>2</td> <td>±1</td> <td><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 {\frac {1}{81{\sqrt {\pi }}}}\left({\frac {Z}{a_{0}}}\right)^{\frac {3}{2}}{\frac {Z^{2}r^{2}}{a_{0}^{2}}}e^{-\textstyle {\frac {Zr}{3a_{0}}}}\sin \theta \cos \theta e^{\pm i\phi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>81</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>π</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Z</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Z</mi> <mi>r</mi> </mrow> <mrow> <mn>3</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> </msup> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> <mi>i</mi> <mi>ϕ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{81{\sqrt {\pi }}}}\left({\frac {Z}{a_{0}}}\right)^{\frac {3}{2}}{\frac {Z^{2}r^{2}}{a_{0}^{2}}}e^{-\textstyle {\frac {Zr}{3a_{0}}}}\sin \theta \cos \theta e^{\pm i\phi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7ab7402d61ff88f8a180f2a3a39fb757682a900" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:40.199ex; height:7.843ex;" alt="{\displaystyle {\frac {1}{81{\sqrt {\pi }}}}\left({\frac {Z}{a_{0}}}\right)^{\frac {3}{2}}{\frac {Z^{2}r^{2}}{a_{0}^{2}}}e^{-\textstyle {\frac {Zr}{3a_{0}}}}\sin \theta \cos \theta e^{\pm i\phi }}"></span> </td> <td style="text-align:center"><span typeof="mw:File"><a href="/wiki/Datei:Hydrogen_eigenstate_n3_l2_m-1.png" class="mw-file-description" title="3d−1-Orbital"><img alt="3d−1-Orbital" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Hydrogen_eigenstate_n3_l2_m-1.png/105px-Hydrogen_eigenstate_n3_l2_m-1.png" decoding="async" width="105" height="105" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Hydrogen_eigenstate_n3_l2_m-1.png/158px-Hydrogen_eigenstate_n3_l2_m-1.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Hydrogen_eigenstate_n3_l2_m-1.png/210px-Hydrogen_eigenstate_n3_l2_m-1.png 2x" data-file-width="2560" data-file-height="2560" /></a></span> <span typeof="mw:File"><a href="/wiki/Datei:Hydrogen_eigenstate_n3_l2_m1.png" class="mw-file-description" title="3d1-Orbital"><img alt="3d1-Orbital" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Hydrogen_eigenstate_n3_l2_m1.png/105px-Hydrogen_eigenstate_n3_l2_m1.png" decoding="async" width="105" height="105" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Hydrogen_eigenstate_n3_l2_m1.png/158px-Hydrogen_eigenstate_n3_l2_m1.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Hydrogen_eigenstate_n3_l2_m1.png/210px-Hydrogen_eigenstate_n3_l2_m1.png 2x" data-file-width="2560" data-file-height="2560" /></a></span> </td></tr> <tr> <td>3d<sub>−2/+2</sub></td> <td>3</td> <td>2</td> <td>±2</td> <td><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 {\frac {1}{162{\sqrt {\pi }}}}\left({\frac {Z}{a_{0}}}\right)^{\frac {3}{2}}{\frac {Z^{2}r^{2}}{a_{0}^{2}}}e^{-\textstyle {\frac {Zr}{3a_{0}}}}\sin ^{2}\theta e^{\pm 2i\phi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>162</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>π</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Z</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Z</mi> <mi>r</mi> </mrow> <mrow> <mn>3</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> </msup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> <mn>2</mn> <mi>i</mi> <mi>ϕ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{162{\sqrt {\pi }}}}\left({\frac {Z}{a_{0}}}\right)^{\frac {3}{2}}{\frac {Z^{2}r^{2}}{a_{0}^{2}}}e^{-\textstyle {\frac {Zr}{3a_{0}}}}\sin ^{2}\theta e^{\pm 2i\phi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a0c88ce3b9852267260ccc4a6c6cb68d39653d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:38.262ex; height:7.843ex;" alt="{\displaystyle {\frac {1}{162{\sqrt {\pi }}}}\left({\frac {Z}{a_{0}}}\right)^{\frac {3}{2}}{\frac {Z^{2}r^{2}}{a_{0}^{2}}}e^{-\textstyle {\frac {Zr}{3a_{0}}}}\sin ^{2}\theta e^{\pm 2i\phi }}"></span> </td> <td style="text-align:center"><span typeof="mw:File"><a href="/wiki/Datei:Hydrogen_eigenstate_n3_l2_m-2.png" class="mw-file-description" title="3d−2-Orbital"><img alt="3d−2-Orbital" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Hydrogen_eigenstate_n3_l2_m-2.png/105px-Hydrogen_eigenstate_n3_l2_m-2.png" decoding="async" width="105" height="105" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Hydrogen_eigenstate_n3_l2_m-2.png/158px-Hydrogen_eigenstate_n3_l2_m-2.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Hydrogen_eigenstate_n3_l2_m-2.png/210px-Hydrogen_eigenstate_n3_l2_m-2.png 2x" data-file-width="2560" data-file-height="2560" /></a></span> <span typeof="mw:File"><a href="/wiki/Datei:Hydrogen_eigenstate_n3_l2_m2.png" class="mw-file-description" title="3d2-Orbital"><img alt="3d2-Orbital" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Hydrogen_eigenstate_n3_l2_m2.png/105px-Hydrogen_eigenstate_n3_l2_m2.png" decoding="async" width="105" height="105" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Hydrogen_eigenstate_n3_l2_m2.png/158px-Hydrogen_eigenstate_n3_l2_m2.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Hydrogen_eigenstate_n3_l2_m2.png/210px-Hydrogen_eigenstate_n3_l2_m2.png 2x" data-file-width="2560" data-file-height="2560" /></a></span> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Natürliches_Orbital"><span id="Nat.C3.BCrliches_Orbital"></span>Natürliches Orbital</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atomorbital&veaction=edit&section=11" title="Abschnitt bearbeiten: Natürliches Orbital" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Atomorbital&action=edit&section=11" title="Quellcode des Abschnitts bearbeiten: Natürliches Orbital"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ein natürliches Orbital ist ein Orbital, das sich nicht als Eigenfunktion eines Hamiltonoperators ergibt, sondern als Eigenfunktion eines <a href="/wiki/Dichteoperator#Einteilchendichteoperator" title="Dichteoperator">Einelektronen-Dichteoperators</a>. Dieser wird aus einem vorgegebenen Vielteilchenzustand gewonnen, der beispielsweise auch Elektronenkorrelationen enthalten kann und damit über den Rahmen eines Einzelteilchenmodells hinausgeht. Die mit den natürlichen Orbitalen gebildete <a href="/wiki/Elektronenkonfiguration" title="Elektronenkonfiguration">Elektronenkonfiguration</a> ergibt die beste Annäherung an den anfangs gegebenen Vielteilchenzustand, die mit einem Einzelteilchenmodell möglich ist. </p> <div class="mw-heading mw-heading2"><h2 id="Zeitabhängigkeit"><span id="Zeitabh.C3.A4ngigkeit"></span>Zeitabhängigkeit</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atomorbital&veaction=edit&section=12" title="Abschnitt bearbeiten: Zeitabhängigkeit" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Atomorbital&action=edit&section=12" title="Quellcode des Abschnitts bearbeiten: Zeitabhängigkeit"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Werden Orbitale als Eigenfunktionen eines Operators definiert, der zu einer Energie korrespondiert, dann sind diese Orbitale im Rahmen des gewählten Modells stationär. Beispiele hierfür sind die <a href="/wiki/Hartree-Fock-Methode" title="Hartree-Fock-Methode">Hartree-Fock</a>-Orbitale als Eigenfunktionen des Fockoperators <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 {\hat {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e22e0749dfc79fd15d8f156203a276fb7092fc51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.805ex; height:2.843ex;" alt="{\displaystyle {\hat {F}}}"></span> und die <a href="/wiki/Kohn-Sham-Gleichung" class="mw-redirect" title="Kohn-Sham-Gleichung">Kohn-Sham</a>-Orbitale, die Eigenfunktionen des Kohn-Sham-Hamilton-Operators sind. Im Gegensatz dazu sind die sogenannten <i>natürlichen Orbitale,</i> als Eigenfunktionen des reduzierten <a href="/wiki/Dichteoperator#Einteilchendichteoperator" title="Dichteoperator">Einelektronen-Dichteoperators</a>, nicht stationär. </p> <div class="mw-heading mw-heading2"><h2 id="Hybridisierung">Hybridisierung</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atomorbital&veaction=edit&section=13" title="Abschnitt bearbeiten: Hybridisierung" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Atomorbital&action=edit&section=13" title="Quellcode des Abschnitts bearbeiten: Hybridisierung"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Einige <a href="/wiki/Symmetrie_(Geometrie)" title="Symmetrie (Geometrie)">Symmetrien</a> von chemischen Bindungen scheinen den charakteristischen Formen der Orbitale zu widersprechen. Diese Bindungen werden durch die Bildung von <a href="/wiki/Hybridorbital" title="Hybridorbital">Hybrid-Orbitalen</a> verständlich, die sich bei Anwesenheit von Elektronen mit verschiedenem Bahndrehimpuls bilden können, wenn sie energetisch nahezu gleichwertig sind (siehe oben). </p> <div class="mw-heading mw-heading2"><h2 id="Mehr-Elektronen-Wellenfunktionen">Mehr-Elektronen-Wellenfunktionen</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atomorbital&veaction=edit&section=14" title="Abschnitt bearbeiten: Mehr-Elektronen-Wellenfunktionen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Atomorbital&action=edit&section=14" title="Quellcode des Abschnitts bearbeiten: Mehr-Elektronen-Wellenfunktionen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Die Interpretation von Orbitalen als Wellenfunktionen je eines Elektrons ist nur bei Einzelelektronensystemen eindeutig möglich. Eine Wellenfunktion für <i>N</i> Elektronen kann dann konstruiert werden, indem man <i>N</i> Orbitale in eine <a href="/wiki/Slater-Determinante" title="Slater-Determinante">Slater-Determinante</a> einsetzt. Dies garantiert die für <a href="/wiki/Fermion" title="Fermion">Fermionen</a> notwendige Antisymmetrie der gesamten Wellenfunktion, kann aber darüber hinaus gehende <a href="/wiki/Quantenverschr%C3%A4nkung" title="Quantenverschränkung">Elektronenkorrelationen</a> nicht darstellen. Um auch die Elektron-Elektron-Wechselwirkung näherungsweise zu berücksichtigen, können die Orbitale durch <a href="/wiki/Hartree-Fock-Methode" title="Hartree-Fock-Methode">Hartree-Fock</a>-, Kohn-Sham-Rechnungen (siehe: <a href="/wiki/Dichtefunktionaltheorie_(Quantenphysik)" title="Dichtefunktionaltheorie (Quantenphysik)">Dichtefunktionaltheorie in der Quantenphysik</a>) oder MCSCF-Rechnungen (MCSCF: Multiconfiguration Self Consistent Field) bestimmt werden. Doch stets bleibt gültig, dass anders gewählte Orbitale, wenn sie linear unabhängige <a href="/wiki/Linearkombination" title="Linearkombination">Linearkombinationen</a> der ursprünglichen sind, mathematisch die gleiche Slater-Determinante ergeben, sodass man aus einer gegebenen Mehrteilchen-Wellenfunktion nicht eindeutig zurückschließen kann, welches die einzelnen besetzten Orbitale sind. </p> <div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atomorbital&veaction=edit&section=15" title="Abschnitt bearbeiten: Literatur" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Atomorbital&action=edit&section=15" title="Quellcode des Abschnitts bearbeiten: Literatur"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Wolfgang_Demtr%C3%B6der" title="Wolfgang Demtröder">Wolfgang Demtröder</a>: <cite style="font-style:italic">Atome, Moleküle und Festkörper</cite>. 3. Auflage. Springer, 2002, <a href="/wiki/Spezial:ISBN-Suche/3540214739" class="internal mw-magiclink-isbn">ISBN 3-540-21473-9</a>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Atomorbital&rft.au=Wolfgang+Demtr%C3%B6der&rft.btitle=Atome%2C+Molek%C3%BCle+und+Festk%C3%B6rper&rft.date=2002&rft.edition=3&rft.genre=book&rft.isbn=3540214739&rft.pub=Springer" style="display:none"> </span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atomorbital&veaction=edit&section=16" title="Abschnitt bearbeiten: Weblinks" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Atomorbital&action=edit&section=16" title="Quellcode des Abschnitts bearbeiten: Weblinks"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="sisterproject" style="margin:0.1em 0 0 0;"><div class="noresize noviewer" style="display:inline-block; 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wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q53860" title="Link zum verbundenen Objekt im Datenrepositorium [g]" accesskey="g"><span>Wikidata-Datenobjekt</span></a></li> </ul> </div> </nav> <nav id="p-lang" class="mw-portlet mw-portlet-lang vector-menu-portal portal vector-menu" aria-labelledby="p-lang-label" > <h3 id="p-lang-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">In anderen Sprachen</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Atoomorbitaal" title="Atoomorbitaal – Afrikaans" lang="af" hreflang="af" data-title="Atoomorbitaal" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Orbital_atomico" title="Orbital atomico – Aragonesisch" lang="an" hreflang="an" data-title="Orbital atomico" data-language-autonym="Aragonés" data-language-local-name="Aragonesisch" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%AF%D8%A7%D8%B1_%D8%B0%D8%B1%D9%8A" title="مدار ذري – Arabisch" lang="ar" hreflang="ar" data-title="مدار ذري" data-language-autonym="العربية" data-language-local-name="Arabisch" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Orbital_at%C3%B3micu" title="Orbital atómicu – Asturisch" lang="ast" hreflang="ast" data-title="Orbital atómicu" data-language-autonym="Asturianu" data-language-local-name="Asturisch" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Atom_orbital%C4%B1" title="Atom orbitalı – Aserbaidschanisch" lang="az" hreflang="az" data-title="Atom orbitalı" data-language-autonym="Azərbaycanca" data-language-local-name="Aserbaidschanisch" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%90%D1%82%D0%B0%D0%BC%D0%BD%D0%B0%D1%8F_%D0%B0%D1%80%D0%B1%D1%96%D1%82%D0%B0%D0%BB%D1%8C" title="Атамная арбіталь – Belarussisch" lang="be" hreflang="be" data-title="Атамная арбіталь" data-language-autonym="Беларуская" data-language-local-name="Belarussisch" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%90%D1%82%D0%BE%D0%BC%D0%BD%D0%B0_%D0%BE%D1%80%D0%B1%D0%B8%D1%82%D0%B0%D0%BB%D0%B0" title="Атомна орбитала – Bulgarisch" lang="bg" hreflang="bg" data-title="Атомна орбитала" data-language-autonym="Български" data-language-local-name="Bulgarisch" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AA%E0%A6%BE%E0%A6%B0%E0%A6%AE%E0%A6%BE%E0%A6%A3%E0%A6%AC%E0%A6%BF%E0%A6%95_%E0%A6%95%E0%A6%95%E0%A7%8D%E0%A6%B7%E0%A6%95" title="পারমাণবিক কক্ষক – Bengalisch" lang="bn" hreflang="bn" data-title="পারমাণবিক কক্ষক" data-language-autonym="বাংলা" data-language-local-name="Bengalisch" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Atomska_orbitala" title="Atomska orbitala – Bosnisch" lang="bs" hreflang="bs" data-title="Atomska orbitala" data-language-autonym="Bosanski" data-language-local-name="Bosnisch" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Orbital_at%C3%B2mic" title="Orbital atòmic – Katalanisch" lang="ca" hreflang="ca" data-title="Orbital atòmic" data-language-autonym="Català" data-language-local-name="Katalanisch" 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/Atomov%C3%BD_orbital" title="Atomový orbital – Tschechisch" lang="cs" hreflang="cs" data-title="Atomový orbital" data-language-autonym="Čeština" data-language-local-name="Tschechisch" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%90%D1%82%D0%BE%D0%BC%D0%BB%D0%B0_%D0%BE%D1%80%D0%B1%D0%B8%D1%82%D0%B0%D0%BB%D1%8C" title="Атомла орбиталь – Tschuwaschisch" lang="cv" hreflang="cv" data-title="Атомла орбиталь" data-language-autonym="Чӑвашла" data-language-local-name="Tschuwaschisch" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Orbital_atomig" title="Orbital atomig – Walisisch" lang="cy" hreflang="cy" data-title="Orbital atomig" data-language-autonym="Cymraeg" data-language-local-name="Walisisch" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Kvantemekanisk_atommodel" title="Kvantemekanisk atommodel – Dänisch" lang="da" hreflang="da" data-title="Kvantemekanisk atommodel" data-language-autonym="Dansk" data-language-local-name="Dänisch" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CF%84%CE%BF%CE%BC%CE%B9%CE%BA%CF%8C_%CF%84%CF%81%CE%BF%CF%87%CE%B9%CE%B1%CE%BA%CF%8C" title="Ατομικό τροχιακό – Griechisch" lang="el" hreflang="el" data-title="Ατομικό τροχιακό" data-language-autonym="Ελληνικά" data-language-local-name="Griechisch" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Atomic_orbital" title="Atomic orbital – Englisch" lang="en" hreflang="en" data-title="Atomic orbital" data-language-autonym="English" data-language-local-name="Englisch" 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/Orbital_at%C3%B3mico" title="Orbital atómico – Spanisch" lang="es" hreflang="es" data-title="Orbital atómico" data-language-autonym="Español" data-language-local-name="Spanisch" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Aatomorbitaal" title="Aatomorbitaal – Estnisch" lang="et" hreflang="et" data-title="Aatomorbitaal" data-language-autonym="Eesti" data-language-local-name="Estnisch" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Orbital_atomiko" title="Orbital atomiko – Baskisch" lang="eu" hreflang="eu" data-title="Orbital atomiko" 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%A7%D9%88%D8%B1%D8%A8%DB%8C%D8%AA%D8%A7%D9%84_%D8%A7%D8%AA%D9%85%DB%8C" title="اوربیتال اتمی – Persisch" lang="fa" hreflang="fa" data-title="اوربیتال اتمی" data-language-autonym="فارسی" data-language-local-name="Persisch" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Atomiorbitaali" title="Atomiorbitaali – Finnisch" lang="fi" hreflang="fi" data-title="Atomiorbitaali" data-language-autonym="Suomi" data-language-local-name="Finnisch" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Orbitale_atomique" title="Orbitale atomique – Französisch" lang="fr" hreflang="fr" data-title="Orbitale atomique" data-language-autonym="Français" data-language-local-name="Französisch" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Orbital_at%C3%B3mico" title="Orbital atómico – Galicisch" lang="gl" hreflang="gl" data-title="Orbital atómico" data-language-autonym="Galego" data-language-local-name="Galicisch" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%95%D7%A8%D7%91%D7%99%D7%98%D7%9C_%D7%90%D7%98%D7%95%D7%9E%D7%99" title="אורביטל אטומי – Hebräisch" lang="he" hreflang="he" data-title="אורביטל אטומי" data-language-autonym="עברית" data-language-local-name="Hebräisch" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A4%B0%E0%A4%AE%E0%A4%BE%E0%A4%A3%E0%A5%81_%E0%A4%95%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A4%95" title="परमाणु कक्षक – Hindi" lang="hi" hreflang="hi" data-title="परमाणु कक्षक" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B1%D5%BF%D5%B8%D5%B4%D5%A1%D5%B5%D5%AB%D5%B6_%D6%85%D6%80%D5%A2%D5%AB%D5%BF%D5%A1%D5%AC" title="Ատոմային օրբիտալ – Armenisch" lang="hy" hreflang="hy" data-title="Ատոմային օրբիտալ" data-language-autonym="Հայերեն" data-language-local-name="Armenisch" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Orbital_atom" title="Orbital atom – Indonesisch" lang="id" hreflang="id" data-title="Orbital atom" 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/Orbitale_atomico" title="Orbitale atomico – Italienisch" lang="it" hreflang="it" data-title="Orbitale atomico" data-language-autonym="Italiano" data-language-local-name="Italienisch" 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/%E5%8E%9F%E5%AD%90%E8%BB%8C%E9%81%93" title="原子軌道 – Japanisch" lang="ja" hreflang="ja" data-title="原子軌道" data-language-autonym="日本語" data-language-local-name="Japanisch" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Amezzay_abelkam" title="Amezzay abelkam – Kabylisch" lang="kab" hreflang="kab" data-title="Amezzay abelkam" data-language-autonym="Taqbaylit" data-language-local-name="Kabylisch" class="interlanguage-link-target"><span>Taqbaylit</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9B%90%EC%9E%90_%EA%B6%A4%EB%8F%84" title="원자 궤도 – Koreanisch" lang="ko" hreflang="ko" data-title="원자 궤도" data-language-autonym="한국어" data-language-local-name="Koreanisch" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Orbitalis_atomica" title="Orbitalis atomica – Latein" lang="la" hreflang="la" data-title="Orbitalis atomica" data-language-autonym="Latina" data-language-local-name="Latein" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lt badge-Q70894304 mw-list-item" title=""><a href="https://lt.wikipedia.org/wiki/Atomo_orbital%C4%97" title="Atomo orbitalė – Litauisch" lang="lt" hreflang="lt" data-title="Atomo orbitalė" data-language-autonym="Lietuvių" data-language-local-name="Litauisch" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%90%D1%82%D0%BE%D0%BC%D1%81%D0%BA%D0%B0_%D0%BE%D1%80%D0%B1%D0%B8%D1%82%D0%B0%D0%BB%D0%B0" title="Атомска орбитала – Mazedonisch" lang="mk" hreflang="mk" data-title="Атомска орбитала" data-language-autonym="Македонски" data-language-local-name="Mazedonisch" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%86%E0%B4%B1%E0%B5%8D%E0%B4%B1%E0%B5%8B%E0%B4%AE%E0%B4%BF%E0%B4%95_%E0%B4%93%E0%B5%BC%E0%B4%AC%E0%B4%BF%E0%B4%B1%E0%B5%8D%E0%B4%B1%E0%B5%BD" title="ആറ്റോമിക ഓർബിറ്റൽ – Malayalam" lang="ml" hreflang="ml" data-title="ആറ്റോമിക ഓർബിറ്റൽ" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Orbital_atom" title="Orbital atom – Malaiisch" lang="ms" hreflang="ms" data-title="Orbital atom" data-language-autonym="Bahasa Melayu" data-language-local-name="Malaiisch" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl badge-Q70894304 mw-list-item" title=""><a href="https://nl.wikipedia.org/wiki/Atomaire_orbitaal" title="Atomaire orbitaal – Niederländisch" lang="nl" hreflang="nl" data-title="Atomaire orbitaal" data-language-autonym="Nederlands" data-language-local-name="Niederländisch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Orbitala_atomica" title="Orbitala atomica – Okzitanisch" lang="oc" hreflang="oc" data-title="Orbitala atomica" data-language-autonym="Occitan" data-language-local-name="Okzitanisch" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%85%E0%A8%9F%E0%A8%BE%E0%A8%AE%E0%A8%BF%E0%A8%95_%E0%A8%86%E0%A8%B0%E0%A8%AC%E0%A9%80%E0%A8%9F%E0%A8%B2" title="ਅਟਾਮਿਕ ਆਰਬੀਟਲ – Punjabi" lang="pa" hreflang="pa" data-title="ਅਟਾਮਿਕ ਆਰਬੀਟਲ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl badge-Q70894304 mw-list-item" title=""><a href="https://pl.wikipedia.org/wiki/Orbital_atomowy" title="Orbital atomowy – Polnisch" lang="pl" hreflang="pl" data-title="Orbital atomowy" data-language-autonym="Polski" data-language-local-name="Polnisch" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%A7%DB%8C%D9%B9%D9%85%DB%8C_%D9%85%D8%AF%D8%A7%D8%B1" title="ایٹمی مدار – Westliches Panjabi" lang="pnb" hreflang="pnb" data-title="ایٹمی مدار" data-language-autonym="پنجابی" data-language-local-name="Westliches Panjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D8%A7%D8%AA%D9%88%D9%85%D9%8A_%D9%85%D8%AF%D8%A7%D8%B1" title="اتومي مدار – Paschtu" lang="ps" hreflang="ps" data-title="اتومي مدار" data-language-autonym="پښتو" data-language-local-name="Paschtu" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Orbital_at%C3%B4mico" title="Orbital atômico – Portugiesisch" lang="pt" hreflang="pt" data-title="Orbital atômico" data-language-autonym="Português" data-language-local-name="Portugiesisch" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Orbital_atomic" title="Orbital atomic – Rumänisch" lang="ro" hreflang="ro" data-title="Orbital atomic" data-language-autonym="Română" data-language-local-name="Rumänisch" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%90%D1%82%D0%BE%D0%BC%D0%BD%D0%B0%D1%8F_%D0%BE%D1%80%D0%B1%D0%B8%D1%82%D0%B0%D0%BB%D1%8C" title="Атомная орбиталь – Russisch" lang="ru" hreflang="ru" data-title="Атомная орбиталь" data-language-autonym="Русский" data-language-local-name="Russisch" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Atomska_orbitala" title="Atomska orbitala – Serbokroatisch" lang="sh" hreflang="sh" data-title="Atomska orbitala" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbokroatisch" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Atomic_orbital" title="Atomic orbital – einfaches Englisch" lang="en-simple" hreflang="en-simple" data-title="Atomic orbital" data-language-autonym="Simple English" data-language-local-name="einfaches Englisch" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/At%C3%B3mov%C3%BD_orbit%C3%A1l" title="Atómový orbitál – Slowakisch" lang="sk" hreflang="sk" data-title="Atómový orbitál" data-language-autonym="Slovenčina" data-language-local-name="Slowakisch" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl badge-Q70894304 mw-list-item" title=""><a href="https://sl.wikipedia.org/wiki/Atomska_orbitala" title="Atomska orbitala – Slowenisch" lang="sl" hreflang="sl" data-title="Atomska orbitala" data-language-autonym="Slovenščina" data-language-local-name="Slowenisch" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%90%D1%82%D0%BE%D0%BC%D1%81%D0%BA%D0%B0_%D0%BE%D1%80%D0%B1%D0%B8%D1%82%D0%B0%D0%BB%D0%B0" title="Атомска орбитала – Serbisch" lang="sr" hreflang="sr" data-title="Атомска орбитала" data-language-autonym="Српски / srpski" data-language-local-name="Serbisch" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Atomorbital" title="Atomorbital – Schwedisch" lang="sv" hreflang="sv" data-title="Atomorbital" data-language-autonym="Svenska" data-language-local-name="Schwedisch" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%8E%E0%AE%B2%E0%AE%95%E0%AF%8D%E0%AE%9F%E0%AF%8D%E0%AE%B0%E0%AE%BE%E0%AE%A9%E0%AF%8D_%E0%AE%9A%E0%AF%81%E0%AE%B1%E0%AF%8D%E0%AE%B1%E0%AF%81%E0%AE%B5%E0%AE%9F%E0%AF%8D%E0%AE%9F%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AE%BE%E0%AE%A4%E0%AF%88" title="எலக்ட்ரான் சுற்றுவட்டப்பாதை – Tamil" lang="ta" hreflang="ta" data-title="எலக்ட்ரான் சுற்றுவட்டப்பாதை" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%85%E0%B0%9F%E0%B0%BE%E0%B0%AE%E0%B0%BF%E0%B0%95%E0%B1%8D_%E0%B0%86%E0%B0%B0%E0%B1%8D%E0%B0%AC%E0%B0%BF%E0%B0%9F%E0%B0%BE%E0%B0%B2%E0%B1%8D" title="అటామిక్ ఆర్బిటాల్ – Telugu" lang="te" hreflang="te" data-title="అటామిక్ ఆర్బిటాల్" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%AD%E0%B8%AD%E0%B8%A3%E0%B9%8C%E0%B8%9A%E0%B8%B4%E0%B8%97%E0%B8%B1%E0%B8%A5%E0%B9%80%E0%B8%8A%E0%B8%B4%E0%B8%87%E0%B8%AD%E0%B8%B0%E0%B8%95%E0%B8%AD%E0%B8%A1" title="ออร์บิทัลเชิงอะตอม – Thailändisch" lang="th" hreflang="th" data-title="ออร์บิทัลเชิงอะตอม" data-language-autonym="ไทย" data-language-local-name="Thailändisch" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Atomikong_orbital" title="Atomikong orbital – Tagalog" lang="tl" hreflang="tl" data-title="Atomikong orbital" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Atomik_orbital" title="Atomik orbital – Türkisch" lang="tr" hreflang="tr" data-title="Atomik orbital" data-language-autonym="Türkçe" data-language-local-name="Türkisch" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%90%D1%82%D0%BE%D0%BC%D0%BD%D0%B0_%D0%BE%D1%80%D0%B1%D1%96%D1%82%D0%B0%D0%BB%D1%8C" title="Атомна орбіталь – Ukrainisch" lang="uk" hreflang="uk" data-title="Атомна орбіталь" data-language-autonym="Українська" data-language-local-name="Ukrainisch" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Atom_orbitali" title="Atom orbitali – Usbekisch" lang="uz" hreflang="uz" data-title="Atom orbitali" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Usbekisch" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Orbital_nguy%C3%AAn_t%E1%BB%AD" title="Orbital nguyên tử – Vietnamesisch" lang="vi" hreflang="vi" data-title="Orbital nguyên tử" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamesisch" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%8E%9F%E5%AD%90%E8%BD%A8%E9%81%93" title="原子轨道 – Wu" lang="wuu" hreflang="wuu" data-title="原子轨道" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%8E%9F%E5%AD%90%E8%BD%A8%E9%81%93" title="原子轨道 – Chinesisch" lang="zh" hreflang="zh" data-title="原子轨道" data-language-autonym="中文" data-language-local-name="Chinesisch" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E5%8E%9F%E5%AD%90%E8%BB%8C%E5%9F%9F" title="原子軌域 – Klassisches Chinesisch" lang="lzh" hreflang="lzh" data-title="原子軌域" data-language-autonym="文言" data-language-local-name="Klassisches Chinesisch" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Go%C3%A2n-ch%C3%BA_k%C3%BAi-t%C5%8D" title="Goân-chú kúi-tō – Min Nan" lang="nan" hreflang="nan" data-title="Goân-chú kúi-tō" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Min Nan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%8E%9F%E5%AD%90%E8%BB%8C%E9%81%93" title="原子軌道 – Kantonesisch" lang="yue" hreflang="yue" data-title="原子軌道" data-language-autonym="粵語" data-language-local-name="Kantonesisch" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q53860#sitelinks-wikipedia" title="Links auf Artikel in anderen Sprachen bearbeiten" class="wbc-editpage">Links bearbeiten</a></span></div> </div> </nav> </div> </div> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Diese Seite wurde zuletzt am 22. 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Atomorbital – Wikipedia