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</button> <ul id="toc-Electron_properties-sublist" class="vector-toc-list"> <li id="toc-Formal_quantum_mechanical_definition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Formal_quantum_mechanical_definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Formal quantum mechanical definition</span> </div> </a> <ul id="toc-Formal_quantum_mechanical_definition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Types_of_orbital" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Types_of_orbital"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Types of orbital</span> </div> </a> <ul id="toc-Types_of_orbital-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>History</span> </div> </a> <button aria-controls="toc-History-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle History subsection</span> </button> <ul id="toc-History-sublist" class="vector-toc-list"> <li id="toc-Early_models" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Early_models"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Early models</span> </div> </a> <ul id="toc-Early_models-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bohr_atom" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bohr_atom"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Bohr atom</span> </div> </a> <ul id="toc-Bohr_atom-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Modern_conceptions_and_connections_to_the_Heisenberg_uncertainty_principle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Modern_conceptions_and_connections_to_the_Heisenberg_uncertainty_principle"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Modern conceptions and connections to the Heisenberg uncertainty principle</span> </div> </a> <ul id="toc-Modern_conceptions_and_connections_to_the_Heisenberg_uncertainty_principle-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Orbital_names" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Orbital_names"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Orbital names</span> </div> </a> <button aria-controls="toc-Orbital_names-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Orbital names subsection</span> </button> <ul id="toc-Orbital_names-sublist" class="vector-toc-list"> <li id="toc-Orbital_notation_and_subshells" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Orbital_notation_and_subshells"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Orbital notation and subshells</span> </div> </a> <ul id="toc-Orbital_notation_and_subshells-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-X-ray_notation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#X-ray_notation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>X-ray notation</span> </div> </a> <ul id="toc-X-ray_notation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Hydrogen-like_orbitals" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Hydrogen-like_orbitals"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Hydrogen-like orbitals</span> </div> </a> <ul id="toc-Hydrogen-like_orbitals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quantum_numbers" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Quantum_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Quantum numbers</span> </div> </a> <button aria-controls="toc-Quantum_numbers-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Quantum numbers subsection</span> </button> <ul id="toc-Quantum_numbers-sublist" class="vector-toc-list"> <li id="toc-Complex_orbitals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Complex_orbitals"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Complex orbitals</span> </div> </a> <ul id="toc-Complex_orbitals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Real_orbitals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Real_orbitals"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Real orbitals</span> </div> </a> <ul id="toc-Real_orbitals-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Shapes_of_orbitals" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Shapes_of_orbitals"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Shapes of orbitals</span> </div> </a> <button aria-controls="toc-Shapes_of_orbitals-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Shapes of orbitals subsection</span> </button> <ul id="toc-Shapes_of_orbitals-sublist" class="vector-toc-list"> <li id="toc-Orbitals_table" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Orbitals_table"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Orbitals table</span> </div> </a> <ul id="toc-Orbitals_table-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Qualitative_understanding_of_shapes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Qualitative_understanding_of_shapes"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Qualitative understanding of shapes</span> </div> </a> <ul id="toc-Qualitative_understanding_of_shapes-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Orbital_energy" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Orbital_energy"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Orbital energy</span> </div> </a> <ul id="toc-Orbital_energy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Electron_placement_and_the_periodic_table" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Electron_placement_and_the_periodic_table"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Electron placement and the periodic table</span> </div> </a> <button aria-controls="toc-Electron_placement_and_the_periodic_table-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Electron placement and the periodic table subsection</span> </button> <ul id="toc-Electron_placement_and_the_periodic_table-sublist" class="vector-toc-list"> <li id="toc-Relativistic_effects" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relativistic_effects"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Relativistic effects</span> </div> </a> <ul id="toc-Relativistic_effects-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-pp_hybridization_(conjectured)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#pp_hybridization_(conjectured)"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>pp hybridization (conjectured)</span> </div> </a> <ul id="toc-pp_hybridization_(conjectured)-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Transitions_between_orbitals" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Transitions_between_orbitals"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Transitions between orbitals</span> </div> </a> <ul id="toc-Transitions_between_orbitals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Atomic orbital</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 64 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-64" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">64 languages</span> </label> <div class="vector-dropdown-content"> <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-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 – Arabic" lang="ar" hreflang="ar" data-title="مدار ذري" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Orbital_atomico" title="Orbital atomico – Aragonese" lang="an" hreflang="an" data-title="Orbital atomico" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target"><span>Aragonés</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 – Asturian" lang="ast" hreflang="ast" data-title="Orbital atómicu" data-language-autonym="Asturianu" data-language-local-name="Asturian" 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 – Azerbaijani" lang="az" hreflang="az" data-title="Atom orbitalı" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</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 – Bangla" lang="bn" hreflang="bn" data-title="পারমাণবিক কক্ষক" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</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 – Minnan" lang="nan" hreflang="nan" data-title="Goân-chú kúi-tō" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</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 – Belarusian" lang="be" hreflang="be" data-title="Атамная арбіталь" data-language-autonym="Беларуская" data-language-local-name="Belarusian" 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 – Bulgarian" lang="bg" hreflang="bg" data-title="Атомна орбитала" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Atomska_orbitala" title="Atomska orbitala – Bosnian" lang="bs" hreflang="bs" data-title="Atomska orbitala" data-language-autonym="Bosanski" data-language-local-name="Bosnian" 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 – Catalan" lang="ca" hreflang="ca" data-title="Orbital atòmic" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</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 – Chuvash" lang="cv" hreflang="cv" data-title="Атомла орбиталь" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</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 – Czech" lang="cs" hreflang="cs" data-title="Atomový orbital" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Orbital_atomig" title="Orbital atomig – Welsh" lang="cy" hreflang="cy" data-title="Orbital atomig" data-language-autonym="Cymraeg" data-language-local-name="Welsh" 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 – Danish" lang="da" hreflang="da" data-title="Kvantemekanisk atommodel" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Atomorbital" title="Atomorbital – German" lang="de" hreflang="de" data-title="Atomorbital" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Aatomorbitaal" title="Aatomorbitaal – Estonian" lang="et" hreflang="et" data-title="Aatomorbitaal" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</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 – Greek" lang="el" hreflang="el" data-title="Ατομικό τροχιακό" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</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 – Spanish" lang="es" hreflang="es" data-title="Orbital atómico" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Orbital_atomiko" title="Orbital atomiko – Basque" lang="eu" hreflang="eu" data-title="Orbital atomiko" data-language-autonym="Euskara" data-language-local-name="Basque" 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 – Persian" lang="fa" hreflang="fa" data-title="اوربیتال اتمی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Orbitale_atomique" title="Orbitale atomique – French" lang="fr" hreflang="fr" data-title="Orbitale atomique" data-language-autonym="Français" data-language-local-name="French" 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 – Galician" lang="gl" hreflang="gl" data-title="Orbital atómico" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</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 – Korean" lang="ko" hreflang="ko" data-title="원자 궤도" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</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 – Armenian" lang="hy" hreflang="hy" data-title="Ատոմային օրբիտալ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</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-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Orbital_atom" title="Orbital atom – Indonesian" lang="id" hreflang="id" data-title="Orbital atom" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" 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 – Italian" lang="it" hreflang="it" data-title="Orbitale atomico" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</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 – Hebrew" lang="he" hreflang="he" data-title="אורביטל אטומי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Orbitalis_atomica" title="Orbitalis atomica – Latin" lang="la" hreflang="la" data-title="Orbitalis atomica" data-language-autonym="Latina" data-language-local-name="Latin" 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 – Lithuanian" lang="lt" hreflang="lt" data-title="Atomo orbitalė" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" 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 – Macedonian" lang="mk" hreflang="mk" data-title="Атомска орбитала" data-language-autonym="Македонски" data-language-local-name="Macedonian" 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 – Malay" lang="ms" hreflang="ms" data-title="Orbital atom" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" 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 – Dutch" lang="nl" hreflang="nl" data-title="Atomaire orbitaal" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</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 – Japanese" lang="ja" hreflang="ja" data-title="原子軌道" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Orbitala_atomica" title="Orbitala atomica – Occitan" lang="oc" hreflang="oc" data-title="Orbitala atomica" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Atom_orbitali" title="Atom orbitali – Uzbek" lang="uz" hreflang="uz" data-title="Atom orbitali" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</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-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 – Western Punjabi" lang="pnb" hreflang="pnb" data-title="ایٹمی مدار" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" 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 – Pashto" lang="ps" hreflang="ps" data-title="اتومي مدار" data-language-autonym="پښتو" data-language-local-name="Pashto" class="interlanguage-link-target"><span>پښتو</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 – Polish" lang="pl" hreflang="pl" data-title="Orbital atomowy" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Orbital_at%C3%B4mico" title="Orbital atômico – Portuguese" lang="pt" hreflang="pt" data-title="Orbital atômico" data-language-autonym="Português" data-language-local-name="Portuguese" 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 – Romanian" lang="ro" hreflang="ro" data-title="Orbital atomic" data-language-autonym="Română" data-language-local-name="Romanian" 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 – Russian" lang="ru" hreflang="ru" data-title="Атомная орбиталь" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Atomic_orbital" title="Atomic orbital – Simple English" lang="en-simple" hreflang="en-simple" data-title="Atomic orbital" data-language-autonym="Simple English" data-language-local-name="Simple English" 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 – Slovak" lang="sk" hreflang="sk" data-title="Atómový orbitál" data-language-autonym="Slovenčina" data-language-local-name="Slovak" 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 – Slovenian" lang="sl" hreflang="sl" data-title="Atomska orbitala" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" 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 – Serbian" lang="sr" hreflang="sr" data-title="Атомска орбитала" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Atomska_orbitala" title="Atomska orbitala – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Atomska orbitala" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Atomiorbitaali" title="Atomiorbitaali – Finnish" lang="fi" hreflang="fi" data-title="Atomiorbitaali" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Atomorbital" title="Atomorbital – Swedish" lang="sv" hreflang="sv" data-title="Atomorbital" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</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-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-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Amezzay_abelkam" title="Amezzay abelkam – Kabyle" lang="kab" hreflang="kab" data-title="Amezzay abelkam" data-language-autonym="Taqbaylit" data-language-local-name="Kabyle" class="interlanguage-link-target"><span>Taqbaylit</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 – Thai" lang="th" hreflang="th" data-title="ออร์บิทัลเชิงอะตอม" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Atomik_orbital" title="Atomik orbital – Turkish" lang="tr" hreflang="tr" data-title="Atomik orbital" data-language-autonym="Türkçe" data-language-local-name="Turkish" 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 – Ukrainian" lang="uk" hreflang="uk" data-title="Атомна орбіталь" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</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 – Vietnamese" lang="vi" hreflang="vi" data-title="Orbital nguyên tử" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</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 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="原子軌域" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</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-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 – Cantonese" lang="yue" hreflang="yue" data-title="原子軌道" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</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 – Chinese" lang="zh" hreflang="zh" data-title="原子轨道" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</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="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div 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data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">"Orbital shell" redirects here. For the collection of spaceflight orbits, see <a href="/wiki/Satellite_constellation" title="Satellite constellation">Orbital shell (spaceflight)</a>.</div> <p class="mw-empty-elt"> </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Neon_orbitals.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/58/Neon_orbitals.png/500px-Neon_orbitals.png" decoding="async" width="440" height="121" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/58/Neon_orbitals.png/960px-Neon_orbitals.png 1.5x" data-file-width="1920" data-file-height="526" /></a><figcaption>The shapes of the first five atomic orbitals are 1s, 2s, 2p<sub>x</sub>, 2p<sub>y</sub>, and 2p<sub>z</sub>. The two colors show the phase or sign of the wave function in each region. Each picture is <a href="/wiki/Domain_coloring" title="Domain coloring">domain coloring</a> of a <span class="texhtml">ψ(<i>x</i>, <i>y</i>, <i>z</i>)</span> function which depends on the coordinates of one electron. To see the elongated shape of <span class="texhtml">ψ(<i>x</i>, <i>y</i>, <i>z</i>)<sup>2</sup></span> functions that show <a href="/wiki/Probability_density" class="mw-redirect" title="Probability density">probability density</a> more directly, see pictures of d-orbitals below.</figcaption></figure> <p>In <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, an <b>atomic orbital</b> (<span class="rt-commentedText nowrap"><span class="IPA nopopups noexcerpt" lang="en-fonipa"><a href="/wiki/Help:IPA/English" title="Help:IPA/English">/<span style="border-bottom:1px dotted"><span title="/ˈ/: primary stress follows">ˈ</span><span title="/ɔːr/: 'ar' in 'war'">ɔːr</span><span title="'b' in 'buy'">b</span><span title="/ɪ/: 'i' in 'kit'">ɪ</span><span title="'t' in 'tie'">t</span><span title="/ə/: 'a' in 'about'">ə</span><span title="'l' in 'lie'">l</span></span>/</a></span> <span class="ext-phonos"><span data-nosnippet="" id="ooui-php-1" class="noexcerpt ext-phonos-PhonosButton ext-phonos-PhonosButton-emptylabel oo-ui-widget oo-ui-widget-enabled oo-ui-buttonElement oo-ui-buttonElement-frameless oo-ui-iconElement oo-ui-buttonWidget" data-ooui="{"_":"mw.Phonos.PhonosButton","href":"\/\/upload.wikimedia.org\/wikipedia\/commons\/transcoded\/e\/e4\/LL-Q1860_%28eng%29-Flame%2C_not_lame-Atomic_orbital.wav\/LL-Q1860_%28eng%29-Flame%2C_not_lame-Atomic_orbital.wav.mp3","rel":["nofollow"],"framed":false,"icon":"volumeUp","data":{"ipa":"","text":"","lang":"en","wikibase":"","file":"LL-Q1860 (eng)-Flame, not lame-Atomic orbital.wav"},"classes":["noexcerpt","ext-phonos-PhonosButton","ext-phonos-PhonosButton-emptylabel"]}"><a role="button" tabindex="0" href="//upload.wikimedia.org/wikipedia/commons/transcoded/e/e4/LL-Q1860_%28eng%29-Flame%2C_not_lame-Atomic_orbital.wav/LL-Q1860_%28eng%29-Flame%2C_not_lame-Atomic_orbital.wav.mp3" rel="nofollow" aria-label="Play audio" title="Play audio" class="oo-ui-buttonElement-button"><span class="oo-ui-iconElement-icon oo-ui-icon-volumeUp"></span><span class="oo-ui-labelElement-label"></span><span class="oo-ui-indicatorElement-indicator oo-ui-indicatorElement-noIndicator"></span></a></span><sup class="ext-phonos-attribution noexcerpt navigation-not-searchable"><a href="/wiki/File:LL-Q1860_(eng)-Flame,_not_lame-Atomic_orbital.wav" title="File:LL-Q1860 (eng)-Flame, not lame-Atomic orbital.wav">ⓘ</a></sup></span></span>) is a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> describing the location and <a href="/wiki/Matter_wave" title="Matter wave">wave-like behavior</a> of an <a href="/wiki/Electron" title="Electron">electron</a> in an <a href="/wiki/Atom" title="Atom">atom</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> This function describes an electron's <a href="/wiki/Charge_density" title="Charge density">charge distribution</a> around the <a href="/wiki/Atomic_nucleus" title="Atomic nucleus">atom's nucleus</a>, and can be used to calculate the <a href="/wiki/Probability" title="Probability">probability</a> of finding an electron in a specific region around the nucleus.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Each orbital in an atom is characterized by a set of values of three <a href="/wiki/Quantum_number" title="Quantum number">quantum numbers</a> <span class="texhtml mvar" style="font-style:italic;">n</span>, <span class="texhtml mvar" style="font-style:italic;">ℓ</span>, and <span class="texhtml mvar" style="font-style:italic;">m<sub>ℓ</sub></span>, which respectively correspond to electron's energy, its <a href="/wiki/Angular_momentum" title="Angular momentum">orbital angular momentum</a>, and its orbital angular momentum projected along a chosen axis (<a href="/wiki/Magnetic_quantum_number" title="Magnetic quantum number">magnetic quantum number</a>). The orbitals with a well-defined magnetic quantum number are generally complex-valued. Real-valued orbitals can be formed as linear combinations of <span class="texhtml mvar" style="font-style:italic;">m<sub>ℓ</sub></span> and <span class="texhtml mvar" style="font-style:italic;">−m<sub>ℓ</sub></span> orbitals, and are often labeled using associated <a href="/wiki/Spherical_harmonics#Harmonic_polynomial_representation" title="Spherical harmonics">harmonic polynomials</a> (e.g., <i>xy</i>, <span class="nowrap"><i>x</i><sup>2</sup> − <i>y</i><sup>2</sup></span>) which describe their angular structure. </p><p>An orbital can be occupied by a maximum of two electrons, each with its own <a href="/wiki/Spin_quantum_number" title="Spin quantum number">projection of spin</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_{s}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{s}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/488560816fccdf62695552ac8bf0611a0d5c09b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.044ex; height:2.009ex;" alt="{\displaystyle m_{s}}" /></span>. The simple names <b>s orbital</b>, <b>p orbital</b>, <b>d orbital</b>, and <b>f orbital</b> refer to orbitals with angular momentum quantum number <span class="texhtml"><i>ℓ</i> = 0, 1, 2,</span> and <span class="texhtml">3</span> respectively. These names, together with their n values, are used to describe <a href="/wiki/Electron_configuration" title="Electron configuration">electron configurations</a> of atoms. They are derived from description by early spectroscopists of certain series of <a href="/wiki/Alkali_metal" title="Alkali metal">alkali metal</a> <a href="/wiki/Spectral_line" title="Spectral line">spectroscopic lines</a> as <a href="/wiki/Sharp_series" title="Sharp series">sharp</a>, <a href="/wiki/Principal_series_(spectroscopy)" title="Principal series (spectroscopy)">principal</a>, <a href="/wiki/Diffuse_series" title="Diffuse series">diffuse</a>, and <a href="/wiki/Fundamental_series" title="Fundamental series">fundamental</a>. Orbitals for <span class="texhtml"><i>ℓ</i> > 3</span> continue alphabetically (g, h, i, k, ...),<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> omitting j<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> because some languages do not distinguish between letters "i" and "j".<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>Atomic orbitals are basic building blocks of the <b>atomic orbital model</b> (or electron cloud or wave mechanics model), a modern framework for visualizing submicroscopic behavior of electrons in matter. In this model, the electron cloud of an atom may be seen as being built up (in approximation) in an electron configuration that is a product of simpler <a href="/wiki/Hydrogen_atom" title="Hydrogen atom">hydrogen-like</a> atomic orbitals. The repeating <i>periodicity</i> of blocks of 2, 6, 10, and 14 <a href="/wiki/Chemical_element" title="Chemical element">elements</a> within sections of <a href="/wiki/Periodic_table" title="Periodic table">periodic table</a> arises naturally from total number of electrons that occupy a complete set of s, p, d, and f orbitals, respectively, though for higher values of quantum number <span class="texhtml mvar" style="font-style:italic;">n</span>, particularly when the atom bears a positive charge, energies of certain sub-shells become very similar and so, <a href="/wiki/Aufbau_principle" title="Aufbau principle">order</a> in which they are said to be populated by electrons (e.g., <a href="/wiki/Chromium" title="Chromium">Cr</a> = [Ar]4s<sup>1</sup>3d<sup>5</sup> and Cr<sup>2+</sup> = [Ar]3d<sup>4</sup>) can be rationalized only somewhat arbitrarily. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Hydrogen_Density_Plots.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Hydrogen_Density_Plots.png/330px-Hydrogen_Density_Plots.png" decoding="async" width="330" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Hydrogen_Density_Plots.png/500px-Hydrogen_Density_Plots.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Hydrogen_Density_Plots.png/960px-Hydrogen_Density_Plots.png 2x" data-file-width="2200" data-file-height="2000" /></a><figcaption>Cross-sections of atomic orbitals of the electron in a hydrogen atom at different energy levels. The probability of finding the electron is given by the color, as shown in the key at upper right.</figcaption></figure> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Electron_properties">Electron properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atomic_orbital&action=edit&section=1" title="Edit section: Electron properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>With the development of <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> and experimental findings (such as the two slit diffraction of electrons), it was found that the electrons orbiting a nucleus could not be fully described as particles, but needed to be explained by <a href="/wiki/Wave%E2%80%93particle_duality" title="Wave–particle duality">wave–particle duality</a>. In this sense, electrons have the following properties: </p><p><b>Wave-like properties:</b> </p> <ol><li>Electrons do not orbit a nucleus in the manner of a planet orbiting a star, but instead exist as <a href="/wiki/Standing_wave" title="Standing wave">standing waves</a>. Thus the lowest possible energy an electron can take is similar to the <a href="/wiki/Fundamental_frequency" title="Fundamental frequency">fundamental frequency</a> of a wave on a string. Higher energy states are similar to <a href="/wiki/Harmonics" class="mw-redirect" title="Harmonics">harmonics</a> of that fundamental frequency.</li> <li>The electrons are never in a single point location, though the probability of interacting with the electron at a single point can be found from the electron's <a href="/wiki/Wave_function" title="Wave function">wave function</a>. The electron's charge acts like it is smeared out in space in a continuous distribution, proportional at any point to the squared magnitude of the electron's wave function.</li></ol> <p><b>Particle-like properties:</b> </p> <ol><li>The number of electrons orbiting a nucleus can be only an integer.</li> <li>Electrons jump between orbitals like particles. For example, if one <a href="/wiki/Photon" title="Photon">photon</a> strikes the electrons, only one electron changes state as a result.</li> <li>Electrons retain particle-like properties such as: each wave state has the same electric charge as its electron particle. Each wave state has a single discrete spin (spin up or spin down) depending on its <a href="/wiki/Quantum_superposition" title="Quantum superposition">superposition</a>.</li></ol> <p>Thus, electrons cannot be described simply as solid particles. An analogy might be that of a large and often oddly shaped "atmosphere" (the electron), distributed around a relatively tiny planet (the nucleus). Atomic orbitals exactly describe the shape of this "atmosphere" only when one electron is present. When more electrons are added, the additional electrons tend to more evenly fill in a volume of space around the nucleus so that the resulting collection ("electron cloud"<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>) tends toward a generally spherical zone of probability describing the electron's location, because of the <a href="/wiki/Uncertainty_principle" title="Uncertainty principle">uncertainty principle</a>. </p><p>One should remember that these orbital 'states', as described here, are merely <a href="/wiki/Eigenstates" class="mw-redirect" title="Eigenstates">eigenstates</a> of an electron in its orbit. An actual electron exists in a superposition of states, which is like a <a href="/wiki/Weighted_average" class="mw-redirect" title="Weighted average">weighted average</a>, but with <a href="/wiki/Complex_number" title="Complex number">complex number</a> weights. So, for instance, an electron could be in a pure eigenstate (2, 1, 0), or a mixed state <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>(2, 1, 0) + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" /><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</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 i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" /></span> (2, 1, 1), or even the mixed state <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" /><span class="sfrac">⁠<span class="tion"><span class="num">2</span><span class="sr-only">/</span><span class="den">5</span></span>⁠</span>(2, 1, 0) + <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" /><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">5</span></span>⁠</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 i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" /></span> (2, 1, 1). For each eigenstate, a property has an <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalue</a>. So, for the three states just mentioned, the value of <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> is 2, and the value of <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> is 1. For the second and third states, the value for <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> is a superposition of 0 and 1. As a superposition of states, it is ambiguous—either exactly 0 or exactly 1—not an intermediate or average value like the fraction <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" /><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>. A superposition of <a href="/wiki/Quantum_state" title="Quantum state">eigenstates</a> (2, 1, 1) and (3, 2, 1) would have an ambiguous <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> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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>, but <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> would definitely be 1. Eigenstates make it easier to deal with the math. You can choose a different <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a> of eigenstates by superimposing eigenstates from any other basis (see <a class="mw-selflink-fragment" href="#Real_orbitals">Real orbitals</a> below). </p> <div class="mw-heading mw-heading3"><h3 id="Formal_quantum_mechanical_definition">Formal quantum mechanical definition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atomic_orbital&action=edit&section=2" title="Edit section: Formal quantum mechanical definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Atomic orbitals may be defined more precisely in formal <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanical</a> language. They are approximate solutions to the <a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a> for the electrons bound to the atom by the <a href="/wiki/Electric_field" title="Electric field">electric field</a> of the atom's <a href="/wiki/Atomic_nucleus" title="Atomic nucleus">nucleus</a>. Specifically, in quantum mechanics, the state of an atom, i.e., an <a href="/wiki/Eigenstate" class="mw-redirect" title="Eigenstate">eigenstate</a> of the atomic <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a>, is approximated by an expansion (see <a href="/wiki/Configuration_interaction" title="Configuration interaction">configuration interaction</a> expansion and <a href="/wiki/Basis_set_(chemistry)" title="Basis set (chemistry)">basis set</a>) into <a href="/wiki/Linear_combination" title="Linear combination">linear combinations</a> of anti-symmetrized products (<a href="/wiki/Slater_determinant" title="Slater determinant">Slater determinants</a>) of one-electron functions. The spatial components of these one-electron functions are called atomic orbitals. (When one considers also their <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin</a> component, one speaks of <b>atomic spin orbitals</b>.) A state is actually a function of the coordinates of all the electrons, so that their motion is correlated, but this is often approximated by this <a href="/wiki/Nuclear_structure#The_independent-particle_model" title="Nuclear structure">independent-particle model</a> of products of single electron wave functions.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> (The <a href="/wiki/London_dispersion_force" title="London dispersion force">London dispersion force</a>, for example, depends on the correlations of the motion of the electrons.) </p><p>In <a href="/wiki/Atomic_physics" title="Atomic physics">atomic physics</a>, the <a href="/wiki/Atomic_spectral_line" class="mw-redirect" title="Atomic spectral line">atomic spectral lines</a> correspond to transitions (<a href="/wiki/Atomic_electron_transition" title="Atomic electron transition">quantum leaps</a>) between <a href="/wiki/Quantum_state" title="Quantum state">quantum states</a> of an atom. These states are labeled by a set of <a href="/wiki/Quantum_number" title="Quantum number">quantum numbers</a> summarized in the <a href="/wiki/Term_symbol" title="Term symbol">term symbol</a> and usually associated with particular electron configurations, i.e., by occupation schemes of atomic orbitals (for example, 1s<sup>2</sup> 2s<sup>2</sup> 2p<sup>6</sup> for the ground state of <a href="/wiki/Neon" title="Neon">neon</a>-term symbol: <sup>1</sup>S<sub>0</sub>). </p><p>This notation means that the corresponding Slater determinants have a clear higher weight in the <a href="/wiki/Configuration_interaction" title="Configuration interaction">configuration interaction</a> expansion. The atomic orbital concept is therefore a key concept for visualizing the excitation process associated with a given <a href="/wiki/Atomic_electron_transition" title="Atomic electron transition">transition</a>. For example, one can say for a given transition that it corresponds to the excitation of an electron from an occupied orbital to a given unoccupied orbital. Nevertheless, one has to keep in mind that electrons are <a href="/wiki/Fermion" title="Fermion">fermions</a> ruled by the <a href="/wiki/Pauli_exclusion_principle" title="Pauli exclusion principle">Pauli exclusion principle</a> and cannot be distinguished from each other.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Moreover, it sometimes happens that the configuration interaction expansion converges very slowly and that one cannot speak about simple one-determinant wave function at all. This is the case when <a href="/wiki/Electron_correlation" class="mw-redirect" title="Electron correlation">electron correlation</a> is large. </p><p>Fundamentally, an atomic orbital is a one-electron wave function, even though many electrons are not in one-electron atoms, and so the one-electron view is an approximation. When thinking about orbitals, we are often given an orbital visualization heavily influenced by the <a href="/wiki/Hartree%E2%80%93Fock" class="mw-redirect" title="Hartree–Fock">Hartree–Fock</a> approximation, which is one way to reduce the complexities of <a href="/wiki/Molecular_orbital_theory" title="Molecular orbital theory">molecular orbital theory</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Types_of_orbital">Types of orbital</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atomic_orbital&action=edit&section=3" title="Edit section: Types of orbital"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Atomic-orbital-clouds_spdf_m0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Atomic-orbital-clouds_spdf_m0.png/330px-Atomic-orbital-clouds_spdf_m0.png" decoding="async" width="330" height="330" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Atomic-orbital-clouds_spdf_m0.png/500px-Atomic-orbital-clouds_spdf_m0.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Atomic-orbital-clouds_spdf_m0.png/960px-Atomic-orbital-clouds_spdf_m0.png 2x" data-file-width="1600" data-file-height="1600" /></a><figcaption>3D views of some <a href="/wiki/Hydrogen-like_atom" title="Hydrogen-like atom">hydrogen-like</a> atomic orbitals showing probability density and phase (<b>g</b> orbitals and higher not shown)</figcaption></figure> <p>Atomic orbitals can be the hydrogen-like "orbitals" which are exact solutions to the <a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a> for a <a href="/wiki/Hydrogen-like_atom" title="Hydrogen-like atom">hydrogen-like "atom"</a> (i.e., atom with one electron). Alternatively, atomic orbitals refer to functions that depend on the coordinates of one electron (i.e., orbitals) but are used as starting points for approximating wave functions that depend on the simultaneous coordinates of all the electrons in an atom or molecule. The <a href="/wiki/Coordinate_system" title="Coordinate system">coordinate systems</a> chosen for orbitals are usually <a href="/wiki/Spherical_coordinates" class="mw-redirect" title="Spherical coordinates">spherical coordinates</a> <span class="texhtml">(<i>r</i>, <i>θ</i>, <i>φ</i>)</span> in atoms and <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian</a> <span class="texhtml">(<i>x</i>, <i>y</i>, <i>z</i>)</span> in polyatomic molecules. The advantage of spherical coordinates here is that an orbital wave function is a product of three factors each dependent on a single coordinate: <span class="texhtml"><i>ψ</i>(<i>r</i>, <i>θ</i>, <i>φ</i>) = <i>R</i>(<i>r</i>) Θ(<i>θ</i>) Φ(<i>φ</i>)</span>. The angular factors of atomic orbitals <span class="texhtml">Θ(<i>θ</i>) Φ(<i>φ</i>)</span> generate s, p, d, etc. functions as <a href="/wiki/Spherical_harmonics#Real_form" title="Spherical harmonics">real combinations</a> of <a href="/wiki/Spherical_harmonics" title="Spherical harmonics">spherical harmonics</a> <span class="texhtml"><i>Y</i><sub><i>ℓm</i></sub>(<i>θ</i>, <i>φ</i>)</span> (where <span class="texhtml mvar" style="font-style:italic;">ℓ</span> and <span class="texhtml mvar" style="font-style:italic;">m</span> are quantum numbers). There are typically three mathematical forms for the radial functions <span class="texhtml"><i>R</i>(<i>r</i>)</span> which can be chosen as a starting point for the calculation of the properties of atoms and molecules with many electrons: </p> <ol><li>The <i>hydrogen-like orbitals</i> are derived from the exact solutions of the Schrödinger equation for one electron and a nucleus, for a <a href="/wiki/Hydrogen-like_atom" title="Hydrogen-like atom">hydrogen-like atom</a>. The part of the function that depends on distance <i>r</i> from the nucleus has radial <a href="/wiki/Node_(physics)" title="Node (physics)">nodes</a> and decays as <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 e^{-\alpha r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> <mi>r</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-\alpha r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7776e1b5bceb04f54b4e64404b0ce543dfd1ff91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.388ex; height:2.509ex;" alt="{\displaystyle e^{-\alpha r}}" /></span>.</li> <li>The <a href="/wiki/Slater-type_orbital" title="Slater-type orbital">Slater-type orbital</a> (STO) is a form without radial nodes but decays from the nucleus as does a hydrogen-like orbital.</li> <li>The form of the <a href="/wiki/Gaussian_orbital" title="Gaussian orbital">Gaussian type orbital</a> (Gaussians) has no radial nodes and decays as <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 e^{-\alpha r^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-\alpha r^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94e7b2b81bb12dea3549c0e684afb95ea33d626d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.219ex; height:3.009ex;" alt="{\displaystyle e^{-\alpha r^{2}}}" /></span>.</li></ol> <p>Although hydrogen-like orbitals are still used as pedagogical tools, the advent of computers has made STOs preferable for atoms and diatomic molecules since combinations of STOs can replace the nodes in hydrogen-like orbitals. Gaussians are typically used in molecules with three or more atoms. Although not as accurate by themselves as STOs, combinations of many Gaussians can attain the accuracy of hydrogen-like orbitals. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atomic_orbital&action=edit&section=4" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Atomic_theory" class="mw-redirect" title="Atomic theory">Atomic theory</a></div> <p>The term <i>orbital</i> was introduced by <a href="/wiki/Robert_S._Mulliken" title="Robert S. Mulliken">Robert S. Mulliken</a> in 1932 as short for <i>one-electron orbital wave function</i>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Niels_Bohr" title="Niels Bohr">Niels Bohr</a> explained around 1913 that electrons might revolve around a compact nucleus with definite angular momentum.<sup id="cite_ref-Bohr_1913_476_12-0" class="reference"><a href="#cite_note-Bohr_1913_476-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Bohr's model was an improvement on the 1911 explanations of <a href="/wiki/Ernest_Rutherford" title="Ernest Rutherford">Ernest Rutherford</a>, that of the electron moving around a nucleus. Japanese physicist <a href="/wiki/Hantaro_Nagaoka" title="Hantaro Nagaoka">Hantaro Nagaoka</a> published an orbit-based hypothesis for electron behavior as early as 1904.<sup id="cite_ref-Nagaoka_1904_445–455_13-0" class="reference"><a href="#cite_note-Nagaoka_1904_445–455-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> These theories were each built upon new observations starting with simple understanding and becoming more correct and complex. Explaining the behavior of these electron "orbits" was one of the driving forces behind the development of <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Early_models">Early models</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atomic_orbital&action=edit&section=5" title="Edit section: Early models"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>With <a href="/wiki/J._J._Thomson" title="J. J. Thomson">J. J. Thomson</a>'s discovery of the electron in 1897,<sup id="cite_ref-referenceC_15-0" class="reference"><a href="#cite_note-referenceC-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> it became clear that atoms were not the <a href="/wiki/Elementary_particle" title="Elementary particle">smallest building blocks of nature</a>, but were rather composite particles. The newly discovered structure within atoms tempted many to imagine how the atom's constituent parts might interact with each other. Thomson theorized that multiple electrons revolve in orbit-like rings within a positively charged jelly-like substance,<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> and between the electron's discovery and 1909, this "<a href="/wiki/Plum_pudding_model" title="Plum pudding model">plum pudding model</a>" was the most widely accepted explanation of atomic structure. </p><p>Shortly after Thomson's discovery, <a href="/wiki/Hantaro_Nagaoka" title="Hantaro Nagaoka">Hantaro Nagaoka</a> predicted a different model for electronic structure.<sup id="cite_ref-Nagaoka_1904_445–455_13-1" class="reference"><a href="#cite_note-Nagaoka_1904_445–455-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> Unlike the plum pudding model, the positive charge in Nagaoka's "Saturnian Model" was concentrated into a central core, pulling the electrons into circular orbits reminiscent of Saturn's rings. Few people took notice of Nagaoka's work at the time,<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> and Nagaoka himself recognized a fundamental defect in the theory even at its conception, namely that a classical charged object cannot sustain orbital motion because it is accelerating and therefore loses energy due to electromagnetic radiation.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> Nevertheless, the <a href="/wiki/Saturnian_model" class="mw-redirect" title="Saturnian model">Saturnian model</a> turned out to have more in common with modern theory than any of its contemporaries. </p> <div class="mw-heading mw-heading3"><h3 id="Bohr_atom">Bohr atom</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atomic_orbital&action=edit&section=6" title="Edit section: Bohr atom"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 1909, <a href="/wiki/Ernest_Rutherford" title="Ernest Rutherford">Ernest Rutherford</a> discovered that the bulk of the atomic mass was tightly condensed into a nucleus, which was also found to be positively charged. It became clear from his analysis in 1911 that the plum pudding model could not explain atomic structure. In 1913, Rutherford's post-doctoral student, <a href="/wiki/Niels_Bohr" title="Niels Bohr">Niels Bohr</a>, proposed a new model of the atom, wherein electrons orbited the nucleus with classical periods, but were permitted to have only discrete values of angular momentum, quantized in units <a href="/wiki/Planck_constant" title="Planck constant">ħ</a>.<sup id="cite_ref-Bohr_1913_476_12-1" class="reference"><a href="#cite_note-Bohr_1913_476-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> This constraint automatically allowed only certain electron energies. The <a href="/wiki/Bohr_model" title="Bohr model">Bohr model</a> of the atom fixed the problem of energy loss from radiation from a ground state (by declaring that there was no state below this), and more importantly explained the origin of spectral lines. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Bohr_atom_model.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Bohr_atom_model.svg/250px-Bohr_atom_model.svg.png" decoding="async" width="220" height="192" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Bohr_atom_model.svg/330px-Bohr_atom_model.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/93/Bohr_atom_model.svg/440px-Bohr_atom_model.svg.png 2x" data-file-width="310" data-file-height="270" /></a><figcaption>The <a href="/wiki/Bohr_model" title="Bohr model">Rutherford–Bohr model</a> of the hydrogen atom</figcaption></figure> <p>After Bohr's use of <a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a>'s explanation of the <a href="/wiki/Photoelectric_effect" title="Photoelectric effect">photoelectric effect</a> to relate energy levels in atoms with the wavelength of emitted light, the connection between the structure of electrons in atoms and the <a href="/wiki/Emission_spectra" class="mw-redirect" title="Emission spectra">emission</a> and <a href="/wiki/Absorption_spectra" class="mw-redirect" title="Absorption spectra">absorption spectra</a> of atoms became an increasingly useful tool in the understanding of electrons in atoms. The most prominent feature of emission and absorption spectra (known experimentally since the middle of the 19th century), was that these atomic spectra contained discrete lines. The significance of the Bohr model was that it related the lines in emission and absorption spectra to the energy differences between the orbits that electrons could take around an atom. This was, however, <i>not</i> achieved by Bohr through giving the electrons some kind of wave-like properties, since the idea that electrons could behave as <a href="/wiki/Matter_waves" class="mw-redirect" title="Matter waves">matter waves</a> was not suggested until eleven years later. Still, the Bohr model's use of quantized angular momenta and therefore quantized energy levels was a significant step toward the understanding of electrons in atoms, and also a significant step towards the development of <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> in suggesting that quantized restraints must account for all discontinuous energy levels and spectra in atoms. </p><p>With <a href="/wiki/Louis_de_Broglie" title="Louis de Broglie">de Broglie</a>'s suggestion of the existence of electron matter waves in 1924, and for a short time before the full 1926 <a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a> treatment of <a href="/wiki/Hydrogen-like_atom" title="Hydrogen-like atom">hydrogen-like atoms</a>, a Bohr electron "wavelength" could be seen to be a function of its momentum; so a Bohr orbiting electron was seen to orbit in a circle at a multiple of its half-wavelength. The Bohr model for a short time could be seen as a classical model with an additional constraint provided by the 'wavelength' argument. However, this period was immediately superseded by the full three-dimensional wave mechanics of 1926. In our current understanding of physics, the Bohr model is called a semi-classical model because of its quantization of angular momentum, not primarily because of its relationship with electron wavelength, which appeared in hindsight a dozen years after the Bohr model was proposed. </p><p>The Bohr model was able to explain the emission and absorption spectra of <a href="/wiki/Hydrogen" title="Hydrogen">hydrogen</a>. The energies of electrons in the <i>n</i> = 1, 2, 3, etc. states in the Bohr model match those of current physics. However, this did not explain similarities between different atoms, as expressed by the periodic table, such as the fact that <a href="/wiki/Helium" title="Helium">helium</a> (two electrons), neon (10 electrons), and <a href="/wiki/Argon" title="Argon">argon</a> (18 electrons) exhibit similar chemical inertness. Modern <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> explains this in terms of <a href="/wiki/Electron_shell" title="Electron shell">electron shells</a> and subshells which can each hold a number of electrons determined by the <a href="/wiki/Pauli_exclusion_principle" title="Pauli exclusion principle">Pauli exclusion principle</a>. Thus the <i>n</i> = 1 state can hold one or two electrons, while the <i>n</i> = 2 state can hold up to eight electrons in 2s and 2p subshells. In helium, all <i>n</i> = 1 states are fully occupied; the same is true for <i>n</i> = 1 and <i>n</i> = 2 in neon. In argon, the 3s and 3p subshells are similarly fully occupied by eight electrons; quantum mechanics also allows a 3d subshell but this is at higher energy than the 3s and 3p in argon (contrary to the situation for hydrogen) and remains empty. </p> <div class="mw-heading mw-heading3"><h3 id="Modern_conceptions_and_connections_to_the_Heisenberg_uncertainty_principle">Modern conceptions and connections to the Heisenberg uncertainty principle</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atomic_orbital&action=edit&section=7" title="Edit section: Modern conceptions and connections to the Heisenberg uncertainty principle"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Immediately after <a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Heisenberg</a> discovered his <a href="/wiki/Uncertainty_principle" title="Uncertainty principle">uncertainty principle</a>,<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Niels_Bohr" title="Niels Bohr">Bohr</a> noted that the existence of any sort of <a href="/wiki/Wave_packet" title="Wave packet">wave packet</a> implies uncertainty in the wave frequency and wavelength, since a spread of frequencies is needed to create the packet itself.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> In quantum mechanics, where all particle momenta are associated with waves, it is the formation of such a wave packet which localizes the wave, and thus the particle, in space. In states where a quantum mechanical particle is bound, it must be localized as a wave packet, and the existence of the packet and its minimum size implies a spread and minimal value in particle wavelength, and thus also momentum and energy. In quantum mechanics, as a particle is localized to a smaller region in space, the associated compressed wave packet requires a larger and larger range of momenta, and thus larger kinetic energy. Thus the binding energy to contain or trap a particle in a smaller region of space increases without bound as the region of space grows smaller. Particles cannot be restricted to a geometric point in space, since this would require infinite particle momentum. </p><p>In chemistry, <a href="/wiki/Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Erwin Schrödinger</a>, <a href="/wiki/Linus_Pauling" title="Linus Pauling">Linus Pauling</a>, Mulliken and others noted that the consequence of Heisenberg's relation was that the electron, as a wave packet, could not be considered to have an exact location in its orbital. <a href="/wiki/Max_Born" title="Max Born">Max Born</a> suggested that the electron's position needed to be described by a <a href="/wiki/Probability_distribution" title="Probability distribution">probability distribution</a> which was connected with finding the electron at some point in the wave-function which described its associated wave packet. The new quantum mechanics did not give exact results, but only the probabilities for the occurrence of a variety of possible such results. Heisenberg held that the path of a moving particle has no meaning if we cannot observe it, as we cannot with electrons in an atom. </p><p>In the quantum picture of Heisenberg, Schrödinger and others, the Bohr atom number <i>n</i> for each orbital became known as an <i>n-sphere</i><sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (January 2013)">citation needed</span></a></i>]</sup> in a three-dimensional atom and was pictured as the most probable energy of the probability cloud of the electron's wave packet which surrounded the atom. </p> <div class="mw-heading mw-heading2"><h2 id="Orbital_names">Orbital names</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atomic_orbital&action=edit&section=8" title="Edit section: Orbital names"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Orbital_notation_and_subshells">Orbital notation and subshells</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atomic_orbital&action=edit&section=9" title="Edit section: Orbital notation and subshells"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Orbitals have been given names, which are usually given in the form: </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 X\,\mathrm {type} \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">y</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">e</mi> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\,\mathrm {type} \ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bae5f2be1604d6d2b4c67af5d3b021dd2a406c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.405ex; height:2.509ex;" alt="{\displaystyle X\,\mathrm {type} \ }" /></span></dd></dl> <p>where <i>X</i> is the energy level corresponding to the <a href="/wiki/Principal_quantum_number" title="Principal quantum number">principal quantum number</a> <span class="texhtml mvar" style="font-style:italic;">n</span>; <b>type</b> is a lower-case letter denoting the shape or <a href="/wiki/Electron_shell#Subshells" title="Electron shell">subshell</a> of the orbital, corresponding to the <a href="/wiki/Angular_momentum_quantum_number" class="mw-redirect" title="Angular momentum quantum number">angular momentum quantum number</a> <span class="texhtml mvar" style="font-style:italic;">ℓ</span>. </p><p>For example, the orbital 1s (pronounced as the individual numbers and letters: "'one' 'ess'") is the lowest energy level (<span class="texhtml"><i>n</i> = 1</span>) and has an angular quantum number of <span class="texhtml"><i>ℓ</i> = 0</span>, denoted as s. Orbitals with <span class="texhtml"><i>ℓ</i> = 1, 2 and 3</span> are denoted as p, d and f respectively. </p><p>The set of orbitals for a given n and <span class="texhtml mvar" style="font-style:italic;">ℓ</span> is called a <i>subshell</i>, denoted </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 X\,\mathrm {type} ^{y}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mspace width="thinmathspace"></mspace> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">y</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msup> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\,\mathrm {type} ^{y}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f5d601893713a5b90fb415b7a54941882727d9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.454ex; height:2.676ex;" alt="{\displaystyle X\,\mathrm {type} ^{y}\ }" /></span>.</dd></dl> <p>The superscript y shows the number of electrons in the subshell. For example, the notation 2p<sup>4</sup> indicates that the 2p subshell of an atom contains 4 electrons. This subshell has 3 orbitals, each with n = 2 and <span class="texhtml mvar" style="font-style:italic;">ℓ</span> = 1. </p> <div class="mw-heading mw-heading3"><h3 id="X-ray_notation">X-ray notation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atomic_orbital&action=edit&section=10" title="Edit section: X-ray notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/X-ray_notation" title="X-ray notation">X-ray notation</a></div> <p>There is also another, less common system still used in X-ray science known as <a href="/wiki/X-ray_notation" title="X-ray notation">X-ray notation</a>, which is a continuation of the notations used before orbital theory was well understood. In this system, the principal quantum number is given a letter associated with it. For <span class="texhtml"><i>n</i> = 1, 2, 3, 4, 5, ...</span>, the letters associated with those numbers are K, L, M, N, O, ... respectively. </p> <div class="mw-heading mw-heading2"><h2 id="Hydrogen-like_orbitals">Hydrogen-like orbitals</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atomic_orbital&action=edit&section=11" title="Edit section: Hydrogen-like orbitals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Hydrogen-like_atom" title="Hydrogen-like atom">Hydrogen-like atom</a></div> <p>The simplest atomic orbitals are those that are calculated for systems with a single electron, such as the <a href="/wiki/Hydrogen_atom" title="Hydrogen atom">hydrogen atom</a>. An atom of any other element <a href="/wiki/Ion" title="Ion">ionized</a> down to a single electron (He<sup>+</sup>, Li<sup>2+</sup>, etc.) is very similar to hydrogen, and the orbitals take the same form. In the Schrödinger equation for this system of one negative and one positive particle, the atomic orbitals are the <a href="/wiki/Eigenstates" class="mw-redirect" title="Eigenstates">eigenstates</a> of the <a href="/wiki/Hamiltonian_operator" class="mw-redirect" title="Hamiltonian operator">Hamiltonian operator</a> for the energy. They can be obtained analytically, meaning that the resulting orbitals are products of a <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> series, and <a href="/wiki/Exponential_function" title="Exponential function">exponential</a> and <a href="/wiki/Trigonometric_functions" title="Trigonometric functions">trigonometric functions</a>. (see <a href="/wiki/Hydrogen_atom" title="Hydrogen atom">hydrogen atom</a>). </p><p>For atoms with two or more electrons, the governing equations can be solved only with the use of methods of iterative approximation. Orbitals of multi-electron atoms are <i>qualitatively</i> similar to those of hydrogen, and in the simplest models, they are taken to have the same form. For more rigorous and precise analysis, numerical approximations must be used. </p><p>A given (hydrogen-like) atomic orbital is identified by unique values of three quantum numbers: <span class="texhtml mvar" style="font-style:italic;"><a href="/wiki/Principal_quantum_number" title="Principal quantum number">n</a></span>, <span class="texhtml mvar" style="font-style:italic;"><a href="/wiki/Azimuthal_quantum_number" title="Azimuthal quantum number">ℓ</a></span>, and <span class="texhtml mvar" style="font-style:italic;"><a href="/wiki/Magnetic_quantum_number" title="Magnetic quantum number">m<sub>ℓ</sub></a></span>. The rules restricting the values of the quantum numbers, and their energies (see below), explain the electron configuration of the atoms and the <a href="/wiki/Periodic_table" title="Periodic table">periodic table</a>. </p><p>The stationary states (<a href="/wiki/Quantum_state" title="Quantum state">quantum states</a>) of a hydrogen-like atom are its atomic orbitals. However, in general, an electron's behavior is not fully described by a single orbital. Electron states are best represented by time-depending "mixtures" (<a href="/wiki/Linear_combination" title="Linear combination">linear combinations</a>) of multiple orbitals. See <a href="/wiki/Linear_combination_of_atomic_orbitals_molecular_orbital_method" class="mw-redirect" title="Linear combination of atomic orbitals molecular orbital method">Linear combination of atomic orbitals molecular orbital method</a>. </p><p>The quantum number <span class="texhtml mvar" style="font-style:italic;">n</span> first appeared in the <a href="/wiki/Bohr_model" title="Bohr model">Bohr model</a> where it determines the radius of each circular electron orbit. In modern quantum mechanics however, <span class="texhtml mvar" style="font-style:italic;">n</span> determines the mean distance of the electron from the nucleus; all electrons with the same value of <i>n</i> lie at the same average distance. For this reason, orbitals with the same value of <i>n</i> are said to comprise a "<a href="/wiki/Electron_shell" title="Electron shell">shell</a>". Orbitals with the same value of <i>n</i> and also the same value of <span class="texhtml mvar" style="font-style:italic;">ℓ</span> are even more closely related, and are said to comprise a "<a href="/wiki/Electron_subshell" class="mw-redirect" title="Electron subshell">subshell</a>". </p> <div class="mw-heading mw-heading2"><h2 id="Quantum_numbers">Quantum numbers</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atomic_orbital&action=edit&section=12" title="Edit section: Quantum numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Quantum_number" title="Quantum number">Quantum number</a></div> <p>Because of the quantum mechanical nature of the electrons around a nucleus, atomic orbitals can be uniquely defined by a set of integers known as quantum numbers. These quantum numbers occur only in certain combinations of values, and their physical interpretation changes depending on whether <a href="/wiki/Real_number" title="Real number">real</a> or <a href="/wiki/Complex_number" title="Complex number">complex</a> versions of the atomic orbitals are employed. </p> <div class="mw-heading mw-heading3"><h3 id="Complex_orbitals">Complex orbitals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atomic_orbital&action=edit&section=13" title="Edit section: Complex orbitals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Electronic_levels.svg" class="mw-file-description"><img alt="Electronic levels" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Electronic_levels.svg/450px-Electronic_levels.svg.png" decoding="async" width="450" height="450" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Electronic_levels.svg/675px-Electronic_levels.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/de/Electronic_levels.svg/900px-Electronic_levels.svg.png 2x" data-file-width="512" data-file-height="512" /></a><figcaption>Energetic levels and sublevels of polyelectronic atoms</figcaption></figure> <p>In physics, the most common orbital descriptions are based on the solutions to the hydrogen atom, where orbitals are given by the product between a radial function and a pure <a href="/wiki/Spherical_harmonic" class="mw-redirect" title="Spherical harmonic">spherical harmonic</a>. The quantum numbers, together with the rules governing their possible values, are as follows: </p><p>The <a href="/wiki/Principal_quantum_number" title="Principal quantum number">principal quantum number</a> <span class="texhtml mvar" style="font-style:italic;">n</span> describes the energy of the electron and is always a <a href="/wiki/Positive_integer" class="mw-redirect" title="Positive integer">positive integer</a>. In fact, it can be any positive integer, but for reasons discussed below, large numbers are seldom encountered. Each atom has, in general, many orbitals associated with each value of <i>n</i>; these orbitals together are sometimes called <i><a href="/wiki/Electron_shells" class="mw-redirect" title="Electron shells">electron shells</a></i>. </p><p>The <a href="/wiki/Azimuthal_quantum_number" title="Azimuthal quantum number">azimuthal quantum number</a> <span class="texhtml mvar" style="font-style:italic;">ℓ</span> describes the orbital angular momentum of each electron and is a non-negative integer. Within a shell where <span class="texhtml mvar" style="font-style:italic;">n</span> is some integer <span class="texhtml"><i>n</i><sub>0</sub></span>, <span class="texhtml mvar" style="font-style:italic;">ℓ</span> ranges across all (integer) values satisfying the relation <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 0\leq \ell \leq n_{0}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>ℓ</mi> <mo>≤</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq \ell \leq n_{0}-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d93f45f0408a4cf8707525718f06c837ef2a350f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.781ex; height:2.509ex;" alt="{\displaystyle 0\leq \ell \leq n_{0}-1}" /></span>. For instance, the <span class="texhtml"><i>n</i> = 1</span> shell has only orbitals with <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 \ell =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 \ell =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73607cac64f11029ccb86b9403c0ec2bd1629ded" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.231ex; height:2.176ex;" alt="{\displaystyle \ell =0}" /></span>, and the <span class="texhtml"><i>n</i> = 2</span> shell has only orbitals with <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 \ell =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 \ell =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73607cac64f11029ccb86b9403c0ec2bd1629ded" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.231ex; height:2.176ex;" alt="{\displaystyle \ell =0}" /></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell =1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell =1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/942a65b8d58f57930318142ad11ca4be77a53460" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.231ex; height:2.176ex;" alt="{\displaystyle \ell =1}" /></span>. The set of orbitals associated with a particular value of <span class="texhtml mvar" style="font-style:italic;">ℓ</span> are sometimes collectively called a <i>subshell</i>. </p><p>The <a href="/wiki/Magnetic_quantum_number" title="Magnetic quantum number">magnetic quantum number</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_{\ell }}"> <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_{\ell }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f002d4bfc27d584d498407bf774e6a7de83e84a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.958ex; height:2.009ex;" alt="{\displaystyle m_{\ell }}" /></span>, describes the projection of the orbital angular momentum along a chosen axis. It determines the magnitude of the current circulating around that axis and the orbital contribution to the <a href="/wiki/Electron_magnetic_moment" title="Electron magnetic moment">magnetic moment of an electron</a> via the <a href="/wiki/Magnetic_moment#Ampèrian_loop_model" title="Magnetic moment">Ampèrian loop</a> model.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> Within a subshell <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 \ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }" /></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 m_{\ell }}"> <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_{\ell }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f002d4bfc27d584d498407bf774e6a7de83e84a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.958ex; height:2.009ex;" alt="{\displaystyle m_{\ell }}" /></span> obtains the integer values in the range <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 -\ell \leq m_{\ell }\leq \ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>ℓ</mi> <mo>≤</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msub> <mo>≤</mo> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\ell \leq m_{\ell }\leq \ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a4168eba9aeb925bec324ca828c573bd84f5213" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.903ex; height:2.509ex;" alt="{\displaystyle -\ell \leq m_{\ell }\leq \ell }" /></span>. </p><p>The above results may be summarized in the following table. Each cell represents a subshell, and lists the values of <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_{\ell }}"> <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_{\ell }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f002d4bfc27d584d498407bf774e6a7de83e84a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.958ex; height:2.009ex;" alt="{\displaystyle m_{\ell }}" /></span> available in that subshell. Empty cells represent subshells that do not exist. </p> <table class="wikitable"> <tbody><tr> <th> </th> <th><span class="texhtml"><i>ℓ</i> = 0 (s)</span> </th> <th><span class="texhtml"><i>ℓ</i> = 1 (p)</span> </th> <th><span class="texhtml"><i>ℓ</i> = 2 (d)</span> </th> <th><span class="texhtml"><i>ℓ</i> = 3 (f)</span> </th> <th><span class="texhtml"><i>ℓ</i> = 4 (g)</span> </th> <th>... </th></tr> <tr> <th><span class="texhtml"><i>n</i> = 1</span> </th> <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 m_{\ell }=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> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{\ell }=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/759e9025ea1a302f2acc720c80aae0c40e590838" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.219ex; height:2.509ex;" alt="{\displaystyle m_{\ell }=0}" /></span> </td> <td></td> <td></td> <td></td> <td></td> <td>... </td></tr> <tr> <th><span class="texhtml"><i>n</i> = 2</span> </th> <td>0</td> <td>−1, 0, 1 </td> <td></td> <td></td> <td></td> <td>... </td></tr> <tr> <th><span class="texhtml"><i>n</i> = 3</span> </th> <td>0</td> <td>−1, 0, 1</td> <td>−2, −1, 0, 1, 2 </td> <td></td> <td></td> <td>... </td></tr> <tr> <th><span class="texhtml"><i>n</i> = 4</span> </th> <td>0</td> <td>−1, 0, 1</td> <td>−2, −1, 0, 1, 2</td> <td>−3, −2, −1, 0, 1, 2, 3 </td> <td></td> <td>... </td></tr> <tr> <th><span class="texhtml"><i>n</i> = 5</span> </th> <td>0</td> <td>−1, 0, 1</td> <td>−2, −1, 0, 1, 2</td> <td>−3, −2, −1, 0, 1, 2, 3</td> <td>−4, −3, −2, −1, 0, 1, 2, 3, 4 </td> <td>... </td></tr> <tr> <th>... </th> <td>...</td> <td>...</td> <td>...</td> <td>...</td> <td>...</td> <td>... </td></tr></tbody></table> <p>Subshells are usually identified by their <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>- and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }" /></span>-values. <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> is represented by its numerical value, but <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 \ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }" /></span> is represented by a letter as follows: 0 is represented by 's', 1 by 'p', 2 by 'd', 3 by 'f', and 4 by 'g'. For instance, one may speak of the subshell with <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> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell =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 \ell =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73607cac64f11029ccb86b9403c0ec2bd1629ded" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.231ex; height:2.176ex;" alt="{\displaystyle \ell =0}" /></span> as a '2s subshell'. </p><p>Each electron also has angular momentum in the form of <a href="/wiki/Spin_(physics)" title="Spin (physics)">quantum mechanical spin</a> given by spin <i>s</i> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" /><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>. Its projection along a specified axis is given by the <a href="/wiki/Spin_magnetic_quantum_number" class="mw-redirect" title="Spin magnetic quantum number">spin magnetic quantum number</a>, <i>m<sub>s</sub></i>, which can be +<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" /><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> or −<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" /><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>. These values are also called "spin up" or "spin down" respectively. </p><p>The <a href="/wiki/Pauli_exclusion_principle" title="Pauli exclusion principle">Pauli exclusion principle</a> states that no two electrons in an atom can have the same values of all four quantum numbers. If there are two electrons in an orbital with given values for three quantum numbers, (<span class="texhtml mvar" style="font-style:italic;">n</span>, <span class="texhtml mvar" style="font-style:italic;">ℓ</span>, <span class="texhtml mvar" style="font-style:italic;">m</span>), these two electrons must differ in their spin projection <i>m<sub>s</sub></i>. </p><p>The above conventions imply a preferred axis (for example, the <i>z</i> direction in Cartesian coordinates), and they also imply a preferred direction along this preferred axis. Otherwise there would be no sense in distinguishing <span class="texhtml"><i>m</i> = +1</span> from <span class="texhtml"><i>m</i> = −1</span>. As such, the model is most useful when applied to physical systems that share these symmetries. The <a href="/wiki/Stern%E2%80%93Gerlach_experiment" title="Stern–Gerlach experiment">Stern–Gerlach experiment</a>—where an atom is exposed to a magnetic field—provides one such example.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Real_orbitals">Real orbitals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atomic_orbital&action=edit&section=14" title="Edit section: Real orbitals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Orbital_p1-px_animation.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/66/Orbital_p1-px_animation.gif" decoding="async" width="220" height="180" class="mw-file-element" data-file-width="220" data-file-height="180" /></a><figcaption>Animation of continuously varying superpositions between the <span class="texhtml">p<sub>1</sub></span> and the <span class="texhtml">p<sub><i>x</i></sub></span> orbitals. This animation does not use the Condon–Shortley phase convention.</figcaption></figure> <p>Instead of the complex orbitals described above, it is common, especially in the chemistry literature, to use <i>real</i> atomic orbitals. These real orbitals arise from simple linear combinations of complex orbitals. Using the <a href="/wiki/Spherical_harmonics#Condon–Shortley_phase" title="Spherical harmonics">Condon–Shortley phase convention</a>, real orbitals are related to complex orbitals in the same way that the real spherical harmonics are related to complex spherical harmonics. Letting <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,\ell ,m}}"> <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> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{n,\ell ,m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c597029c999eca3b0af2e95eaf7b9166b0b3814" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.775ex; height:2.843ex;" alt="{\displaystyle \psi _{n,\ell ,m}}" /></span> denote a complex orbital with quantum numbers <span class="texhtml mvar" style="font-style:italic;">n</span>, <span class="texhtml mvar" style="font-style:italic;">ℓ</span>, and <span class="texhtml mvar" style="font-style:italic;">m</span>, the real 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 _{n,\ell ,m}^{\text{real}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>ℓ</mi> <mo>,</mo> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>real</mtext> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{n,\ell ,m}^{\text{real}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7bbaba9206394b23ddd7baf066aba983725f95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.775ex; height:3.509ex;" alt="{\displaystyle \psi _{n,\ell ,m}^{\text{real}}}" /></span> may be defined by<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\psi _{n,\ell ,m}^{\text{real}}&={\begin{cases}{\sqrt {2}}(-1)^{m}{\text{Im}}\left\{\psi _{n,\ell ,|m|}\right\}&{\text{ for }}m<0\\[2pt]\psi _{n,\ell ,|m|}&{\text{ for }}m=0\\[2pt]{\sqrt {2}}(-1)^{m}{\text{Re}}\left\{\psi _{n,\ell ,|m|}\right\}&{\text{ for }}m>0\end{cases}}\\[4pt]&={\begin{cases}{\frac {i}{\sqrt {2}}}\left(\psi _{n,\ell ,-|m|}-(-1)^{m}\psi _{n,\ell ,|m|}\right)&{\text{ for }}m<0\\[2pt]\psi _{n,\ell ,|m|}&{\text{ for }}m=0\\[4pt]{\frac {1}{\sqrt {2}}}\left(\psi _{n,\ell ,-|m|}+(-1)^{m}\psi _{n,\ell ,|m|}\right)&{\text{ for }}m>0\end{cases}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>ℓ</mi> <mo>,</mo> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>real</mtext> </mrow> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing="0.4em 0.4em 0.2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mtext>Im</mtext> </mrow> <mrow> <mo>{</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>ℓ</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msub> <mo>}</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>m</mi> <mo><</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>ℓ</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msub> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mtext>Re</mtext> </mrow> <mrow> <mo>{</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>ℓ</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msub> <mo>}</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>m</mi> <mo>></mo> <mn>0</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing="0.4em 0.6em 0.2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>ℓ</mi> <mo>,</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msub> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>ℓ</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>m</mi> <mo><</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>ℓ</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msub> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>ℓ</mi> <mo>,</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>ℓ</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mi>m</mi> <mo>></mo> <mn>0</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\psi _{n,\ell ,m}^{\text{real}}&={\begin{cases}{\sqrt {2}}(-1)^{m}{\text{Im}}\left\{\psi _{n,\ell ,|m|}\right\}&{\text{ for }}m<0\\[2pt]\psi _{n,\ell ,|m|}&{\text{ for }}m=0\\[2pt]{\sqrt {2}}(-1)^{m}{\text{Re}}\left\{\psi _{n,\ell ,|m|}\right\}&{\text{ for }}m>0\end{cases}}\\[4pt]&={\begin{cases}{\frac {i}{\sqrt {2}}}\left(\psi _{n,\ell ,-|m|}-(-1)^{m}\psi _{n,\ell ,|m|}\right)&{\text{ for }}m<0\\[2pt]\psi _{n,\ell ,|m|}&{\text{ for }}m=0\\[4pt]{\frac {1}{\sqrt {2}}}\left(\psi _{n,\ell ,-|m|}+(-1)^{m}\psi _{n,\ell ,|m|}\right)&{\text{ for }}m>0\end{cases}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f0eeddfce0870e5cd8044250644d286927595b2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.709ex; margin-bottom: -0.295ex; width:54.51ex; height:25.176ex;" alt="{\displaystyle {\begin{aligned}\psi _{n,\ell ,m}^{\text{real}}&={\begin{cases}{\sqrt {2}}(-1)^{m}{\text{Im}}\left\{\psi _{n,\ell ,|m|}\right\}&{\text{ for }}m<0\\[2pt]\psi _{n,\ell ,|m|}&{\text{ for }}m=0\\[2pt]{\sqrt {2}}(-1)^{m}{\text{Re}}\left\{\psi _{n,\ell ,|m|}\right\}&{\text{ for }}m>0\end{cases}}\\[4pt]&={\begin{cases}{\frac {i}{\sqrt {2}}}\left(\psi _{n,\ell ,-|m|}-(-1)^{m}\psi _{n,\ell ,|m|}\right)&{\text{ for }}m<0\\[2pt]\psi _{n,\ell ,|m|}&{\text{ for }}m=0\\[4pt]{\frac {1}{\sqrt {2}}}\left(\psi _{n,\ell ,-|m|}+(-1)^{m}\psi _{n,\ell ,|m|}\right)&{\text{ for }}m>0\end{cases}}\end{aligned}}}" /></span> </p><p>If <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,\ell ,m}(r,\theta ,\phi )=R_{nl}(r)Y_{\ell }^{m}(\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> <mi>m</mi> </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>R</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> <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 _{n,\ell ,m}(r,\theta ,\phi )=R_{nl}(r)Y_{\ell }^{m}(\theta ,\phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bfced276963c67b90c60905d7f14ad48711fd61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.5ex; height:3.009ex;" alt="{\displaystyle \psi _{n,\ell ,m}(r,\theta ,\phi )=R_{nl}(r)Y_{\ell }^{m}(\theta ,\phi )}" /></span>, with <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_{nl}(r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</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 R_{nl}(r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6a373dc45077d5d9fde7ef5b8bc5eaecce554e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.331ex; height:2.843ex;" alt="{\displaystyle R_{nl}(r)}" /></span> the radial part of the orbital, this definition is equivalent to <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,\ell ,m}^{\text{real}}(r,\theta ,\phi )=R_{nl}(r)Y_{\ell m}(\theta ,\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>ℓ</mi> <mo>,</mo> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>real</mtext> </mrow> </msubsup> <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>R</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> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> <mi>m</mi> </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 \psi _{n,\ell ,m}^{\text{real}}(r,\theta ,\phi )=R_{nl}(r)Y_{\ell m}(\theta ,\phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48bcda0959480be21e58cbcfdeba966a4b705a68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:31.636ex; height:3.509ex;" alt="{\displaystyle \psi _{n,\ell ,m}^{\text{real}}(r,\theta ,\phi )=R_{nl}(r)Y_{\ell m}(\theta ,\phi )}" /></span> where <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_{\ell m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{\ell m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dae6b1931bc4b14b0af5021c5bc88cb4471d0cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.711ex; height:2.509ex;" alt="{\displaystyle Y_{\ell m}}" /></span> is the real spherical harmonic related to either the real or imaginary part of the complex spherical harmonic <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_{\ell }^{m}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{\ell }^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9004e5e8982bc3908955fc97e0ed801544f9ce34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.575ex; height:2.843ex;" alt="{\displaystyle Y_{\ell }^{m}}" /></span>. </p><p>Real spherical harmonics are physically relevant when an atom is embedded in a crystalline solid, in which case there are multiple preferred symmetry axes but no single preferred direction.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (February 2022)">citation needed</span></a></i>]</sup> Real atomic orbitals are also more frequently encountered in introductory chemistry textbooks and shown in common orbital visualizations.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> In real hydrogen-like orbitals, quantum numbers <span class="texhtml mvar" style="font-style:italic;">n</span> and <span class="texhtml mvar" style="font-style:italic;">ℓ</span> have the same interpretation and significance as their complex counterparts, but <span class="texhtml mvar" style="font-style:italic;">m</span> is no longer a good quantum number (but its absolute value is). </p><p>Some real orbitals are given specific names beyond the simple <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,\ell ,m}}"> <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> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{n,\ell ,m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c597029c999eca3b0af2e95eaf7b9166b0b3814" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.775ex; height:2.843ex;" alt="{\displaystyle \psi _{n,\ell ,m}}" /></span> designation. Orbitals with quantum number <span class="texhtml"><i>ℓ</i> = 0, 1, 2, 3, 4, 5, 6...</span> are called <span class="texhtml">s, p, d, f, g, h, i, ...</span> orbitals. With this one can already assign names to complex orbitals such as <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{\text{p}}_{\pm 1}=\psi _{2,1,\pm 1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>p</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>±</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2{\text{p}}_{\pm 1}=\psi _{2,1,\pm 1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6edf29a433b613807dda53db85cf5dee6d8ad08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.291ex; height:2.843ex;" alt="{\displaystyle 2{\text{p}}_{\pm 1}=\psi _{2,1,\pm 1}}" /></span>; the first symbol is the <span class="texhtml mvar" style="font-style:italic;">n</span> quantum number, the second character is the symbol for that particular <span class="texhtml mvar" style="font-style:italic;">ℓ</span> quantum number and the subscript is the <span class="texhtml mvar" style="font-style:italic;">m</span> quantum number. </p><p>As an example of how the full orbital names are generated for real orbitals, one may calculate <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,1,\pm 1}^{\text{real}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>±</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>real</mtext> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{n,1,\pm 1}^{\text{real}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40348a7349698acf950765a929d62fac963fc2f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:6.569ex; height:3.509ex;" alt="{\displaystyle \psi _{n,1,\pm 1}^{\text{real}}}" /></span>. From the <a href="/wiki/Table_of_spherical_harmonics" title="Table of spherical harmonics">table of spherical harmonics</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="{\textstyle \psi _{n,1,\pm 1}=R_{n,1}Y_{1}^{\pm 1}=\mp R_{n,1}{\sqrt {3/8\pi }}\cdot (x\pm iy)/r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>±</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> <msubsup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <mo>∓</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>8</mn> <mi>π</mi> </msqrt> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>±</mo> <mi>i</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \psi _{n,1,\pm 1}=R_{n,1}Y_{1}^{\pm 1}=\mp R_{n,1}{\sqrt {3/8\pi }}\cdot (x\pm iy)/r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4de7e0a9656bc8231bc8534cd93300e7d10f7702" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:46.301ex; height:3.343ex;" alt="{\textstyle \psi _{n,1,\pm 1}=R_{n,1}Y_{1}^{\pm 1}=\mp R_{n,1}{\sqrt {3/8\pi }}\cdot (x\pm iy)/r}" /></span> with <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="{\textstyle r={\sqrt {x^{2}+y^{2}+z^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle r={\sqrt {x^{2}+y^{2}+z^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/128dbe4aa26a600bb69db22ec47d44e2bab44e62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.895ex; height:3.509ex;" alt="{\textstyle r={\sqrt {x^{2}+y^{2}+z^{2}}}}" /></span>. Then </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\psi _{n,1,+1}^{\text{real}}&=R_{n,1}{\sqrt {\frac {3}{4\pi }}}\cdot {\frac {x}{r}}\\\psi _{n,1,-1}^{\text{real}}&=R_{n,1}{\sqrt {\frac {3}{4\pi }}}\cdot {\frac {y}{r}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>+</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>real</mtext> </mrow> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>3</mn> <mrow> <mn>4</mn> <mi>π</mi> </mrow> </mfrac> </msqrt> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>r</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>real</mtext> </mrow> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>3</mn> <mrow> <mn>4</mn> <mi>π</mi> </mrow> </mfrac> </msqrt> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mi>r</mi> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\psi _{n,1,+1}^{\text{real}}&=R_{n,1}{\sqrt {\frac {3}{4\pi }}}\cdot {\frac {x}{r}}\\\psi _{n,1,-1}^{\text{real}}&=R_{n,1}{\sqrt {\frac {3}{4\pi }}}\cdot {\frac {y}{r}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcc95b9485c758041c29ae76663715abade19479" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:24.18ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}\psi _{n,1,+1}^{\text{real}}&=R_{n,1}{\sqrt {\frac {3}{4\pi }}}\cdot {\frac {x}{r}}\\\psi _{n,1,-1}^{\text{real}}&=R_{n,1}{\sqrt {\frac {3}{4\pi }}}\cdot {\frac {y}{r}}\end{aligned}}}" /></span> </p><p>Likewise <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="{\textstyle \psi _{n,1,0}=R_{n,1}{\sqrt {3/4\pi }}\cdot z/r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mi>π</mi> </msqrt> </mrow> <mo>⋅</mo> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \psi _{n,1,0}=R_{n,1}{\sqrt {3/4\pi }}\cdot z/r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73673847a7dc10f5ecb26b0c5f8cbac48cc047c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.772ex; height:3.343ex;" alt="{\textstyle \psi _{n,1,0}=R_{n,1}{\sqrt {3/4\pi }}\cdot z/r}" /></span>. As a more complicated example: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{n,3,+1}^{\text{real}}=R_{n,3}{\frac {1}{4}}{\sqrt {\frac {21}{2\pi }}}\cdot {\frac {x\cdot (5z^{2}-r^{2})}{r^{3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>+</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>real</mtext> </mrow> </msubsup> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mn>3</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>21</mn> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </msqrt> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>5</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{n,3,+1}^{\text{real}}=R_{n,3}{\frac {1}{4}}{\sqrt {\frac {21}{2\pi }}}\cdot {\frac {x\cdot (5z^{2}-r^{2})}{r^{3}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0ed3f03b8fa43860eb0acc35c644b626702824a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:37.166ex; height:6.343ex;" alt="{\displaystyle \psi _{n,3,+1}^{\text{real}}=R_{n,3}{\frac {1}{4}}{\sqrt {\frac {21}{2\pi }}}\cdot {\frac {x\cdot (5z^{2}-r^{2})}{r^{3}}}}" /></span> </p><p>In all these cases we generate a Cartesian label for the orbital by examining, and abbreviating, the polynomial in <span class="texhtml"><i>x</i>, <i>y</i>, <i>z</i></span> appearing in the numerator. We ignore any terms in the <span class="texhtml"><i>z</i>, <i>r</i></span> polynomial except for the term with the highest exponent in <span class="texhtml mvar" style="font-style:italic;">z</span>. We then use the abbreviated polynomial as a subscript label for the atomic state, using the same nomenclature as above to indicate the <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> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }" /></span> quantum numbers.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (March 2024)">citation needed</span></a></i>]</sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\psi _{n,1,-1}^{\text{real}}&=n{\text{p}}_{y}={\frac {i}{\sqrt {2}}}\left(n{\text{p}}_{-1}+n{\text{p}}_{+1}\right)\\\psi _{n,1,0}^{\text{real}}&=n{\text{p}}_{z}=2{\text{p}}_{0}\\\psi _{n,1,+1}^{\text{real}}&=n{\text{p}}_{x}={\frac {1}{\sqrt {2}}}\left(n{\text{p}}_{-1}-n{\text{p}}_{+1}\right)\\\psi _{n,3,+1}^{\text{real}}&=nf_{xz^{2}}={\frac {1}{\sqrt {2}}}\left(nf_{-1}-nf_{+1}\right)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>real</mtext> </mrow> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>n</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>p</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>p</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>n</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>p</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>real</mtext> </mrow> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>n</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>p</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>p</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>+</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>real</mtext> </mrow> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>n</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>p</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>p</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mi>n</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>p</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>+</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>real</mtext> </mrow> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>n</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mi>n</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\psi _{n,1,-1}^{\text{real}}&=n{\text{p}}_{y}={\frac {i}{\sqrt {2}}}\left(n{\text{p}}_{-1}+n{\text{p}}_{+1}\right)\\\psi _{n,1,0}^{\text{real}}&=n{\text{p}}_{z}=2{\text{p}}_{0}\\\psi _{n,1,+1}^{\text{real}}&=n{\text{p}}_{x}={\frac {1}{\sqrt {2}}}\left(n{\text{p}}_{-1}-n{\text{p}}_{+1}\right)\\\psi _{n,3,+1}^{\text{real}}&=nf_{xz^{2}}={\frac {1}{\sqrt {2}}}\left(nf_{-1}-nf_{+1}\right)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5be58fabaac609e0200413baa4bc1bf80eabb744" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.671ex; width:37.531ex; height:22.509ex;" alt="{\displaystyle {\begin{aligned}\psi _{n,1,-1}^{\text{real}}&=n{\text{p}}_{y}={\frac {i}{\sqrt {2}}}\left(n{\text{p}}_{-1}+n{\text{p}}_{+1}\right)\\\psi _{n,1,0}^{\text{real}}&=n{\text{p}}_{z}=2{\text{p}}_{0}\\\psi _{n,1,+1}^{\text{real}}&=n{\text{p}}_{x}={\frac {1}{\sqrt {2}}}\left(n{\text{p}}_{-1}-n{\text{p}}_{+1}\right)\\\psi _{n,3,+1}^{\text{real}}&=nf_{xz^{2}}={\frac {1}{\sqrt {2}}}\left(nf_{-1}-nf_{+1}\right)\end{aligned}}}" /></span> </p><p>The expression above all use the <a href="/wiki/Spherical_harmonics#Condon–Shortley_phase" title="Spherical harmonics">Condon–Shortley phase convention</a> which is favored by quantum physicists.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> Other conventions exist for the phase of the spherical harmonics.<sup id="cite_ref-Levine7ed_27-0" class="reference"><a href="#cite_note-Levine7ed-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> Under these different conventions the <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{p}}_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>p</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{p}}_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5186915c8e7bbc4aca060c0143fc181e9698c82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.465ex; height:2.176ex;" alt="{\displaystyle {\text{p}}_{x}}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{p}}_{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>p</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{p}}_{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff9ad9a2439bff7bb148b3fdca942285eded21f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.342ex; height:2.509ex;" alt="{\displaystyle {\text{p}}_{y}}" /></span> orbitals may appear, for example, as the sum and difference of <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{p}}_{+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>p</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{p}}_{+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c93fd72b60d7c8a71690a92e82c2f7a2a22e4e55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.625ex; height:2.343ex;" alt="{\displaystyle {\text{p}}_{+1}}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{p}}_{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>p</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{p}}_{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/159bca7465bf28d99f0586121cc071f69d31a685" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.625ex; height:2.343ex;" alt="{\displaystyle {\text{p}}_{-1}}" /></span>, contrary to what is shown above. </p><p>Below is a list of these Cartesian polynomial names for the atomic orbitals.<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> There does not seem to be reference in the literature as to how to abbreviate the long Cartesian spherical harmonic polynomials for <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 \ell >3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> <mo>></mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell >3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e91b5f299aefad76e097dcf7b13d5557d4fbcc2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.231ex; height:2.176ex;" alt="{\displaystyle \ell >3}" /></span> so there does not seem be consensus on the naming of <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 g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" /></span> orbitals or higher according to this nomenclature.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (March 2024)">citation needed</span></a></i>]</sup> </p> <table class="wikitable"> <tbody><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 \psi _{m=-3}+\psi _{m=+3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mo>−</mo> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mo>+</mo> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{m=-3}+\psi _{m=+3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/affb9b04fd39e87564c94f0c24037453468142b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.975ex; height:2.509ex;" alt="{\displaystyle \psi _{m=-3}+\psi _{m=+3}}" /></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 _{m=-2}+\psi _{m=+2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mo>+</mo> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{m=-2}+\psi _{m=+2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a07aee365301a56102c0c6e2abad509b090c460" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.975ex; height:2.509ex;" alt="{\displaystyle \psi _{m=-2}+\psi _{m=+2}}" /></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 _{m=-1}+\psi _{m=+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{m=-1}+\psi _{m=+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/707b5a04e6337e6816c6fc04551db0410359d8fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.975ex; height:2.509ex;" alt="{\displaystyle \psi _{m=-1}+\psi _{m=+1}}" /></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 _{m=0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{m=0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/992747ef0096c6db4260eb0abd677e85fd441665" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.289ex; height:2.509ex;" alt="{\displaystyle \psi _{m=0}}" /></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 _{m=-1}-\psi _{m=+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{m=-1}-\psi _{m=+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d1758a6aae3378937601faf5fd9ada384a8eed5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.975ex; height:2.509ex;" alt="{\displaystyle \psi _{m=-1}-\psi _{m=+1}}" /></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 _{m=-2}-\psi _{m=+2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mo>+</mo> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{m=-2}-\psi _{m=+2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d79f4f5be143daef4e6b272f1fc859c251daae15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.975ex; height:2.509ex;" alt="{\displaystyle \psi _{m=-2}-\psi _{m=+2}}" /></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 _{m=-3}-\psi _{m=+3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mo>−</mo> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mo>+</mo> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{m=-3}-\psi _{m=+3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d1d3d1b9307821c67bddebaabab95c7015cf06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.975ex; height:2.509ex;" alt="{\displaystyle \psi _{m=-3}-\psi _{m=+3}}" /></span> </th></tr> <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 \ell =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 \ell =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73607cac64f11029ccb86b9403c0ec2bd1629ded" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.231ex; height:2.176ex;" alt="{\displaystyle \ell =0}" /></span> </th> <td></td> <td></td> <td></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 {\text{s}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{s}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d2cf6483154452c8e8717cef7d81428a125ba06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.916ex; height:1.676ex;" alt="{\displaystyle {\text{s}}}" /></span></td> <td></td> <td></td> <td> </td></tr> <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 \ell =1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell =1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/942a65b8d58f57930318142ad11ca4be77a53460" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.231ex; height:2.176ex;" alt="{\displaystyle \ell =1}" /></span> </th> <td></td> <td></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 {\text{p}}_{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>p</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{p}}_{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff9ad9a2439bff7bb148b3fdca942285eded21f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.342ex; height:2.509ex;" alt="{\displaystyle {\text{p}}_{y}}" /></span></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 {\text{p}}_{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>p</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{p}}_{z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0691c1683731993d4de68836c9a0421513c4e2e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.294ex; height:2.176ex;" alt="{\displaystyle {\text{p}}_{z}}" /></span></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 {\text{p}}_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>p</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{p}}_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5186915c8e7bbc4aca060c0143fc181e9698c82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.465ex; height:2.176ex;" alt="{\displaystyle {\text{p}}_{x}}" /></span></td> <td></td> <td> </td></tr> <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 \ell =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 \ell =2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/682399b5a2ae37a3b5b13a0e4e3f7348fa3c0ab4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.231ex; height:2.176ex;" alt="{\displaystyle \ell =2}" /></span> </th> <td></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 {\text{d}}_{xy}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{d}}_{xy}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d3cf3fd96c0f96548397cc79e8e231b0fca7f34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.282ex; height:2.843ex;" alt="{\displaystyle {\text{d}}_{xy}}" /></span></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 {\text{d}}_{yz}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{d}}_{yz}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d502c7ba6dbaaf2921b25982d6e06d81292e5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.111ex; height:2.843ex;" alt="{\displaystyle {\text{d}}_{yz}}" /></span></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 {\text{d}}_{z^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{d}}_{z^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aec264363e5b8c96d5cb041fe1dced9ced78fc89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.127ex; height:2.676ex;" alt="{\displaystyle {\text{d}}_{z^{2}}}" /></span></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 {\text{d}}_{xz}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{d}}_{xz}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9531cd8a3521817884177056b822b9ad00ce364" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.234ex; height:2.509ex;" alt="{\displaystyle {\text{d}}_{xz}}" /></span></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 {\text{d}}_{x^{2}-y^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{d}}_{x^{2}-y^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28b98f69a7f92787fa529e93ea1ce23958b82a32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.227ex; height:3.009ex;" alt="{\displaystyle {\text{d}}_{x^{2}-y^{2}}}" /></span></td> <td> </td></tr> <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 \ell =3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell =3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4358eabf9aa70bc959f066c915df0f14efaea5ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.231ex; height:2.176ex;" alt="{\displaystyle \ell =3}" /></span> </th> <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 {\text{f}}_{y(3x^{2}-y^{2})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>f</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mo stretchy="false">(</mo> <mn>3</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{f}}_{y(3x^{2}-y^{2})}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84d8f946c13605ec40014d3439cfa6aca8a6c647" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.565ex; height:3.009ex;" alt="{\displaystyle {\text{f}}_{y(3x^{2}-y^{2})}}" /></span></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 {\text{f}}_{xyz}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>f</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{f}}_{xyz}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25f476966edfb96fad986932fdf7814621baaef0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.471ex; height:2.843ex;" alt="{\displaystyle {\text{f}}_{xyz}}" /></span></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 {\text{f}}_{yz^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>f</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{f}}_{yz^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a969ff6d991c137d39a1af5775de397a6b645fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.364ex; height:3.009ex;" alt="{\displaystyle {\text{f}}_{yz^{2}}}" /></span></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 {\text{f}}_{z^{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>f</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{f}}_{z^{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d7eff2945598bdb0c61f1783e82031ab422688e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.547ex; height:2.676ex;" alt="{\displaystyle {\text{f}}_{z^{3}}}" /></span></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 {\text{f}}_{xz^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>f</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{f}}_{xz^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6fc19210c725f81823ef042a69f9681b4cafedd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.676ex;" alt="{\displaystyle {\text{f}}_{xz^{2}}}" /></span></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 {\text{f}}_{z(x^{2}-y^{2})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>f</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{f}}_{z(x^{2}-y^{2})}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d5c0b75d736448e3afd5fc481042b6e83cd35e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.695ex; height:3.009ex;" alt="{\displaystyle {\text{f}}_{z(x^{2}-y^{2})}}" /></span></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 {\text{f}}_{x(x^{2}-3y^{2})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>f</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{f}}_{x(x^{2}-3y^{2})}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96a218cd0c395d4eb0e2848aa826fc7fd07e6bd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.688ex; height:3.009ex;" alt="{\displaystyle {\text{f}}_{x(x^{2}-3y^{2})}}" /></span> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Shapes_of_orbitals">Shapes of orbitals</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atomic_orbital&action=edit&section=15" title="Edit section: Shapes of orbitals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Atomic-orbital-cloud_n6_l0_m0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bf/Atomic-orbital-cloud_n6_l0_m0.png/220px-Atomic-orbital-cloud_n6_l0_m0.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bf/Atomic-orbital-cloud_n6_l0_m0.png/330px-Atomic-orbital-cloud_n6_l0_m0.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bf/Atomic-orbital-cloud_n6_l0_m0.png/440px-Atomic-orbital-cloud_n6_l0_m0.png 2x" data-file-width="1024" data-file-height="1024" /></a><figcaption>Transparent cloud view of a computed 6s <span class="texhtml">(<i>n</i> = 6, <i>ℓ</i> = 0, <i>m</i> = 0)</span> hydrogen orbital. The s orbitals, though spherically symmetric, have radially placed wave-nodes for <span class="texhtml"><i>n</i> > 1</span>. Only s orbitals invariably have a center anti-node; the other types never do.</figcaption></figure> <p>Simple pictures showing orbital shapes are intended to describe the angular forms of regions in space where the electrons occupying the orbital are likely to be found. The diagrams cannot show the entire region where an electron can be found, since according to quantum mechanics there is a non-zero probability of finding the electron (almost) anywhere in space. Instead the diagrams are approximate representations of boundary or <a href="/wiki/Isosurface" title="Isosurface">contour surfaces</a> where the probability density <span class="texhtml">| ψ(<i>r</i>, <i>θ</i>, <i>φ</i>) |<sup>2</sup></span> has a constant value, chosen so that there is a certain probability (for example 90%) of finding the electron within the contour. Although <span class="texhtml">| <i>ψ</i> |<sup>2</sup></span> as the square of an <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a> is everywhere non-negative, the sign of the <a href="/wiki/Wave_function" title="Wave function">wave function</a> <span class="texhtml">ψ(<i>r</i>, <i>θ</i>, <i>φ</i>)</span> is often indicated in each subregion of the orbital picture. </p><p>Sometimes the <span class="texhtml">ψ</span> function is graphed to show its phases, rather than <span class="texhtml">| ψ(<i>r</i>, <i>θ</i>, <i>φ</i>) |<sup>2</sup></span> which shows probability density but has no phase (which is lost when taking absolute value, since <span class="texhtml">ψ(<i>r</i>, <i>θ</i>, <i>φ</i>)</span> is a <a href="/wiki/Complex_number" title="Complex number">complex number</a>). <span class="texhtml">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;">ψ(<i>r</i>, <i>θ</i>, <i>φ</i>)</span>|<sup>2</sup></span> orbital graphs tend to have less spherical, thinner lobes than <span class="texhtml">ψ(<i>r</i>, <i>θ</i>, <i>φ</i>)</span> graphs, but have the same number of lobes in the same places, and otherwise are recognizable. This article, to show wave function phase, shows mostly <span class="texhtml">ψ(<i>r</i>, <i>θ</i>, <i>φ</i>)</span> graphs. </p><p>The lobes can be seen as <a href="/wiki/Standing_wave" title="Standing wave">standing wave</a> <a href="/wiki/Wave_interference" title="Wave interference">interference</a> patterns between the two counter-rotating, ring-resonant <a href="/wiki/Traveling_wave" class="mw-redirect" title="Traveling wave">traveling wave</a> <span class="texhtml mvar" style="font-style:italic;">m</span> and <span class="texhtml">−<i>m</i></span> modes; the projection of the orbital onto the xy plane has a resonant <span class="texhtml mvar" style="font-style:italic;">m</span> wavelength around the circumference. Although rarely shown, the traveling wave solutions can be seen as rotating banded tori; the bands represent phase information. For each <span class="texhtml mvar" style="font-style:italic;">m</span> there are two standing wave solutions <span class="texhtml">⟨<i>m</i>⟩ + ⟨−<i>m</i>⟩</span> and <span class="texhtml">⟨<i>m</i>⟩ − ⟨−<i>m</i>⟩</span>. If <span class="texhtml"><i>m</i> = 0</span>, the orbital is vertical, counter rotating information is unknown, and the orbital is <i>z</i>-axis symmetric. If <span class="texhtml"><i>ℓ</i> = 0</span> there are no counter rotating modes. There are only radial modes and the shape is spherically symmetric. </p><p><i><a href="/wiki/Node_(physics)" title="Node (physics)">Nodal</a> planes</i> and <i>nodal spheres</i> are surfaces on which the probability density vanishes. The number of nodal surfaces is controlled by the quantum numbers <span class="texhtml mvar" style="font-style:italic;">n</span> and <span class="texhtml mvar" style="font-style:italic;">ℓ</span>. An orbital with azimuthal quantum number <span class="texhtml mvar" style="font-style:italic;">ℓ</span> has <span class="texhtml mvar" style="font-style:italic;">ℓ</span> radial nodal planes passing through the origin. For example, the s orbitals (<span class="texhtml"><i>ℓ</i> = 0</span>) are spherically symmetric and have no nodal planes, whereas the p orbitals (<span class="texhtml"><i>ℓ</i> = 1</span>) have a single nodal plane between the lobes. The number of nodal spheres equals <span class="texhtml mvar" style="font-style:italic;">n-ℓ-1</span>, consistent with the restriction <span class="texhtml mvar" style="font-style:italic;">ℓ ≤ n-1</span> on the quantum numbers. The principal quantum number controls the total number of nodal surfaces which is <span class="texhtml mvar" style="font-style:italic;">n-1</span>.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> Loosely speaking, <span class="texhtml mvar" style="font-style:italic;">n</span> is energy, <span class="texhtml mvar" style="font-style:italic;">ℓ</span> is analogous to <a href="/wiki/Orbital_eccentricity" title="Orbital eccentricity">eccentricity</a>, and <span class="texhtml mvar" style="font-style:italic;">m</span> is orientation. </p><p>In general, <span class="texhtml mvar" style="font-style:italic;">n</span> determines size and energy of the orbital for a given nucleus; as <span class="texhtml mvar" style="font-style:italic;">n</span> increases, the size of the orbital increases. The higher nuclear charge <span class="texhtml mvar" style="font-style:italic;">Z</span> of heavier elements causes their orbitals to contract by comparison to lighter ones, so that the size of the atom remains very roughly constant, even as the number of electrons increases. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Sr_core-electron_orbitals_for_Wiki.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Sr_core-electron_orbitals_for_Wiki.jpg/220px-Sr_core-electron_orbitals_for_Wiki.jpg" decoding="async" width="220" height="439" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Sr_core-electron_orbitals_for_Wiki.jpg/330px-Sr_core-electron_orbitals_for_Wiki.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/2/28/Sr_core-electron_orbitals_for_Wiki.jpg 2x" data-file-width="431" data-file-height="860" /></a><figcaption>Experimentally imaged 1s and 2p core-electron orbitals of Sr, including the effects of atomic thermal vibration and excitation broadening, retrieved from energy dispersive x-ray spectroscopy (EDX) in scanning transmission electron microscopy (STEM).<sup id="cite_ref-Jeong_165140_32-0" class="reference"><a href="#cite_note-Jeong_165140-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>Also in general terms, <span class="texhtml mvar" style="font-style:italic;">ℓ</span> determines an orbital's shape, and <span class="texhtml mvar" style="font-style:italic;">m<sub>ℓ</sub></span> its orientation. However, since some orbitals are described by equations in <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>, the shape sometimes depends on <span class="texhtml mvar" style="font-style:italic;">m<sub>ℓ</sub></span> also. Together, the whole set of orbitals for a given <span class="texhtml mvar" style="font-style:italic;">ℓ</span> and <span class="texhtml mvar" style="font-style:italic;">n</span> fill space as symmetrically as possible, though with increasingly complex sets of lobes and nodes. </p><p>The single s 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 \ell =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 \ell =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73607cac64f11029ccb86b9403c0ec2bd1629ded" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.231ex; height:2.176ex;" alt="{\displaystyle \ell =0}" /></span>) are shaped like spheres. For <span class="texhtml"><i>n</i> = 1</span> it is roughly a <a href="/wiki/Ball_(mathematics)" title="Ball (mathematics)">solid ball</a> (densest at center and fades outward exponentially), but for <span class="texhtml"><i>n</i> ≥ 2</span>, each single s orbital is made of spherically symmetric surfaces which are nested shells (i.e., the "wave-structure" is radial, following a sinusoidal radial component as well). See illustration of a cross-section of these nested shells, at right. The s orbitals for all <span class="texhtml mvar" style="font-style:italic;">n</span> numbers are the only orbitals with an anti-node (a region of high wave function density) at the center of the nucleus. All other orbitals (p, d, f, etc.) have angular momentum, and thus avoid the nucleus (having a wave node <i>at</i> the nucleus). Recently, there has been an effort to experimentally image the 1s and 2p orbitals in a SrTiO<sub>3</sub> crystal using scanning transmission electron microscopy with energy dispersive x-ray spectroscopy.<sup id="cite_ref-Jeong_165140_32-1" class="reference"><a href="#cite_note-Jeong_165140-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> Because the imaging was conducted using an electron beam, Coulombic beam-orbital interaction that is often termed as the impact parameter effect is included in the outcome (see the figure at right). </p><p>The shapes of p, d and f orbitals are described verbally here and shown graphically in the <i>Orbitals table</i> below. The three p orbitals for <span class="texhtml"><i>n</i> = 2</span> have the form of two <a href="/wiki/Ellipsoid" title="Ellipsoid">ellipsoids</a> with a <a href="/wiki/Point_of_tangency" class="mw-redirect" title="Point of tangency">point of tangency</a> at the <a href="/wiki/Atomic_nucleus" title="Atomic nucleus">nucleus</a> (the two-lobed shape is sometimes referred to as a "<a href="/wiki/Dumbbell" title="Dumbbell">dumbbell</a>"—there are two lobes pointing in opposite directions from each other). The three p orbitals in each <a href="/wiki/Electron_shell" title="Electron shell">shell</a> are oriented at right angles to each other, as determined by their respective linear combination of values of <span class="texhtml mvar" style="font-style:italic;">m<sub>ℓ</sub></span>. The overall result is a lobe pointing along each direction of the primary axes. </p><p>Four of the five d orbitals for <span class="texhtml"><i>n</i> = 3</span> look similar, each with four pear-shaped lobes, each lobe tangent at right angles to two others, and the centers of all four lying in one plane. Three of these planes are the xy-, xz-, and yz-planes—the lobes are between the pairs of primary axes—and the fourth has the center along the x and y axes themselves. The fifth and final d orbital consists of three regions of high probability density: a <a href="/wiki/Torus" title="Torus">torus</a> in between two pear-shaped regions placed symmetrically on its z axis. The overall total of 18 directional lobes point in every primary axis direction and between every pair. </p><p>There are seven f orbitals, each with shapes more complex than those of the d orbitals. </p><p>Additionally, as is the case with the s orbitals, individual p, d, f and g orbitals with <span class="texhtml mvar" style="font-style:italic;">n</span> values higher than the lowest possible value, exhibit an additional radial node structure which is reminiscent of harmonic waves of the same type, as compared with the lowest (or fundamental) mode of the wave. As with s orbitals, this phenomenon provides p, d, f, and g orbitals at the next higher possible value of <span class="texhtml mvar" style="font-style:italic;">n</span> (for example, 3p orbitals vs. the fundamental 2p), an additional node in each lobe. Still higher values of <span class="texhtml mvar" style="font-style:italic;">n</span> further increase the number of radial nodes, for each type of orbital. </p><p>The shapes of atomic orbitals in one-electron atom are related to 3-dimensional <a href="/wiki/Spherical_harmonics" title="Spherical harmonics">spherical harmonics</a>. These shapes are not unique, and any linear combination is valid, like a transformation to <a href="/wiki/Cubic_harmonic" title="Cubic harmonic">cubic harmonics</a>, in fact it is possible to generate sets where all the d's are the same shape, just like the <span class="texhtml">p<sub><i>x</i></sub>, p<sub><i>y</i></sub>,</span> and <span class="texhtml">p<sub><i>z</i></sub></span> are the same shape.<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Schrodinger_model_of_the_atom.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Schrodinger_model_of_the_atom.svg/250px-Schrodinger_model_of_the_atom.svg.png" decoding="async" width="220" height="167" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Schrodinger_model_of_the_atom.svg/330px-Schrodinger_model_of_the_atom.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Schrodinger_model_of_the_atom.svg/500px-Schrodinger_model_of_the_atom.svg.png 2x" data-file-width="693" data-file-height="526" /></a><figcaption>The 1s, 2s, and 2p orbitals of a sodium atom</figcaption></figure> <p>Although individual orbitals are most often shown independent of each other, the orbitals coexist around the nucleus at the same time. Also, in 1927, <a href="/wiki/Albrecht_Uns%C3%B6ld" title="Albrecht Unsöld">Albrecht Unsöld</a> proved that if one sums the electron density of all orbitals of a particular azimuthal quantum number <span class="texhtml mvar" style="font-style:italic;">ℓ</span> of the same shell <span class="texhtml mvar" style="font-style:italic;">n</span> (e.g., all three 2p orbitals, or all five 3d orbitals) where each orbital is occupied by an electron or each is occupied by an electron pair, then all angular dependence disappears; that is, the resulting total density of all the atomic orbitals in that subshell (those with the same <span class="texhtml mvar" style="font-style:italic;">ℓ</span>) is spherical. This is known as <a href="/wiki/Spherical_harmonics#Addition_theorem" title="Spherical harmonics">Unsöld's theorem</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Orbitals_table">Orbitals table</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atomic_orbital&action=edit&section=16" title="Edit section: Orbitals table"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This table shows the real hydrogen-like wave functions for all atomic orbitals up to 7s, and therefore covers the occupied orbitals in the ground state of all elements in the periodic table up to <a href="/wiki/Radium" title="Radium">radium</a> and some beyond. "ψ" graphs are shown with <b>−</b> and <b>+</b> <a href="/wiki/Wave_function" title="Wave function">wave function</a> phases shown in two different colors (arbitrarily red and blue). The <span class="texhtml">p<sub><i>z</i></sub></span> orbital is the same as the <span class="texhtml">p<sub>0</sub></span> orbital, but the <span class="texhtml">p<sub><i>x</i></sub></span> and <span class="texhtml">p<sub><i>y</i></sub></span> are formed by taking linear combinations of the <span class="texhtml">p<sub>+1</sub></span> and <span class="texhtml">p<sub>−1</sub></span> orbitals (which is why they are listed under the <span class="texhtml"><i>m</i> = ±1</span> label). Also, the <span class="texhtml">p<sub>+1</sub></span> and <span class="texhtml">p<sub>−1</sub></span> are not the same shape as the <span class="texhtml">p<sub>0</sub></span>, since they are pure <a href="/wiki/Spherical_harmonics" title="Spherical harmonics">spherical harmonics</a>. </p> <table class="wikitable"> <tbody><tr> <th> </th> <th>s (<span class="texhtml"><i>ℓ</i> = 0</span>) </th> <th colspan="3">p (<span class="texhtml"><i>ℓ</i> = 1</span>) </th> <th colspan="5">d (<span class="texhtml"><i>ℓ</i> = 2</span>) </th> <th colspan="7">f (<span class="texhtml"><i>ℓ</i> = 3</span>) </th></tr> <tr> <th> </th> <th><span class="texhtml"><i>m</i> = 0</span> </th> <th><span class="texhtml"><i>m</i> = 0</span> </th> <th colspan="2"><span class="texhtml"><i>m</i> = ±1</span> </th> <th><span class="texhtml"><i>m</i> = 0</span> </th> <th colspan="2"><span class="texhtml"><i>m</i> = ±1</span> </th> <th colspan="2"><span class="texhtml"><i>m</i> = ±2</span> </th> <th><span class="texhtml"><i>m</i> = 0</span> </th> <th colspan="2"><span class="texhtml"><i>m</i> = ±1</span> </th> <th colspan="2"><span class="texhtml"><i>m</i> = ±2</span> </th> <th colspan="2"><span class="texhtml"><i>m</i> = ±3</span> </th></tr> <tr> <th> </th> <th>s </th> <th>p<sub><i>z</i></sub> </th> <th>p<sub><i>x</i></sub> </th> <th>p<sub><i>y</i></sub> </th> <th>d<sub><i>z</i><sup>2</sup></sub> </th> <th>d<sub><i>xz</i></sub> </th> <th>d<sub><i>yz</i></sub> </th> <th>d<sub><i>xy</i></sub> </th> <th>d<sub><i>x</i><sup>2</sup>−<i>y</i><sup>2</sup></sub> </th> <th>f<sub><i>z</i><sup>3</sup></sub> </th> <th>f<sub><i>xz</i><sup>2</sup></sub> </th> <th>f<sub><i>yz</i><sup>2</sup></sub> </th> <th>f<sub><i>xyz</i></sub> </th> <th>f<sub><i>z</i>(<i>x</i><sup>2</sup>−<i>y</i><sup>2</sup>)</sub> </th> <th>f<sub><i>x</i>(<i>x</i><sup>2</sup>−3<i>y</i><sup>2</sup>)</sub> </th> <th>f<sub><i>y</i>(3<i>x</i><sup>2</sup>−<i>y</i><sup>2</sup>)</sub> </th></tr> <tr> <th><span class="texhtml"><i>n</i> = 1</span> </th> <td><span typeof="mw:File"><a href="/wiki/File:S1M0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/S1M0.png/50px-S1M0.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/S1M0.png/75px-S1M0.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1d/S1M0.png/100px-S1M0.png 2x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <th><span class="texhtml"><i>n</i> = 2</span> </th> <td><span typeof="mw:File"><a href="/wiki/File:S2M0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/S2M0.png/60px-S2M0.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/S2M0.png/120px-S2M0.png 1.5x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:P2M0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/P2M0.png/60px-P2M0.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/P2M0.png/120px-P2M0.png 1.5x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Px_orbital.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/Px_orbital.png/60px-Px_orbital.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/Px_orbital.png/120px-Px_orbital.png 1.5x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Py_orbital.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Py_orbital.png/60px-Py_orbital.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Py_orbital.png/120px-Py_orbital.png 1.5x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <th><span class="texhtml"><i>n</i> = 3</span> </th> <td><span typeof="mw:File"><a href="/wiki/File:S3M0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/S3M0.png/50px-S3M0.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/S3M0.png/75px-S3M0.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ae/S3M0.png/100px-S3M0.png 2x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:P3M0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2d/P3M0.png/50px-P3M0.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2d/P3M0.png/75px-P3M0.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2d/P3M0.png/100px-P3M0.png 2x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:P3x.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/P3x.png/50px-P3x.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/P3x.png/75px-P3x.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ed/P3x.png/100px-P3x.png 2x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:P3y.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/01/P3y.png/50px-P3y.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/01/P3y.png/75px-P3y.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/01/P3y.png/100px-P3y.png 2x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:D3M0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/D3M0.png/50px-D3M0.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/D3M0.png/75px-D3M0.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/57/D3M0.png/100px-D3M0.png 2x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Dxz_orbital.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/Dxz_orbital.png/50px-Dxz_orbital.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/Dxz_orbital.png/75px-Dxz_orbital.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/35/Dxz_orbital.png/100px-Dxz_orbital.png 2x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Dyz_orbital.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bf/Dyz_orbital.png/60px-Dyz_orbital.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bf/Dyz_orbital.png/120px-Dyz_orbital.png 1.5x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Dxy_orbital.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Dxy_orbital.png/60px-Dxy_orbital.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Dxy_orbital.png/120px-Dxy_orbital.png 1.5x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Dx2-y2_orbital.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Dx2-y2_orbital.png/50px-Dx2-y2_orbital.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Dx2-y2_orbital.png/75px-Dx2-y2_orbital.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Dx2-y2_orbital.png/100px-Dx2-y2_orbital.png 2x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <th><span class="texhtml"><i>n</i> = 4</span> </th> <td><span typeof="mw:File"><a href="/wiki/File:S4M0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/S4M0.png/50px-S4M0.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/S4M0.png/75px-S4M0.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/S4M0.png/100px-S4M0.png 2x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:P4M0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/P4M0.png/50px-P4M0.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/P4M0.png/75px-P4M0.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8f/P4M0.png/100px-P4M0.png 2x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:P4x.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/P4x.png/60px-P4x.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/P4x.png/120px-P4x.png 1.5x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:P4y.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/P4y.png/60px-P4y.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/P4y.png/120px-P4y.png 1.5x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:D4M0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/D4M0.png/50px-D4M0.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/D4M0.png/75px-D4M0.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/14/D4M0.png/100px-D4M0.png 2x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:D4xz.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/D4xz.png/60px-D4xz.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/D4xz.png/120px-D4xz.png 1.5x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:D4yz2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/D4yz2.png/50px-D4yz2.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/D4yz2.png/75px-D4yz2.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/86/D4yz2.png/100px-D4yz2.png 2x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:D4xy.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/D4xy.png/60px-D4xy.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/D4xy.png/120px-D4xy.png 1.5x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:D4x2-y2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/D4x2-y2.png/50px-D4x2-y2.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/D4x2-y2.png/75px-D4x2-y2.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/50/D4x2-y2.png/100px-D4x2-y2.png 2x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:F4M0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/F4M0.png/60px-F4M0.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/F4M0.png/120px-F4M0.png 1.5x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Fxz2_orbital.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Fxz2_orbital.png/50px-Fxz2_orbital.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Fxz2_orbital.png/75px-Fxz2_orbital.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/08/Fxz2_orbital.png/100px-Fxz2_orbital.png 2x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Fyz2_orbital.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/Fyz2_orbital.png/60px-Fyz2_orbital.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/Fyz2_orbital.png/120px-Fyz2_orbital.png 1.5x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Fxyz_orbital.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/Fxyz_orbital.png/50px-Fxyz_orbital.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/Fxyz_orbital.png/75px-Fxyz_orbital.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/40/Fxyz_orbital.png/100px-Fxyz_orbital.png 2x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Fz(x2-y2)_orbital.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/25/Fz%28x2-y2%29_orbital.png/60px-Fz%28x2-y2%29_orbital.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/25/Fz%28x2-y2%29_orbital.png/120px-Fz%28x2-y2%29_orbital.png 1.5x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Fx(x2-3y2)_orbital.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Fx%28x2-3y2%29_orbital.png/50px-Fx%28x2-3y2%29_orbital.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Fx%28x2-3y2%29_orbital.png/75px-Fx%28x2-3y2%29_orbital.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/24/Fx%28x2-3y2%29_orbital.png/100px-Fx%28x2-3y2%29_orbital.png 2x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Fy(3x2-y2)_orbital.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Fy%283x2-y2%29_orbital.png/50px-Fy%283x2-y2%29_orbital.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Fy%283x2-y2%29_orbital.png/75px-Fy%283x2-y2%29_orbital.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Fy%283x2-y2%29_orbital.png/100px-Fy%283x2-y2%29_orbital.png 2x" data-file-width="1000" data-file-height="1000" /></a></span> </td></tr> <tr> <th><span class="texhtml"><i>n</i> = 5</span> </th> <td><span typeof="mw:File"><a href="/wiki/File:S5M0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/S5M0.png/50px-S5M0.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/S5M0.png/75px-S5M0.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/68/S5M0.png/100px-S5M0.png 2x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:P5M0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/P5M0.png/50px-P5M0.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/P5M0.png/75px-P5M0.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/20/P5M0.png/100px-P5M0.png 2x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:P5x.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/P5x.png/60px-P5x.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/P5x.png/120px-P5x.png 1.5x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:P5y.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/P5y.png/50px-P5y.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/P5y.png/75px-P5y.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fc/P5y.png/100px-P5y.png 2x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:D5M0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/D5M0.png/60px-D5M0.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/D5M0.png/120px-D5M0.png 1.5x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:D5xz.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/D5xz.png/60px-D5xz.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/D5xz.png/120px-D5xz.png 1.5x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:D5yz.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/D5yz.png/50px-D5yz.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/D5yz.png/75px-D5yz.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/28/D5yz.png/100px-D5yz.png 2x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:D5xy.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/D5xy.png/50px-D5xy.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/D5xy.png/75px-D5xy.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1b/D5xy.png/100px-D5xy.png 2x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:D5x2-y2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/D5x2-y2.png/60px-D5x2-y2.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/D5x2-y2.png/120px-D5x2-y2.png 1.5x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><b>. . .</b> </td> <td><b>. . .</b> </td> <td><b>. . .</b> </td> <td><b>. . .</b> </td> <td><b>. . .</b> </td> <td><b>. . .</b> </td> <td><b>. . .</b> </td></tr> <tr> <th><span class="texhtml"><i>n</i> = 6</span> </th> <td><span typeof="mw:File"><a href="/wiki/File:S6M0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/S6M0.png/50px-S6M0.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/S6M0.png/75px-S6M0.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8a/S6M0.png/100px-S6M0.png 2x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:P6M0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/P6M0.png/50px-P6M0.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/P6M0.png/75px-P6M0.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2f/P6M0.png/100px-P6M0.png 2x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:P6x.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/P6x.png/50px-P6x.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/P6x.png/75px-P6x.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3d/P6x.png/100px-P6x.png 2x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:P6y.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/P6y.png/60px-P6y.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/P6y.png/120px-P6y.png 1.5x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><b>. . .</b> ‡ </td> <td><b>. . .</b> ‡ </td> <td><b>. . .</b> ‡ </td> <td><b>. . .</b> ‡ </td> <td><b>. . .</b> ‡ </td> <td><b>. . .</b> * </td> <td><b>. . .</b> * </td> <td><b>. . .</b> * </td> <td><b>. . .</b> * </td> <td><b>. . .</b> * </td> <td><b>. . .</b> * </td> <td><b>. . .</b> * </td></tr> <tr> <th><span class="texhtml"><i>n</i> = 7</span> </th> <td><span typeof="mw:File"><a href="/wiki/File:S7M0.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/S7M0.png/60px-S7M0.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/S7M0.png/120px-S7M0.png 1.5x" data-file-width="1000" data-file-height="1000" /></a></span> </td> <td><b>. . .</b> † </td> <td><b>. . .</b> † </td> <td><b>. . .</b> † </td> <td><b>. . .</b> * </td> <td><b>. . .</b> * </td> <td><b>. . .</b> * </td> <td><b>. . .</b> * </td> <td><b>. . .</b> * </td> <td><b>. . .</b> * </td> <td><b>. . .</b> * </td> <td><b>. . .</b> * </td> <td><b>. . .</b> * </td> <td><b>. . .</b> * </td> <td><b>. . .</b> * </td> <td><b>. . .</b> * </td></tr></tbody></table> <p>* <i>No elements with 6f, 7d or 7f electrons have been discovered yet.</i> </p><p>† <i>Elements with 7p electrons have been discovered, but their <a href="/wiki/Electronic_configuration" class="mw-redirect" title="Electronic configuration">electronic configurations</a> are only predicted – save the exceptional <a href="/wiki/Lawrencium" title="Lawrencium">Lr</a>, which fills 7p<sup>1</sup> instead of 6d<sup>1</sup>.</i> </p><p>‡ <i>For the elements whose highest occupied orbital is a 6d orbital, only some electronic configurations have been confirmed.</i> (<a href="/wiki/Meitnerium" title="Meitnerium">Mt</a>, <a href="/wiki/Darmstadtium" title="Darmstadtium">Ds</a>, <a href="/wiki/Roentgenium" title="Roentgenium">Rg</a> and <a href="/wiki/Copernicium" title="Copernicium">Cn</a> are still missing). </p><p> These are the real-valued orbitals commonly used in chemistry. Only the <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=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e57f21007575fd03e3be0da20af34d25829cc9a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=0}" /></span> orbitals where are eigenstates of the orbital angular momentum operator, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\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/f890c759326e0b9e75b24931dcf2a53862ab309c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.585ex; height:3.176ex;" alt="{\displaystyle {\hat {L}}_{z}}" /></span>. The columns with <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=\pm 1,\pm 2,\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mo>±</mo> <mn>1</mn> <mo>,</mo> <mo>±</mo> <mn>2</mn> <mo>,</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=\pm 1,\pm 2,\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd3150849b69d9966653d6f13eb3c65d4e36600d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.871ex; height:2.509ex;" alt="{\displaystyle m=\pm 1,\pm 2,\cdots }" /></span> are combinations of two eigenstates. See <a href="/wiki/File:Atomic_orbitals_spdf_m-eigenstates_and_superpositions.png" title="File:Atomic orbitals spdf m-eigenstates and superpositions.png">comparison in the following picture</a>: </p><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Atomic_orbitals_spdf_m-eigenstates_and_superpositions.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Atomic_orbitals_spdf_m-eigenstates_and_superpositions.png/220px-Atomic_orbitals_spdf_m-eigenstates_and_superpositions.png" decoding="async" width="220" height="126" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Atomic_orbitals_spdf_m-eigenstates_and_superpositions.png/330px-Atomic_orbitals_spdf_m-eigenstates_and_superpositions.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Atomic_orbitals_spdf_m-eigenstates_and_superpositions.png/440px-Atomic_orbitals_spdf_m-eigenstates_and_superpositions.png 2x" data-file-width="2800" data-file-height="1600" /></a><figcaption>Atomic orbitals spdf m-eigenstates and superpositions</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Qualitative_understanding_of_shapes">Qualitative understanding of shapes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atomic_orbital&action=edit&section=17" title="Edit section: Qualitative understanding of shapes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The shapes of atomic orbitals can be qualitatively understood by considering the analogous case of <a href="/wiki/Vibrations_of_a_circular_drum" class="mw-redirect" title="Vibrations of a circular drum">standing waves on a circular drum</a>.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> To see the analogy, the mean vibrational displacement of each bit of drum membrane from the equilibrium point over many cycles (a measure of average drum membrane velocity and momentum at that point) must be considered relative to that point's distance from the center of the drum head. If this displacement is taken as being analogous to the probability of finding an electron at a given distance from the nucleus, then it will be seen that the many modes of the vibrating disk form patterns that trace the various shapes of atomic orbitals. The basic reason for this correspondence lies in the fact that the distribution of kinetic energy and momentum in a matter-wave is predictive of where the particle associated with the wave will be. That is, the probability of finding an electron at a given place is also a function of the electron's average momentum at that point, since high electron momentum at a given position tends to "localize" the electron in that position, via the properties of electron wave-packets (see the <a href="/wiki/Uncertainty_principle" title="Uncertainty principle">Heisenberg uncertainty principle</a> for details of the mechanism). </p><p>This relationship means that certain key features can be observed in both drum membrane modes and atomic orbitals. For example, in all of the modes analogous to <b>s</b> orbitals (the top row in the animated illustration below), it can be seen that the very center of the drum membrane vibrates most strongly, corresponding to the <a href="/wiki/Antinode" class="mw-redirect" title="Antinode">antinode</a> in all <b>s</b> orbitals in an atom. This antinode means the electron is most likely to be at the physical position of the nucleus (which it passes straight through without scattering or striking it), since it is moving (on average) most rapidly at that point, giving it maximal momentum. </p><p>A mental "planetary orbit" picture closest to the behavior of electrons in <b>s</b> orbitals, all of which have no angular momentum, might perhaps be that of a <a href="/wiki/Keplerian_orbit" class="mw-redirect" title="Keplerian orbit">Keplerian orbit</a> with the <a href="/wiki/Orbital_eccentricity" title="Orbital eccentricity">orbital eccentricity</a> of 1 but a finite major axis, not physically possible (because <a href="/wiki/Particle" title="Particle">particles</a> were to collide), but can be imagined as a <a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">limit</a> of orbits with equal major axes but increasing eccentricity. </p><p>Below, a number of drum membrane vibration modes and the respective wave functions of the hydrogen atom are shown. A correspondence can be considered where the wave functions of a vibrating drum head are for a two-coordinate system <span class="texhtml">ψ(<i>r</i>, <i>θ</i>)</span> and the wave functions for a vibrating sphere are three-coordinate <span class="texhtml">ψ(<i>r</i>, <i>θ</i>, <i>φ</i>)</span>. </p> <ul class="gallery mw-gallery-nolines" style="max-width: 627px;"> <li class="gallerycaption">s-type drum modes and wave functions</li> <li class="gallerybox" style="width: 205px"> <div class="thumb" style="width: 200px;"><span typeof="mw:File"><a href="/wiki/File:Drum_vibration_mode01.gif" class="mw-file-description" title="Drum mode '"`UNIQ--postMath-00000056-QINU`"'"><img alt="Drum mode '"`UNIQ--postMath-00000056-QINU`"'" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/15/Drum_vibration_mode01.gif/186px-Drum_vibration_mode01.gif" decoding="async" width="186" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/1/15/Drum_vibration_mode01.gif 1.5x" data-file-width="249" data-file-height="161" /></a></span></div> <div class="gallerytext">Drum mode <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 u_{01}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>01</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{01}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a3f65d14fa8f56cc1f09dea5855c564adcf415b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.206ex; height:2.009ex;" alt="{\displaystyle u_{01}}" /></span></div> </li> <li class="gallerybox" style="width: 205px"> <div class="thumb" style="width: 200px;"><span typeof="mw:File"><a href="/wiki/File:Drum_vibration_mode02.gif" class="mw-file-description" title="Drum mode '"`UNIQ--postMath-00000057-QINU`"'"><img alt="Drum mode '"`UNIQ--postMath-00000057-QINU`"'" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Drum_vibration_mode02.gif/250px-Drum_vibration_mode02.gif" decoding="async" width="179" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/7/7a/Drum_vibration_mode02.gif 1.5x" data-file-width="252" data-file-height="169" /></a></span></div> <div class="gallerytext">Drum mode <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 u_{02}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>02</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{02}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8649f7c8f3455043855549b3416d5c8696dcafa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.206ex; height:2.009ex;" alt="{\displaystyle u_{02}}" /></span></div> </li> <li class="gallerybox" style="width: 205px"> <div class="thumb" style="width: 200px;"><span typeof="mw:File"><a href="/wiki/File:Drum_vibration_mode03.gif" class="mw-file-description" title="Drum mode '"`UNIQ--postMath-00000058-QINU`"'"><img alt="Drum mode '"`UNIQ--postMath-00000058-QINU`"'" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/23/Drum_vibration_mode03.gif/200px-Drum_vibration_mode03.gif" decoding="async" width="200" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/2/23/Drum_vibration_mode03.gif 1.5x" data-file-width="250" data-file-height="130" /></a></span></div> <div class="gallerytext">Drum mode <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 u_{03}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>03</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{03}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6c00df74336b0f739267e17af34b7782e814b8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.206ex; height:2.009ex;" alt="{\displaystyle u_{03}}" /></span></div> </li> <li class="gallerybox" style="width: 205px"> <div class="thumb" style="width: 200px;"><span typeof="mw:File"><a href="/wiki/File:Phi_1s.gif" class="mw-file-description" title="Wave function of 1s orbital (real part, 2D-cut, '"`UNIQ--postMath-00000059-QINU`"')"><img alt="Wave function of 1s orbital (real part, 2D-cut, '"`UNIQ--postMath-00000059-QINU`"')" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Phi_1s.gif/250px-Phi_1s.gif" decoding="async" width="200" height="105" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Phi_1s.gif/330px-Phi_1s.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/33/Phi_1s.gif/400px-Phi_1s.gif 2x" data-file-width="496" data-file-height="260" /></a></span></div> <div class="gallerytext">Wave function of 1s orbital (real part, 2D-cut, <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_{\mathrm {max} }=2a_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">x</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>2</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{\mathrm {max} }=2a_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9ed5896d63e2140f4d6967ceb0ff9681ce3c746" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.885ex; height:2.509ex;" alt="{\displaystyle r_{\mathrm {max} }=2a_{0}}" /></span>)</div> </li> <li class="gallerybox" style="width: 205px"> <div class="thumb" style="width: 200px;"><span typeof="mw:File"><a href="/wiki/File:Phi_2s.gif" class="mw-file-description" title="Wave function of 2s orbital (real part, 2D-cut, '"`UNIQ--postMath-0000005A-QINU`"')"><img alt="Wave function of 2s orbital (real part, 2D-cut, '"`UNIQ--postMath-0000005A-QINU`"')" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/Phi_2s.gif/250px-Phi_2s.gif" decoding="async" width="200" height="105" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/Phi_2s.gif/330px-Phi_2s.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/10/Phi_2s.gif/400px-Phi_2s.gif 2x" data-file-width="496" data-file-height="260" /></a></span></div> <div class="gallerytext">Wave function of 2s orbital (real part, 2D-cut, <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_{\mathrm {max} }=10a_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">x</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>10</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{\mathrm {max} }=10a_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c3f4dbd074399a7d5cf841bcedd2c65b6dd3b53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.047ex; height:2.509ex;" alt="{\displaystyle r_{\mathrm {max} }=10a_{0}}" /></span>)</div> </li> <li class="gallerybox" style="width: 205px"> <div class="thumb" style="width: 200px;"><span typeof="mw:File"><a href="/wiki/File:Phi_3s.gif" class="mw-file-description" title="Wave function of 3s orbital (real part, 2D-cut, '"`UNIQ--postMath-0000005B-QINU`"')"><img alt="Wave function of 3s orbital (real part, 2D-cut, '"`UNIQ--postMath-0000005B-QINU`"')" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Phi_3s.gif/250px-Phi_3s.gif" decoding="async" width="200" height="105" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Phi_3s.gif/330px-Phi_3s.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/44/Phi_3s.gif/400px-Phi_3s.gif 2x" data-file-width="496" data-file-height="260" /></a></span></div> <div class="gallerytext">Wave function of 3s orbital (real part, 2D-cut, <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_{\mathrm {max} }=20a_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">x</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>20</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{\mathrm {max} }=20a_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08aa52b236eaa6de62038cd9fa96d08a8eebf411" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.047ex; height:2.509ex;" alt="{\displaystyle r_{\mathrm {max} }=20a_{0}}" /></span>)</div> </li> </ul> <p>None of the other sets of modes in a drum membrane have a central antinode, and in all of them the center of the drum does not move. These correspond to a node at the nucleus for all non-<b>s</b> orbitals in an atom. These orbitals all have some angular momentum, and in the planetary model, they correspond to particles in orbit with eccentricity less than 1.0, so that they do not pass straight through the center of the primary body, but keep somewhat away from it. </p><p>In addition, the drum modes analogous to <b>p</b> and <b>d</b> modes in an atom show spatial irregularity along the different radial directions from the center of the drum, whereas all of the modes analogous to <b>s</b> modes are perfectly symmetrical in radial direction. The non-radial-symmetry properties of non-<b>s</b> orbitals are necessary to localize a particle with angular momentum and a wave nature in an orbital where it must tend to stay away from the central attraction force, since any particle localized at the point of central attraction could have no angular momentum. For these modes, waves in the drum head tend to avoid the central point. Such features again emphasize that the shapes of atomic orbitals are a direct consequence of the wave nature of electrons. </p> <ul class="gallery mw-gallery-nolines" style="max-width: 627px;"> <li class="gallerycaption">p-type drum modes and wave functions</li> <li class="gallerybox" style="width: 205px"> <div class="thumb" style="width: 200px;"><span typeof="mw:File"><a href="/wiki/File:Drum_vibration_mode11.gif" class="mw-file-description" title="Drum mode '"`UNIQ--postMath-0000005C-QINU`"'"><img alt="Drum mode '"`UNIQ--postMath-0000005C-QINU`"'" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Drum_vibration_mode11.gif/200px-Drum_vibration_mode11.gif" decoding="async" width="200" height="105" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/3/3c/Drum_vibration_mode11.gif 1.5x" data-file-width="248" data-file-height="130" /></a></span></div> <div class="gallerytext">Drum mode <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 u_{11}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{11}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19a88ec5a96eb124e3134862684656933fbaade1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.206ex; height:2.009ex;" alt="{\displaystyle u_{11}}" /></span></div> </li> <li class="gallerybox" style="width: 205px"> <div class="thumb" style="width: 200px;"><span typeof="mw:File"><a href="/wiki/File:Drum_vibration_mode12.gif" class="mw-file-description" title="Drum mode '"`UNIQ--postMath-0000005D-QINU`"'"><img alt="Drum mode '"`UNIQ--postMath-0000005D-QINU`"'" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Drum_vibration_mode12.gif/176px-Drum_vibration_mode12.gif" decoding="async" width="176" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/e/e9/Drum_vibration_mode12.gif 1.5x" data-file-width="249" data-file-height="170" /></a></span></div> <div class="gallerytext">Drum mode <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 u_{12}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{12}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a73ef949223ef790e7745aec6d5b346c8be49cc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.206ex; height:2.009ex;" alt="{\displaystyle u_{12}}" /></span></div> </li> <li class="gallerybox" style="width: 205px"> <div class="thumb" style="width: 200px;"><span typeof="mw:File"><a href="/wiki/File:Drum_vibration_mode13.gif" class="mw-file-description" title="Drum mode '"`UNIQ--postMath-0000005E-QINU`"'"><img alt="Drum mode '"`UNIQ--postMath-0000005E-QINU`"'" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Drum_vibration_mode13.gif/250px-Drum_vibration_mode13.gif" decoding="async" width="200" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/c/c4/Drum_vibration_mode13.gif 1.5x" data-file-width="250" data-file-height="130" /></a></span></div> <div class="gallerytext">Drum mode <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 u_{13}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{13}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55680be611153b4e8c14dce3481bb85ac86e1b23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.206ex; height:2.009ex;" alt="{\displaystyle u_{13}}" /></span></div> </li> <li class="gallerybox" style="width: 205px"> <div class="thumb" style="width: 200px;"><span typeof="mw:File"><a href="/wiki/File:Phi_2p.gif" class="mw-file-description" title="Wave function of 2p orbital (real part, 2D-cut, '"`UNIQ--postMath-0000005F-QINU`"')"><img alt="Wave function of 2p orbital (real part, 2D-cut, '"`UNIQ--postMath-0000005F-QINU`"')" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Phi_2p.gif/250px-Phi_2p.gif" decoding="async" width="200" height="105" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Phi_2p.gif/330px-Phi_2p.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/08/Phi_2p.gif/400px-Phi_2p.gif 2x" data-file-width="496" data-file-height="260" /></a></span></div> <div class="gallerytext">Wave function of 2p orbital (real part, 2D-cut, <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_{\mathrm {max} }=10a_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">x</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>10</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{\mathrm {max} }=10a_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c3f4dbd074399a7d5cf841bcedd2c65b6dd3b53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.047ex; height:2.509ex;" alt="{\displaystyle r_{\mathrm {max} }=10a_{0}}" /></span>)</div> </li> <li class="gallerybox" style="width: 205px"> <div class="thumb" style="width: 200px;"><span typeof="mw:File"><a href="/wiki/File:Phi_3p.gif" class="mw-file-description" title="Wave function of 3p orbital (real part, 2D-cut, '"`UNIQ--postMath-00000060-QINU`"')"><img alt="Wave function of 3p orbital (real part, 2D-cut, '"`UNIQ--postMath-00000060-QINU`"')" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Phi_3p.gif/250px-Phi_3p.gif" decoding="async" width="200" height="105" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Phi_3p.gif/330px-Phi_3p.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Phi_3p.gif/400px-Phi_3p.gif 2x" data-file-width="496" data-file-height="260" /></a></span></div> <div class="gallerytext">Wave function of 3p orbital (real part, 2D-cut, <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_{\mathrm {max} }=20a_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">x</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>20</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{\mathrm {max} }=20a_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08aa52b236eaa6de62038cd9fa96d08a8eebf411" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.047ex; height:2.509ex;" alt="{\displaystyle r_{\mathrm {max} }=20a_{0}}" /></span>)</div> </li> <li class="gallerybox" style="width: 205px"> <div class="thumb" style="width: 200px;"><span typeof="mw:File"><a href="/wiki/File:Phi_4p.gif" class="mw-file-description" title="Wave function of 4p orbital (real part, 2D-cut, '"`UNIQ--postMath-00000061-QINU`"')"><img alt="Wave function of 4p orbital (real part, 2D-cut, '"`UNIQ--postMath-00000061-QINU`"')" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Phi_4p.gif/200px-Phi_4p.gif" decoding="async" width="200" height="105" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Phi_4p.gif/300px-Phi_4p.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Phi_4p.gif/400px-Phi_4p.gif 2x" data-file-width="496" data-file-height="260" /></a></span></div> <div class="gallerytext">Wave function of 4p orbital (real part, 2D-cut, <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_{\mathrm {max} }=25a_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">x</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>25</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{\mathrm {max} }=25a_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09a3bc40b98d01908f26129eaff791d26609c86b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.047ex; height:2.509ex;" alt="{\displaystyle r_{\mathrm {max} }=25a_{0}}" /></span>)</div> </li> </ul> <ul class="gallery mw-gallery-nolines" style="max-width: 627px;"> <li class="gallerycaption">d-type drum modes</li> <li class="gallerybox" style="width: 205px"> <div class="thumb" style="width: 200px;"><span typeof="mw:File"><a href="/wiki/File:Drum_vibration_mode21.gif" class="mw-file-description" title="Drum mode '"`UNIQ--postMath-00000062-QINU`"'"><img alt="Drum mode '"`UNIQ--postMath-00000062-QINU`"'" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6e/Drum_vibration_mode21.gif/200px-Drum_vibration_mode21.gif" decoding="async" width="200" height="105" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/6/6e/Drum_vibration_mode21.gif 1.5x" data-file-width="248" data-file-height="130" /></a></span></div> <div class="gallerytext">Drum mode <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 u_{21}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{21}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf978c9e6d38884739ee4c185cb097ed4633c835" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.206ex; height:2.009ex;" alt="{\displaystyle u_{21}}" /></span> </div> </li> <li class="gallerybox" style="width: 205px"> <div class="thumb" style="width: 200px;"><span typeof="mw:File"><a href="/wiki/File:Drum_vibration_mode22.gif" class="mw-file-description" title="Drum mode '"`UNIQ--postMath-00000063-QINU`"'"><img alt="Drum mode '"`UNIQ--postMath-00000063-QINU`"'" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Drum_vibration_mode22.gif/200px-Drum_vibration_mode22.gif" decoding="async" width="200" height="105" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/6/6f/Drum_vibration_mode22.gif 1.5x" data-file-width="248" data-file-height="130" /></a></span></div> <div class="gallerytext">Drum mode <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 u_{22}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{22}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f9492a867d06345ee64be43885c3e20e4064e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.206ex; height:2.009ex;" alt="{\displaystyle u_{22}}" /></span></div> </li> <li class="gallerybox" style="width: 205px"> <div class="thumb" style="width: 200px;"><span typeof="mw:File"><a href="/wiki/File:Drum_vibration_mode23.gif" class="mw-file-description" title="Drum mode '"`UNIQ--postMath-00000064-QINU`"'"><img alt="Drum mode '"`UNIQ--postMath-00000064-QINU`"'" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Drum_vibration_mode23.gif/250px-Drum_vibration_mode23.gif" decoding="async" width="200" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/1/1f/Drum_vibration_mode23.gif 1.5x" data-file-width="250" data-file-height="130" /></a></span></div> <div class="gallerytext">Drum mode <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 u_{23}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{23}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3627b016df390414dfef66d83bdb72c40fcc35d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.206ex; height:2.009ex;" alt="{\displaystyle u_{23}}" /></span></div> </li> </ul> <div class="mw-heading mw-heading2"><h2 id="Orbital_energy">Orbital energy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atomic_orbital&action=edit&section=18" title="Edit section: Orbital energy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Electron_shell" title="Electron shell">Electron shell</a></div> <p>In atoms with one electron (<a href="/wiki/Hydrogen-like_atom" title="Hydrogen-like atom">hydrogen-like atom</a>), the energy of an orbital (and, consequently, any electron in the orbital) is determined mainly by <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>. The <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> orbital has the lowest possible energy in the atom. Each successively higher value of <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> has a higher energy, but the difference decreases as <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> increases. For high <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>, the energy becomes so high that the electron can easily escape the atom. In single electron atoms, all levels with different <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 \ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }" /></span> within a given <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> are degenerate in the Schrödinger approximation, and have the same energy. This approximation is broken slightly in the solution to the Dirac equation (where energy depends on <span class="texhtml mvar" style="font-style:italic;">n</span> and another quantum number <span class="texhtml mvar" style="font-style:italic;">j</span>), and by the effect of the magnetic field of the nucleus and <a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">quantum electrodynamics</a> effects. The latter induce tiny binding energy differences especially for <b>s</b> electrons that go nearer the nucleus, since these feel a very slightly different nuclear charge, even in one-electron atoms; see <a href="/wiki/Lamb_shift" title="Lamb shift">Lamb shift</a>. </p><p>In atoms with multiple electrons, the energy of an electron depends not only on its orbital, but also on its interactions with other electrons. These interactions depend on the detail of its spatial probability distribution, and so the <a href="/wiki/Energy_level" title="Energy level">energy levels</a> of orbitals depend not only on <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> but also on <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 \ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }" /></span>. Higher values of <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 \ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }" /></span> are associated with higher values of energy; for instance, the 2p state is higher than the 2s state. When <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 \ell =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 \ell =2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/682399b5a2ae37a3b5b13a0e4e3f7348fa3c0ab4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.231ex; height:2.176ex;" alt="{\displaystyle \ell =2}" /></span>, the increase in energy of the orbital becomes so large as to push the energy of orbital above the energy of the s orbital in the next higher shell; when <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 \ell =3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell =3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4358eabf9aa70bc959f066c915df0f14efaea5ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.231ex; height:2.176ex;" alt="{\displaystyle \ell =3}" /></span> the energy is pushed into the shell two steps higher. The filling of the 3d orbitals does not occur until the 4s orbitals have been filled. </p><p>The increase in energy for subshells of increasing angular momentum in larger atoms is due to electron–electron interaction effects, and it is specifically related to the ability of low angular momentum electrons to penetrate more effectively toward the nucleus, where they are subject to less screening from the charge of intervening electrons. Thus, in atoms with higher atomic number, the <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 \ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }" /></span> of electrons becomes more and more of a determining factor in their energy, and the principal quantum numbers <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> of electrons becomes less and less important in their energy placement. </p><p>The energy sequence of the first 35 subshells (e.g., 1s, 2p, 3d, etc.) is given in the following table. Each cell represents a subshell with <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> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }" /></span> given by its row and column indices, respectively. The number in the cell is the subshell's position in the sequence. For a linear listing of the subshells in terms of increasing energies in multielectron atoms, see the section below. </p> <table class="wikitable"> <tbody><tr> <th style="background:var(--background-color-neutral,#eaecf0);color:inherit;background:linear-gradient(to top right,var(--background-color-neutral,#eaecf0) 49%,var(--border-color-base,#a2a9b1) 49.5%,var(--border-color-base,#a2a9b1) 50.5%,var(--background-color-neutral,#eaecf0) 51%);line-height:1.2;padding:0.1em 0.4em;"><div style="margin-left:2em;text-align:right"><i>l</i></div><div style="margin-right:2em;text-align:left"><i>n</i></div> </th> <th>s </th> <th>p </th> <th>d </th> <th>f </th> <th>g </th> <th>h </th></tr> <tr> <th><i>1</i> </th> <td>1</td> <td></td> <td></td> <td></td> <td></td> <td> </td></tr> <tr> <th><i>2</i> </th> <td>2</td> <td>3</td> <td></td> <td></td> <td></td> <td> </td></tr> <tr> <th><i>3</i> </th> <td>4</td> <td>5</td> <td>7</td> <td></td> <td></td> <td> </td></tr> <tr> <th><i>4</i> </th> <td>6</td> <td>8</td> <td>10</td> <td>13</td> <td></td> <td> </td></tr> <tr> <th><i>5</i> </th> <td>9</td> <td>11</td> <td>14</td> <td>17</td> <td><i>21</i></td> <td> </td></tr> <tr> <th><i>6</i> </th> <td>12</td> <td>15</td> <td>18</td> <td><i>22</i></td> <td><i>26</i></td> <td><i>31</i> </td></tr> <tr> <th><i>7</i> </th> <td>16</td> <td>19</td> <td><i>23</i></td> <td><i>27</i></td> <td><i>32</i></td> <td><i>37</i> </td></tr> <tr> <th><i>8</i> </th> <td><i>20</i></td> <td><i>24</i></td> <td><i>28</i></td> <td><i>33</i></td> <td><i>38</i></td> <td><i>44</i> </td></tr> <tr> <th><i>9</i> </th> <td><i>25</i></td> <td><i>29</i></td> <td><i>34</i></td> <td><i>39</i></td> <td><i>45</i></td> <td><i>51</i> </td></tr> <tr> <th><i>10</i> </th> <td><i>30</i></td> <td><i>35</i></td> <td><i>40</i></td> <td><i>46</i></td> <td><i>52</i></td> <td><i>59</i> </td></tr></tbody></table> <p><i>Note: empty cells indicate non-existent sublevels, while numbers in italics indicate sublevels that could (potentially) exist, but which do not hold electrons in any element currently known.</i> </p> <div class="mw-heading mw-heading2"><h2 id="Electron_placement_and_the_periodic_table">Electron placement and the periodic table</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atomic_orbital&action=edit&section=19" title="Edit section: Electron placement and the periodic table"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Electron_orbitals.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/Electron_orbitals.svg/500px-Electron_orbitals.svg.png" decoding="async" width="350" height="227" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/Electron_orbitals.svg/525px-Electron_orbitals.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/11/Electron_orbitals.svg/700px-Electron_orbitals.svg.png 2x" data-file-width="857" data-file-height="556" /></a><figcaption>Electron atomic and <a href="/wiki/Molecular_orbital" title="Molecular orbital">molecular</a> orbitals. The chart of orbitals (<b>left</b>) is arranged by increasing energy (see <a href="/wiki/Madelung_rule" class="mw-redirect" title="Madelung rule">Madelung rule</a>). <i>Atomic orbits are functions of three variables (two angles, and the distance <span class="texhtml mvar" style="font-style:italic;">r</span> from the nucleus). These images are faithful to the angular component of the orbital, but not entirely representative of the orbital as a whole.</i></figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><span><video id="mwe_player_0" poster="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Atomic_orbitals_and_periodic_table_construction.ogv/220px--Atomic_orbitals_and_periodic_table_construction.ogv.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="220" height="124" data-durationhint="64" data-mwtitle="Atomic_orbitals_and_periodic_table_construction.ogv" data-mwprovider="wikimediacommons" resource="/wiki/File:Atomic_orbitals_and_periodic_table_construction.ogv"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/61/Atomic_orbitals_and_periodic_table_construction.ogv/Atomic_orbitals_and_periodic_table_construction.ogv.480p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="480p.vp9.webm" data-width="854" data-height="480" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/61/Atomic_orbitals_and_periodic_table_construction.ogv/Atomic_orbitals_and_periodic_table_construction.ogv.720p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="720p.vp9.webm" data-width="1280" data-height="720" /><source src="//upload.wikimedia.org/wikipedia/commons/6/61/Atomic_orbitals_and_periodic_table_construction.ogv" type="video/ogg; codecs="theora, vorbis"" data-width="1280" data-height="720" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/61/Atomic_orbitals_and_periodic_table_construction.ogv/Atomic_orbitals_and_periodic_table_construction.ogv.144p.mjpeg.mov" type="video/quicktime" data-transcodekey="144p.mjpeg.mov" data-width="256" data-height="144" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/61/Atomic_orbitals_and_periodic_table_construction.ogv/Atomic_orbitals_and_periodic_table_construction.ogv.240p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="240p.vp9.webm" data-width="426" data-height="240" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/61/Atomic_orbitals_and_periodic_table_construction.ogv/Atomic_orbitals_and_periodic_table_construction.ogv.360p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="360p.vp9.webm" data-width="640" data-height="360" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/61/Atomic_orbitals_and_periodic_table_construction.ogv/Atomic_orbitals_and_periodic_table_construction.ogv.360p.webm" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="640" data-height="360" /><track src="https://commons.wikimedia.org/w/api.php?action=timedtext&title=File%3AAtomic_orbitals_and_periodic_table_construction.ogv&lang=es&trackformat=vtt&origin=%2A" kind="subtitles" type="text/vtt" srclang="es" label="español ‪(es)‬" data-dir="ltr" /></video></span><figcaption>Atomic orbitals and periodic table construction</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Electron_configuration" title="Electron configuration">Electron configuration</a> and <a href="/wiki/Electron_shell" title="Electron shell">Electron shell</a></div> <p>Several rules govern the placement of electrons in orbitals (<i><a href="/wiki/Electron_configuration" title="Electron configuration">electron configuration</a></i>). The first dictates that no two electrons in an atom may have the same set of values of quantum numbers (this is the <a href="/wiki/Pauli_exclusion_principle" title="Pauli exclusion principle">Pauli exclusion principle</a>). These quantum numbers include the three that define orbitals, as well as the <a href="/wiki/Spin_magnetic_quantum_number" class="mw-redirect" title="Spin magnetic quantum number">spin magnetic quantum number</a> <span class="texhtml mvar" style="font-style:italic;">m<sub>s</sub></span>. Thus, two electrons may occupy a single orbital, so long as they have different values of <span class="texhtml mvar" style="font-style:italic;">m<sub>s</sub></span>. Because <span class="texhtml mvar" style="font-style:italic;">m<sub>s</sub></span> takes one of only two values (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" /><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> or <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" /><span class="sfrac">⁠<span class="tion"><span class="num">-1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>), at most two electrons can occupy each orbital. </p><p>Additionally, an electron always tends to fall to the lowest possible energy state. It is possible for it to occupy any orbital so long as it does not violate the Pauli exclusion principle, but if lower-energy orbitals are available, this condition is unstable. The electron will eventually lose energy (by releasing a <a href="/wiki/Photon" title="Photon">photon</a>) and drop into the lower orbital. Thus, electrons fill orbitals in the order specified by the energy sequence given above. </p><p>This behavior is responsible for the structure of the <a href="/wiki/Periodic_table" title="Periodic table">periodic table</a>. The table may be divided into several rows (called 'periods'), numbered starting with 1 at the top. The presently known elements occupy seven periods. If a certain period has number <i>i</i>, it consists of elements whose outermost electrons fall in the <i>i</i>th shell. <a href="/wiki/Niels_Bohr" title="Niels Bohr">Niels Bohr</a> was the first to propose (1923) that the <a href="/wiki/Periodic_table" title="Periodic table">periodicity</a> in the properties of the elements might be explained by the periodic filling of the electron energy levels, resulting in the electronic structure of the atom.<sup id="cite_ref-Bohr_36-0" class="reference"><a href="#cite_note-Bohr-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> </p><p>The periodic table may also be divided into several numbered rectangular '<a href="/wiki/Periodic_table_block" class="mw-redirect" title="Periodic table block">blocks</a>'. The elements belonging to a given block have this common feature: their highest-energy electrons all belong to the same <span class="texhtml mvar" style="font-style:italic;">ℓ</span>-state (but the <span class="texhtml mvar" style="font-style:italic;">n</span> associated with that <span class="texhtml mvar" style="font-style:italic;">ℓ</span>-state depends upon the period). For instance, the leftmost two columns constitute the 's-block'. The outermost electrons of <a href="/wiki/Lithium" title="Lithium">Li</a> and <a href="/wiki/Beryllium" title="Beryllium">Be</a> respectively belong to the 2s subshell, and those of <a href="/wiki/Sodium" title="Sodium">Na</a> and <a href="/wiki/Magnesium" title="Magnesium">Mg</a> to the 3s subshell. </p><p>The following is the order for filling the "subshell" orbitals, which also gives the order of the "blocks" in the periodic table: </p> <dl><dd><b>1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p</b></dd></dl> <p>The "periodic" nature of the filling of orbitals, as well as emergence of the <b>s</b>, <b>p</b>, <b>d</b>, and <b>f</b> "blocks", is more obvious if this order of filling is given in matrix form, with increasing principal quantum numbers starting the new rows ("periods") in the matrix. Then, each subshell (composed of the first two quantum numbers) is repeated as many times as required for each pair of electrons it may contain. The result is a compressed periodic table, with each entry representing two successive elements: </p> <table class="wikitable"> <tbody><tr> <td>1s</td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td> </td></tr> <tr> <td>2s</td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td>2p</td> <td>2p</td> <td>2p </td></tr> <tr> <td>3s</td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td>3p</td> <td>3p</td> <td>3p </td></tr> <tr> <td>4s</td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td>3d</td> <td>3d</td> <td>3d</td> <td>3d</td> <td>3d</td> <td>4p</td> <td>4p</td> <td>4p </td></tr> <tr> <td>5s</td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td>4d</td> <td>4d</td> <td>4d</td> <td>4d</td> <td>4d</td> <td>5p</td> <td>5p</td> <td>5p </td></tr> <tr> <td>6s</td> <td>4f</td> <td>4f</td> <td>4f</td> <td>4f</td> <td>4f</td> <td>4f</td> <td>4f</td> <td>5d</td> <td>5d</td> <td>5d</td> <td>5d</td> <td>5d</td> <td>6p</td> <td>6p</td> <td>6p </td></tr> <tr> <td>7s</td> <td>5f</td> <td>5f</td> <td>5f</td> <td>5f</td> <td>5f</td> <td>5f</td> <td>5f</td> <td>6d</td> <td>6d</td> <td>6d</td> <td>6d</td> <td>6d</td> <td>7p</td> <td>7p</td> <td>7p </td></tr></tbody></table> <p>Although this is the general order of orbital filling according to the Madelung rule, there are exceptions, and the actual electronic energies of each element are also dependent upon additional details of the atoms (see <a href="/wiki/Electron_configuration#Atoms:_Aufbau_principle_and_Madelung_rule" title="Electron configuration">Electron configuration § Atoms: Aufbau principle and Madelung rule</a>). </p><p>The number of electrons in an electrically neutral atom increases with the <a href="/wiki/Atomic_number" title="Atomic number">atomic number</a>. The electrons in the outermost shell, or <i><a href="/wiki/Valence_electron" title="Valence electron">valence electrons</a></i>, tend to be responsible for an element's chemical behavior. Elements that contain the same number of valence electrons can be grouped together and display similar chemical properties. </p> <div class="mw-heading mw-heading3"><h3 id="Relativistic_effects">Relativistic effects</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atomic_orbital&action=edit&section=20" title="Edit section: Relativistic effects"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Relativistic_quantum_chemistry" title="Relativistic quantum chemistry">Relativistic quantum chemistry</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Extended_periodic_table" title="Extended periodic table">Extended periodic table</a></div> <p>For elements with high atomic number <span class="texhtml mvar" style="font-style:italic;">Z</span>, the effects of relativity become more pronounced, and especially so for s electrons, which move at relativistic velocities as they penetrate the screening electrons near the core of high-<span class="texhtml mvar" style="font-style:italic;">Z</span> atoms. This relativistic increase in momentum for high speed electrons causes a corresponding decrease in wavelength and contraction of 6s orbitals relative to 5d orbitals (by comparison to corresponding s and d electrons in lighter elements in the same column of the periodic table); this results in 6s valence electrons becoming lowered in energy. </p><p>Examples of significant physical outcomes of this effect include the lowered melting temperature of <a href="/wiki/Mercury_(element)" title="Mercury (element)">mercury</a> (which results from 6s electrons not being available for metal bonding) and the golden color of gold and <a href="/wiki/Caesium" title="Caesium">caesium</a>.<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> </p><p>In the <a href="/wiki/Bohr_model" title="Bohr model">Bohr model</a>, an <span class="texhtml"><i>n</i> = 1</span> electron has a velocity given by <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=Z\alpha c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mi>Z</mi> <mi>α</mi> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=Z\alpha c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0498ab72bdf5e7fdf8bbf92dfdc271ef2b15076" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.401ex; height:2.176ex;" alt="{\displaystyle v=Z\alpha c}" /></span>, where <span class="texhtml mvar" style="font-style:italic;">Z</span> is the atomic number, <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /></span> is the <a href="/wiki/Fine-structure_constant" title="Fine-structure constant">fine-structure constant</a>, and <span class="texhtml"><i>c</i></span> is the speed of light. In non-relativistic quantum mechanics, therefore, any atom with an atomic number greater than 137 would require its 1s electrons to be traveling faster than the speed of light. Even in the <a href="/wiki/Dirac_equation" title="Dirac equation">Dirac equation</a>, which accounts for relativistic effects, the wave function of the electron for atoms with <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>137}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo>></mo> <mn>137</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z>137}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5e1614dcfe75f6096afe244e3d5ba573b2f462e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.266ex; height:2.176ex;" alt="{\displaystyle Z>137}" /></span> is oscillatory and <a href="/wiki/Unbounded_function" class="mw-redirect" title="Unbounded function">unbounded</a>. The significance of element 137, also known as <a href="/wiki/Untriseptium" class="mw-redirect" title="Untriseptium">untriseptium</a>, was first pointed out by the physicist <a href="/wiki/Richard_Feynman" title="Richard Feynman">Richard Feynman</a>. Element 137 is sometimes informally called <a href="/wiki/Feynmanium" class="mw-redirect" title="Feynmanium">feynmanium</a> (symbol Fy).<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> However, Feynman's approximation fails to predict the exact critical value of <span class="texhtml mvar" style="font-style:italic;">Z</span> due to the non-point-charge nature of the nucleus and very small orbital radius of inner electrons, resulting in a potential seen by inner electrons which is effectively less than <span class="texhtml mvar" style="font-style:italic;">Z</span>. The critical <span class="texhtml mvar" style="font-style:italic;">Z</span> value, which makes the atom unstable with regard to high-field breakdown of the vacuum and production of electron-positron pairs, does not occur until <span class="texhtml mvar" style="font-style:italic;">Z</span> is about 173. These conditions are not seen except transiently in collisions of very heavy nuclei such as lead or uranium in accelerators, where such electron-positron production from these effects has been claimed to be observed. </p><p>There are no nodes in relativistic orbital densities, although individual components of the wave function will have nodes.<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="pp_hybridization_(conjectured)"><span id="pp_hybridization_.28conjectured.29"></span>pp hybridization (conjectured)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atomic_orbital&action=edit&section=21" title="Edit section: pp hybridization (conjectured)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In late <a href="/wiki/Extended_periodic_table" title="Extended periodic table">period 8 elements</a>, a <a href="/wiki/Orbital_hybridisation" title="Orbital hybridisation">hybrid</a> of 8p<sub>3/2</sub> and 9p<sub>1/2</sub> is expected to exist,<sup id="cite_ref-BFricke_40-0" class="reference"><a href="#cite_note-BFricke-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> where "3/2" and "1/2" refer to the <a href="/wiki/Total_angular_momentum_quantum_number" title="Total angular momentum quantum number">total angular momentum quantum number</a>. This "pp" hybrid may be responsible for the <a href="/wiki/P-block" class="mw-redirect" title="P-block">p-block</a> of the period due to properties similar to p subshells in ordinary <a href="/wiki/Valence_shell" class="mw-redirect" title="Valence shell">valence shells</a>. Energy levels of 8p<sub>3/2</sub> and 9p<sub>1/2</sub> come close due to relativistic <a href="/wiki/Spin%E2%80%93orbit_interaction" title="Spin–orbit interaction">spin–orbit effects</a>; the 9s subshell should also participate, as these elements are expected to be analogous to the respective 5p elements <a href="/wiki/Indium" title="Indium">indium</a> through <a href="/wiki/Xenon" title="Xenon">xenon</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Transitions_between_orbitals">Transitions between orbitals</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atomic_orbital&action=edit&section=22" title="Edit section: Transitions between orbitals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Atomic_electron_transition" title="Atomic electron transition">Atomic electron transition</a></div> <p>Bound quantum states have discrete energy levels. When applied to atomic orbitals, this means that the energy differences between states are also discrete. A transition between these states (i.e., an electron absorbing or emitting a photon) can thus happen only if the photon has an energy corresponding with the exact energy difference between said states. </p><p>Consider two states of the hydrogen atom: </p> <ol><li>State <span class="texhtml"><i>n</i> = 1</span>, <span class="texhtml"><i>ℓ</i> = 0</span>, <span class="texhtml"><i>m</i><sub><i>ℓ</i></sub> = 0</span> and <span class="texhtml"><i>m<sub>s</sub></i> = +<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" /><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span></li> <li>State <span class="texhtml"><i>n</i> = 2</span>, <span class="texhtml"><i>ℓ</i> = 0</span>, <span class="texhtml"><i>m</i><sub><i>ℓ</i></sub> = 0</span> and <span class="texhtml"><i>m<sub>s</sub></i> = −<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035" /><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span></li></ol> <p>By quantum theory, state 1 has a fixed energy of <span class="texhtml"><i>E</i><sub>1</sub></span>, and state 2 has a fixed energy of <span class="texhtml"><i>E</i><sub>2</sub></span>. Now, what would happen if an electron in state 1 were to move to state 2? For this to happen, the electron would need to gain an energy of exactly <span class="texhtml"><i>E</i><sub>2</sub> − <i>E</i><sub>1</sub></span>. If the electron receives energy that is less than or greater than this value, it cannot jump from state 1 to state 2. Now, suppose we irradiate the atom with a broad-spectrum of light. Photons that reach the atom that have an energy of exactly <span class="texhtml"><i>E</i><sub>2</sub> − <i>E</i><sub>1</sub></span> will be absorbed by the electron in state 1, and that electron will jump to state 2. However, photons that are greater or lower in energy cannot be absorbed by the electron, because the electron can jump only to one of the orbitals, it cannot jump to a state between orbitals. The result is that only photons of a specific frequency will be absorbed by the atom. This creates a line in the spectrum, known as an absorption line, which corresponds to the energy difference between states 1 and 2. </p><p>The atomic orbital model thus predicts line spectra, which are observed experimentally. This is one of the main validations of the atomic orbital model. </p><p>The atomic orbital model is nevertheless an approximation to the full quantum theory, which only recognizes many electron states. The predictions of line spectra are qualitatively useful but are not quantitatively accurate for atoms and ions other than those containing only one electron. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atomic_orbital&action=edit&section=23" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 20em;"> <ul><li><a href="/wiki/Atomic_electron_configuration_table" class="mw-redirect" title="Atomic electron configuration table">Atomic electron configuration table</a></li></ul> <ul><li><a href="/wiki/Condensed_matter_physics" title="Condensed matter physics">Condensed matter physics</a></li> <li><a href="/wiki/Electron_configuration" title="Electron configuration">Electron configuration</a></li> <li><a href="/wiki/Energy_level" title="Energy level">Energy level</a></li> <li><a href="/wiki/Hund%27s_rules" title="Hund's rules">Hund's rules</a></li> <li><a href="/wiki/Molecular_orbital" title="Molecular orbital">Molecular orbital</a></li> <li><a href="/wiki/Orbital_overlap" title="Orbital overlap">Orbital overlap</a></li> <li><a href="/wiki/Quantum_chemistry" title="Quantum chemistry">Quantum chemistry</a></li> <li><a href="/wiki/Quantum_chemistry_computer_programs" class="mw-redirect" title="Quantum chemistry computer programs">Quantum chemistry computer programs</a></li> <li><a href="/wiki/Solid-state_physics" title="Solid-state physics">Solid-state physics</a></li> <li><a href="/wiki/Wave_function_collapse" title="Wave function collapse">Wave function collapse</a></li> <li><a href="/wiki/Wiswesser%27s_rule" class="mw-redirect" title="Wiswesser's rule">Wiswesser's rule</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Atomic_orbital&action=edit&section=24" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFOrchinMacomberPinhasWilson2005" class="citation book cs1">Orchin, Milton; Macomber, Roger S.; Pinhas, Allan; Wilson, R. Marshall (2005). <a rel="nofollow" class="external text" href="http://media.wiley.com/product_data/excerpt/81/04716802/0471680281.pdf">"1. Atomic Orbital Theory"</a> <span class="cs1-format">(PDF)</span>. <i>The Vocabulary and Concepts of Organic Chemistry</i> (2nd ed.). Wiley. <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://media.wiley.com/product_data/excerpt/81/04716802/0471680281.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 9 October 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=1.+Atomic+Orbital+Theory&rft.btitle=The+Vocabulary+and+Concepts+of+Organic+Chemistry&rft.edition=2nd&rft.pub=Wiley&rft.date=2005&rft.aulast=Orchin&rft.aufirst=Milton&rft.au=Macomber%2C+Roger+S.&rft.au=Pinhas%2C+Allan&rft.au=Wilson%2C+R.+Marshall&rft_id=http%3A%2F%2Fmedia.wiley.com%2Fproduct_data%2Fexcerpt%2F81%2F04716802%2F0471680281.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAtomic+orbital" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDaintith2004" class="citation book cs1">Daintith, J. (2004). <a rel="nofollow" class="external text" href="https://archive.org/details/dictionaryofchem0000unse_r3p4/page/408/mode/2up?view=theater"><i>Oxford Dictionary of Chemistry</i></a>. New York: Oxford University Press. pp. <span class="nowrap">407–</span>409. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-860918-6" title="Special:BookSources/978-0-19-860918-6"><bdi>978-0-19-860918-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Oxford+Dictionary+of+Chemistry&rft.place=New+York&rft.pages=%3Cspan+class%3D%22nowrap%22%3E407-%3C%2Fspan%3E409&rft.pub=Oxford+University+Press&rft.date=2004&rft.isbn=978-0-19-860918-6&rft.aulast=Daintith&rft.aufirst=J.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fdictionaryofchem0000unse_r3p4%2Fpage%2F408%2Fmode%2F2up%3Fview%3Dtheater&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAtomic+orbital" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGriffiths1995" class="citation book cs1">Griffiths, David (1995). <i>Introduction to Quantum Mechanics</i>. Prentice Hall. pp. <span class="nowrap">190–</span>191. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-124405-4" title="Special:BookSources/978-0-13-124405-4"><bdi>978-0-13-124405-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Quantum+Mechanics&rft.pages=%3Cspan+class%3D%22nowrap%22%3E190-%3C%2Fspan%3E191&rft.pub=Prentice+Hall&rft.date=1995&rft.isbn=978-0-13-124405-4&rft.aulast=Griffiths&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAtomic+orbital" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLevine2000" class="citation book cs1">Levine, Ira (2000). <a rel="nofollow" class="external text" href="https://archive.org/details/quantumchemistry00levi_0/page/144"><i>Quantum Chemistry</i></a> (5 ed.). Prentice Hall. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/quantumchemistry00levi_0/page/144">144–145</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-685512-5" title="Special:BookSources/978-0-13-685512-5"><bdi>978-0-13-685512-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Chemistry&rft.pages=144-145&rft.edition=5&rft.pub=Prentice+Hall&rft.date=2000&rft.isbn=978-0-13-685512-5&rft.aulast=Levine&rft.aufirst=Ira&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fquantumchemistry00levi_0%2Fpage%2F144&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAtomic+orbital" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLaidlerMeiser1982" class="citation book cs1">Laidler, Keith J.; Meiser, John H. (1982). <i>Physical Chemistry</i>. Benjamin/Cummings. p. 488. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8053-5682-3" title="Special:BookSources/978-0-8053-5682-3"><bdi>978-0-8053-5682-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physical+Chemistry&rft.pages=488&rft.pub=Benjamin%2FCummings&rft.date=1982&rft.isbn=978-0-8053-5682-3&rft.aulast=Laidler&rft.aufirst=Keith+J.&rft.au=Meiser%2C+John+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAtomic+orbital" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAtkinsde_PaulaFriedman2009" class="citation book cs1">Atkins, Peter; de Paula, Julio; Friedman, Ronald (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=QbQJAgAAQBAJ&pg=PA106"><i>Quanta, Matter, and Change: A Molecular Approach to Physical Chemistry</i></a>. Oxford University Press. p. 106. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-920606-3" title="Special:BookSources/978-0-19-920606-3"><bdi>978-0-19-920606-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quanta%2C+Matter%2C+and+Change%3A+A+Molecular+Approach+to+Physical+Chemistry&rft.pages=106&rft.pub=Oxford+University+Press&rft.date=2009&rft.isbn=978-0-19-920606-3&rft.aulast=Atkins&rft.aufirst=Peter&rft.au=de+Paula%2C+Julio&rft.au=Friedman%2C+Ronald&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DQbQJAgAAQBAJ%26pg%3DPA106&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAtomic+orbital" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFFeynmanLeightonSands2006" class="citation book cs1">Feynman, Richard; Leighton, Robert B.; Sands, Matthew (2006). <i>The Feynman Lectures on Physics – The Definitive Edition, Vol 1 lect 6</i>. Pearson PLC, Addison Wesley. p. 11. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8053-9046-9" title="Special:BookSources/978-0-8053-9046-9"><bdi>978-0-8053-9046-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Feynman+Lectures+on+Physics+%E2%80%93+The+Definitive+Edition%2C+Vol+1+lect+6&rft.pages=11&rft.pub=Pearson+PLC%2C+Addison+Wesley&rft.date=2006&rft.isbn=978-0-8053-9046-9&rft.aulast=Feynman&rft.aufirst=Richard&rft.au=Leighton%2C+Robert+B.&rft.au=Sands%2C+Matthew&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAtomic+orbital" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><a href="/wiki/Roger_Penrose" title="Roger Penrose">Roger Penrose</a>, <i><a href="/wiki/The_Road_to_Reality" title="The Road to Reality">The Road to Reality</a></i></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLevine1991" class="citation book cs1">Levine, Ira N. (1991). <i>Quantum Chemistry</i> (4th ed.). Prentice-Hall. p. 262. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-205-12770-3" title="Special:BookSources/0-205-12770-3"><bdi>0-205-12770-3</bdi></a>. <q>Therefore, the wave function of a system of identical interacting particles must not distinguish among the particles.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Chemistry&rft.pages=262&rft.edition=4th&rft.pub=Prentice-Hall&rft.date=1991&rft.isbn=0-205-12770-3&rft.aulast=Levine&rft.aufirst=Ira+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAtomic+orbital" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMulliken1932" class="citation journal cs1">Mulliken, Robert S. (July 1932). "Electronic Structures of Polyatomic Molecules and Valence. II. 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New York: Springer/TELOS. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0387207773" title="Special:BookSources/978-0387207773"><bdi>978-0387207773</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Advanced+visual+quantum+mechanics&rft.place=New+York&rft.pub=Springer%2FTELOS&rft.date=2004&rft.isbn=978-0387207773&rft.aulast=Thaller&rft.aufirst=Bernd&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAtomic+orbital" class="Z3988"></span></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPetrucci,_RalphHerring,_F.Madura,_JeffryBissonnette,_Carey2016" class="citation book cs1">Petrucci, Ralph; Herring, F.; Madura, Jeffry; Bissonnette, Carey (2016). <i>General chemistry: principles and modern applications</i> (11th ed.). 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Recent Impact of Physics on Inorganic Chemistry. Vol. 21. pp. <span class="nowrap">89–</span>144. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBFb0116498">10.1007/BFb0116498</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-07109-9" title="Special:BookSources/978-3-540-07109-9"><bdi>978-3-540-07109-9</bdi></a><span class="reference-accessdate">. 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New York: Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-530573-9" title="Special:BookSources/978-0-19-530573-9"><bdi>978-0-19-530573-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Periodic+Table%2C+Its+Story+and+Its+Significance&rft.place=New+York&rft.pub=Oxford+University+Press&rft.date=2007&rft.isbn=978-0-19-530573-9&rft.aulast=Scerri&rft.aufirst=Eric&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fperiodictableits0000scer&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAtomic+orbital" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLevine2014" class="citation book cs1">Levine, Ira (2014). <i>Quantum Chemistry</i> (7th ed.). 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Educ</i>. <b>43</b> (4): 187. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1966JChEd..43..187C">1966JChEd..43..187C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1021%2Fed043p187">10.1021/ed043p187</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=J.+Chem.+Educ.&rft.atitle=Atomic+Orbitals%3A+Limitations+and+Variations&rft.volume=43&rft.issue=4&rft.pages=187&rft.date=1966&rft_id=info%3Adoi%2F10.1021%2Fed043p187&rft_id=info%3Abibcode%2F1966JChEd..43..187C&rft.aulast=Cohen&rft.aufirst=Irwin&rft.au=Bustard%2C+Thomas&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAtomic+orbital" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External 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href="http://www.shef.ac.uk/chemistry/orbitron/">The Orbitron</a>, a visualization of all common and uncommon atomic orbitals, from 1s to 7g</li> <li><a rel="nofollow" class="external text" href="http://www.orbitals.com/orb/orbtable.htm">Grand table</a> Still images of many orbitals</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul 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.navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Atomic_models290" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Atomic_models" title="Template:Atomic models"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Atomic_models" title="Template talk:Atomic models"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Atomic_models" title="Special:EditPage/Template:Atomic models"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Atomic_models290" style="font-size:114%;margin:0 4em"><a href="/wiki/History_of_atomic_theory" title="History of atomic theory">Atomic models</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Historic models</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>1804 <a href="/wiki/John_Dalton#Atomic_theory" title="John Dalton">Dalton model</a> (billiard ball model)</li> <li>1867 <a href="/wiki/Vortex_theory_of_the_atom" title="Vortex theory of the atom">vortex theory of the atom</a> (knot model)</li> <li>1902 <a href="/wiki/Cubical_atom" title="Cubical atom">Lewis model</a> (cubical atom model)</li> <li>1904 <a href="/wiki/Hantaro_Nagaoka#Saturnian_model_of_the_atom" title="Hantaro Nagaoka">Nagaoka model</a> (Saturnian model)</li> <li>1904 <a href="/wiki/Plum_pudding_model" title="Plum pudding model">plum pudding model</a></li> <li>1911 <a href="/wiki/Rutherford_model" title="Rutherford model">Rutherford model</a> (planetary model)</li> <li>1913 <a href="/wiki/Bohr_model" title="Bohr model">Bohr model</a> (old quantum model)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Current models</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>1926 <a class="mw-selflink selflink">electron cloud model</a></li> <li>1928 <a href="/wiki/Hydrogen-like_atom" title="Hydrogen-like atom">Dirac–Gordon model</a> (relativistic quantum model)</li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Atoms" title="Category:Atoms">Category:Atoms</a></li> <li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Symbol_portal_class.svg" class="mw-file-description" title="Portal"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/e/e2/Symbol_portal_class.svg/20px-Symbol_portal_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/e/e2/Symbol_portal_class.svg/40px-Symbol_portal_class.svg.png 1.5x" data-file-width="180" data-file-height="185" /></a></span> <a href="/wiki/Portal:Physics" title="Portal:Physics">Portal:Physics</a> / <a href="/wiki/Portal:Chemistry" title="Portal:Chemistry">Chemistry</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235" /></div><div role="navigation" class="navbox" aria-labelledby="Electron_configuration258" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231" /><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Electron_configuration_navbox" title="Template:Electron configuration navbox"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Electron_configuration_navbox" title="Template talk:Electron configuration navbox"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Electron_configuration_navbox" title="Special:EditPage/Template:Electron configuration navbox"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Electron_configuration258" style="font-size:114%;margin:0 4em"><a href="/wiki/Electron_configuration" title="Electron configuration">Electron configuration</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/Electron_shell" title="Electron shell">Electron shell</a></li> <li><a class="mw-selflink selflink">Atomic orbital</a></li> <li><a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a> <ul><li><a href="/wiki/Introduction_to_quantum_mechanics" title="Introduction to quantum mechanics">Introduction to quantum mechanics</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Quantum_number" title="Quantum number">Quantum numbers</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Principal_quantum_number" title="Principal quantum number">Principal quantum number (<span class="texhtml mvar" style="font-style:italic;">n</span>)</a></li> <li><a href="/wiki/Azimuthal_quantum_number" title="Azimuthal quantum number">Azimuthal quantum number (<span class="texhtml mvar" style="font-style:italic;">ℓ</span>)</a></li> <li><a href="/wiki/Magnetic_quantum_number" title="Magnetic quantum number">Magnetic quantum number (<span class="texhtml mvar" style="font-style:italic;">m</span>)</a></li> <li><a href="/wiki/Spin_quantum_number" title="Spin quantum number">Spin quantum number (<span class="texhtml mvar" style="font-style:italic;">s</span>)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Ground-state configurations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Periodic_table_(electron_configurations)" title="Periodic table (electron configurations)">Periodic table (electron configurations)</a></li> <li><a href="/wiki/Electron_configurations_of_the_elements_(data_page)" title="Electron configurations of the elements (data page)">Electron configurations of the elements (data page)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Electron filling</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pauli_exclusion_principle" title="Pauli exclusion principle">Pauli exclusion principle</a></li> <li><a href="/wiki/Hund%27s_rule_of_maximum_multiplicity" title="Hund's rule of maximum multiplicity">Hund's rule</a></li> <li><a href="/wiki/Aufbau_principle" title="Aufbau principle">Aufbau principle</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Electron pairing</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Electron_pair" title="Electron pair">Electron pair</a></li> <li><a href="/wiki/Unpaired_electron" title="Unpaired electron">Unpaired electron</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Bonding participation</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Valence_electron" title="Valence electron">Valence electron</a></li> <li><a href="/wiki/Core_electron" title="Core electron">Core electron</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Electron_counting" title="Electron counting">Electron counting</a> rules</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Octet_rule" title="Octet rule">Octet rule</a></li> <li><a href="/wiki/18-electron_rule" title="18-electron rule">18-electron rule</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235" /></div><div role="navigation" class="navbox" aria-labelledby="Chemical_bonding_theory174" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231" /><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Chemical_bonding_theory" title="Template:Chemical bonding theory"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Chemical_bonding_theory" title="Template talk:Chemical bonding theory"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Chemical_bonding_theory" title="Special:EditPage/Template:Chemical bonding theory"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Chemical_bonding_theory174" style="font-size:114%;margin:0 4em"><a href="/wiki/Chemical_bond" title="Chemical bond">Chemical bonding</a> theory</div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a class="mw-selflink selflink">Atomic orbital</a></li> <li><a href="/wiki/Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a> <ul><li><a href="/wiki/Introduction_to_quantum_mechanics" title="Introduction to quantum mechanics">Introduction to quantum mechanics</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of bonds</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">By symmetry</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Sigma_bond" title="Sigma bond">Sigma (σ)</a></li> <li><a href="/wiki/Pi_bond" title="Pi bond">Pi (π)</a></li> <li><a href="/wiki/Delta_bond" title="Delta bond">Delta (δ)</a></li> <li><a href="/wiki/Phi_bond" title="Phi bond">Phi (φ)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By <a href="/wiki/Bond_order" title="Bond order">multiplicity</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Single_bond" title="Single bond">1 (single)</a></li> <li><a href="/wiki/Double_bond" title="Double bond">2 (double)</a></li> <li><a href="/wiki/Triple_bond" title="Triple bond">3 (triple)</a></li> <li><a href="/wiki/Quadruple_bond" title="Quadruple bond">4 (quadruple)</a></li> <li><a href="/wiki/Quintuple_bond" title="Quintuple bond">5 (quintuple)</a></li> <li><a href="/wiki/Sextuple_bond" title="Sextuple bond">6 (sextuple)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Triplet_state" title="Triplet state">Triplet</a></li> <li><a href="/wiki/Singlet_state" title="Singlet state">Singlet</a></li> <li><a href="/wiki/Exchange_interaction" title="Exchange interaction">Exchange-coupled</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Valence_bond_theory" title="Valence bond theory">Valence bond theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Orbital_hybridisation" title="Orbital hybridisation">Hybrid orbital</a></li> <li><a href="/wiki/Resonance_(chemistry)" title="Resonance (chemistry)">Resonance</a></li> <li><a href="/wiki/Lewis_structure" title="Lewis structure">Lewis structure</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Constituent units</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Covalent_bond" title="Covalent bond">Covalent bond</a></li> <li><a href="/wiki/Lone_pair" title="Lone pair">Lone pair</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Molecular_orbital_theory" title="Molecular orbital theory">Molecular orbital theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Molecular_orbital" title="Molecular orbital">Molecular orbital</a></li> <li><a href="/wiki/Linear_combination_of_atomic_orbitals" title="Linear combination of atomic orbitals">LCAO</a></li> <li><a href="/wiki/Molecular_orbital_diagram" title="Molecular orbital diagram">MO diagram</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Constituent units</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bonding_molecular_orbital" title="Bonding molecular orbital">Bonding MO</a></li> <li><a href="/wiki/Non-bonding_orbital" title="Non-bonding orbital">Non-bonding MO</a></li> <li><a href="/wiki/Antibonding_molecular_orbital" title="Antibonding molecular 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