Molecola - Wikipedia

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class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">sposta nella barra laterale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">nascondi</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Inizio</div> </a> </li> <li id="toc-Dinamica_molecolare" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Dinamica_molecolare"> <div class="vector-toc-text"> 1 Dinamica molecolare </div> </a> <ul id="toc-Dinamica_molecolare-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Molecole_biatomiche" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Molecole_biatomiche"> <div class="vector-toc-text"> 2 Molecole biatomiche </div> </a> <button aria-controls="toc-Molecole_biatomiche-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Attiva/disattiva la sottosezione Molecole biatomiche </button> <ul id="toc-Molecole_biatomiche-sublist" class="vector-toc-list"> <li id="toc-Lo_ione_molecolare_H2+" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lo_ione_molecolare_H2+"> <div class="vector-toc-text"> 2.1 Lo ione molecolare H2+ </div> </a> <ul id="toc-Lo_ione_molecolare_H2+-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-La_molecola_H2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#La_molecola_H2"> <div class="vector-toc-text"> 2.2 La molecola H2 </div> </a> <ul id="toc-La_molecola_H2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Molecole_eteronucleari" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Molecole_eteronucleari"> <div class="vector-toc-text"> 2.3 Molecole eteronucleari </div> </a> <ul id="toc-Molecole_eteronucleari-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Molecole_poliatomiche" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Molecole_poliatomiche"> <div class="vector-toc-text"> 3 Molecole poliatomiche </div> </a> <button aria-controls="toc-Molecole_poliatomiche-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Attiva/disattiva la sottosezione Molecole poliatomiche </button> <ul id="toc-Molecole_poliatomiche-sublist" class="vector-toc-list"> <li id="toc-La_molecola_H2O" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#La_molecola_H2O"> <div class="vector-toc-text"> 3.1 La molecola H2O </div> </a> <ul id="toc-La_molecola_H2O-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-La_molecola_CH4" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#La_molecola_CH4"> <div class="vector-toc-text"> 3.2 La molecola CH4 </div> </a> <ul id="toc-La_molecola_CH4-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Orbitali_e_legami_molecolari" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Orbitali_e_legami_molecolari"> <div class="vector-toc-text"> 4 Orbitali e legami molecolari </div> </a> <button aria-controls="toc-Orbitali_e_legami_molecolari-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Attiva/disattiva la sottosezione Orbitali e legami molecolari </button> <ul id="toc-Orbitali_e_legami_molecolari-sublist" class="vector-toc-list"> <li id="toc-Teoria_degli_orbitali_molecolari" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Teoria_degli_orbitali_molecolari"> <div class="vector-toc-text"> 4.1 Teoria degli orbitali molecolari </div> </a> <ul id="toc-Teoria_degli_orbitali_molecolari-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rappresentazione_degli_orbitali_molecolari" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rappresentazione_degli_orbitali_molecolari"> <div class="vector-toc-text"> 4.2 Rappresentazione degli orbitali molecolari </div> </a> <ul id="toc-Rappresentazione_degli_orbitali_molecolari-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Moti_interni_nelle_molecole_biatomiche" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Moti_interni_nelle_molecole_biatomiche"> <div class="vector-toc-text"> 5 Moti interni nelle molecole biatomiche </div> </a> <ul id="toc-Moti_interni_nelle_molecole_biatomiche-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Moti_interni_nelle_molecole_poliatomiche" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Moti_interni_nelle_molecole_poliatomiche"> <div class="vector-toc-text"> 6 Moti interni nelle molecole poliatomiche </div> </a> <button aria-controls="toc-Moti_interni_nelle_molecole_poliatomiche-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Attiva/disattiva la sottosezione Moti interni nelle molecole poliatomiche </button> <ul id="toc-Moti_interni_nelle_molecole_poliatomiche-sublist" class="vector-toc-list"> <li id="toc-Moto_vibrazionale" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Moto_vibrazionale"> <div class="vector-toc-text"> 6.1 Moto vibrazionale </div> </a> <ul id="toc-Moto_vibrazionale-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Moto_rotazionale" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Moto_rotazionale"> <div class="vector-toc-text"> 6.2 Moto rotazionale </div> </a> <ul id="toc-Moto_rotazionale-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Spettro_elettromagnetico_molecolare" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Spettro_elettromagnetico_molecolare"> <div class="vector-toc-text"> 7 Spettro elettromagnetico molecolare </div> </a> <button aria-controls="toc-Spettro_elettromagnetico_molecolare-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Attiva/disattiva la sottosezione Spettro elettromagnetico molecolare </button> <ul id="toc-Spettro_elettromagnetico_molecolare-sublist" class="vector-toc-list"> <li id="toc-Spettro_nucleare" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spettro_nucleare"> <div class="vector-toc-text"> 7.1 Spettro nucleare </div> </a> <ul id="toc-Spettro_nucleare-sublist" class="vector-toc-list"> <li id="toc-Spettro_nelle_molecole_biatomiche" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Spettro_nelle_molecole_biatomiche"> <div class="vector-toc-text"> 7.1.1 Spettro nelle molecole biatomiche </div> </a> <ul id="toc-Spettro_nelle_molecole_biatomiche-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spettro_nelle_molecole_poliatomiche" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Spettro_nelle_molecole_poliatomiche"> <div class="vector-toc-text"> 7.1.2 Spettro nelle molecole poliatomiche </div> </a> <ul id="toc-Spettro_nelle_molecole_poliatomiche-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Spettro_elettronico" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spettro_elettronico"> <div class="vector-toc-text"> 7.2 Spettro elettronico </div> </a> <ul id="toc-Spettro_elettronico-sublist" class="vector-toc-list"> <li id="toc-Il_principio_di_Franck_Condon" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Il_principio_di_Franck_Condon"> <div class="vector-toc-text"> 7.2.1 Il principio di Franck Condon </div> </a> <ul id="toc-Il_principio_di_Franck_Condon-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Note" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Note"> <div class="vector-toc-text"> 8 Note </div> </a> <ul id="toc-Note-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliografia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliografia"> <div class="vector-toc-text"> 9 Bibliografia </div> </a> <ul id="toc-Bibliografia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Voci_correlate" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Voci_correlate"> <div class="vector-toc-text"> 10 Voci correlate </div> </a> <ul id="toc-Voci_correlate-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Altri_progetti" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Altri_progetti"> <div class="vector-toc-text"> 11 Altri progetti </div> </a> <ul id="toc-Altri_progetti-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Collegamenti_esterni" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Collegamenti_esterni"> <div class="vector-toc-text"> 12 Collegamenti esterni </div> </a> <ul id="toc-Collegamenti_esterni-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="Indice" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <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="Mostra/Nascondi l'indice" > <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" > Mostra/Nascondi l'indice </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">Molecola</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="Vai a una voce in un'altra lingua. Disponibile in 165 lingue" > <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-165" aria-hidden="true" > 165 lingue </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/Molekuul" title="Molekuul - afrikaans" lang="af" hreflang="af" data-title="Molekuul" data-language-autonym="Afrikaans" data-language-local-name="afrikaans" class="interlanguage-link-target">Afrikaans</a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Molek%C3%BCl" title="Molekül - tedesco svizzero" lang="gsw" hreflang="gsw" data-title="Molekül" data-language-autonym="Alemannisch" data-language-local-name="tedesco svizzero" class="interlanguage-link-target">Alemannisch</a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%88%9E%E1%88%88%E1%8A%AA%E1%8B%8D%E1%88%8D" title="ሞለኪውል - amarico" lang="am" hreflang="am" data-title="ሞለኪውል" data-language-autonym="አማርኛ" data-language-local-name="amarico" class="interlanguage-link-target">አማርኛ</a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Molecula" title="Molecula - aragonese" lang="an" hreflang="an" data-title="Molecula" data-language-autonym="Aragonés" data-language-local-name="aragonese" class="interlanguage-link-target">Aragonés</a></li><li class="interlanguage-link interwiki-ang mw-list-item"><a href="https://ang.wikipedia.org/wiki/Gemot" title="Gemot - inglese antico" lang="ang" hreflang="ang" data-title="Gemot" data-language-autonym="Ænglisc" data-language-local-name="inglese antico" class="interlanguage-link-target">Ænglisc</a></li><li class="interlanguage-link interwiki-anp mw-list-item"><a href="https://anp.wikipedia.org/wiki/%E0%A4%85%E0%A4%A3%E0%A5%81" title="अणु - angika" lang="anp" hreflang="anp" data-title="अणु" data-language-autonym="अंगिका" data-language-local-name="angika" class="interlanguage-link-target">अंगिका</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AC%D8%B2%D9%8A%D8%A1" title="جزيء - arabo" lang="ar" hreflang="ar" data-title="جزيء" data-language-autonym="العربية" data-language-local-name="arabo" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%85%E0%A6%A3%E0%A7%81" title="অণু - assamese" lang="as" hreflang="as" data-title="অণু" data-language-autonym="অসমীয়া" data-language-local-name="assamese" class="interlanguage-link-target">অসমীয়া</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Mol%C3%A9cula" title="Molécula - asturiano" lang="ast" hreflang="ast" data-title="Molécula" data-language-autonym="Asturianu" data-language-local-name="asturiano" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Molekul" title="Molekul - azerbaigiano" lang="az" hreflang="az" data-title="Molekul" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaigiano" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D9%85%D9%88%D9%84%DA%A9%D9%88%D9%84" title="مولکول - South Azerbaijani" lang="azb" hreflang="azb" data-title="مولکول" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target">تۆرکجه</a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9C%D0%BE%D0%BB%D0%B5%D0%BA%D1%83%D0%BB%D0%B0" title="Молекула - baschiro" lang="ba" hreflang="ba" data-title="Молекула" data-language-autonym="Башҡортса" data-language-local-name="baschiro" class="interlanguage-link-target">Башҡортса</a></li><li class="interlanguage-link interwiki-bar mw-list-item"><a href="https://bar.wikipedia.org/wiki/Molekui" title="Molekui - bavarese" lang="bar" hreflang="bar" data-title="Molekui" data-language-autonym="Boarisch" data-language-local-name="bavarese" class="interlanguage-link-target">Boarisch</a></li><li class="interlanguage-link interwiki-bat-smg mw-list-item"><a href="https://bat-smg.wikipedia.org/wiki/Muolekol%C4%97" title="Muolekolė - samogitico" lang="sgs" hreflang="sgs" data-title="Muolekolė" data-language-autonym="Žemaitėška" data-language-local-name="samogitico" class="interlanguage-link-target">Žemaitėška</a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Molekula" title="Molekula - Central Bikol" lang="bcl" hreflang="bcl" data-title="Molekula" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target">Bikol Central</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9C%D0%B0%D0%BB%D0%B5%D0%BA%D1%83%D0%BB%D0%B0" title="Малекула - bielorusso" lang="be" hreflang="be" data-title="Малекула" data-language-autonym="Беларуская" data-language-local-name="bielorusso" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9C%D0%B0%D0%BB%D0%B5%D0%BA%D1%83%D0%BB%D0%B0" title="Малекула - Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Малекула" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target">Беларуская (тарашкевіца)</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9C%D0%BE%D0%BB%D0%B5%D0%BA%D1%83%D0%BB%D0%B0" title="Молекула - bulgaro" lang="bg" hreflang="bg" data-title="Молекула" data-language-autonym="Български" data-language-local-name="bulgaro" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%85%E0%A6%A3%E0%A7%81" title="অণু - bengalese" lang="bn" hreflang="bn" data-title="অণু" data-language-autonym="বাংলা" data-language-local-name="bengalese" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Molekulenn" title="Molekulenn - bretone" lang="br" hreflang="br" data-title="Molekulenn" data-language-autonym="Brezhoneg" data-language-local-name="bretone" class="interlanguage-link-target">Brezhoneg</a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Molekula" title="Molekula - bosniaco" lang="bs" hreflang="bs" data-title="Molekula" data-language-autonym="Bosanski" data-language-local-name="bosniaco" class="interlanguage-link-target">Bosanski</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Mol%C3%A8cula" title="Molècula - catalano" lang="ca" hreflang="ca" data-title="Molècula" data-language-autonym="Català" data-language-local-name="catalano" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cdo mw-list-item"><a href="https://cdo.wikipedia.org/wiki/H%C5%ADng-c%E1%B9%B3%CC%84" title="Hŭng-cṳ̄ - Mindong" lang="cdo" hreflang="cdo" data-title="Hŭng-cṳ̄" data-language-autonym="閩東語 / Mìng-dĕ̤ng-ngṳ̄" data-language-local-name="Mindong" class="interlanguage-link-target">閩東語 / Mìng-dĕ̤ng-ngṳ̄</a></li><li class="interlanguage-link interwiki-ceb mw-list-item"><a href="https://ceb.wikipedia.org/wiki/Molekula" title="Molekula - cebuano" lang="ceb" hreflang="ceb" data-title="Molekula" data-language-autonym="Cebuano" data-language-local-name="cebuano" class="interlanguage-link-target">Cebuano</a></li><li class="interlanguage-link interwiki-chr mw-list-item"><a href="https://chr.wikipedia.org/wiki/%E1%8F%A7%E1%8F%93%E1%8E%B4%E1%8F%85%E1%8F%93_%E1%8F%A7%E1%8F%99%E1%8F%A2%E1%8F%85" title="ᏧᏓᎴᏅᏓ ᏧᏙᏢᏅ - cherokee" lang="chr" hreflang="chr" data-title="ᏧᏓᎴᏅᏓ ᏧᏙᏢᏅ" data-language-autonym="ᏣᎳᎩ" data-language-local-name="cherokee" class="interlanguage-link-target">ᏣᎳᎩ</a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%AF%DB%95%D8%B1%D8%AF" title="گەرد - curdo centrale" lang="ckb" hreflang="ckb" data-title="گەرد" data-language-autonym="کوردی" data-language-local-name="curdo centrale" class="interlanguage-link-target">کوردی</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Molekula" title="Molekula - ceco" lang="cs" hreflang="cs" data-title="Molekula" data-language-autonym="Čeština" data-language-local-name="ceco" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9C%D0%BE%D0%BB%D0%B5%D0%BA%D1%83%D0%BB%D0%B0" title="Молекула - ciuvascio" lang="cv" hreflang="cv" data-title="Молекула" data-language-autonym="Чӑвашла" data-language-local-name="ciuvascio" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Moleciwl" title="Moleciwl - gallese" lang="cy" hreflang="cy" data-title="Moleciwl" data-language-autonym="Cymraeg" data-language-local-name="gallese" class="interlanguage-link-target">Cymraeg</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Molekyle" title="Molekyle - danese" lang="da" hreflang="da" data-title="Molekyle" data-language-autonym="Dansk" data-language-local-name="danese" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Molek%C3%BCl" title="Molekül - tedesco" lang="de" hreflang="de" data-title="Molekül" data-language-autonym="Deutsch" data-language-local-name="tedesco" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-dsb mw-list-item"><a href="https://dsb.wikipedia.org/wiki/Molekul" title="Molekul - basso sorabo" lang="dsb" hreflang="dsb" data-title="Molekul" data-language-autonym="Dolnoserbski" data-language-local-name="basso sorabo" class="interlanguage-link-target">Dolnoserbski</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9C%CF%8C%CF%81%CE%B9%CE%BF" title="Μόριο - greco" lang="el" hreflang="el" data-title="Μόριο" data-language-autonym="Ελληνικά" data-language-local-name="greco" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-eml mw-list-item"><a href="https://eml.wikipedia.org/wiki/Mol%C3%A8cula" title="Molècula - Emiliano-Romagnolo" lang="egl" hreflang="egl" data-title="Molècula" data-language-autonym="Emiliàn e rumagnòl" data-language-local-name="Emiliano-Romagnolo" class="interlanguage-link-target">Emiliàn e rumagnòl</a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Molecule" title="Molecule - inglese" lang="en" hreflang="en" data-title="Molecule" data-language-autonym="English" data-language-local-name="inglese" class="interlanguage-link-target">English</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Molekulo" title="Molekulo - esperanto" lang="eo" hreflang="eo" data-title="Molekulo" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Mol%C3%A9cula" title="Molécula - spagnolo" lang="es" hreflang="es" data-title="Molécula" data-language-autonym="Español" data-language-local-name="spagnolo" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Molekul" title="Molekul - estone" lang="et" hreflang="et" data-title="Molekul" data-language-autonym="Eesti" data-language-local-name="estone" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Molekula" title="Molekula - basco" lang="eu" hreflang="eu" data-title="Molekula" data-language-autonym="Euskara" data-language-local-name="basco" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-ext mw-list-item"><a href="https://ext.wikipedia.org/wiki/Mol%C3%A9cula" title="Molécula - estremegno" lang="ext" hreflang="ext" data-title="Molécula" data-language-autonym="Estremeñu" data-language-local-name="estremegno" class="interlanguage-link-target">Estremeñu</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D9%88%D9%84%DA%A9%D9%88%D9%84" title="مولکول - persiano" lang="fa" hreflang="fa" data-title="مولکول" data-language-autonym="فارسی" data-language-local-name="persiano" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Molekyyli" title="Molekyyli - finlandese" lang="fi" hreflang="fi" data-title="Molekyyli" data-language-autonym="Suomi" data-language-local-name="finlandese" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Mol%C3%B5kul" title="Molõkul - võro" lang="vro" hreflang="vro" data-title="Molõkul" data-language-autonym="Võro" data-language-local-name="võro" class="interlanguage-link-target">Võro</a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/M%C3%BDl" title="Mýl - faroese" lang="fo" hreflang="fo" data-title="Mýl" data-language-autonym="Føroyskt" data-language-local-name="faroese" class="interlanguage-link-target">Føroyskt</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Mol%C3%A9cule" title="Molécule - francese" lang="fr" hreflang="fr" data-title="Molécule" data-language-autonym="Français" data-language-local-name="francese" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Molek%C3%BC%C3%BCl" title="Moleküül - frisone settentrionale" lang="frr" hreflang="frr" data-title="Moleküül" data-language-autonym="Nordfriisk" data-language-local-name="frisone settentrionale" class="interlanguage-link-target">Nordfriisk</a></li><li class="interlanguage-link interwiki-fy mw-list-item"><a href="https://fy.wikipedia.org/wiki/Molekule" title="Molekule - frisone occidentale" lang="fy" hreflang="fy" data-title="Molekule" data-language-autonym="Frysk" data-language-local-name="frisone occidentale" class="interlanguage-link-target">Frysk</a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/M%C3%B3il%C3%ADn" title="Móilín - irlandese" lang="ga" hreflang="ga" data-title="Móilín" data-language-autonym="Gaeilge" data-language-local-name="irlandese" class="interlanguage-link-target">Gaeilge</a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Mol%C3%A9kil" title="Molékil - Guianan Creole" lang="gcr" hreflang="gcr" data-title="Molékil" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target">Kriyòl gwiyannen</a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Moileciuil" title="Moileciuil - gaelico scozzese" lang="gd" hreflang="gd" data-title="Moileciuil" data-language-autonym="Gàidhlig" data-language-local-name="gaelico scozzese" class="interlanguage-link-target">Gàidhlig</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Mol%C3%A9cula" title="Molécula - galiziano" lang="gl" hreflang="gl" data-title="Molécula" data-language-autonym="Galego" data-language-local-name="galiziano" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-gn mw-list-item"><a href="https://gn.wikipedia.org/wiki/Mol%C3%A9kula" title="Molékula - guaraní" lang="gn" hreflang="gn" data-title="Molékula" data-language-autonym="Avañe'ẽ" data-language-local-name="guaraní" class="interlanguage-link-target">Avañe'ẽ</a></li><li class="interlanguage-link interwiki-gor mw-list-item"><a href="https://gor.wikipedia.org/wiki/Molekul" title="Molekul - gorontalo" lang="gor" hreflang="gor" data-title="Molekul" data-language-autonym="Bahasa Hulontalo" data-language-local-name="gorontalo" class="interlanguage-link-target">Bahasa Hulontalo</a></li><li class="interlanguage-link interwiki-gv mw-list-item"><a href="https://gv.wikipedia.org/wiki/Co-vreneen" title="Co-vreneen - mannese" lang="gv" hreflang="gv" data-title="Co-vreneen" data-language-autonym="Gaelg" data-language-local-name="mannese" class="interlanguage-link-target">Gaelg</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%95%D7%9C%D7%A7%D7%95%D7%9C%D7%94" title="מולקולה - ebraico" lang="he" hreflang="he" data-title="מולקולה" data-language-autonym="עברית" data-language-local-name="ebraico" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%85%E0%A4%A3%E0%A5%81" title="अणु - hindi" lang="hi" hreflang="hi" data-title="अणु" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Molecule" title="Molecule - hindi figiano" lang="hif" hreflang="hif" data-title="Molecule" data-language-autonym="Fiji Hindi" data-language-local-name="hindi figiano" class="interlanguage-link-target">Fiji Hindi</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Molekula" title="Molekula - croato" lang="hr" hreflang="hr" data-title="Molekula" data-language-autonym="Hrvatski" data-language-local-name="croato" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-hsb mw-list-item"><a href="https://hsb.wikipedia.org/wiki/Molekul" title="Molekul - alto sorabo" lang="hsb" hreflang="hsb" data-title="Molekul" data-language-autonym="Hornjoserbsce" data-language-local-name="alto sorabo" class="interlanguage-link-target">Hornjoserbsce</a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Molekil" title="Molekil - creolo haitiano" lang="ht" hreflang="ht" data-title="Molekil" data-language-autonym="Kreyòl ayisyen" data-language-local-name="creolo haitiano" class="interlanguage-link-target">Kreyòl ayisyen</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Molekula" title="Molekula - ungherese" lang="hu" hreflang="hu" data-title="Molekula" data-language-autonym="Magyar" data-language-local-name="ungherese" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%84%D5%B8%D5%AC%D5%A5%D5%AF%D5%B8%D6%82%D5%AC" title="Մոլեկուլ - armeno" lang="hy" hreflang="hy" data-title="Մոլեկուլ" data-language-autonym="Հայերեն" data-language-local-name="armeno" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Molecula" title="Molecula - interlingua" lang="ia" hreflang="ia" data-title="Molecula" data-language-autonym="Interlingua" data-language-local-name="interlingua" class="interlanguage-link-target">Interlingua</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Molekul" title="Molekul - indonesiano" lang="id" hreflang="id" data-title="Molekul" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesiano" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-ie mw-list-item"><a href="https://ie.wikipedia.org/wiki/Molecul" title="Molecul - interlingue" lang="ie" hreflang="ie" data-title="Molecul" data-language-autonym="Interlingue" data-language-local-name="interlingue" class="interlanguage-link-target">Interlingue</a></li><li class="interlanguage-link interwiki-ilo mw-list-item"><a href="https://ilo.wikipedia.org/wiki/Molekula" title="Molekula - ilocano" lang="ilo" hreflang="ilo" data-title="Molekula" data-language-autonym="Ilokano" data-language-local-name="ilocano" class="interlanguage-link-target">Ilokano</a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Molekulo" title="Molekulo - ido" lang="io" hreflang="io" data-title="Molekulo" data-language-autonym="Ido" data-language-local-name="ido" class="interlanguage-link-target">Ido</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Sameind" title="Sameind - islandese" lang="is" hreflang="is" data-title="Sameind" data-language-autonym="Íslenska" data-language-local-name="islandese" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%88%86%E5%AD%90" title="分子 - giapponese" lang="ja" hreflang="ja" data-title="分子" data-language-autonym="日本語" data-language-local-name="giapponese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Malikyuul" title="Malikyuul - creolo giamaicano" lang="jam" hreflang="jam" data-title="Malikyuul" data-language-autonym="Patois" data-language-local-name="creolo giamaicano" class="interlanguage-link-target">Patois</a></li><li class="interlanguage-link interwiki-jbo mw-list-item"><a href="https://jbo.wikipedia.org/wiki/xumsle" title="xumsle - lojban" lang="jbo" hreflang="jbo" data-title="xumsle" data-language-autonym="La .lojban." data-language-local-name="lojban" class="interlanguage-link-target">La .lojban.</a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Molekul" title="Molekul - giavanese" lang="jv" hreflang="jv" data-title="Molekul" data-language-autonym="Jawa" data-language-local-name="giavanese" class="interlanguage-link-target">Jawa</a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9B%E1%83%9D%E1%83%9A%E1%83%94%E1%83%99%E1%83%A3%E1%83%9A%E1%83%90" title="მოლეკულა - georgiano" lang="ka" hreflang="ka" data-title="მოლეკულა" data-language-autonym="ქართული" data-language-local-name="georgiano" class="interlanguage-link-target">ქართული</a></li><li class="interlanguage-link interwiki-kaa mw-list-item"><a href="https://kaa.wikipedia.org/wiki/Molekula" title="Molekula - kara-kalpak" lang="kaa" hreflang="kaa" data-title="Molekula" data-language-autonym="Qaraqalpaqsha" data-language-local-name="kara-kalpak" class="interlanguage-link-target">Qaraqalpaqsha</a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Taxeclawt" title="Taxeclawt - cabilo" lang="kab" hreflang="kab" data-title="Taxeclawt" data-language-autonym="Taqbaylit" data-language-local-name="cabilo" class="interlanguage-link-target">Taqbaylit</a></li><li class="interlanguage-link interwiki-kge mw-list-item"><a href="https://kge.wikipedia.org/wiki/Molekul" title="Molekul - Komering" lang="kge" hreflang="kge" data-title="Molekul" data-language-autonym="Kumoring" data-language-local-name="Komering" class="interlanguage-link-target">Kumoring</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9C%D0%BE%D0%BB%D0%B5%D0%BA%D1%83%D0%BB%D0%B0" title="Молекула - kazako" lang="kk" hreflang="kk" data-title="Молекула" data-language-autonym="Қазақша" data-language-local-name="kazako" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%98%E1%9F%89%E1%9E%BC%E1%9E%9B%E1%9F%81%E1%9E%82%E1%9E%BB%E1%9E%9B" title="ម៉ូលេគុល - khmer" lang="km" hreflang="km" data-title="ម៉ូលេគុល" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="khmer" class="interlanguage-link-target">ភាសាខ្មែរ</a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%85%E0%B2%A3%E0%B3%81" title="ಅಣು - kannada" lang="kn" hreflang="kn" data-title="ಅಣು" data-language-autonym="ಕನ್ನಡ" data-language-local-name="kannada" class="interlanguage-link-target">ಕನ್ನಡ</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B6%84%EC%9E%90" title="분자 - coreano" lang="ko" hreflang="ko" data-title="분자" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-ks mw-list-item"><a href="https://ks.wikipedia.org/wiki/%DA%A9%D9%86%D9%95%D9%88%D9%8E%D9%B9" title="کنٕوَٹ - kashmiri" lang="ks" hreflang="ks" data-title="کنٕوَٹ" data-language-autonym="कॉशुर / کٲشُر" data-language-local-name="kashmiri" class="interlanguage-link-target">कॉशुर / کٲشُر</a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Molekul" title="Molekul - curdo" lang="ku" hreflang="ku" data-title="Molekul" data-language-autonym="Kurdî" data-language-local-name="curdo" class="interlanguage-link-target">Kurdî</a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9C%D0%BE%D0%BB%D0%B5%D0%BA%D1%83%D0%BB%D0%B0" title="Молекула - kirghiso" lang="ky" hreflang="ky" data-title="Молекула" data-language-autonym="Кыргызча" data-language-local-name="kirghiso" class="interlanguage-link-target">Кыргызча</a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Molecula" title="Molecula - latino" lang="la" hreflang="la" data-title="Molecula" data-language-autonym="Latina" data-language-local-name="latino" class="interlanguage-link-target">Latina</a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Molek%C3%BCll" title="Moleküll - lussemburghese" lang="lb" hreflang="lb" data-title="Moleküll" data-language-autonym="Lëtzebuergesch" data-language-local-name="lussemburghese" class="interlanguage-link-target">Lëtzebuergesch</a></li><li class="interlanguage-link interwiki-lg mw-list-item"><a href="https://lg.wikipedia.org/wiki/Molekyo" title="Molekyo - ganda" lang="lg" hreflang="lg" data-title="Molekyo" data-language-autonym="Luganda" data-language-local-name="ganda" class="interlanguage-link-target">Luganda</a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Molecuul" title="Molecuul - limburghese" lang="li" hreflang="li" data-title="Molecuul" data-language-autonym="Limburgs" data-language-local-name="limburghese" class="interlanguage-link-target">Limburgs</a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Mulecula" title="Mulecula - lombardo" lang="lmo" hreflang="lmo" data-title="Mulecula" data-language-autonym="Lombard" data-language-local-name="lombardo" class="interlanguage-link-target">Lombard</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Molekul%C4%97" title="Molekulė - lituano" lang="lt" hreflang="lt" data-title="Molekulė" data-language-autonym="Lietuvių" data-language-local-name="lituano" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Molekula" title="Molekula - lettone" lang="lv" hreflang="lv" data-title="Molekula" data-language-autonym="Latviešu" data-language-local-name="lettone" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-mdf mw-list-item"><a href="https://mdf.wikipedia.org/wiki/%D0%9C%D0%BE%D0%BB%D0%B5%D0%BA%D1%83%D0%BB%D0%B0%D1%81%D1%8C" title="Молекулась - moksha" lang="mdf" hreflang="mdf" data-title="Молекулась" data-language-autonym="Мокшень" data-language-local-name="moksha" class="interlanguage-link-target">Мокшень</a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/M%C3%B4lekiola" title="Môlekiola - malgascio" lang="mg" hreflang="mg" data-title="Môlekiola" data-language-autonym="Malagasy" data-language-local-name="malgascio" class="interlanguage-link-target">Malagasy</a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9C%D0%BE%D0%BB%D0%B5%D0%BA%D1%83%D0%BB%D0%B0" title="Молекула - macedone" lang="mk" hreflang="mk" data-title="Молекула" data-language-autonym="Македонски" data-language-local-name="macedone" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%A4%E0%B4%A8%E0%B5%8D%E0%B4%AE%E0%B4%BE%E0%B4%A4%E0%B5%8D%E0%B4%B0" title="തന്മാത്ര - malayalam" lang="ml" hreflang="ml" data-title="തന്മാത്ര" data-language-autonym="മലയാളം" data-language-local-name="malayalam" class="interlanguage-link-target">മലയാളം</a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%9C%D0%BE%D0%BB%D0%B5%D0%BA%D1%83%D0%BB" title="Молекул - mongolo" lang="mn" hreflang="mn" data-title="Молекул" data-language-autonym="Монгол" data-language-local-name="mongolo" class="interlanguage-link-target">Монгол</a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B0%E0%A5%87%E0%A4%A3%E0%A5%82" title="रेणू - marathi" lang="mr" hreflang="mr" data-title="रेणू" data-language-autonym="मराठी" data-language-local-name="marathi" class="interlanguage-link-target">मराठी</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Molekul" title="Molekul - malese" lang="ms" hreflang="ms" data-title="Molekul" data-language-autonym="Bahasa Melayu" data-language-local-name="malese" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%99%E1%80%B1%E1%80%AC%E1%80%BA%E1%80%9C%E1%80%AE%E1%80%80%E1%80%BB%E1%80%B0%E1%80%B8" title="မော်လီကျူး - birmano" lang="my" hreflang="my" data-title="မော်လီကျူး" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="birmano" class="interlanguage-link-target">မြန်မာဘာသာ</a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Molek%C3%BCl" title="Molekül - basso tedesco" lang="nds" hreflang="nds" data-title="Molekül" data-language-autonym="Plattdüütsch" data-language-local-name="basso tedesco" class="interlanguage-link-target">Plattdüütsch</a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%85%E0%A4%A3%E0%A5%81" title="अणु - nepalese" lang="ne" hreflang="ne" data-title="अणु" data-language-autonym="नेपाली" data-language-local-name="nepalese" class="interlanguage-link-target">नेपाली</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Molecuul" title="Molecuul - olandese" lang="nl" hreflang="nl" data-title="Molecuul" data-language-autonym="Nederlands" data-language-local-name="olandese" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Molekyl" title="Molekyl - norvegese nynorsk" lang="nn" hreflang="nn" data-title="Molekyl" data-language-autonym="Norsk nynorsk" data-language-local-name="norvegese nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Molekyl" title="Molekyl - norvegese bokmål" lang="nb" hreflang="nb" data-title="Molekyl" data-language-autonym="Norsk bokmål" data-language-local-name="norvegese bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nov mw-list-item"><a href="https://nov.wikipedia.org/wiki/Molekule" title="Molekule - novial" lang="nov" hreflang="nov" data-title="Molekule" data-language-autonym="Novial" data-language-local-name="novial" class="interlanguage-link-target">Novial</a></li><li class="interlanguage-link interwiki-nrm mw-list-item"><a href="https://nrm.wikipedia.org/wiki/Mol%C3%A9tchule" title="Molétchule - Norman" lang="nrf" hreflang="nrf" data-title="Molétchule" data-language-autonym="Nouormand" data-language-local-name="Norman" class="interlanguage-link-target">Nouormand</a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Molecula" title="Molecula - occitano" lang="oc" hreflang="oc" data-title="Molecula" data-language-autonym="Occitan" data-language-local-name="occitano" class="interlanguage-link-target">Occitan</a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Maddarroo(molakiyulii)" title="Maddarroo(molakiyulii) - oromo" lang="om" hreflang="om" data-title="Maddarroo(molakiyulii)" data-language-autonym="Oromoo" data-language-local-name="oromo" class="interlanguage-link-target">Oromoo</a></li><li class="interlanguage-link interwiki-os mw-list-item"><a href="https://os.wikipedia.org/wiki/%D0%9C%D0%BE%D0%BB%D0%B5%D0%BA%D1%83%D0%BB%C3%A6" title="Молекулæ - ossetico" lang="os" hreflang="os" data-title="Молекулæ" data-language-autonym="Ирон" data-language-local-name="ossetico" class="interlanguage-link-target">Ирон</a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%85%E0%A8%A3%E0%A9%82" title="ਅਣੂ - punjabi" lang="pa" hreflang="pa" data-title="ਅਣੂ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="punjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-pam mw-list-item"><a href="https://pam.wikipedia.org/wiki/Molekula" title="Molekula - pampanga" lang="pam" hreflang="pam" data-title="Molekula" data-language-autonym="Kapampangan" data-language-local-name="pampanga" class="interlanguage-link-target">Kapampangan</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Cz%C4%85steczka" title="Cząsteczka - polacco" lang="pl" hreflang="pl" data-title="Cząsteczka" data-language-autonym="Polski" data-language-local-name="polacco" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Mol%C3%A9cola" title="Molécola - piemontese" lang="pms" hreflang="pms" data-title="Molécola" data-language-autonym="Piemontèis" data-language-local-name="piemontese" class="interlanguage-link-target">Piemontèis</a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%85%D8%A7%D9%84%DB%8C%DA%A9%DB%8C%D9%88%D9%84" title="مالیکیول - Western Punjabi" lang="pnb" hreflang="pnb" data-title="مالیکیول" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target">پنجابی</a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D9%85%D8%A7%D9%84%DB%8C%DA%A9%D9%88%D9%84" title="مالیکول - pashto" lang="ps" hreflang="ps" data-title="مالیکول" data-language-autonym="پښتو" data-language-local-name="pashto" class="interlanguage-link-target">پښتو</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Mol%C3%A9cula" title="Molécula - portoghese" lang="pt" hreflang="pt" data-title="Molécula" data-language-autonym="Português" data-language-local-name="portoghese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/I%C3%B1uwa" title="Iñuwa - quechua" lang="qu" hreflang="qu" data-title="Iñuwa" data-language-autonym="Runa Simi" data-language-local-name="quechua" class="interlanguage-link-target">Runa Simi</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Molecul%C4%83" title="Moleculă - rumeno" lang="ro" hreflang="ro" data-title="Moleculă" data-language-autonym="Română" data-language-local-name="rumeno" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%BE%D0%BB%D0%B5%D0%BA%D1%83%D0%BB%D0%B0" title="Молекула - russo" lang="ru" hreflang="ru" data-title="Молекула" data-language-autonym="Русский" data-language-local-name="russo" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D0%9C%D0%BE%D0%BB%D0%B5%D0%BA%D1%83%D0%BB%D0%B0" title="Молекула - ruteno" lang="rue" hreflang="rue" data-title="Молекула" data-language-autonym="Русиньскый" data-language-local-name="ruteno" class="interlanguage-link-target">Русиньскый</a></li><li class="interlanguage-link interwiki-sa mw-list-item"><a href="https://sa.wikipedia.org/wiki/%E0%A4%B0%E0%A5%87%E0%A4%A3%E0%A5%81_:" title="रेणु : - sanscrito" lang="sa" hreflang="sa" data-title="रेणु :" data-language-autonym="संस्कृतम्" data-language-local-name="sanscrito" class="interlanguage-link-target">संस्कृतम्</a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%9C%D3%A9%D0%BB%D2%AF%D3%A9%D0%BA%D2%AF%D0%BB%D1%8D" title="Мөлүөкүлэ - sacha" lang="sah" hreflang="sah" data-title="Мөлүөкүлэ" data-language-autonym="Саха тыла" data-language-local-name="sacha" class="interlanguage-link-target">Саха тыла</a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Molecule" title="Molecule - scozzese" lang="sco" hreflang="sco" data-title="Molecule" data-language-autonym="Scots" data-language-local-name="scozzese" class="interlanguage-link-target">Scots</a></li><li class="interlanguage-link interwiki-sd mw-list-item"><a href="https://sd.wikipedia.org/wiki/%D9%85%D8%A7%D9%84%D9%8A%DA%AA%D9%8A%D9%88%D9%84" title="ماليڪيول - sindhi" lang="sd" hreflang="sd" data-title="ماليڪيول" data-language-autonym="سنڌي" data-language-local-name="sindhi" class="interlanguage-link-target">سنڌي</a></li><li class="interlanguage-link interwiki-se mw-list-item"><a href="https://se.wikipedia.org/wiki/Molekyla" title="Molekyla - sami del nord" lang="se" hreflang="se" data-title="Molekyla" data-language-autonym="Davvisámegiella" data-language-local-name="sami del nord" class="interlanguage-link-target">Davvisámegiella</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Molekul" title="Molekul - serbo-croato" lang="sh" hreflang="sh" data-title="Molekul" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbo-croato" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%85%E0%B6%AB%E0%B7%94%E0%B7%80" title="අණුව - singalese" lang="si" hreflang="si" data-title="අණුව" data-language-autonym="සිංහල" data-language-local-name="singalese" class="interlanguage-link-target">සිංහල</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Molecule" title="Molecule - Simple English" lang="en-simple" hreflang="en-simple" data-title="Molecule" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Molekula" title="Molekula - slovacco" lang="sk" hreflang="sk" data-title="Molekula" data-language-autonym="Slovenčina" data-language-local-name="slovacco" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-skr mw-list-item"><a href="https://skr.wikipedia.org/wiki/%D9%85%D8%A7%D9%84%DB%8C%DA%A9%DB%8C%D9%88%D9%84" title="مالیکیول - Saraiki" lang="skr" hreflang="skr" data-title="مالیکیول" data-language-autonym="سرائیکی" data-language-local-name="Saraiki" class="interlanguage-link-target">سرائیکی</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Molekula" title="Molekula - sloveno" lang="sl" hreflang="sl" data-title="Molekula" data-language-autonym="Slovenščina" data-language-local-name="sloveno" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-sm mw-list-item"><a href="https://sm.wikipedia.org/wiki/Molecule" title="Molecule - samoano" lang="sm" hreflang="sm" data-title="Molecule" data-language-autonym="Gagana Samoa" data-language-local-name="samoano" class="interlanguage-link-target">Gagana Samoa</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Molekula" title="Molekula - albanese" lang="sq" hreflang="sq" data-title="Molekula" data-language-autonym="Shqip" data-language-local-name="albanese" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9C%D0%BE%D0%BB%D0%B5%D0%BA%D1%83%D0%BB" title="Молекул - serbo" lang="sr" hreflang="sr" data-title="Молекул" data-language-autonym="Српски / srpski" data-language-local-name="serbo" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Molekul" title="Molekul - sundanese" lang="su" hreflang="su" data-title="Molekul" data-language-autonym="Sunda" data-language-local-name="sundanese" class="interlanguage-link-target">Sunda</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Molekyl" title="Molekyl - svedese" lang="sv" hreflang="sv" data-title="Molekyl" data-language-autonym="Svenska" data-language-local-name="svedese" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Molekuli" title="Molekuli - swahili" lang="sw" hreflang="sw" data-title="Molekuli" data-language-autonym="Kiswahili" data-language-local-name="swahili" class="interlanguage-link-target">Kiswahili</a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/Tajleczka" title="Tajleczka - slesiano" lang="szl" hreflang="szl" data-title="Tajleczka" data-language-autonym="Ślůnski" data-language-local-name="slesiano" class="interlanguage-link-target">Ślůnski</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AE%E0%AF%82%E0%AE%B2%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AF%82%E0%AE%B1%E0%AF%81" title="மூலக்கூறு - tamil" lang="ta" hreflang="ta" data-title="மூலக்கூறு" data-language-autonym="தமிழ்" data-language-local-name="tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%85%E0%B0%A3%E0%B1%81%E0%B0%B5%E0%B1%81" title="అణువు - telugu" lang="te" hreflang="te" data-title="అణువు" data-language-autonym="తెలుగు" data-language-local-name="telugu" class="interlanguage-link-target">తెలుగు</a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%9C%D0%BE%D0%BB%D0%B5%D0%BA%D1%83%D0%BB%D0%B0" title="Молекула - tagico" lang="tg" hreflang="tg" data-title="Молекула" data-language-autonym="Тоҷикӣ" data-language-local-name="tagico" class="interlanguage-link-target">Тоҷикӣ</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%82%E0%B8%A1%E0%B9%80%E0%B8%A5%E0%B8%81%E0%B8%B8%E0%B8%A5" title="โมเลกุล - thailandese" lang="th" hreflang="th" data-title="โมเลกุล" data-language-autonym="ไทย" data-language-local-name="thailandese" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tk mw-list-item"><a href="https://tk.wikipedia.org/wiki/Molekula" title="Molekula - turcomanno" lang="tk" hreflang="tk" data-title="Molekula" data-language-autonym="Türkmençe" data-language-local-name="turcomanno" class="interlanguage-link-target">Türkmençe</a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Molekula" title="Molekula - tagalog" lang="tl" hreflang="tl" data-title="Molekula" data-language-autonym="Tagalog" data-language-local-name="tagalog" class="interlanguage-link-target">Tagalog</a></li><li class="interlanguage-link interwiki-tpi mw-list-item"><a href="https://tpi.wikipedia.org/wiki/Molekel" title="Molekel - tok pisin" lang="tpi" hreflang="tpi" data-title="Molekel" data-language-autonym="Tok Pisin" data-language-local-name="tok pisin" class="interlanguage-link-target">Tok Pisin</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Molek%C3%BCl" title="Molekül - turco" lang="tr" hreflang="tr" data-title="Molekül" data-language-autonym="Türkçe" data-language-local-name="turco" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%9C%D0%BE%D0%BB%D0%B5%D0%BA%D1%83%D0%BB%D0%B0" title="Молекула - tataro" lang="tt" hreflang="tt" data-title="Молекула" data-language-autonym="Татарча / tatarça" data-language-local-name="tataro" class="interlanguage-link-target">Татарча / tatarça</a></li><li class="interlanguage-link interwiki-ug mw-list-item"><a href="https://ug.wikipedia.org/wiki/%D9%85%D9%88%D9%84%DB%90%D9%83%DB%87%D9%84%D8%A7" title="مولېكۇلا - uiguro" lang="ug" hreflang="ug" data-title="مولېكۇلا" data-language-autonym="ئۇيغۇرچە / Uyghurche" data-language-local-name="uiguro" class="interlanguage-link-target">ئۇيغۇرچە / Uyghurche</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D0%BE%D0%BB%D0%B5%D0%BA%D1%83%D0%BB%D0%B0" title="Молекула - ucraino" lang="uk" hreflang="uk" data-title="Молекула" data-language-autonym="Українська" data-language-local-name="ucraino" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%B3%D8%A7%D9%84%D9%85%DB%81" title="سالمہ - urdu" lang="ur" hreflang="ur" data-title="سالمہ" data-language-autonym="اردو" data-language-local-name="urdu" class="interlanguage-link-target">اردو</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Molekula" title="Molekula - uzbeco" lang="uz" hreflang="uz" data-title="Molekula" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbeco" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-vep mw-list-item"><a href="https://vep.wikipedia.org/wiki/Molekul" title="Molekul - vepso" lang="vep" hreflang="vep" data-title="Molekul" data-language-autonym="Vepsän kel’" data-language-local-name="vepso" class="interlanguage-link-target">Vepsän kel’</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C3%A2n_t%E1%BB%AD" title="Phân tử - vietnamita" lang="vi" hreflang="vi" data-title="Phân tử" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamita" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-vls mw-list-item"><a href="https://vls.wikipedia.org/wiki/Molecule" title="Molecule - fiammingo occidentale" lang="vls" hreflang="vls" data-title="Molecule" data-language-autonym="West-Vlams" data-language-local-name="fiammingo occidentale" class="interlanguage-link-target">West-Vlams</a></li><li class="interlanguage-link interwiki-wa mw-list-item"><a href="https://wa.wikipedia.org/wiki/Molecule" title="Molecule - vallone" lang="wa" hreflang="wa" data-title="Molecule" data-language-autonym="Walon" data-language-local-name="vallone" class="interlanguage-link-target">Walon</a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Molekula" title="Molekula - waray" lang="war" hreflang="war" data-title="Molekula" data-language-autonym="Winaray" data-language-local-name="waray" class="interlanguage-link-target">Winaray</a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%88%86%E5%AD%90" title="分子 - wu" lang="wuu" hreflang="wuu" data-title="分子" data-language-autonym="吴语" data-language-local-name="wu" class="interlanguage-link-target">吴语</a></li><li class="interlanguage-link interwiki-xal mw-list-item"><a href="https://xal.wikipedia.org/wiki/%D0%9C%D0%BE%D0%BB%D0%B5%D0%BA%D1%83%D0%BB" title="Молекул - kalmyk" lang="xal" hreflang="xal" data-title="Молекул" data-language-autonym="Хальмг" data-language-local-name="kalmyk" class="interlanguage-link-target">Хальмг</a></li><li class="interlanguage-link interwiki-xh mw-list-item"><a href="https://xh.wikipedia.org/wiki/I-Molecule" title="I-Molecule - xhosa" lang="xh" hreflang="xh" data-title="I-Molecule" data-language-autonym="IsiXhosa" data-language-local-name="xhosa" class="interlanguage-link-target">IsiXhosa</a></li><li class="interlanguage-link interwiki-xmf mw-list-item"><a href="https://xmf.wikipedia.org/wiki/%E1%83%9B%E1%83%9D%E1%83%9A%E1%83%94%E1%83%99%E1%83%A3%E1%83%9A%E1%83%90" title="მოლეკულა - mengrelio" lang="xmf" hreflang="xmf" data-title="მოლეკულა" data-language-autonym="მარგალური" data-language-local-name="mengrelio" class="interlanguage-link-target">მარგალური</a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%9E%D7%90%D7%9C%D7%A2%D7%A7%D7%95%D7%9C" title="מאלעקול - yiddish" lang="yi" hreflang="yi" data-title="מאלעקול" data-language-autonym="ייִדיש" data-language-local-name="yiddish" class="interlanguage-link-target">ייִדיש</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%88%86%E5%AD%90" title="分子 - cinese" lang="zh" hreflang="zh" data-title="分子" data-language-autonym="中文" data-language-local-name="cinese" class="interlanguage-link-target">中文</a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E5%88%86%E5%AD%90" title="分子 - cinese classico" lang="lzh" hreflang="lzh" data-title="分子" data-language-autonym="文言" data-language-local-name="cinese classico" class="interlanguage-link-target">文言</a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Hun-ch%C3%BA" title="Hun-chú - min nan" lang="nan" hreflang="nan" data-title="Hun-chú" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="min nan" class="interlanguage-link-target">閩南語 / Bân-lâm-gú</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%88%86%E5%AD%90" title="分子 - cantonese" lang="yue" hreflang="yue" data-title="分子" data-language-autonym="粵語" data-language-local-name="cantonese" class="interlanguage-link-target">粵語</a></li><li class="interlanguage-link interwiki-zu mw-list-item"><a href="https://zu.wikipedia.org/wiki/Imizwayi" title="Imizwayi - zulu" lang="zu" hreflang="zu" data-title="Imizwayi" data-language-autonym="IsiZulu" data-language-local-name="zulu" class="interlanguage-link-target">IsiZulu</a></li> </ul> <div class="after-portlet after-portlet-lang"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q11369#sitelinks-wikipedia" title="Modifica collegamenti interlinguistici" class="wbc-editpage">Modifica collegamenti</a></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespace"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Molecola" title="Vedi la voce [c]" accesskey="c">Voce</a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Discussione:Molecola" rel="discussion" title="Vedi le discussioni relative a questa pagina [t]" accesskey="t">Discussione</a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Cambia versione linguistica" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" >italiano </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Visite"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Molecola">Leggi</a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Molecola&veaction=edit" title="Modifica questa pagina [v]" accesskey="v">Modifica</a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Molecola&action=edit" title="Modifica il wikitesto di questa pagina [e]" accesskey="e">Modifica wikitesto</a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Molecola&action=history" title="Versioni precedenti di questa pagina [h]" accesskey="h">Cronologia</a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Strumenti pagine"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Strumenti" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" >Strumenti </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Strumenti</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">sposta nella barra laterale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">nascondi</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Altre opzioni" > <div class="vector-menu-heading"> Azioni </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Molecola">Leggi</a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Molecola&veaction=edit" title="Modifica questa pagina [v]" accesskey="v">Modifica</a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Molecola&action=edit" title="Modifica il wikitesto di questa pagina [e]" accesskey="e">Modifica wikitesto</a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Molecola&action=history">Cronologia</a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Generale </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Speciale:PuntanoQui/Molecola" title="Elenco di tutte le pagine che sono collegate a questa [j]" accesskey="j">Puntano qui</a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Speciale:ModificheCorrelate/Molecola" rel="nofollow" title="Elenco delle ultime modifiche alle pagine collegate a questa [k]" accesskey="k">Modifiche correlate</a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Speciale:PagineSpeciali" title="Elenco di tutte le pagine speciali [q]" accesskey="q">Pagine speciali</a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Molecola&oldid=142133299" title="Collegamento permanente a questa versione di questa pagina">Link permanente</a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Molecola&action=info" title="Ulteriori informazioni su questa pagina">Informazioni pagina</a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Speciale:Cita&page=Molecola&id=142133299&wpFormIdentifier=titleform" title="Informazioni su come citare questa pagina">Cita questa voce</a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Speciale:UrlShortener&url=https%3A%2F%2Fit.wikipedia.org%2Fwiki%2FMolecola">Ottieni URL breve</a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Speciale:QrCode&url=https%3A%2F%2Fit.wikipedia.org%2Fwiki%2FMolecola">Scarica codice QR</a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Stampa/esporta </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Speciale:Libro&bookcmd=book_creator&referer=Molecola">Crea un libro</a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Speciale:DownloadAsPdf&page=Molecola&action=show-download-screen">Scarica come PDF</a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Molecola&printable=yes" title="Versione stampabile di questa pagina [p]" accesskey="p">Versione stampabile</a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In altri progetti </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Molecules" hreflang="en">Wikimedia Commons</a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q11369" title="Collegamento all'elemento connesso dell'archivio dati [g]" accesskey="g">Elemento Wikidata</a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Strumenti pagine"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Aspetto"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Aspetto</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">sposta nella barra laterale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">nascondi</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Da Wikipedia, l'enciclopedia libera.</div> </div> <div id="contentSub"><div id="mw-content-subtitle">(Reindirizzamento da <a href="/w/index.php?title=Molecole&redirect=no" class="mw-redirect" title="Molecole">Molecole</a>)</div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="it" dir="ltr"><style data-mw-deduplicate="TemplateStyles:r130658281">body:not(.skin-minerva) .mw-parser-output .hatnote.nota-disambigua{clear:both;margin-top:0;padding:.05em .5em}</style> <style data-mw-deduplicate="TemplateStyles:r139142988">.mw-parser-output .hatnote-content{align-items:center;display:flex}.mw-parser-output .hatnote-icon{flex-shrink:0}.mw-parser-output .hatnote-icon img{display:flex}.mw-parser-output .hatnote-text{font-style:italic}body:not(.skin-minerva) .mw-parser-output .hatnote{border:1px solid #CCC;display:flex;margin:.5em 0;padding:.2em .5em}body:not(.skin-minerva) .mw-parser-output .hatnote-text{padding-left:.5em}body.skin-minerva .mw-parser-output .hatnote-icon{padding-right:8px}body.skin-minerva .mw-parser-output .hatnote-icon img{height:auto;width:16px}body.skin--responsive .mw-parser-output .hatnote a.new{color:#d73333}body.skin--responsive .mw-parser-output .hatnote a.new:visited{color:#a55858}</style> <div class="hatnote noprint nota-disambigua"> <div class="hatnote-content"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Nota_disambigua.svg/18px-Nota_disambigua.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Nota_disambigua.svg/27px-Nota_disambigua.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Nota_disambigua.svg/36px-Nota_disambigua.svg.png 2x" data-file-width="200" data-file-height="200" /> <a href="/wiki/Aiuto:Disambiguazione" title="Aiuto:Disambiguazione">Disambiguazione</a> – Se stai cercando il personaggio dei fumetti, vedi <a href="/wiki/Molecola_(personaggio)" class="mw-redirect" title="Molecola (personaggio)">Molecola (personaggio)</a>.</div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r130658281"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r139142988"> <div class="hatnote noprint nota-disambigua"> <div class="hatnote-content"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Nota_disambigua.svg/18px-Nota_disambigua.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Nota_disambigua.svg/27px-Nota_disambigua.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Nota_disambigua.svg/36px-Nota_disambigua.svg.png 2x" data-file-width="200" data-file-height="200" /> <a href="/wiki/Aiuto:Disambiguazione" title="Aiuto:Disambiguazione">Disambiguazione</a> – Se stai cercando la bibita alla cola, vedi <a href="/wiki/MoleCola_(bibita)" title="MoleCola (bibita)">MoleCola (bibita)</a>.</div> </div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Sucrose_molecule.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Sucrose_molecule.svg/220px-Sucrose_molecule.svg.png" decoding="async" width="220" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Sucrose_molecule.svg/330px-Sucrose_molecule.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1a/Sucrose_molecule.svg/440px-Sucrose_molecule.svg.png 2x" data-file-width="512" data-file-height="373" /></a><figcaption>Modello della molecola del <a href="/wiki/Saccarosio" title="Saccarosio">saccarosio</a> (principale componente dello <a href="/wiki/Zucchero" class="mw-redirect" title="Zucchero">zucchero</a> da tavola)</figcaption></figure> In <a href="/wiki/Fisica" title="Fisica">fisica</a> e <a href="/wiki/Chimica" title="Chimica">chimica</a>, la molecola (dal <a href="/wiki/Lingua_latina" title="Lingua latina">latino</a> <a href="/wiki/Scienza" title="Scienza">scientifico</a> molecula, derivato a sua volta da moles, che significa "mole", cioè "piccola quantità") è un'entità <a href="/wiki/Carica_elettrica" title="Carica elettrica">elettricamente neutra</a> composta da due o più <a href="/wiki/Atomo" title="Atomo">atomi</a> uniti da un <a href="/wiki/Legame_covalente" title="Legame covalente">legame covalente</a>.<a href="#cite_note-1">[1]</a><a href="#cite_note-2">[2]</a> Nella definizione del <a href="/wiki/Compendium_of_Chemical_Terminology" title="Compendium of Chemical Terminology">compendium of Chemical Terminology</a> della <a href="/wiki/Unione_internazionale_di_chimica_pura_e_applicata" title="Unione internazionale di chimica pura e applicata">IUPAC</a> gli atomi formano una <a href="/wiki/Buca_di_potenziale" title="Buca di potenziale">buca di potenziale</a> coulombiano sufficientemente profonda da consentire la presenza di almeno uno stato vibrazionale.<a href="#cite_note-3">[3]</a> Può essere composta da più atomi dello stesso <a href="/wiki/Elemento_chimico" title="Elemento chimico">elemento</a> o di elementi diversi e identifica una <a href="/wiki/Sostanza_pura" title="Sostanza pura">sostanza</a>, di cui costituisce l'unità fondamentale. Molecole costituite dagli stessi atomi con una diversa disposizione nello spazio sono dette <a href="/wiki/Isomeria" title="Isomeria">isomeri</a> di una sostanza e si differenziano per le proprietà fisiche. In <a href="/wiki/Chimica_organica" title="Chimica organica">chimica organica</a> e <a href="/wiki/Biochimica" title="Biochimica">biochimica</a>, il termine molecola identifica talvolta anche <a href="/wiki/Ione_molecolare" title="Ione molecolare">ioni poliatomici</a>, mentre nella <a href="/wiki/Teoria_cinetica_dei_gas" title="Teoria cinetica dei gas">teoria cinetica dei gas</a> è spesso utilizzato per ogni particella gassosa, indipendentemente dalla sua composizione: con tale definizione anche i singoli atomi nella famiglia dei <a href="/wiki/Gas_nobile" class="mw-redirect" title="Gas nobile">gas nobili</a> possono essere considerati molecole.<a href="#cite_note-4">[4]</a> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Dinamica_molecolare">Dinamica molecolare</h2>[<a href="/w/index.php?title=Molecola&veaction=edit&section=1" title="Modifica la sezione Dinamica molecolare" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Molecola&action=edit&section=1" title="Edit section's source code: Dinamica molecolare">modifica wikitesto</a>]</div> La descrizione a livello atomico della materia utilizza il formalismo della <a href="/wiki/Meccanica_quantistica" title="Meccanica quantistica">meccanica quantistica</a>, che attraverso la caratterizzazione probabilistica di una particella fornita dalla <a href="/wiki/Funzione_d%27onda" title="Funzione d'onda">funzione d'onda</a> permette di spiegare la natura <a href="/wiki/Interazione_elettromagnetica" title="Interazione elettromagnetica">elettromagnetica</a> dei <a href="/wiki/Interazioni_fondamentali" title="Interazioni fondamentali">legami fisici</a> e <a href="/wiki/Legame_chimico" title="Legame chimico">chimici</a> che governano il comportamento delle molecole e dei loro costituenti. In tale contesto, lo studio della <a href="/wiki/Dinamica_molecolare" title="Dinamica molecolare">dinamica molecolare</a> si basa sull'<a href="/wiki/Approssimazione_di_Born-Oppenheimer" title="Approssimazione di Born-Oppenheimer">approssimazione di Born-Oppenheimer</a>, anche detta approssimazione adiabatica, che considera il moto dei <a href="/wiki/Nucleo_atomico" title="Nucleo atomico">nuclei</a> indipendente da quello degli <a href="/wiki/Elettrone" title="Elettrone">elettroni</a>, dal momento che i primi sono estremamente più pesanti e quindi più lenti dei secondi. Questo rende possibile la fattorizzazione della funzione d'onda totale della molecola:<a href="#cite_note-5">[5]</a><a href="#cite_note-cimiraglia-6">[6]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi _{\mathrm {total} }=\psi _{\mathrm {n} }(\mathbf {R} )\psi _{\mathrm {e} }(\mathbf {R} ,\mathbf {r} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi _{\mathrm {total} }=\psi _{\mathrm {n} }(\mathbf {R} )\psi _{\mathrm {e} }(\mathbf {R} ,\mathbf {r} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35fb9add7398f99b9aa58036946a7242aa927930" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.415ex; height:2.843ex;" alt="{\displaystyle \Psi _{\mathrm {total} }=\psi _{\mathrm {n} }(\mathbf {R} )\psi _{\mathrm {e} }(\mathbf {R} ,\mathbf {r} )}"></dd></dl> dove il pedice e indica la funzione d'onda degli elettroni, il pedice n dei nuclei, ed <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} }"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }"> sono rispettivamente le posizioni di nuclei ed elettroni. Tale funzione d'onda soddisfa l'equazione agli <a href="/wiki/Autovettore_e_autovalore" title="Autovettore e autovalore">autovalori</a>: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[T_{\mathrm {e} }+T_{\mathrm {n} }+V_{\mathrm {ne} }(\mathbf {R} ,\mathbf {r} )+V_{\mathrm {ee} }(\mathbf {r} )+V_{\mathrm {nn} }(\mathbf {R} )\right]\Psi _{\mathrm {total} }(\mathbf {R} ,\mathbf {r} )=E(\mathbf {R} )\Psi _{\mathrm {total} }(\mathbf {R} ,\mathbf {r} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[T_{\mathrm {e} }+T_{\mathrm {n} }+V_{\mathrm {ne} }(\mathbf {R} ,\mathbf {r} )+V_{\mathrm {ee} }(\mathbf {r} )+V_{\mathrm {nn} }(\mathbf {R} )\right]\Psi _{\mathrm {total} }(\mathbf {R} ,\mathbf {r} )=E(\mathbf {R} )\Psi _{\mathrm {total} }(\mathbf {R} ,\mathbf {r} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fe6c950c0c256deb9804be59b253d5b0b159d0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:71.659ex; height:2.843ex;" alt="{\displaystyle \left[T_{\mathrm {e} }+T_{\mathrm {n} }+V_{\mathrm {ne} }(\mathbf {R} ,\mathbf {r} )+V_{\mathrm {ee} }(\mathbf {r} )+V_{\mathrm {nn} }(\mathbf {R} )\right]\Psi _{\mathrm {total} }(\mathbf {R} ,\mathbf {r} )=E(\mathbf {R} )\Psi _{\mathrm {total} }(\mathbf {R} ,\mathbf {r} )}"></dd></dl> dove <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\mathrm {e} }\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\mathrm {e} }\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/689b1c925e704501f450542312d29032bf1b25bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.9ex; height:2.509ex;" alt="{\displaystyle T_{\mathrm {e} }\ }"> è l'energia cinetica degli elettroni, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\mathrm {n} }\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\mathrm {n} }\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7a02ea1a8162a9af5196507766db0bca055241f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.084ex; height:2.509ex;" alt="{\displaystyle T_{\mathrm {n} }\ }"> quella dei nuclei, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\mathrm {ne} }(\mathbf {R} ,\mathbf {r} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\mathrm {ne} }(\mathbf {R} ,\mathbf {r} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16a8782002b1e62d83946bea394120361a596402" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.18ex; height:2.843ex;" alt="{\displaystyle V_{\mathrm {ne} }(\mathbf {R} ,\mathbf {r} )}"> l'<a href="/wiki/Forza_di_Coulomb" title="Forza di Coulomb">interazione coulombiana</a> tra nuclei ed elettroni, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\mathrm {ee} }(\mathbf {r} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\mathrm {ee} }(\mathbf {r} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e6d8a23c467ef09f573ee7d5aadbeaeab18fc7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.959ex; height:2.843ex;" alt="{\displaystyle V_{\mathrm {ee} }(\mathbf {r} )}"> l'interazione coulombiana tra gli elettroni e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\mathrm {nn} }(\mathbf {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\mathrm {nn} }(\mathbf {R} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e7fb40b49ac5d8c4406d37db2c30e356c8e7248" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.228ex; height:2.843ex;" alt="{\displaystyle V_{\mathrm {nn} }(\mathbf {R} )}"> quella tra i nuclei. Nell'approssimazione adiabatica, si richiede che la funzione d'onda elettronica soddisfi l'equazione agli autovalori: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[T_{\mathrm {e} }+V_{\mathrm {ne} }(\mathbf {R} ,\mathbf {r} )+V_{\mathrm {ee} }(\mathbf {r} )\right]\psi _{\mathrm {e} }(\mathbf {R} ,\mathbf {r} )=E_{\mathrm {e} }(\mathbf {R} )\psi _{\mathrm {e} }(\mathbf {R} ,\mathbf {r} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[T_{\mathrm {e} }+V_{\mathrm {ne} }(\mathbf {R} ,\mathbf {r} )+V_{\mathrm {ee} }(\mathbf {r} )\right]\psi _{\mathrm {e} }(\mathbf {R} ,\mathbf {r} )=E_{\mathrm {e} }(\mathbf {R} )\psi _{\mathrm {e} }(\mathbf {R} ,\mathbf {r} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad4ab644a0a73006f448d26d3dea2c5b0e598731" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:51.257ex; height:2.843ex;" alt="{\displaystyle \left[T_{\mathrm {e} }+V_{\mathrm {ne} }(\mathbf {R} ,\mathbf {r} )+V_{\mathrm {ee} }(\mathbf {r} )\right]\psi _{\mathrm {e} }(\mathbf {R} ,\mathbf {r} )=E_{\mathrm {e} }(\mathbf {R} )\psi _{\mathrm {e} }(\mathbf {R} ,\mathbf {r} )}"></dd></dl> La precedente espressione è ottenuta grazie al fatto che l'<a href="/wiki/Operatore_(matematica)" title="Operatore (matematica)">operatore</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla _{r}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla _{r}^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba358b38532d9fdd5a456a31dbd1b789052a54c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.99ex; height:2.843ex;" alt="{\displaystyle \nabla _{r}^{2}}">, contenuto nel termine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\mathrm {e} }\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\mathrm {e} }\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/689b1c925e704501f450542312d29032bf1b25bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.9ex; height:2.509ex;" alt="{\displaystyle T_{\mathrm {e} }\ }">, non agisce sulle coordinate dei nuclei, così che la funzione d'onda dei nuclei si possa raccogliere a fattor comune. La funzione d'onda dei nuclei, invece, è ricavata a partire dall'equazione totale, che esplicitando l'<a href="/wiki/Operatore_impulso" title="Operatore impulso">operatore impulso</a> diventa: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(-\sum _{i}{{\frac {\hbar ^{2}}{2M_{i}}}\nabla _{R}^{2}}+\left[T_{\mathrm {e} }+V_{\mathrm {ne} }(\mathbf {R} ,\mathbf {r} )+V_{\mathrm {ee} }(\mathbf {r} )+V_{\mathrm {nn} }(\mathbf {R} )\right]\right)\psi _{\mathrm {e} }(\mathbf {r} ,\mathbf {R} )\psi _{\mathrm {n} }(\mathbf {R} )=E(\mathbf {R} )\psi _{\mathrm {e} }(\mathbf {r} ,\mathbf {R} )\psi _{\mathrm {n} }(\mathbf {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msubsup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>+</mo> <mrow> <mo>[</mo> <mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> </mrow> <mo>)</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(-\sum _{i}{{\frac {\hbar ^{2}}{2M_{i}}}\nabla _{R}^{2}}+\left[T_{\mathrm {e} }+V_{\mathrm {ne} }(\mathbf {R} ,\mathbf {r} )+V_{\mathrm {ee} }(\mathbf {r} )+V_{\mathrm {nn} }(\mathbf {R} )\right]\right)\psi _{\mathrm {e} }(\mathbf {r} ,\mathbf {R} )\psi _{\mathrm {n} }(\mathbf {R} )=E(\mathbf {R} )\psi _{\mathrm {e} }(\mathbf {r} ,\mathbf {R} )\psi _{\mathrm {n} }(\mathbf {R} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bcccd002e0dc7b9a90ae914fa9145213fec19f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:94.294ex; height:7.509ex;" alt="{\displaystyle \left(-\sum _{i}{{\frac {\hbar ^{2}}{2M_{i}}}\nabla _{R}^{2}}+\left[T_{\mathrm {e} }+V_{\mathrm {ne} }(\mathbf {R} ,\mathbf {r} )+V_{\mathrm {ee} }(\mathbf {r} )+V_{\mathrm {nn} }(\mathbf {R} )\right]\right)\psi _{\mathrm {e} }(\mathbf {r} ,\mathbf {R} )\psi _{\mathrm {n} }(\mathbf {R} )=E(\mathbf {R} )\psi _{\mathrm {e} }(\mathbf {r} ,\mathbf {R} )\psi _{\mathrm {n} }(\mathbf {R} )}"></dd></dl> Essendo che: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla _{R}^{2}\left[\psi _{\mathrm {e} }(\mathbf {r} ,\mathbf {R} )\psi _{\mathrm {n} }(\mathbf {R} )\right]=\psi _{\mathrm {e} }(\mathbf {r} ,\mathbf {R} )\nabla _{R}^{2}\psi _{\mathrm {n} }(\mathbf {R} )+2\left[\nabla _{R}\psi _{\mathrm {e} }(\mathbf {r} ,\mathbf {R} )\right]\nabla _{R}\psi _{\mathrm {n} }(\mathbf {R} )+\psi _{\mathrm {n} }(\mathbf {R} )\nabla _{R}^{2}\psi _{\mathrm {e} }(\mathbf {r} ,\mathbf {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow> <mo>[</mo> <mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <msubsup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <mrow> <mo>[</mo> <mrow> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <msubsup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla _{R}^{2}\left[\psi _{\mathrm {e} }(\mathbf {r} ,\mathbf {R} )\psi _{\mathrm {n} }(\mathbf {R} )\right]=\psi _{\mathrm {e} }(\mathbf {r} ,\mathbf {R} )\nabla _{R}^{2}\psi _{\mathrm {n} }(\mathbf {R} )+2\left[\nabla _{R}\psi _{\mathrm {e} }(\mathbf {r} ,\mathbf {R} )\right]\nabla _{R}\psi _{\mathrm {n} }(\mathbf {R} )+\psi _{\mathrm {n} }(\mathbf {R} )\nabla _{R}^{2}\psi _{\mathrm {e} }(\mathbf {r} ,\mathbf {R} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ea2510d14abcee8538b1cb4344aace0ab9e8db4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:90.351ex; height:3.176ex;" alt="{\displaystyle \nabla _{R}^{2}\left[\psi _{\mathrm {e} }(\mathbf {r} ,\mathbf {R} )\psi _{\mathrm {n} }(\mathbf {R} )\right]=\psi _{\mathrm {e} }(\mathbf {r} ,\mathbf {R} )\nabla _{R}^{2}\psi _{\mathrm {n} }(\mathbf {R} )+2\left[\nabla _{R}\psi _{\mathrm {e} }(\mathbf {r} ,\mathbf {R} )\right]\nabla _{R}\psi _{\mathrm {n} }(\mathbf {R} )+\psi _{\mathrm {n} }(\mathbf {R} )\nabla _{R}^{2}\psi _{\mathrm {e} }(\mathbf {r} ,\mathbf {R} )}"></dd></dl> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Potenziale_intermolecolare.PNG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Potenziale_intermolecolare.PNG/310px-Potenziale_intermolecolare.PNG" decoding="async" width="310" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Potenziale_intermolecolare.PNG/465px-Potenziale_intermolecolare.PNG 1.5x, //upload.wikimedia.org/wikipedia/commons/b/b3/Potenziale_intermolecolare.PNG 2x" data-file-width="481" data-file-height="342" /></a><figcaption>Andamento del potenziale adiabatico in funzione della distanza di separazione tra i nuclei in una <a href="/wiki/Molecola_biatomica" title="Molecola biatomica">molecola biatomica</a></figcaption></figure> Si ottiene: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\sum _{i}{\frac {\hbar ^{2}}{2M_{i}}}\left(\psi _{\mathrm {e} }(\mathbf {r} ,\mathbf {R} )\nabla _{R}^{2}\psi _{\mathrm {n} }(\mathbf {R} )+2\left[\nabla _{R}\psi _{\mathrm {e} }(\mathbf {r} ,\mathbf {R} )\right]\nabla _{R}\psi _{\mathrm {n} }(\mathbf {R} )+\psi _{\mathrm {n} }(\mathbf {R} )\nabla _{R}^{2}\psi _{\mathrm {e} }(\mathbf {r} ,\mathbf {R} )\right)+}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <msubsup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <mrow> <mo>[</mo> <mrow> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <msubsup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\sum _{i}{\frac {\hbar ^{2}}{2M_{i}}}\left(\psi _{\mathrm {e} }(\mathbf {r} ,\mathbf {R} )\nabla _{R}^{2}\psi _{\mathrm {n} }(\mathbf {R} )+2\left[\nabla _{R}\psi _{\mathrm {e} }(\mathbf {r} ,\mathbf {R} )\right]\nabla _{R}\psi _{\mathrm {n} }(\mathbf {R} )+\psi _{\mathrm {n} }(\mathbf {R} )\nabla _{R}^{2}\psi _{\mathrm {e} }(\mathbf {r} ,\mathbf {R} )\right)+}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89ffcd328734f831bafdb86ae13f429cd577208f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:82.962ex; height:6.843ex;" alt="{\displaystyle -\sum _{i}{\frac {\hbar ^{2}}{2M_{i}}}\left(\psi _{\mathrm {e} }(\mathbf {r} ,\mathbf {R} )\nabla _{R}^{2}\psi _{\mathrm {n} }(\mathbf {R} )+2\left[\nabla _{R}\psi _{\mathrm {e} }(\mathbf {r} ,\mathbf {R} )\right]\nabla _{R}\psi _{\mathrm {n} }(\mathbf {R} )+\psi _{\mathrm {n} }(\mathbf {R} )\nabla _{R}^{2}\psi _{\mathrm {e} }(\mathbf {r} ,\mathbf {R} )\right)+}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +\left[T_{\mathrm {e} }+V_{\mathrm {ne} }(\mathbf {R} ,\mathbf {r} )+V_{\mathrm {ee} }(\mathbf {r} )+V_{\mathrm {nn} }(\mathbf {R} )\right]\psi _{\mathrm {e} }(\mathbf {r} ,\mathbf {R} )\psi _{\mathrm {n} }(\mathbf {R} )=E(\mathbf {R} )\psi _{\mathrm {e} }(\mathbf {r} ,\mathbf {R} )\psi _{\mathrm {n} }(\mathbf {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mrow> <mo>[</mo> <mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +\left[T_{\mathrm {e} }+V_{\mathrm {ne} }(\mathbf {R} ,\mathbf {r} )+V_{\mathrm {ee} }(\mathbf {r} )+V_{\mathrm {nn} }(\mathbf {R} )\right]\psi _{\mathrm {e} }(\mathbf {r} ,\mathbf {R} )\psi _{\mathrm {n} }(\mathbf {R} )=E(\mathbf {R} )\psi _{\mathrm {e} }(\mathbf {r} ,\mathbf {R} )\psi _{\mathrm {n} }(\mathbf {R} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ca3c9a16306a86ac00cd73be598b132d0f3b6aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:75.562ex; height:2.843ex;" alt="{\displaystyle +\left[T_{\mathrm {e} }+V_{\mathrm {ne} }(\mathbf {R} ,\mathbf {r} )+V_{\mathrm {ee} }(\mathbf {r} )+V_{\mathrm {nn} }(\mathbf {R} )\right]\psi _{\mathrm {e} }(\mathbf {r} ,\mathbf {R} )\psi _{\mathrm {n} }(\mathbf {R} )=E(\mathbf {R} )\psi _{\mathrm {e} }(\mathbf {r} ,\mathbf {R} )\psi _{\mathrm {n} }(\mathbf {R} )}"></dd></dl> che, trascurando per l'approssimazione adiabatica il termine: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\left[\nabla _{R}\psi _{\mathrm {e} }(\mathbf {r} ,\mathbf {R} )\right]\nabla _{R}\psi _{\mathrm {n} }(\mathbf {R} )+\psi _{\mathrm {n} }(\mathbf {R} )\nabla _{R}^{2}\psi _{\mathrm {e} }(\mathbf {r} ,\mathbf {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow> <mo>[</mo> <mrow> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <msubsup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\left[\nabla _{R}\psi _{\mathrm {e} }(\mathbf {r} ,\mathbf {R} )\right]\nabla _{R}\psi _{\mathrm {n} }(\mathbf {R} )+\psi _{\mathrm {n} }(\mathbf {R} )\nabla _{R}^{2}\psi _{\mathrm {e} }(\mathbf {r} ,\mathbf {R} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/802313832f8fd89f3dc299d9b4208fd68370a527" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:46.109ex; height:3.176ex;" alt="{\displaystyle 2\left[\nabla _{R}\psi _{\mathrm {e} }(\mathbf {r} ,\mathbf {R} )\right]\nabla _{R}\psi _{\mathrm {n} }(\mathbf {R} )+\psi _{\mathrm {n} }(\mathbf {R} )\nabla _{R}^{2}\psi _{\mathrm {e} }(\mathbf {r} ,\mathbf {R} )}"></dd></dl> diventa, inserendo la soluzione <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\mathrm {e} }(\mathbf {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\mathrm {e} }(\mathbf {R} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/325a9187a02cf17f38ab25f5375eb98573e50ed2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.49ex; height:2.843ex;" alt="{\displaystyle E_{\mathrm {e} }(\mathbf {R} )}"> dell'equazione elettronica: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[-\sum _{i}{\frac {\hbar ^{2}}{2M_{i}}}\nabla _{R}^{2}+E_{\mathrm {e} }(\mathbf {R} )+V_{\mathrm {nn} }(\mathbf {R} )\right]\psi _{\mathrm {n} }(\mathbf {R} )=E_{\mathrm {n} }\psi _{\mathrm {n} }(\mathbf {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msubsup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[-\sum _{i}{\frac {\hbar ^{2}}{2M_{i}}}\nabla _{R}^{2}+E_{\mathrm {e} }(\mathbf {R} )+V_{\mathrm {nn} }(\mathbf {R} )\right]\psi _{\mathrm {n} }(\mathbf {R} )=E_{\mathrm {n} }\psi _{\mathrm {n} }(\mathbf {R} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd9d24946969a24f4827c1eee728265335cda01c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:55.805ex; height:7.509ex;" alt="{\displaystyle \left[-\sum _{i}{\frac {\hbar ^{2}}{2M_{i}}}\nabla _{R}^{2}+E_{\mathrm {e} }(\mathbf {R} )+V_{\mathrm {nn} }(\mathbf {R} )\right]\psi _{\mathrm {n} }(\mathbf {R} )=E_{\mathrm {n} }\psi _{\mathrm {n} }(\mathbf {R} )}"></dd></dl> che è l'equazione del moto dei nuclei. Il potenziale che guida il moto dei nuclei: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\mathrm {ad} }(\mathbf {R} )=E_{\mathrm {e} }(\mathbf {R} )+V_{\mathrm {nn} }(\mathbf {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\mathrm {ad} }(\mathbf {R} )=E_{\mathrm {e} }(\mathbf {R} )+V_{\mathrm {nn} }(\mathbf {R} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09d66375b2e46b857cba0eca1e813f436db82e34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.793ex; height:2.843ex;" alt="{\displaystyle V_{\mathrm {ad} }(\mathbf {R} )=E_{\mathrm {e} }(\mathbf {R} )+V_{\mathrm {nn} }(\mathbf {R} )}"></dd></dl> è detto potenziale adiabatico o potenziale intermolecolare, e sta alla base della dinamica della molecola. Dall'espressione del potenziale adiabatico si evince che la dinamica dei nuclei è guidata dall'energia <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\mathrm {e} }(\mathbf {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\mathrm {e} }(\mathbf {R} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/325a9187a02cf17f38ab25f5375eb98573e50ed2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.49ex; height:2.843ex;" alt="{\displaystyle E_{\mathrm {e} }(\mathbf {R} )}"> fornita dall'equazione elettronica: questo termine è fondamentale, dal momento che rappresenta il "collante" che tiene uniti i nuclei degli atomi che compongono la molecola.<a href="#cite_note-7">[7]</a> Per le molecole biatomiche il potenziale adiabatico è un <a href="/wiki/Oscillatore_armonico_quantistico" title="Oscillatore armonico quantistico">potenziale armonico</a>, e può essere approssimato dal <a href="/wiki/Potenziale_di_Morse" title="Potenziale di Morse">potenziale di Morse</a>, che a differenza dell'oscillatore armonico quantistico include esplicitamente gli effetti della rottura del <a href="/wiki/Legame_chimico" title="Legame chimico">legame chimico</a>, come l'esistenza di stati non legati. <div class="mw-heading mw-heading2"><h2 id="Molecole_biatomiche">Molecole biatomiche</h2>[<a href="/w/index.php?title=Molecola&veaction=edit&section=2" title="Modifica la sezione Molecole biatomiche" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Molecola&action=edit&section=2" title="Edit section's source code: Molecole biatomiche">modifica wikitesto</a>]</div> <style data-mw-deduplicate="TemplateStyles:r130657691">body:not(.skin-minerva) .mw-parser-output .vedi-anche{font-size:95%}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r139142988"> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /> Lo stesso argomento in dettaglio: <a href="/wiki/Molecola_biatomica" title="Molecola biatomica">Molecola biatomica</a>.</div> </div> Le molecole biatomiche sono composte da due atomi, e si distinguono in molecole omonucleari, quando gli atomi sono dello stesso <a href="/wiki/Elemento_chimico" title="Elemento chimico">elemento chimico</a>, ed eteronucleari, quando invece gli atomi differiscono. <div class="mw-heading mw-heading3"><h3 id="Lo_ione_molecolare_H2+">Lo ione molecolare H2+</h3>[<a href="/w/index.php?title=Molecola&veaction=edit&section=3" title="Modifica la sezione Lo ione molecolare H2+" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Molecola&action=edit&section=3" title="Edit section's source code: Lo ione molecolare H2+">modifica wikitesto</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Wave_functions_binding.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/42/Wave_functions_binding.svg/200px-Wave_functions_binding.svg.png" decoding="async" width="200" height="113" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/42/Wave_functions_binding.svg/300px-Wave_functions_binding.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/42/Wave_functions_binding.svg/400px-Wave_functions_binding.svg.png 2x" data-file-width="320" data-file-height="180" /></a><figcaption>Orbitale molecolare di legame</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Wave_functions_anti-binding.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Wave_functions_anti-binding.svg/200px-Wave_functions_anti-binding.svg.png" decoding="async" width="200" height="159" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Wave_functions_anti-binding.svg/300px-Wave_functions_anti-binding.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/Wave_functions_anti-binding.svg/400px-Wave_functions_anti-binding.svg.png 2x" data-file-width="320" data-file-height="255" /></a><figcaption>Orbitale molecolare di antilegame</figcaption></figure> Le molecole diatomiche omonucleari sono composte da due atomi dello stesso elemento chimico; la più semplice di queste è H2+, per la quale l'equazione elettronica assume la forma:<a href="#cite_note-8">[8]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[{\frac {\hbar ^{2}}{2m_{e}}}\nabla _{r}^{2}-{\frac {ke^{2}}{|\mathbf {r} +\mathbf {R} /2|}}-{\frac {ke^{2}}{|\mathbf {r} -\mathbf {R} /2|}}+{\frac {ke^{2}}{R}}\right]\psi _{\mathrm {e} }(\mathbf {r} )=E_{\mathrm {e} }(\mathbf {r} )\psi _{\mathrm {e} }(\mathbf {r} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msubsup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>R</mi> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\frac {\hbar ^{2}}{2m_{e}}}\nabla _{r}^{2}-{\frac {ke^{2}}{|\mathbf {r} +\mathbf {R} /2|}}-{\frac {ke^{2}}{|\mathbf {r} -\mathbf {R} /2|}}+{\frac {ke^{2}}{R}}\right]\psi _{\mathrm {e} }(\mathbf {r} )=E_{\mathrm {e} }(\mathbf {r} )\psi _{\mathrm {e} }(\mathbf {r} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbdcc15a178e5f1487ae8efb1746b1a94d4898bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:63.838ex; height:6.509ex;" alt="{\displaystyle \left[{\frac {\hbar ^{2}}{2m_{e}}}\nabla _{r}^{2}-{\frac {ke^{2}}{|\mathbf {r} +\mathbf {R} /2|}}-{\frac {ke^{2}}{|\mathbf {r} -\mathbf {R} /2|}}+{\frac {ke^{2}}{R}}\right]\psi _{\mathrm {e} }(\mathbf {r} )=E_{\mathrm {e} }(\mathbf {r} )\psi _{\mathrm {e} }(\mathbf {r} )}"></dd></dl> dove <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1/4\pi \epsilon _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=1/4\pi \epsilon _{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03b35c611ef14695fa52c549cb6e966463d27d95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.127ex; height:2.843ex;" alt="{\displaystyle k=1/4\pi \epsilon _{0}}">, il secondo ed il terzo termine rappresentano l'attrazione Vne dell'elettrone nei confronti dei nuclei ed il quarto la repulsione dei due nuclei. I due protoni formano due buche di potenziale, e la funzione d'onda dell'elettrone è la combinazione lineare di due <a href="/wiki/Atomo_di_idrogeno" title="Atomo di idrogeno">funzioni d'onda idrogenoidi</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{1s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{1s}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6810c6baf4035a37e6086b6fee518f0e36a16fe5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.338ex; height:2.509ex;" alt="{\displaystyle \psi _{1s}}">:<a href="#cite_note-9">[9]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\mathrm {\pm } }(\mathbf {r} )={\frac {1}{\sqrt {2}}}[\psi _{1s}(\mathbf {r} +\mathbf {R} /2)\pm \psi _{1s}(\mathbf {r} -\mathbf {R} /2)]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mo stretchy="false">[</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo>±</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\mathrm {\pm } }(\mathbf {r} )={\frac {1}{\sqrt {2}}}[\psi _{1s}(\mathbf {r} +\mathbf {R} /2)\pm \psi _{1s}(\mathbf {r} -\mathbf {R} /2)]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf6c56809c63e4c6bfd05f1afb629af6a11f059" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:43.939ex; height:6.176ex;" alt="{\displaystyle \psi _{\mathrm {\pm } }(\mathbf {r} )={\frac {1}{\sqrt {2}}}[\psi _{1s}(\mathbf {r} +\mathbf {R} /2)\pm \psi _{1s}(\mathbf {r} -\mathbf {R} /2)]}"></dd></dl> La funzione d'onda <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{+}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6713c31818343696bbba6f140c4d008c692935c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.024ex; height:2.509ex;" alt="{\displaystyle \psi _{+}}"> costituisce l'<a href="/wiki/Orbitale_molecolare" title="Orbitale molecolare">orbitale molecolare</a> di legame, la funzione <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{-}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abe855238bc20489dc7735844c7039062b613092" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.024ex; height:2.509ex;" alt="{\displaystyle \psi _{-}}"> costituisce l'orbitale di antilegame.<a href="#cite_note-10">[10]</a> L'orbitale di legame ha energia minore dell'orbitale di antilegame. Le funzioni <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\mathrm {\pm } }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\mathrm {\pm } }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b208854aa53c5cdeffee40a12e87008a1acf6e63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.024ex; height:2.509ex;" alt="{\displaystyle \psi _{\mathrm {\pm } }}">, sebbene descrivano bene la distribuzione di probabilità dell'elettrone nello stato fondamentale, non sono soluzioni esatte dell'equazione elettronica. La funzione d'onda <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{+}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6713c31818343696bbba6f140c4d008c692935c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.024ex; height:2.509ex;" alt="{\displaystyle \psi _{+}}">, nello spazio tra i due nuclei, è maggiore delle singole funzioni d'onda idrogenoidi <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\mathrm {1s} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\mathrm {1s} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0785f3d6522b4a37d9f615e1821202889edadeab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.215ex; height:2.509ex;" alt="{\displaystyle \psi _{\mathrm {1s} }}">, ed è questo fatto che genera il <a href="/wiki/Legame_covalente" title="Legame covalente">legame covalente</a> tra i due nuclei. Si nota infatti che la densità di probabilità associata alla funzione d'onda: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi _{\mathrm {\pm } }|^{2}={\frac {1}{2}}[\psi _{1s}^{2}(\mathbf {r} +\mathbf {R} /2)+\psi _{1s}^{2}(\mathbf {r} -\mathbf {R} /2)\pm 2\psi _{1s}(\mathbf {r} +\mathbf {R} /2)\psi _{1s}(\mathbf {r} -\mathbf {R} /2)]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> </mrow> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">[</mo> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo>+</mo> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo>±</mo> <mn>2</mn> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi _{\mathrm {\pm } }|^{2}={\frac {1}{2}}[\psi _{1s}^{2}(\mathbf {r} +\mathbf {R} /2)+\psi _{1s}^{2}(\mathbf {r} -\mathbf {R} /2)\pm 2\psi _{1s}(\mathbf {r} +\mathbf {R} /2)\psi _{1s}(\mathbf {r} -\mathbf {R} /2)]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50a75eda255a84496516016b11905999d90085a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:72.279ex; height:5.176ex;" alt="{\displaystyle |\psi _{\mathrm {\pm } }|^{2}={\frac {1}{2}}[\psi _{1s}^{2}(\mathbf {r} +\mathbf {R} /2)+\psi _{1s}^{2}(\mathbf {r} -\mathbf {R} /2)\pm 2\psi _{1s}(\mathbf {r} +\mathbf {R} /2)\psi _{1s}(\mathbf {r} -\mathbf {R} /2)]}"></dd></dl> contiene un termine di interazione, il doppio prodotto, che rappresenta la sovrapposizione delle due funzioni d'onda: si tratta di una regione di carica negativa che unisce i due nuclei di carica opposta. Per quanto riguarda l'orbitale di antilegame <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{-}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abe855238bc20489dc7735844c7039062b613092" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.024ex; height:2.509ex;" alt="{\displaystyle \psi _{-}}">, esso si annulla a metà tra i due nuclei, dove genera una densità di probabilità minore di quella che avrebbe senza il termine di sovrapposizione. <div class="mw-heading mw-heading3"><h3 id="La_molecola_H2">La molecola H2</h3>[<a href="/w/index.php?title=Molecola&veaction=edit&section=4" title="Modifica la sezione La molecola H2" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Molecola&action=edit&section=4" title="Edit section's source code: La molecola H2">modifica wikitesto</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Dihydrogen-HOMO-phase-3D-balls.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Dihydrogen-HOMO-phase-3D-balls.svg/150px-Dihydrogen-HOMO-phase-3D-balls.svg.png" decoding="async" width="150" height="136" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Dihydrogen-HOMO-phase-3D-balls.svg/225px-Dihydrogen-HOMO-phase-3D-balls.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/19/Dihydrogen-HOMO-phase-3D-balls.svg/300px-Dihydrogen-HOMO-phase-3D-balls.svg.png 2x" data-file-width="1100" data-file-height="994" /></a><figcaption>Orbitale di legame di H2</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Dihydrogen-LUMO-phase-3D-balls.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Dihydrogen-LUMO-phase-3D-balls.png/150px-Dihydrogen-LUMO-phase-3D-balls.png" decoding="async" width="150" height="110" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Dihydrogen-LUMO-phase-3D-balls.png/225px-Dihydrogen-LUMO-phase-3D-balls.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Dihydrogen-LUMO-phase-3D-balls.png/300px-Dihydrogen-LUMO-phase-3D-balls.png 2x" data-file-width="1100" data-file-height="809" /></a><figcaption>Orbitale di antilegame di H2</figcaption></figure> Si consideri ora la molecola H2, la più semplice molecola neutra. Avendo due elettroni, la funzione d'onda elettronica di singoletto è data da:<a href="#cite_note-11">[11]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\mathrm {S} }(1,2)={\frac {1}{\sqrt {2}}}[\psi _{1s}(\mathbf {r_{1}} -\mathbf {R} /2)\psi _{1s}(\mathbf {r_{2}} +\mathbf {R} /2)+\psi _{1s}(\mathbf {r_{2}} -\mathbf {R} /2)\psi _{1s}(\mathbf {r_{1}} +\mathbf {R} /2)]\chi ^{A}(1,2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mo stretchy="false">[</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\mathrm {S} }(1,2)={\frac {1}{\sqrt {2}}}[\psi _{1s}(\mathbf {r_{1}} -\mathbf {R} /2)\psi _{1s}(\mathbf {r_{2}} +\mathbf {R} /2)+\psi _{1s}(\mathbf {r_{2}} -\mathbf {R} /2)\psi _{1s}(\mathbf {r_{1}} +\mathbf {R} /2)]\chi ^{A}(1,2)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/909792d0f57537b94dea0a27378c910077e1d41c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:85.466ex; height:6.176ex;" alt="{\displaystyle \psi _{\mathrm {S} }(1,2)={\frac {1}{\sqrt {2}}}[\psi _{1s}(\mathbf {r_{1}} -\mathbf {R} /2)\psi _{1s}(\mathbf {r_{2}} +\mathbf {R} /2)+\psi _{1s}(\mathbf {r_{2}} -\mathbf {R} /2)\psi _{1s}(\mathbf {r_{1}} +\mathbf {R} /2)]\chi ^{A}(1,2)}"></dd></dl> e rappresenta l'orbitale di legame, mentre quella di tripletto da:<a href="#cite_note-cinquecentodieci-12">[12]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\mathrm {T} }(1,2)={\frac {1}{\sqrt {2}}}[\psi _{1s}(\mathbf {r_{1}} -\mathbf {R} /2)\psi _{1s}(\mathbf {r_{2}} +\mathbf {R} /2)-\psi _{1s}(\mathbf {r_{2}} -\mathbf {R} /2)\psi _{1s}(\mathbf {r_{1}} +\mathbf {R} /2)]\chi ^{S}(1,2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mo stretchy="false">[</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\mathrm {T} }(1,2)={\frac {1}{\sqrt {2}}}[\psi _{1s}(\mathbf {r_{1}} -\mathbf {R} /2)\psi _{1s}(\mathbf {r_{2}} +\mathbf {R} /2)-\psi _{1s}(\mathbf {r_{2}} -\mathbf {R} /2)\psi _{1s}(\mathbf {r_{1}} +\mathbf {R} /2)]\chi ^{S}(1,2)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efa707e0f683775b3cc7ef364bdf94eb4370d291" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:85.566ex; height:6.176ex;" alt="{\displaystyle \psi _{\mathrm {T} }(1,2)={\frac {1}{\sqrt {2}}}[\psi _{1s}(\mathbf {r_{1}} -\mathbf {R} /2)\psi _{1s}(\mathbf {r_{2}} +\mathbf {R} /2)-\psi _{1s}(\mathbf {r_{2}} -\mathbf {R} /2)\psi _{1s}(\mathbf {r_{1}} +\mathbf {R} /2)]\chi ^{S}(1,2)}"></dd></dl> che rappresenta l'orbitale di antilegame, dove: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi ^{A}(1,2)={\frac {1}{\sqrt {2}}}\left(\left|+;-\right\rangle -\left|-;+\right\rangle \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mo>|</mo> <mrow> <mo>+</mo> <mo>;</mo> <mo>−</mo> </mrow> <mo>⟩</mo> </mrow> <mo>−</mo> <mrow> <mo>|</mo> <mrow> <mo>−</mo> <mo>;</mo> <mo>+</mo> </mrow> <mo>⟩</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi ^{A}(1,2)={\frac {1}{\sqrt {2}}}\left(\left|+;-\right\rangle -\left|-;+\right\rangle \right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a68f5b681584d4fcb63c76b2866e2d6484d70f34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:32.561ex; height:6.176ex;" alt="{\displaystyle \chi ^{A}(1,2)={\frac {1}{\sqrt {2}}}\left(\left|+;-\right\rangle -\left|-;+\right\rangle \right)}"></dd></dl> e <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi ^{S}(1,2)={\begin{cases}\left|+;+\right\rangle \\{\frac {1}{\sqrt {2}}}\left(\left|+;-\right\rangle +\left|-;+\right\rangle \right)\\\left|-;-\right\rangle \end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow> <mo>|</mo> <mrow> <mo>+</mo> <mo>;</mo> <mo>+</mo> </mrow> <mo>⟩</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mo>|</mo> <mrow> <mo>+</mo> <mo>;</mo> <mo>−</mo> </mrow> <mo>⟩</mo> </mrow> <mo>+</mo> <mrow> <mo>|</mo> <mrow> <mo>−</mo> <mo>;</mo> <mo>+</mo> </mrow> <mo>⟩</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>|</mo> <mrow> <mo>−</mo> <mo>;</mo> <mo>−</mo> </mrow> <mo>⟩</mo> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi ^{S}(1,2)={\begin{cases}\left|+;+\right\rangle \\{\frac {1}{\sqrt {2}}}\left(\left|+;-\right\rangle +\left|-;+\right\rangle \right)\\\left|-;-\right\rangle \end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92b2acffb08eeb2f9536cf1de0a56dbfa5a90838" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.197ex; margin-bottom: -0.307ex; width:34.299ex; height:10.176ex;" alt="{\displaystyle \chi ^{S}(1,2)={\begin{cases}\left|+;+\right\rangle \\{\frac {1}{\sqrt {2}}}\left(\left|+;-\right\rangle +\left|-;+\right\rangle \right)\\\left|-;-\right\rangle \end{cases}}}"></dd></dl> sono gli stati di <a href="/wiki/Spin" title="Spin">spin</a>, in cui + rappresenta lo spin-up, - lo spin-down. La densità di probabilità spaziale è:<a href="#cite_note-cinquecentodieci-12">[12]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\psi _{\mathrm {S,T} }|^{2}={\frac {1}{2}}[\psi _{1s}^{2}(\mathbf {r_{1}} -\mathbf {R} /2)\psi _{1s}^{2}(\mathbf {r_{2}} +\mathbf {R} /2)+\psi _{1s}^{2}(\mathbf {r_{2}} -\mathbf {R} /2)\psi _{1s}^{2}(\mathbf {r_{1}} +\mathbf {R} /2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mo>,</mo> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">[</mo> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo>+</mo> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi _{\mathrm {S,T} }|^{2}={\frac {1}{2}}[\psi _{1s}^{2}(\mathbf {r_{1}} -\mathbf {R} /2)\psi _{1s}^{2}(\mathbf {r_{2}} +\mathbf {R} /2)+\psi _{1s}^{2}(\mathbf {r_{2}} -\mathbf {R} /2)\psi _{1s}^{2}(\mathbf {r_{1}} +\mathbf {R} /2)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b74fe4b4fa3530e89ce0bdbb131ea6c09d0c9cce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:73.619ex; height:5.176ex;" alt="{\displaystyle |\psi _{\mathrm {S,T} }|^{2}={\frac {1}{2}}[\psi _{1s}^{2}(\mathbf {r_{1}} -\mathbf {R} /2)\psi _{1s}^{2}(\mathbf {r_{2}} +\mathbf {R} /2)+\psi _{1s}^{2}(\mathbf {r_{2}} -\mathbf {R} /2)\psi _{1s}^{2}(\mathbf {r_{1}} +\mathbf {R} /2)}"></dd></dl> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm 2\psi _{1s}(\mathbf {r_{1}} -\mathbf {R} /2)\psi _{1s}(\mathbf {r_{1}} +\mathbf {R} /2)\psi _{1s}(\mathbf {r_{2}} -\mathbf {R} /2)\psi _{1s}(\mathbf {r_{2}} +\mathbf {R} /2)]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mn>2</mn> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm 2\psi _{1s}(\mathbf {r_{1}} -\mathbf {R} /2)\psi _{1s}(\mathbf {r_{1}} +\mathbf {R} /2)\psi _{1s}(\mathbf {r_{2}} -\mathbf {R} /2)\psi _{1s}(\mathbf {r_{2}} +\mathbf {R} /2)]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acbf4078b38a04bc2233a417ca8382bb3bc4f732" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:62.001ex; height:2.843ex;" alt="{\displaystyle \pm 2\psi _{1s}(\mathbf {r_{1}} -\mathbf {R} /2)\psi _{1s}(\mathbf {r_{1}} +\mathbf {R} /2)\psi _{1s}(\mathbf {r_{2}} -\mathbf {R} /2)\psi _{1s}(\mathbf {r_{2}} +\mathbf {R} /2)]}"></dd></dl> Anche in questo caso il termine di interferenza rappresenta la sovrapposizione delle funzioni d'onda idrogenoidi nella regione tra i nuclei, e comporta un aumento di carica nel caso di singoletto (segno +), ed una diminuzione di carica nel tripletto (segno -). <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:H1F1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/H1F1.png/170px-H1F1.png" decoding="async" width="170" height="161" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/H1F1.png/255px-H1F1.png 1.5x, //upload.wikimedia.org/wikipedia/commons/e/e6/H1F1.png 2x" data-file-width="328" data-file-height="310" /></a><figcaption>La molecola biatomica eteronucleare dell'<a href="/wiki/Acido_fluoridrico" title="Acido fluoridrico">acido fluoridrico</a></figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Molecole_eteronucleari">Molecole eteronucleari</h3>[<a href="/w/index.php?title=Molecola&veaction=edit&section=5" title="Modifica la sezione Molecole eteronucleari" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Molecola&action=edit&section=5" title="Edit section's source code: Molecole eteronucleari">modifica wikitesto</a>]</div> Nelle molecole eteronucleari la simmetria che caratterizzava le molecole omonucleari viene a mancare, e gli orbitali non sono una pura combinazione simmetrica e antisimmetrica degli orbitali atomici. In tali molecole gli orbitali possono essere approssimati con gli autostati di una matrice quadrata di dimensione 2:<a href="#cite_note-settanta-13">[13]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{Bmatrix}E_{L}&-\Delta \\-\Delta &E_{L}\end{Bmatrix}}\equiv {\begin{Bmatrix}\langle L|H_{1}|L\rangle &\langle L|H_{1}|R\rangle \\\langle R|H_{1}|L\rangle &\langle R|H_{1}|R\rangle \end{Bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mtd> <mtd> <mo>−</mo> <mi mathvariant="normal">Δ</mi> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi mathvariant="normal">Δ</mi> </mtd> <mtd> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>}</mo> </mrow> </mrow> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo fence="false" stretchy="false">⟨</mo> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>L</mi> <mo fence="false" stretchy="false">⟩</mo> </mtd> <mtd> <mo fence="false" stretchy="false">⟨</mo> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>R</mi> <mo fence="false" stretchy="false">⟩</mo> </mtd> </mtr> <mtr> <mtd> <mo fence="false" stretchy="false">⟨</mo> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>L</mi> <mo fence="false" stretchy="false">⟩</mo> </mtd> <mtd> <mo fence="false" stretchy="false">⟨</mo> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>R</mi> <mo fence="false" stretchy="false">⟩</mo> </mtd> </mtr> </mtable> <mo>}</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{Bmatrix}E_{L}&-\Delta \\-\Delta &E_{L}\end{Bmatrix}}\equiv {\begin{Bmatrix}\langle L|H_{1}|L\rangle &\langle L|H_{1}|R\rangle \\\langle R|H_{1}|L\rangle &\langle R|H_{1}|R\rangle \end{Bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afa92651b7541431a39abd1d0d559ef5305ea30c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.759ex; height:6.176ex;" alt="{\displaystyle {\begin{Bmatrix}E_{L}&-\Delta \\-\Delta &E_{L}\end{Bmatrix}}\equiv {\begin{Bmatrix}\langle L|H_{1}|L\rangle &\langle L|H_{1}|R\rangle \\\langle R|H_{1}|L\rangle &\langle R|H_{1}|R\rangle \end{Bmatrix}}}"></dd></dl> dove: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{1}=T_{e}+V_{eff}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mi>f</mi> <mi>f</mi> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{1}=T_{e}+V_{eff}\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ddb65361ea48768614236ca5252769fd14b7898" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.023ex; height:2.843ex;" alt="{\displaystyle H_{1}=T_{e}+V_{eff}\ }"></dd></dl> è l'effettiva <a href="/wiki/Operatore_hamiltoniano" title="Operatore hamiltoniano">hamiltoniana</a> di singolo elettrone mentre gli stati <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |L\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>L</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |L\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21a3af6dfba8e668f838ce8f273ae461119d9e3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.134ex; height:2.843ex;" alt="{\displaystyle |L\rangle }"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |R\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>R</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |R\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a7007650f556eb1933983a2c884331a27d792d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.315ex; height:2.843ex;" alt="{\displaystyle |R\rangle }"> sono gli orbitali corrispondenti rispettivamente all'atomo sinistro e destro. Gli autovalori associati alla matrice sono: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {E} _{a,b}={\frac {E_{L}+E_{R}}{2}}\mp {\sqrt {\left({\frac {E_{L}-E_{R}}{2}}\right)^{2}+\Delta ^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>∓</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {E} _{a,b}={\frac {E_{L}+E_{R}}{2}}\mp {\sqrt {\left({\frac {E_{L}-E_{R}}{2}}\right)^{2}+\Delta ^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2f5336d05d1a1f039d80f3859aba64add8efdbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:42.292ex; height:7.676ex;" alt="{\displaystyle \mathrm {E} _{a,b}={\frac {E_{L}+E_{R}}{2}}\mp {\sqrt {\left({\frac {E_{L}-E_{R}}{2}}\right)^{2}+\Delta ^{2}}}}"></dd></dl> Gli orbitali di legame <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |b\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>b</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |b\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/610fad1026bcd71efddcdecfc69cea76715259a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.549ex; height:2.843ex;" alt="{\displaystyle |b\rangle }"> e antilegame <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |a\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |a\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/022d80b506eff10c19feb8b5b70bc8c72baa1fa1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.781ex; height:2.843ex;" alt="{\displaystyle |a\rangle }"> sono dati dagli autostati: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |b,a\rangle ={\sqrt {{\frac {1}{2}}\left(1\mp {\frac {u}{\sqrt {1+u^{2}}}}\right)}}|L\rangle \pm {\sqrt {{\frac {1}{2}}\left(1\pm {\frac {u}{\sqrt {1+u^{2}}}}\right)}}|R\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>b</mi> <mo>,</mo> <mi>a</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>∓</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>u</mi> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>L</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>u</mi> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>R</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |b,a\rangle ={\sqrt {{\frac {1}{2}}\left(1\mp {\frac {u}{\sqrt {1+u^{2}}}}\right)}}|L\rangle \pm {\sqrt {{\frac {1}{2}}\left(1\pm {\frac {u}{\sqrt {1+u^{2}}}}\right)}}|R\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9031ddab83dc12bfb607dfbc546a5c2cd5c016a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:61.342ex; height:8.176ex;" alt="{\displaystyle |b,a\rangle ={\sqrt {{\frac {1}{2}}\left(1\mp {\frac {u}{\sqrt {1+u^{2}}}}\right)}}|L\rangle \pm {\sqrt {{\frac {1}{2}}\left(1\pm {\frac {u}{\sqrt {1+u^{2}}}}\right)}}|R\rangle }"></dd></dl> con: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u={\frac {E_{L}-E_{R}}{2\Delta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mrow> <mrow> <mn>2</mn> <mi mathvariant="normal">Δ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u={\frac {E_{L}-E_{R}}{2\Delta }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3d14fe669e31fe2b1bc4b812e7551abad594af5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.366ex; height:5.509ex;" alt="{\displaystyle u={\frac {E_{L}-E_{R}}{2\Delta }}}"></dd></dl> per <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41c2a269988ef0c55e0449b74950a4976e35a067" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle u=0}"> si ottiene la molecola omonucleare, ed il termine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\Delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi mathvariant="normal">Δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\Delta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d38103fc4f6937e0d6a75d77b143addca6375ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.098ex; height:2.176ex;" alt="{\displaystyle 2\Delta }"> rappresenta lo splitting tra l'orbitale di legame e di antilegame di una molecola omonucleare, ovvero lo splitting tra le combinazioni simmetriche ed antisimmetriche.<a href="#cite_note-settanta-13">[13]</a> Al crescere di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}"> gli autostati di legame e di antilegame assomigliano sempre più agli orbitali <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |L\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>L</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |L\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21a3af6dfba8e668f838ce8f273ae461119d9e3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.134ex; height:2.843ex;" alt="{\displaystyle |L\rangle }"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |R\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>R</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |R\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a7007650f556eb1933983a2c884331a27d792d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.315ex; height:2.843ex;" alt="{\displaystyle |R\rangle }"> dei singoli atomi, e lo stesso avviene per i rispettivi autovalori dell'energia.<a href="#cite_note-14">[14]</a> Quando la differenza <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{L}-E_{R}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{L}-E_{R}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5192db8a11d283c0554db248db082496ecc9f601" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.102ex; height:2.509ex;" alt="{\displaystyle E_{L}-E_{R}}"> è tale da comportare un trasferimento completo di carica tra i due atomi, il legame si dice <a href="/wiki/Legame_ionico" title="Legame ionico">ionico</a>. <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:H2o.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/H2o.png/150px-H2o.png" decoding="async" width="150" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/H2o.png/225px-H2o.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9d/H2o.png/300px-H2o.png 2x" data-file-width="386" data-file-height="310" /></a><figcaption>La molecola dell'<a href="/wiki/Acqua" title="Acqua">acqua</a></figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Ch4-hybridisation.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Ch4-hybridisation.png/150px-Ch4-hybridisation.png" decoding="async" width="150" height="138" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/1/13/Ch4-hybridisation.png 1.5x" data-file-width="172" data-file-height="158" /></a><figcaption>Gli orbitali ibridi nella molecola di <a href="/wiki/Metano" title="Metano">metano</a></figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Methaan.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Methaan.png/150px-Methaan.png" decoding="async" width="150" height="196" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Methaan.png/225px-Methaan.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Methaan.png/300px-Methaan.png 2x" data-file-width="377" data-file-height="492" /></a><figcaption>La struttura della molecola di metano</figcaption></figure> <div class="mw-heading mw-heading2"><h2 id="Molecole_poliatomiche">Molecole poliatomiche</h2>[<a href="/w/index.php?title=Molecola&veaction=edit&section=6" title="Modifica la sezione Molecole poliatomiche" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Molecola&action=edit&section=6" title="Edit section's source code: Molecole poliatomiche">modifica wikitesto</a>]</div> Le molecole poliatomiche possiedono più di due atomi, che nella maggior parte dei casi sono diversi fra loro. La loro struttura è estremamente diversificata poiché le possibili combinazioni tra gli orbitali atomici che formano gli orbitali molecolari sono estremamente numerose. Oltre al legame che caratterizza le molecole biatomiche, nelle molecole poliatomiche gli orbitali atomici s e p si possono combinare fra loro per formare orbitali detti ibridi. Si riportano di seguito due esempi di molecole poliatomiche, l'<a href="/wiki/Acqua" title="Acqua">acqua</a> ed il <a href="/wiki/Metano" title="Metano">metano</a>: <div class="mw-heading mw-heading3"><h3 id="La_molecola_H2O">La molecola H2O</h3>[<a href="/w/index.php?title=Molecola&veaction=edit&section=7" title="Modifica la sezione La molecola H2O" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Molecola&action=edit&section=7" title="Edit section's source code: La molecola H2O">modifica wikitesto</a>]</div> Una delle più semplici molecole poliatomiche è quella dell'<a href="/wiki/Acqua" title="Acqua">acqua</a>, in cui l'<a href="/wiki/Ossigeno" title="Ossigeno">ossigeno</a> ha un orbitale p caratterizzato da una tripla degenerazione sui tre assi cartesiani, che genera due possibili configurazioni elettroniche: la prima è il caso in cui i 4 elettroni riempiono completamente due lobi dell'orbitale, lasciando il terzo vuoto, mentre la seconda è il caso in cui si abbiano due elettroni su un lobo, ed uno su ognuno dei restanti due. Tale orbitale può essere quindi scritto come 2pxpypz2, in cui si è supposto che il lobo diretto lungo l'asse z contenga due elettroni, e questo rende possibile la formazione di due legami covalenti, in cui ai lobi x e y si legano i due atomi di <a href="/wiki/Idrogeno" title="Idrogeno">idrogeno</a>.<a href="#cite_note-15">[15]</a> <div class="mw-heading mw-heading3"><h3 id="La_molecola_CH4">La molecola CH4</h3>[<a href="/w/index.php?title=Molecola&veaction=edit&section=8" title="Modifica la sezione La molecola CH4" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Molecola&action=edit&section=8" title="Edit section's source code: La molecola CH4">modifica wikitesto</a>]</div> Il <a href="/wiki/Metano" title="Metano">metano</a> è una molecola con un orbitale ibrido. Il carbonio ha configurazione elettronica 1s22s22p2, e l'orbitale p e nel suo stato fondamentale può quindi legarsi con solo due atomi di idrogeno. La molecola di metano esiste, tuttavia, dal momento che un elettrone dell'orbitale 2s2 viene promosso all'orbitale p, sicché la configurazione elettronica diventa 1s22s2pxpypz, generando quattro elettroni disaccoppiati che possono legarsi ad altrettanti atomi di idrogeno. I quattro orbitali molecolari ibridi sono quindi una combinazione lineare degli stati <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\mathrm {2s} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\mathrm {2s} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d162357bb5722d69c43635dc06f61439678d391a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.215ex; height:2.509ex;" alt="{\displaystyle \psi _{\mathrm {2s} }}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\mathrm {2px} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">x</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\mathrm {2px} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04c83f99cdf3ed021de11c99431f3e0aea6a5809" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.349ex; height:2.843ex;" alt="{\displaystyle \psi _{\mathrm {2px} }}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\mathrm {2py} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">y</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\mathrm {2py} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1140c1843603632da80ddb5d3d6e6a27da8f92da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.349ex; height:2.843ex;" alt="{\displaystyle \psi _{\mathrm {2py} }}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\mathrm {2pz} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">z</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\mathrm {2pz} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb42174d3981cd2b5f44a5fbd6bcd3c4e16eb0b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.211ex; height:2.843ex;" alt="{\displaystyle \psi _{\mathrm {2pz} }}"> della forma:<a href="#cite_note-16">[16]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{i}={\frac {1}{2}}(\psi _{\mathrm {2s} }\pm \psi _{\mathrm {2px} }\pm \psi _{\mathrm {2py} }\pm \psi _{\mathrm {2pz} })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mo>±</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">x</mi> </mrow> </mrow> </msub> <mo>±</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">y</mi> </mrow> </mrow> </msub> <mo>±</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">z</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{i}={\frac {1}{2}}(\psi _{\mathrm {2s} }\pm \psi _{\mathrm {2px} }\pm \psi _{\mathrm {2py} }\pm \psi _{\mathrm {2pz} })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c36a45e53a898709a9fcab6f0a162846c4d3f6e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:33.866ex; height:5.176ex;" alt="{\displaystyle \psi _{i}={\frac {1}{2}}(\psi _{\mathrm {2s} }\pm \psi _{\mathrm {2px} }\pm \psi _{\mathrm {2py} }\pm \psi _{\mathrm {2pz} })}"></dd></dl> e formano un <a href="/wiki/Tetraedro" title="Tetraedro">tetraedro</a> con l'atomo di carbonio al centro. <div class="mw-heading mw-heading2"><h2 id="Orbitali_e_legami_molecolari">Orbitali e legami molecolari</h2>[<a href="/w/index.php?title=Molecola&veaction=edit&section=9" title="Modifica la sezione Orbitali e legami molecolari" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Molecola&action=edit&section=9" title="Edit section's source code: Orbitali e legami molecolari">modifica wikitesto</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r130657691"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r139142988"> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /> Lo stesso argomento in dettaglio: <a href="/wiki/Orbitale_molecolare" title="Orbitale molecolare">Orbitale molecolare</a>.</div> </div> L'orbitale molecolare caratterizza la configurazione elettronica di una molecola, definendo la distribuzione spaziale e l'energia degli elettroni, ed è stato introdotto da <a href="/wiki/Friedrich_Hund" title="Friedrich Hund">Friedrich Hund</a><a href="#cite_note-17">[17]</a><a href="#cite_note-18">[18]</a> e <a href="/wiki/Robert_S._Mulliken" class="mw-redirect" title="Robert S. Mulliken">Robert S. Mulliken</a><a href="#cite_note-19">[19]</a><a href="#cite_note-20">[20]</a> nel 1927 e 1928.<a href="#cite_note-21">[21]</a><a href="#cite_note-22">[22]</a> Un orbitale molecolare è rappresentato da una funzione d'onda il cui quadrato descrive la distribuzione di probabilità relativa alla posizione dell'elettrone. Tale funzione d'onda si ottiene dall'equazione d'onda che descrive l'intera molecola, che in generale non è di facile soluzione: questa problematica viene risolta mediante un'approssimazione che consiste nello scrivere l'orbitale molecolare come <a href="/wiki/Combinazione_lineare" title="Combinazione lineare">combinazione lineare</a> degli orbitali atomici dei singoli atomi. Tale approssimazione è descritta dalla <a href="/wiki/Teoria_degli_orbitali_molecolari" title="Teoria degli orbitali molecolari">teoria degli orbitali molecolari</a>. L'<a href="/wiki/Ordine_di_legame" title="Ordine di legame">ordine di legame</a> è inoltre la semidifferenza tra il numero di elettroni leganti e il numero di elettroni antileganti. L'ordine di legame è un indice della forza del legame stesso e viene utilizzato estensivamente anche nella teoria del <a href="/wiki/Legame_di_valenza" title="Legame di valenza">legame di valenza</a>. <div class="mw-heading mw-heading3"><h3 id="Teoria_degli_orbitali_molecolari">Teoria degli orbitali molecolari</h3>[<a href="/w/index.php?title=Molecola&veaction=edit&section=10" title="Modifica la sezione Teoria degli orbitali molecolari" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Molecola&action=edit&section=10" title="Edit section's source code: Teoria degli orbitali molecolari">modifica wikitesto</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r130657691"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r139142988"> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /> Lo stesso argomento in dettaglio: <a href="/wiki/Teoria_degli_orbitali_molecolari" title="Teoria degli orbitali molecolari">Teoria degli orbitali molecolari</a> e <a href="/wiki/Combinazione_lineare_di_orbitali_atomici" title="Combinazione lineare di orbitali atomici">Combinazione lineare di orbitali atomici</a>.</div> </div> La teoria degli orbitali molecolari è una tecnica per determinare la struttura molecolare in cui si pone che gli elettroni non siano assegnati a particolari legami chimici, ma siano trattati come oggetti che si muovono sotto l'influenza dei nuclei all'interno dell'intera molecola.<a href="#cite_note-23">[23]</a> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:He2_antibonding_orbital.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/He2_antibonding_orbital.png/200px-He2_antibonding_orbital.png" decoding="async" width="200" height="174" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/3/3e/He2_antibonding_orbital.png 1.5x" data-file-width="294" data-file-height="256" /></a><figcaption>Combinazione degli orbitali atomici 1s nella molecola biatomica omonucleare E2. In alto vi è la combinazioni antisimmetrica, che costituisce l'orbitale antilegante, in basso quella simmetrica, meno energetica, che costituisce l'orbitale legante.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:O2_orbitali_molecolari.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/54/O2_orbitali_molecolari.jpg/200px-O2_orbitali_molecolari.jpg" decoding="async" width="200" height="278" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/54/O2_orbitali_molecolari.jpg/300px-O2_orbitali_molecolari.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/5/54/O2_orbitali_molecolari.jpg 2x" data-file-width="340" data-file-height="473" /></a><figcaption>Combinazione degli orbitali atomici 2p e 2s nella molecola biatomica omonucleare O2. In alto vi sono le combinazioni degli orbitali atomici 2p, in basso quelle degli orbitali 2s.</figcaption></figure> La funzione d'onda totale degli elettroni <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"> è scritta come <a href="/wiki/Combinazione_lineare_di_orbitali_atomici" title="Combinazione lineare di orbitali atomici">combinazione lineare</a>:<a href="#cite_note-24">[24]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{j}=\sum _{i=1}^{n}c_{ij}\chi _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{j}=\sum _{i=1}^{n}c_{ij}\chi _{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3b657c317106d3a0fac9c8e6211f144d89775ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.002ex; height:6.843ex;" alt="{\displaystyle \psi _{j}=\sum _{i=1}^{n}c_{ij}\chi _{i}}"></dd></dl> dove <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi _{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88a2f6b3480849cdeaf9ca8d5c628ac3a3eb2ad9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.255ex; height:2.009ex;" alt="{\displaystyle \chi _{i}}"> sono gli orbitali atomici, e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{ij}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a106b8753c0948250bbc2e03df3207799beaedb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.484ex; height:2.343ex;" alt="{\displaystyle c_{ij}}"> i coefficienti della <a href="/wiki/Sommatoria" title="Sommatoria">sommatoria</a>, ricavati risolvendo l'<a href="/wiki/Equazione_di_Schr%C3%B6dinger" title="Equazione di Schrödinger">equazione di Schrödinger</a> per <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{j}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/462f216ac0e553f203e2fb98fd833022fd662c0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.423ex; height:2.843ex;" alt="{\displaystyle \psi _{j}}"> ed applicando il <a href="/wiki/Principio_variazionale" title="Principio variazionale">principio variazionale</a>. Le proprietà principali degli orbitali molecolari così definiti sono: <ul><li>Il numero degli orbitali molecolari è pari al numero di orbitali atomici contenuti nella combinazione lineare dalla quale sono costituiti, poiché gli stati stazionari non si creano né si distruggono.<a href="#cite_note-25">[25]</a></li> <li>Se la molecola possiede simmetrie, gli orbitali atomici degeneri, caratterizzati dalla stessa energia, sono raggruppati in combinazioni lineari che appartengono alla <a href="/wiki/Rappresentazione_dei_gruppi" title="Rappresentazione dei gruppi">rappresentazione</a> del <a href="/wiki/Gruppo_(matematica)" title="Gruppo (matematica)">gruppo</a> di <a href="/wiki/Simmetria_(matematica)" title="Simmetria (matematica)">simmetria</a>.</li> <li>Il numero di orbitali molecolari appartenenti alla rappresentazione di un gruppo è pari al numero di orbitali atomici appartenenti a tale rappresentazione.</li> <li>All'interno di una particolare rappresentazione, gli orbitali atomici si mischiano maggiormente tanto più i loro livelli di energia atomici sono vicini.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Rappresentazione_degli_orbitali_molecolari">Rappresentazione degli orbitali molecolari</h3>[<a href="/w/index.php?title=Molecola&veaction=edit&section=11" title="Modifica la sezione Rappresentazione degli orbitali molecolari" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Molecola&action=edit&section=11" title="Edit section's source code: Rappresentazione degli orbitali molecolari">modifica wikitesto</a>]</div> La nomenclatura degli orbitali molecolari ricalca quella degli orbitali atomici: quando un orbitale ha simmetria cilindrica rispetto alla congiungente dei due nuclei, detta direzione di legame, viene indicato con la <a href="/wiki/Alfabeto_greco" title="Alfabeto greco">lettera greca</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }">; quando si trova da parti opposte rispetto alla direzione di legame viene indicato con <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }">. Accanto alla lettera si scrive un indice che indica da quale tipologia di legame atomico è formato l'orbitale molecolare.<a href="#cite_note-26">[26]</a> Vi è inoltre una terza tipologia di legame, denotato con <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5321cfa797202b3e1f8620663ff43c4660ea03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }">, ottenuto dalla sovrapposizione di quattro lobi di due orbitali atomici. Esistono in questo caso due <a href="/wiki/Piano_nodale" title="Piano nodale">piani nodali</a> siti fra i due nuclei che contraggono tale legame. Il legame δ è riscontrato nel <a href="/wiki/Legame_quadruplo" title="Legame quadruplo">legame quadruplo</a>, legame multiplo importante in <a href="/wiki/Chimica_inorganica" title="Chimica inorganica">chimica inorganica</a> e che caratterizza <a href="/wiki/Complesso_(chimica)" title="Complesso (chimica)">complessi</a> quale [Re2Cl10]4- o altri tipi di <a href="/wiki/Cluster_(chimica)" title="Cluster (chimica)">cluster</a>. L'orbitale di antilegame si denota inoltre con un asterisco, ad esempio la molecola H2 possiede un orbitale di legame <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{1s}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>s</mi> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{1s}\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc7719c7cfe67c9360c3d52f2f6c2998da95b7f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.733ex; height:2.009ex;" alt="{\displaystyle \sigma _{1s}\ }"> ed un orbitale di antilegame <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{1s}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{1s}^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f193ce701db86d1f9eb721bac8313dfb9ec6b5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.153ex; height:2.843ex;" alt="{\displaystyle \sigma _{1s}^{*}}">. <ul><li>Nelle molecole biatomiche omonucleari gli elettroni riempiono gli orbitali con lo stesso schema con cui avviene il riempimento degli orbitali atomici, con l'uinica eccezione che tra gli orbitali derivanti dagli orbitali atomici 2p, gli orbitali <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }">, hanno energia minore degli orbitali <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"> a causa del fatto che la repulsione coulombiana degli orbitali derivati dagli orbitali atomici 1s e 2s aumenta l'energia degli stati <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }">. Questo è dovuto al fatto che gli elettroni dei due legami <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"> sono situati nella regione tra i due nuclei, e pertanto si respingono; nelle molecole più pesanti dell'ossigeno gli orbitali <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{2s}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>s</mi> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{2s}\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e772cc6cd63bc15dba28ad8d998c7168cffb036" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.731ex; height:2.009ex;" alt="{\displaystyle \pi _{2s}\ }"> hanno energia minore e sono situati in prossimità dei nuclei, pertanto il naturale ordinamento energetico è ristabilito. La combinazione lineare delle funzioni d'onda che forma l'orbitale molecolare è rappresentata a lato, dove sono schematizzate la molecola He2 e la molecola O2, la quale ha configurazione elettronica: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{1s}^{2}\sigma _{1s}^{*2}\sigma _{2s}^{2}\sigma _{2s}^{*2}\sigma _{2p_{x}}^{2}\sigma _{2p_{x}}^{*}\pi _{2p_{y}}^{2}\pi _{2p_{z}}^{2}\pi _{2p_{y}}^{*1}\pi _{2p_{z}}^{*1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <msubsup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> <mn>1</mn> </mrow> </msubsup> <msubsup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{1s}^{2}\sigma _{1s}^{*2}\sigma _{2s}^{2}\sigma _{2s}^{*2}\sigma _{2p_{x}}^{2}\sigma _{2p_{x}}^{*}\pi _{2p_{y}}^{2}\pi _{2p_{z}}^{2}\pi _{2p_{y}}^{*1}\pi _{2p_{z}}^{*1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dcfcf98306025ef61cd38d73f7069083653b758" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:37.049ex; height:3.843ex;" alt="{\displaystyle \sigma _{1s}^{2}\sigma _{1s}^{*2}\sigma _{2s}^{2}\sigma _{2s}^{*2}\sigma _{2p_{x}}^{2}\sigma _{2p_{x}}^{*}\pi _{2p_{y}}^{2}\pi _{2p_{z}}^{2}\pi _{2p_{y}}^{*1}\pi _{2p_{z}}^{*1}}">.<a href="#cite_note-27">[27]</a></li> <li>Nel caso di molecole biatomiche eteronucleari, se il <a href="/wiki/Numero_atomico" title="Numero atomico">numero atomico</a> dei due atomi differisce di poco il procedimento che forma gli orbitali è lo stesso delle molecole omonucleari. Vi è tuttavia una differenza di <a href="/wiki/Elettronegativit%C3%A0" title="Elettronegatività">elettronegatività</a> tra i due atomi, e ciò implica la presenza di un <a href="/wiki/Dipolo_elettrico" title="Dipolo elettrico">dipolo elettrico</a> tra di essi dovuto al fatto che gli elettroni si distribuiscono nelle vicinanze dell'atomo più elettronegativo:<a href="#cite_note-28">[28]</a> il legame che si viene a formare prende il nome di covalente polare. <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:MOdiagramCO.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/MOdiagramCO.png/350px-MOdiagramCO.png" decoding="async" width="350" height="233" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/MOdiagramCO.png/525px-MOdiagramCO.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c6/MOdiagramCO.png/700px-MOdiagramCO.png 2x" data-file-width="822" data-file-height="548" /></a><figcaption>La molecola del <a href="/wiki/Monossido_di_carbonio" title="Monossido di carbonio">monossido di carbonio</a> CO</figcaption></figure></li></ul> Tale legame viene rappresentato come in figura a lato, e si può notare che gli elettroni di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{2p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{2p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cb532261bd2df5a65235b29f7959ca2da242941" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.209ex; height:2.343ex;" alt="{\displaystyle \sigma _{2p}}"> hanno energia maggiore, e costituiscono un orbitale detto HOMO (Highest Occupied Molecular Orbital), mentre gli elettroni di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{2p_{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{2p_{x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52b1e258caed1da9782cd278d89b9487caca8188" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.136ex; height:2.343ex;" alt="{\displaystyle \sigma _{2p_{x}}}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{2p_{z}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{2p_{z}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40f28c48eb23d8908a824b73053b1525f490d8bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.997ex; height:2.343ex;" alt="{\displaystyle \sigma _{2p_{z}}}"> costituiscono gli orbitali vuoti a minore energia detti LUMO (Lowest Unoccupied Molecular Orbital). L'orbitale LUMO è il centro in cui la molecola può subire un attacco nucleofilo di una <a href="/wiki/Base_di_Lewis" title="Base di Lewis">base di Lewis</a>, e si tratta quindi del centro di acidità di Lewis. Viceversa, HOMO è il centro di basicità di Lewis della molecola, e può subire un attacco elettrofilo. Se la differenza di elettronegatività è maggiore di un valore convenzionale fissato a 1,9 vi è un trasferimento completo di carica tra i due atomi, cioè la nuvola elettronica può considerarsi come spostata completamente sull'elemento più elettronegativo. Tale legame prende il nome di <a href="/wiki/Legame_ionico" title="Legame ionico">legame ionico</a>. Se il numero atomico dei due atomi differisce di molto accade che gli orbitali molecolari si formino tra orbitali atomici con energia simile, invece che dello stesso tipo.<a href="#cite_note-29">[29]</a> <ul><li>All'aumentare del numero di atomi coinvolti diventa complessa la caratterizzazione degli orbitali, a nell'ambito della teoria degli orbitali molecolari sono stati sviluppati diversi metodi di calcolo degli orbitali, tra i quali vi sono il <a href="/wiki/Metodo_di_H%C3%BCckel" title="Metodo di Hückel">Metodo di Hückel</a>, proposto da <a href="/wiki/Erich_H%C3%BCckel" title="Erich Hückel">Erich Hückel</a> nel <a href="/wiki/1930" title="1930">1930</a>, consiste in un semplice <a href="/wiki/LCAO" class="mw-redirect" title="LCAO">metodo LCAO</a> utilizzato per la determinazione delle energie degli <a href="/wiki/Orbitali_molecolari" class="mw-redirect" title="Orbitali molecolari">orbitali molecolari</a> di sistemi π rappresentati da <a href="/wiki/Idrocarburi" title="Idrocarburi">idrocarburi</a> con <a href="/wiki/Legame_chimico" title="Legame chimico">legami</a> coniugati, risultando applicabile a molecole quali ad esempio l'<a href="/wiki/Etilene" title="Etilene">etilene</a>, il <a href="/wiki/Benzene" title="Benzene">benzene</a> e il <a href="/wiki/Butadiene" title="Butadiene">butadiene</a>.<a href="#cite_note-30">[30]</a><a href="#cite_note-31">[31]</a> La nota <a href="/wiki/Regola_di_H%C3%BCckel" title="Regola di Hückel">regola di Hückel</a> trae origine da queste basi. Il <a href="/wiki/Metodo_di_H%C3%BCckel_esteso" title="Metodo di Hückel esteso">metodo di Hückel esteso</a>, sviluppato da <a href="/wiki/Roald_Hoffmann" title="Roald Hoffmann">Roald Hoffmann</a>, rappresenta invece la base delle <a href="/wiki/Regole_di_Woodward-Hoffmann" title="Regole di Woodward-Hoffmann">regole di Woodward-Hoffmann</a><a href="#cite_note-32">[32]</a> ed è un'estensione a tutti gli orbitali di <a href="/wiki/Valenza_(chimica)" title="Valenza (chimica)">valenza</a>. Negli anni successivi il metodo fu reso applicabile anche agli <a href="/wiki/Composto_eterociclico" title="Composto eterociclico">eterocicli</a> come la <a href="/wiki/Piridina" title="Piridina">piridina</a>, il <a href="/wiki/Pirrolo" title="Pirrolo">pirrolo</a> e il <a href="/wiki/Furano" title="Furano">furano</a>.<a href="#cite_note-33">[33]</a> Vi è infine il <a href="/w/index.php?title=Metodo_di_Pariser%E2%80%93Parr%E2%80%93Pople&action=edit&redlink=1" class="new" title="Metodo di Pariser–Parr–Pople (la pagina non esiste)">metodo di Pariser–Parr–Pople</a>, che sfrutta metodi semi-empirici della <a href="/wiki/Chimica_quantistica" title="Chimica quantistica">chimica quantistica</a> nell'ambito della <a href="/wiki/Chimica_organica" title="Chimica organica">chimica organica</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Moti_interni_nelle_molecole_biatomiche">Moti interni nelle molecole biatomiche</h2>[<a href="/w/index.php?title=Molecola&veaction=edit&section=12" title="Modifica la sezione Moti interni nelle molecole biatomiche" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Molecola&action=edit&section=12" title="Edit section's source code: Moti interni nelle molecole biatomiche">modifica wikitesto</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r130657691"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r139142988"> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /> Lo stesso argomento in dettaglio: <a href="/wiki/Moto_in_un_campo_centrale" title="Moto in un campo centrale">Moto in un campo centrale</a>.</div> </div> I nuclei sono soggetti al potenziale adiabatico definito in precedenza, che nelle molecole biatomiche è indipendente dalla posizione del centro di massa della molecola e dall'orientazione della retta congiungente i due nuclei. Il potenziale gode quindi di invarianza rispetto alle traslazioni ed alle rotazioni, e il moto dei nuclei può essere studiato come un <a href="/wiki/Problema_dei_due_corpi" title="Problema dei due corpi">problema a due corpi</a>, sicché l'<a href="/wiki/Equazione_di_Schr%C3%B6dinger" title="Equazione di Schrödinger">equazione di Schrödinger</a> può essere separata in moto radiale, dipendente dalla distanza tra i due nuclei, e moto orbitale, dipendente dal <a href="/wiki/Numero_quantico_orbitale" title="Numero quantico orbitale">numero quantico orbitale</a>. L'equazione di Schrödinger nel caso di un <a href="/wiki/Moto_in_un_campo_centrale" title="Moto in un campo centrale">moto in un campo centrale</a> è: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[-{\frac {\hbar ^{2}}{2(M+m)}}\nabla _{\mathbf {r} _{cm}}^{2}-{\frac {\hbar ^{2}}{2\mu }}\nabla ^{2}+V_{\mathrm {ad} }(|\mathbf {r} _{rel}|)\right]\psi _{\mathrm {n} }(\mathbf {r} _{cm},\mathbf {r} _{rel})=E_{tot}\psi _{\mathrm {n} }(\mathbf {r} _{cm},\mathbf {r} _{rel})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mi>M</mi> <mo>+</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msubsup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>m</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>μ</mi> </mrow> </mfrac> </mrow> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>e</mi> <mi>l</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>m</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>e</mi> <mi>l</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>m</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>e</mi> <mi>l</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[-{\frac {\hbar ^{2}}{2(M+m)}}\nabla _{\mathbf {r} _{cm}}^{2}-{\frac {\hbar ^{2}}{2\mu }}\nabla ^{2}+V_{\mathrm {ad} }(|\mathbf {r} _{rel}|)\right]\psi _{\mathrm {n} }(\mathbf {r} _{cm},\mathbf {r} _{rel})=E_{tot}\psi _{\mathrm {n} }(\mathbf {r} _{cm},\mathbf {r} _{rel})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f63a966a7e40307058884854406f9068bf7029e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:74.15ex; height:6.509ex;" alt="{\displaystyle \left[-{\frac {\hbar ^{2}}{2(M+m)}}\nabla _{\mathbf {r} _{cm}}^{2}-{\frac {\hbar ^{2}}{2\mu }}\nabla ^{2}+V_{\mathrm {ad} }(|\mathbf {r} _{rel}|)\right]\psi _{\mathrm {n} }(\mathbf {r} _{cm},\mathbf {r} _{rel})=E_{tot}\psi _{\mathrm {n} }(\mathbf {r} _{cm},\mathbf {r} _{rel})}"></dd></dl> dove <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} _{cm}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} _{cm}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f6bc68b9857a7515b8faeceb64f1b9ffa0adad1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.489ex; height:2.009ex;" alt="{\displaystyle \mathbf {r} _{cm}}"> indica la posizione del centro di massa e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} _{rel}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>e</mi> <mi>l</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} _{rel}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f8ead6ca8d89297b5b0433fe4491d8f5ec56e06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.332ex; height:2.009ex;" alt="{\displaystyle \mathbf {r} _{rel}}"> la posizione relativa dei due nuclei, differenza delle rispettive posizioni. Il problema può essere quindi separato in due equazioni, una per il centro di massa ed una per la particella di massa μ che si muove in un campo centrale rispetto al centro di massa. La funzione d'onda si può quindi fattorizzare nel seguente modo: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{\mathrm {n} }(\mathbf {r} _{cm},\mathbf {r} _{rel})=\psi _{cm}(\mathbf {r} _{cm})\psi _{rel}(\mathbf {r} _{rel})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>m</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>e</mi> <mi>l</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>e</mi> <mi>l</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>e</mi> <mi>l</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{\mathrm {n} }(\mathbf {r} _{cm},\mathbf {r} _{rel})=\psi _{cm}(\mathbf {r} _{cm})\psi _{rel}(\mathbf {r} _{rel})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5294d04afad5cfda0bcd8d5923e8b2014e1fafc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.506ex; height:2.843ex;" alt="{\displaystyle \psi _{\mathrm {n} }(\mathbf {r} _{cm},\mathbf {r} _{rel})=\psi _{cm}(\mathbf {r} _{cm})\psi _{rel}(\mathbf {r} _{rel})}">. L'equazione per <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{cm}(\mathbf {r} _{cm})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>m</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{cm}(\mathbf {r} _{cm})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df6ae2342ac01acb6cd6044f9b3c5e6b47161764" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.198ex; height:2.843ex;" alt="{\displaystyle \psi _{cm}(\mathbf {r} _{cm})}">, che rappresenta il problema della <a href="/wiki/Particella_libera" title="Particella libera">particella libera</a>, fornisce l'energia traslazionale della molecola. L'equazione per <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{rel}(\mathbf {r} _{rel})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>e</mi> <mi>l</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>e</mi> <mi>l</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{rel}(\mathbf {r} _{rel})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6e72aa0c6148a02b471593b8115eb23fe0535a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.885ex; height:2.843ex;" alt="{\displaystyle \psi _{rel}(\mathbf {r} _{rel})}"> si può ulteriormente fattorizzare in parte radiale, dipendente da r, e parte angolare, dipendente dalle coordinate angolari: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{rel}(\mathbf {r} _{rel})=R(r)f(\theta ,\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>e</mi> <mi>l</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>e</mi> <mi>l</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>R</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mi>f</mi> <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 _{rel}(\mathbf {r} _{rel})=R(r)f(\theta ,\phi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ccefa30081e968f38866128616e2ba5458c08e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.203ex; height:2.843ex;" alt="{\displaystyle \psi _{rel}(\mathbf {r} _{rel})=R(r)f(\theta ,\phi )}">. La soluzione per <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\theta ,\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\theta ,\phi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b09584135fea5049be8213c6f87e96e0313e7085" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.598ex; height:2.843ex;" alt="{\displaystyle f(\theta ,\phi )}"> sono le <a href="/wiki/Armoniche_sferiche" title="Armoniche sferiche">armoniche sferiche</a>, ed i rispettivi stati sono autostati del <a href="/wiki/Momento_angolare_orbitale_(meccanica_quantistica)" class="mw-redirect" title="Momento angolare orbitale (meccanica quantistica)">momento angolare orbitale</a> e della sua componente lungo l'asse z. L'equazione per <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R(r)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7b97817d8756302ef44f910ec5e76346e8d4f4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.622ex; height:2.843ex;" alt="{\displaystyle R(r)}"> è invece, detta <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g=rR(r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>=</mo> <mi>r</mi> <mi>R</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g=rR(r)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f01337ff0e428e9142ac99b4567d652220657b3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.885ex; height:2.843ex;" alt="{\displaystyle g=rR(r)}">:<a href="#cite_note-34">[34]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[-{\frac {\hbar ^{2}}{2\mu }}{\frac {d^{2}}{dr^{2}}}+{\frac {\hbar ^{2}l(l+1)}{2\mu r^{2}}}+V_{\mathrm {ad} }(|\mathbf {r} _{rel}|)\right]g=Eg}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>μ</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi>d</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>l</mi> <mo stretchy="false">(</mo> <mi>l</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mi>μ</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>e</mi> <mi>l</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mi>g</mi> <mo>=</mo> <mi>E</mi> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[-{\frac {\hbar ^{2}}{2\mu }}{\frac {d^{2}}{dr^{2}}}+{\frac {\hbar ^{2}l(l+1)}{2\mu r^{2}}}+V_{\mathrm {ad} }(|\mathbf {r} _{rel}|)\right]g=Eg}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0d8b7c5343baef9266e16f45363c9237debc159" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:45.417ex; height:7.509ex;" alt="{\displaystyle \left[-{\frac {\hbar ^{2}}{2\mu }}{\frac {d^{2}}{dr^{2}}}+{\frac {\hbar ^{2}l(l+1)}{2\mu r^{2}}}+V_{\mathrm {ad} }(|\mathbf {r} _{rel}|)\right]g=Eg}"></dd></dl> dove il secondo termine rappresenta il contributo energetico rotazionale <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{rot}(l)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>l</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{rot}(l)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dae467f37fce9645f0e08b0549c77880822c9cb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.583ex; height:2.843ex;" alt="{\displaystyle E_{rot}(l)}">, che dipende dal <a href="/wiki/Numero_quantico_orbitale" title="Numero quantico orbitale">numero quantico orbitale</a> l. Il potenziale adiabatico può essere inoltre sviluppato in <a href="/wiki/Serie_di_Taylor" title="Serie di Taylor">serie di Taylor</a>, che troncata al secondo ordine è:<a href="#cite_note-cimiraglia-6">[6]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\mathrm {ad} }(|\mathbf {r} _{rel}|)=V_{\mathrm {ad} }(R_{M})+{\frac {1}{2}}\left({\frac {d^{2}V_{\mathrm {ad} }(|\mathbf {r} '_{rel}|)}{dR'^{2}}}\right)_{R=R_{M}}(|\mathbf {r} _{rel}|-R_{M})^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>e</mi> <mi>l</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>e</mi> <mi>l</mi> </mrow> <mo>′</mo> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <msup> <mi>R</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>e</mi> <mi>l</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>−</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\mathrm {ad} }(|\mathbf {r} _{rel}|)=V_{\mathrm {ad} }(R_{M})+{\frac {1}{2}}\left({\frac {d^{2}V_{\mathrm {ad} }(|\mathbf {r} '_{rel}|)}{dR'^{2}}}\right)_{R=R_{M}}(|\mathbf {r} _{rel}|-R_{M})^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7dc99984e4cf9cccf44931951a18e9c3989e4e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:62.725ex; height:8.009ex;" alt="{\displaystyle V_{\mathrm {ad} }(|\mathbf {r} _{rel}|)=V_{\mathrm {ad} }(R_{M})+{\frac {1}{2}}\left({\frac {d^{2}V_{\mathrm {ad} }(|\mathbf {r} '_{rel}|)}{dR'^{2}}}\right)_{R=R_{M}}(|\mathbf {r} _{rel}|-R_{M})^{2}}"></dd></dl> dove <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{M}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{M}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1a600516077fb78afb09d1aa981513e4d60dc16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.723ex; height:2.509ex;" alt="{\displaystyle R_{M}}"> è il valore di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mathbf {r} _{rel}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>e</mi> <mi>l</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\mathbf {r} _{rel}|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb51f5b03022d539beee5fc560044577063be61d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.626ex; height:2.843ex;" alt="{\displaystyle |\mathbf {r} _{rel}|}"> che minimizza <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\mathrm {ad} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\mathrm {ad} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59155eee06b843d2f27f39f782e53958225bfe02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.323ex; height:2.509ex;" alt="{\displaystyle V_{\mathrm {ad} }}">, e rappresenta la posizione di equilibrio dei due nuclei. Tale espressione rappresenta un <a href="/wiki/Moto_armonico" title="Moto armonico">moto armonico</a> attorno a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{M}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{M}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1a600516077fb78afb09d1aa981513e4d60dc16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.723ex; height:2.509ex;" alt="{\displaystyle R_{M}}"> che fornisce un contributo energetico dato dall'energia dell'equazione elettronica contenuta in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\mathrm {ad} }(R_{M})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\mathrm {ad} }(R_{M})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d8fb51676c98e809203a772a72e0b4637424d7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.856ex; height:2.843ex;" alt="{\displaystyle V_{\mathrm {ad} }(R_{M})}"> e dall'energia vibrazionale <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{vib}(v)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mi>i</mi> <mi>b</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{vib}(v)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3ec13fac2b88d5817df26676c03fef77a0d9c2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.955ex; height:2.843ex;" alt="{\displaystyle E_{vib}(v)}">. <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Molecular_energy_levels_en.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/Molecular_energy_levels_en.svg/310px-Molecular_energy_levels_en.svg.png" decoding="async" width="310" height="225" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/Molecular_energy_levels_en.svg/465px-Molecular_energy_levels_en.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/89/Molecular_energy_levels_en.svg/620px-Molecular_energy_levels_en.svg.png 2x" data-file-width="709" data-file-height="515" /></a><figcaption>Livelli energetici di una molecola: per ogni livello elettronico, associato ad una superficie adiabatica, vi sono diversi livelli vibrazionali, e per ogni livello vibrazionale vi sono diversi livelli rotazionali.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Morse-potential.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Morse-potential.png/310px-Morse-potential.png" decoding="async" width="310" height="286" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Morse-potential.png/465px-Morse-potential.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Morse-potential.png/620px-Morse-potential.png 2x" data-file-width="2828" data-file-height="2612" /></a><figcaption>Rappresentazione dei livelli energetici vibrazionali all'interno di una superficie adiabatica, approssimata dal <a href="/wiki/Potenziale_di_Morse" title="Potenziale di Morse">potenziale di Morse</a>. In verde il potenziale ed i rispettivi livelli eccitati dell'oscillatore armonico corrispondente.</figcaption></figure> Detta <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"> la lunghezza caratteristica data dalla relazione <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar ^{2}/2\mu x_{0}^{2}=1/2kx_{0}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mi>μ</mi> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mi>k</mi> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar ^{2}/2\mu x_{0}^{2}=1/2kx_{0}^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20115c8c0e6c6d624d8a22326346facc2ee94be4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.668ex; height:3.343ex;" alt="{\displaystyle \hbar ^{2}/2\mu x_{0}^{2}=1/2kx_{0}^{2}}"> e detta <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=x/x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=x/x_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecd46b8b3ad1db122ca15c450ffb1766061fb04e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.13ex; height:2.843ex;" alt="{\displaystyle y=x/x_{0}}">, le soluzioni dell'equazione per <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><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}"> sono: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{v}(y)=c_{v}H_{v}(y)e^{-{\frac {1}{2}}y^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{v}(y)=c_{v}H_{v}(y)e^{-{\frac {1}{2}}y^{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b44f6a106a2539e84efc3859906c3fe17863aada" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.914ex; height:4.009ex;" alt="{\displaystyle g_{v}(y)=c_{v}H_{v}(y)e^{-{\frac {1}{2}}y^{2}}}"></dd></dl> dove <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{v}(y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{v}(y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/730441e60531b442c01029c4bd8e599db5225292" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.926ex; height:2.843ex;" alt="{\displaystyle H_{v}(y)}"> è il <a href="/wiki/Polinomi_di_Hermite" title="Polinomi di Hermite">polinomio di Hermite</a> di grado <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}">. Lo spettro energetico contiene in definitiva tre termini: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=V_{\mathrm {ad} }(R_{M})+E_{vib}(v)+E_{rot}(l)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mi>i</mi> <mi>b</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>l</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=V_{\mathrm {ad} }(R_{M})+E_{vib}(v)+E_{rot}(l)\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f79e284e78aa5fdca32dae6ff8a3508a8be09de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.529ex; height:2.843ex;" alt="{\displaystyle E=V_{\mathrm {ad} }(R_{M})+E_{vib}(v)+E_{rot}(l)\ }"></dd></dl> Tali termini sono i contributi energetici che caratterizzano la dinamica della molecola biatomica, e nello specifico sono:<a href="#cite_note-cimiraglia-6">[6]</a><a href="#cite_note-35">[35]</a> <ul><li>Il contributo elettronico, dato dal termine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\mathrm {e} }(R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\mathrm {e} }(R)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e399592cc6f5d298393005e781e997646f5b536e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.251ex; height:2.843ex;" alt="{\displaystyle E_{\mathrm {e} }(R)}"> di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\mathrm {ad} }(R_{M})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\mathrm {ad} }(R_{M})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d8fb51676c98e809203a772a72e0b4637424d7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.856ex; height:2.843ex;" alt="{\displaystyle V_{\mathrm {ad} }(R_{M})}">, definisce la profondità <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\mathrm {ad} }(\infty )-V_{\mathrm {ad} }(R_{M})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\mathrm {ad} }(\infty )-V_{\mathrm {ad} }(R_{M})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62b027f284eff364616eaef905a6ac1a05461e6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.153ex; height:2.843ex;" alt="{\displaystyle V_{\mathrm {ad} }(\infty )-V_{\mathrm {ad} }(R_{M})}"> della <a href="/wiki/Buca_di_potenziale" title="Buca di potenziale">buca di potenziale</a> generata dai due nuclei, responsabile del legame chimico. I livelli energetici associati a questo termine sono detti <a href="/wiki/Superficie_di_energia_potenziale" title="Superficie di energia potenziale">superfici adiabatiche</a>, e corrispondono ai diversi stati energetici degli elettroni. Gli elettroni che vengono promossi da un orbitale ad un altro, ad esempio da un orbitale di legame ad uno di antilegame, effettuano una transizione tra due valori <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\mathrm {ad} }'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mo>′</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\mathrm {ad} }'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/228c7adbd74e4d9cacb77409d816ff90685cb21a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.323ex; height:2.843ex;" alt="{\displaystyle V_{\mathrm {ad} }'}"> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\mathrm {ad} }''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mo>″</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\mathrm {ad} }''}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49440a17fd3f1b11a801eb4e0fdb5a7077dae4bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.323ex; height:2.843ex;" alt="{\displaystyle V_{\mathrm {ad} }''}"> del potenziale adiabatico. Tali transizioni sono dell'ordine di 10 <a href="/wiki/Elettronvolt" title="Elettronvolt">eV</a>, e a differenti superfici adiabatiche corrispondono anche diversi valori di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{M}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{M}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1a600516077fb78afb09d1aa981513e4d60dc16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.723ex; height:2.509ex;" alt="{\displaystyle R_{M}}">. Le transizioni elettroniche tra due di tali superfici sono inoltre accompagnate da transizioni tra diversi stati vibrazionali e rotazionali.</li> <li>Il contributo vibrazionale, meno energetico del precedente, nell'approssimazione di moto armonico fornita dall'esclusione dei termini superiori al secondo ordine nel precedente sviluppo di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\mathrm {ad} }(R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\mathrm {ad} }(R)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fff3381560f2ac4e189238e8fd66dda91cd23491" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.897ex; height:2.843ex;" alt="{\displaystyle V_{\mathrm {ad} }(R)}"> è dato dagli autovalori dell'<a href="/wiki/Oscillatore_armonico_quantistico" title="Oscillatore armonico quantistico">oscillatore armonico quantistico</a>:</li></ul> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{vib}(v)=\hbar \omega \left(v+{\frac {1}{2}}\right)\qquad v=0,1,2,...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mi>i</mi> <mi>b</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi class="MJX-variant">ℏ</mi> <mi>ω</mi> <mrow> <mo>(</mo> <mrow> <mi>v</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="2em" /> <mi>v</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{vib}(v)=\hbar \omega \left(v+{\frac {1}{2}}\right)\qquad v=0,1,2,...}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3f24805e450339859e56fef06b6c61cae2da404" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:41.142ex; height:6.176ex;" alt="{\displaystyle E_{vib}(v)=\hbar \omega \left(v+{\frac {1}{2}}\right)\qquad v=0,1,2,...}"></dd></dl> <dl><dd>dove <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">ℏ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de68de3a92517953436c93b5a76461d49160cc41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle \hbar }"> è la <a href="/wiki/Costante_di_Planck" title="Costante di Planck">costante di Planck</a> e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"> la <a href="/wiki/Frequenza_angolare" class="mw-redirect" title="Frequenza angolare">frequenza angolare</a> dell'oscillazione intorno a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{M}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{M}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1a600516077fb78afb09d1aa981513e4d60dc16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.723ex; height:2.509ex;" alt="{\displaystyle R_{M}}">.</dd> <dd>La frequenza è data da:</dd></dl> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ={\sqrt {\frac {k}{\mu }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>k</mi> <mi>μ</mi> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ={\sqrt {\frac {k}{\mu }}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53abb6b731bd0e6e8f5a7aa352bc2397a3253100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:9.106ex; height:7.509ex;" alt="{\displaystyle \omega ={\sqrt {\frac {k}{\mu }}}}"></dd></dl> <dl><dd>con</dd></dl> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=\left({\frac {d^{2}V_{\mathrm {ad} }(R')}{dR'^{2}}}\right)_{R=R_{M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>R</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <msup> <mi>R</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=\left({\frac {d^{2}V_{\mathrm {ad} }(R')}{dR'^{2}}}\right)_{R=R_{M}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/224d5dba81e8d114c96533727cfe280f13df117b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:24.253ex; height:8.009ex;" alt="{\displaystyle k=\left({\frac {d^{2}V_{\mathrm {ad} }(R')}{dR'^{2}}}\right)_{R=R_{M}}}"></dd></dl> <dl><dd>e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"> la <a href="/wiki/Massa_ridotta" title="Massa ridotta">massa ridotta</a> dell'oscillatore a due corpi, data dal rapporto tra il prodotto e la somma delle masse dei due nuclei.</dd> <dd>Tale contributo descrive il moto armonico dei due nuclei intorno alla posizione di equilibrio, e transizioni tra due livelli vibrazionali sono dell'ordine del decimo di eV.</dd></dl> <ul><li>Il contributo rotazionale, il meno energetico dei tre, è fornito dall'equazione angolare dell'<a href="/wiki/Atomo_di_idrogeno" title="Atomo di idrogeno">atomo di idrogeno</a>, pari a:</li></ul> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{rot}(l)={\frac {L^{2}}{2I}}={\frac {\hbar ^{2}l(l+1)}{2\mu R_{M}^{2}}}\ \ \ \ \ l=0,1,2,...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>l</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>I</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>l</mi> <mo stretchy="false">(</mo> <mi>l</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mi>μ</mi> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mi>l</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{rot}(l)={\frac {L^{2}}{2I}}={\frac {\hbar ^{2}l(l+1)}{2\mu R_{M}^{2}}}\ \ \ \ \ l=0,1,2,...}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90540818e86cb44844d6f8d3462558e67795318c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:42.663ex; height:7.009ex;" alt="{\displaystyle E_{rot}(l)={\frac {L^{2}}{2I}}={\frac {\hbar ^{2}l(l+1)}{2\mu R_{M}^{2}}}\ \ \ \ \ l=0,1,2,...}"></dd></dl> <dl><dd>dove <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><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}"> è il momento angolare orbitale e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=\mu R_{M}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>=</mo> <mi>μ</mi> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I=\mu R_{M}^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9e7159eda3014a254919f49cd51367c27aec496" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.395ex; height:3.176ex;" alt="{\displaystyle I=\mu R_{M}^{2}}"> il <a href="/wiki/Momento_d%27inerzia" class="mw-redirect" title="Momento d'inerzia">momento d'inerzia</a>.</dd> <dd>Tale contributo è generalmente dell'ordine dei meV, ed è calcolato assumendo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=R_{M}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=R_{M}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51f99fc803572f86f61b23b21baaba1ba0df2238" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.586ex; height:2.509ex;" alt="{\displaystyle R=R_{M}}">.</dd></dl> In conclusione, quindi, l'energia interna di una molecola biatomica è: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=V_{\mathrm {ad} }(R_{M})+\hbar \omega \left(v+{\frac {1}{2}}\right)+{\frac {\hbar ^{2}l(l+1)}{2\mu R_{M}^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mi class="MJX-variant">ℏ</mi> <mi>ω</mi> <mrow> <mo>(</mo> <mrow> <mi>v</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>l</mi> <mo stretchy="false">(</mo> <mi>l</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mi>μ</mi> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=V_{\mathrm {ad} }(R_{M})+\hbar \omega \left(v+{\frac {1}{2}}\right)+{\frac {\hbar ^{2}l(l+1)}{2\mu R_{M}^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10ecf6817b7d5d5990f6c7cba3b49891278b9747" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:42.349ex; height:7.009ex;" alt="{\displaystyle E=V_{\mathrm {ad} }(R_{M})+\hbar \omega \left(v+{\frac {1}{2}}\right)+{\frac {\hbar ^{2}l(l+1)}{2\mu R_{M}^{2}}}}"></dd></dl> dove i termini sono elencati in ordine di importanza. <div class="mw-heading mw-heading2"><h2 id="Moti_interni_nelle_molecole_poliatomiche">Moti interni nelle molecole poliatomiche</h2>[<a href="/w/index.php?title=Molecola&veaction=edit&section=13" title="Modifica la sezione Moti interni nelle molecole poliatomiche" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Molecola&action=edit&section=13" title="Edit section's source code: Moti interni nelle molecole poliatomiche">modifica wikitesto</a>]</div> Nelle molecole poliatomiche il calcolo dello spettro energetico può essere molto complesso. Le simmetrie della molecola giocano spesso un ruolo determinante al fine di ottenere gli autovalori dell'energia vibrazionale e rotazionale. <div class="mw-heading mw-heading3"><h3 id="Moto_vibrazionale">Moto vibrazionale</h3>[<a href="/w/index.php?title=Molecola&veaction=edit&section=14" title="Modifica la sezione Moto vibrazionale" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Molecola&action=edit&section=14" title="Edit section's source code: Moto vibrazionale">modifica wikitesto</a>]</div> Nelle molecole poliatomiche l'energia cinetica data dal moto vibrazionale è espressa come: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{v}={\frac {1}{2}}\sum _{\alpha }m_{\alpha }({\dot {x}}_{\alpha }^{2}+{\dot {y}}_{\alpha }^{2}+{\dot {z}}_{\alpha }^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{v}={\frac {1}{2}}\sum _{\alpha }m_{\alpha }({\dot {x}}_{\alpha }^{2}+{\dot {y}}_{\alpha }^{2}+{\dot {z}}_{\alpha }^{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e7c80a4e8ccc466a0400f619e7c284506c56c4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.209ex; height:6.343ex;" alt="{\displaystyle T_{v}={\frac {1}{2}}\sum _{\alpha }m_{\alpha }({\dot {x}}_{\alpha }^{2}+{\dot {y}}_{\alpha }^{2}+{\dot {z}}_{\alpha }^{2})}"></dd></dl> dove le coordinate cartesiane sono le posizioni del nucleo α-esimo rispetto alla posizione di equilibrio. Utilizzando coordinate mass–weighted: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{ij}={\sqrt {m}}_{i}x_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>m</mi> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{ij}={\sqrt {m}}_{i}x_{j}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/803102927cb0e6788c65d2b653a4884c3026cb6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.628ex; height:3.009ex;" alt="{\displaystyle q_{ij}={\sqrt {m}}_{i}x_{j}}"></dd></dl> è possibile definire la matrice <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {U} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {U} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2141bec2344e3dc5241ff50b0fd366755e00223" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.057ex; height:2.176ex;" alt="{\displaystyle \mathbf {U} }"> di elementi: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{\alpha \beta }=\left({\frac {\partial ^{2}V_{\mathrm {ad} }}{\partial q_{\alpha }\,\partial q_{\beta }}}\right)=\left({\frac {\partial ^{2}V_{\mathrm {ad} }}{\partial q_{ij}\,\partial q_{mn}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{\alpha \beta }=\left({\frac {\partial ^{2}V_{\mathrm {ad} }}{\partial q_{\alpha }\,\partial q_{\beta }}}\right)=\left({\frac {\partial ^{2}V_{\mathrm {ad} }}{\partial q_{ij}\,\partial q_{mn}}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d231b1830e056e18c5d1dbfbe168c38664bb2bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:35.059ex; height:6.509ex;" alt="{\displaystyle u_{\alpha \beta }=\left({\frac {\partial ^{2}V_{\mathrm {ad} }}{\partial q_{\alpha }\,\partial q_{\beta }}}\right)=\left({\frac {\partial ^{2}V_{\mathrm {ad} }}{\partial q_{ij}\,\partial q_{mn}}}\right)}"></dd></dl> E quindi, come nelle molecole biatomiche, l'energia vibrazionale può essere espressa come: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{vib}=V_{\mathrm {ad} }(R_{eq})+{\frac {1}{2}}\mathbf {q} ^{T}\mathbf {U} \mathbf {q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mi>i</mi> <mi>b</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mi>q</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{vib}=V_{\mathrm {ad} }(R_{eq})+{\frac {1}{2}}\mathbf {q} ^{T}\mathbf {U} \mathbf {q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa5514a8ae012590ccc47e9820ab7957af15f04e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.885ex; height:5.176ex;" alt="{\displaystyle E_{vib}=V_{\mathrm {ad} }(R_{eq})+{\frac {1}{2}}\mathbf {q} ^{T}\mathbf {U} \mathbf {q} }"></dd></dl> dove <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7be005a326b7ac3fe4c24bca391369f44c4c2876" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.416ex; height:2.009ex;" alt="{\displaystyle \mathbf {q} }"> è il vettore che ha per componenti <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{ij}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b08ec83005828a8789b639e4944b11905e9b18b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.514ex; height:2.343ex;" alt="{\displaystyle q_{ij}}"> Le <a href="/wiki/Equazione_del_moto" title="Equazione del moto">equazioni del moto</a> sono date dal sistema di equazioni differenziali: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\ddot {q}} =-\mathbf {U} \mathbf {q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">q</mi> <mo mathvariant="bold">¨</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\ddot {q}} =-\mathbf {U} \mathbf {q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c6a9e482209ad3142b955f4a279d08be65c07a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.794ex; height:2.676ex;" alt="{\displaystyle \mathbf {\ddot {q}} =-\mathbf {U} \mathbf {q} }"></dd></dl> Ogni atomo vibra con la stessa frequenza angolare, e tali frequenze sono dette modi normali di vibrazione, che si ottengono dalle radici dell'equazione caratteristica per la matrice <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {U} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {U} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2141bec2344e3dc5241ff50b0fd366755e00223" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.057ex; height:2.176ex;" alt="{\displaystyle \mathbf {U} }">: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(u_{\alpha \beta }-\lambda \delta _{\alpha \beta })=0\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mo>−</mo> <mi>λ</mi> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(u_{\alpha \beta }-\lambda \delta _{\alpha \beta })=0\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de4f8a3dbf3f8e0c6db3d5619087e5567c3da4f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.89ex; height:3.009ex;" alt="{\displaystyle \det(u_{\alpha \beta }-\lambda \delta _{\alpha \beta })=0\ }"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Moto_rotazionale">Moto rotazionale</h3>[<a href="/w/index.php?title=Molecola&veaction=edit&section=15" title="Modifica la sezione Moto rotazionale" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Molecola&action=edit&section=15" title="Edit section's source code: Moto rotazionale">modifica wikitesto</a>]</div> Considerando la molecola un corpo rigido, è possibile definire il momento d'inerzia attorno a un asse a come: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{a}=\sum _{\alpha }m_{\alpha }r_{\alpha }^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{a}=\sum _{\alpha }m_{\alpha }r_{\alpha }^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8492ea4d8faf8fbe5a033f2179444e77f4e691bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.623ex; height:5.509ex;" alt="{\displaystyle I_{a}=\sum _{\alpha }m_{\alpha }r_{\alpha }^{2}}"></dd></dl> Gli assi d'inerzia di una molecola sono tre, e i rispettivi momenti d'inerzia sono <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{a}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83b5cb24870dadbb0efdf2e6d864ef853a0f1f89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.125ex; height:2.509ex;" alt="{\displaystyle I_{a}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{b}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5468195bb8c7d158a80efe38ceadcdb2b8fe4d4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.961ex; height:2.509ex;" alt="{\displaystyle I_{b}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{c}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7463de5fff5fee0c540ac1d8cd1aca43f4852c65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.967ex; height:2.509ex;" alt="{\displaystyle I_{c}}">. Se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{a}\neq I_{b}\neq I_{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>≠</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>≠</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{a}\neq I_{b}\neq I_{c}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bc99c2927a390d3aebd2997b7d6aa5212935765" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.25ex; height:2.676ex;" alt="{\displaystyle I_{a}\neq I_{b}\neq I_{c}}">, il corpo rigido è detto asymmetrical top, se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{a}=I_{b}\neq I_{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>≠</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{a}=I_{b}\neq I_{c}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb03f53e65e98769893f9a1f89a7d73f22dd547b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.25ex; height:2.676ex;" alt="{\displaystyle I_{a}=I_{b}\neq I_{c}}"> è detto symmetrical top, mentre se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{a}=I_{b}=I_{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{a}=I_{b}=I_{c}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef33862cef61f5430e9e4a59689c6f2c9a0eca98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.25ex; height:2.509ex;" alt="{\displaystyle I_{a}=I_{b}=I_{c}}"> è detto spherical top. All'interno dei corpi rigidi symmetrical top, se <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{a}=I_{b}<I_{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo><</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{a}=I_{b}<I_{c}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f079ed66da531458122fcd5ef5ed22fc26095cf8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.25ex; height:2.509ex;" alt="{\displaystyle I_{a}=I_{b}<I_{c}}"> il corpo è detto oblato , si tratta di una molecola piatta, come il <a href="/wiki/Benzene" title="Benzene">benzene</a>, se invece <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{a}<I_{b}=I_{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo><</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{a}<I_{b}=I_{c}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2db7035eb167f43979748278c0e9348a12cfc34c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.25ex; height:2.509ex;" alt="{\displaystyle I_{a}<I_{b}=I_{c}}"> è detto prolato, e si tratta di una molecola allungata, come il <a href="/wiki/Pentacloruro_di_fosforo" title="Pentacloruro di fosforo">pentacloruro di fosforo</a>. L'energia cinetica è data da: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{r}={\frac {L_{a}^{2}}{2I_{a}}}+{\frac {L_{b}^{2}}{2I_{b}}}+{\frac {L_{c}^{2}}{2I_{c}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow> <mn>2</mn> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow> <mn>2</mn> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow> <mn>2</mn> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{r}={\frac {L_{a}^{2}}{2I_{a}}}+{\frac {L_{b}^{2}}{2I_{b}}}+{\frac {L_{c}^{2}}{2I_{c}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/deb49c3de3b87ec776aa04344b7a29f146e0f42f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.159ex; height:6.509ex;" alt="{\displaystyle T_{r}={\frac {L_{a}^{2}}{2I_{a}}}+{\frac {L_{b}^{2}}{2I_{b}}}+{\frac {L_{c}^{2}}{2I_{c}}}}"></dd></dl> dove <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{a}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87a7bf511194a88867e63c66594f8c06f106bbbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.685ex; height:2.509ex;" alt="{\displaystyle L_{a}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{b}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c329e421a62c4d4f0b24faa03f451908bad311" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.52ex; height:2.509ex;" alt="{\displaystyle L_{b}}"> ed <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{c}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40dfe0c821ab1ddeb7ab27ddaa97764086f42789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.527ex; height:2.509ex;" alt="{\displaystyle L_{c}}"> sono le tre componenti dell'operatore momento angolare totale di rotazione della molecola lungo gli assi di inerzia a, b e c. <ul><li>Nel caso di uno spherical top si ottiene immediatamente che gli autovalori dell'energia rotazionale sono:</li></ul> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{rot}(L)={\frac {L^{2}}{2I}}={\frac {\hbar ^{2}L(L+1)}{2I}}\ \ \ \ \ L=0,1,2,...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>L</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>I</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>L</mi> <mo stretchy="false">(</mo> <mi>L</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mi>I</mi> </mrow> </mfrac> </mrow> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mi>L</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{rot}(L)={\frac {L^{2}}{2I}}={\frac {\hbar ^{2}L(L+1)}{2I}}\ \ \ \ \ L=0,1,2,...}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/850eb946e547bd8045f1750337bb8d2df0fd4d29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:46.221ex; height:5.843ex;" alt="{\displaystyle E_{rot}(L)={\frac {L^{2}}{2I}}={\frac {\hbar ^{2}L(L+1)}{2I}}\ \ \ \ \ L=0,1,2,...}"></dd></dl> <dl><dd>e la degenerazione degli autovalori è <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2L+1)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>L</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2L+1)^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d09fb63208ef07c73708dbf11792a1de5164d08a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.612ex; height:3.176ex;" alt="{\displaystyle (2L+1)^{2}}">.</dd></dl> <ul><li>Nel caso di un symmetrical top si ha:</li></ul> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{rot}(L)={\frac {L_{a}^{2}+L_{b}^{2}}{2I_{a}}}+{\frac {L_{c}^{2}}{2I_{c}}}={\frac {L^{2}}{2I_{a}}}+\left({\frac {1}{2I_{c}}}-{\frac {1}{2I_{a}}}\right)L_{c}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>L</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <mn>2</mn> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow> <mn>2</mn> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msubsup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{rot}(L)={\frac {L_{a}^{2}+L_{b}^{2}}{2I_{a}}}+{\frac {L_{c}^{2}}{2I_{c}}}={\frac {L^{2}}{2I_{a}}}+\left({\frac {1}{2I_{c}}}-{\frac {1}{2I_{a}}}\right)L_{c}^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe04d186919081ddbf23bacdef08b6029cc2573b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:53.813ex; height:6.676ex;" alt="{\displaystyle E_{rot}(L)={\frac {L_{a}^{2}+L_{b}^{2}}{2I_{a}}}+{\frac {L_{c}^{2}}{2I_{c}}}={\frac {L^{2}}{2I_{a}}}+\left({\frac {1}{2I_{c}}}-{\frac {1}{2I_{a}}}\right)L_{c}^{2}}"></dd></dl> <dl><dd>e dal momento che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba162c66ca85776c83557af5088cc6f8584d1912" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{2}}"> commuta con ogni sua componente e con <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{z}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77a5b940110c5e1fe03782a31c5e700939ae20e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.585ex; height:2.509ex;" alt="{\displaystyle L_{z}}">, l'autofunzione associata all'energia vibrazionale è simultanea a questi tre operatori.</dd> <dd>L'energia rotazionale è data allora da:</dd></dl> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{rot}(L,m)={\frac {\hbar ^{2}L(L+1)}{2I_{a}}}+\left({\frac {1}{2I_{c}}}-{\frac {1}{2I_{a}}}\right)\hbar ^{2}m^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>L</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>L</mi> <mo stretchy="false">(</mo> <mi>L</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{rot}(L,m)={\frac {\hbar ^{2}L(L+1)}{2I_{a}}}+\left({\frac {1}{2I_{c}}}-{\frac {1}{2I_{a}}}\right)\hbar ^{2}m^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f3fa6b048bf630afe7c394ccea9764370e4cb2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:48.884ex; height:6.509ex;" alt="{\displaystyle E_{rot}(L,m)={\frac {\hbar ^{2}L(L+1)}{2I_{a}}}+\left({\frac {1}{2I_{c}}}-{\frac {1}{2I_{a}}}\right)\hbar ^{2}m^{2}}"></dd></dl> <dl><dd>con degenerazione <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2(2L+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo stretchy="false">(</mo> <mn>2</mn> <mi>L</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2(2L+1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70601d3695aa755d301126d3dad49d6ad46837f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.72ex; height:2.843ex;" alt="{\displaystyle 2(2L+1)}"> se m è diverso da zero, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2L+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>L</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2L+1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9aa6df5ad70917b8f36b81d05d75bd34e140ad17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.557ex; height:2.843ex;" alt="{\displaystyle (2L+1)}"> se è invece nullo.</dd></dl> <ul><li>Il caso di asymmetrical top è più complesso, ed è necessario diagonalizzare la matrice di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{r}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d42a2f96162536ac26ef7cefedb159170597124" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.331ex; height:2.509ex;" alt="{\displaystyle T_{r}}"> nella base delle autofunzioni di L e Lz.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Spettro_elettromagnetico_molecolare">Spettro elettromagnetico molecolare</h2>[<a href="/w/index.php?title=Molecola&veaction=edit&section=16" title="Modifica la sezione Spettro elettromagnetico molecolare" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Molecola&action=edit&section=16" title="Edit section's source code: Spettro elettromagnetico molecolare">modifica wikitesto</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r130657691"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r139142988"> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /> Lo stesso argomento in dettaglio: <a href="/wiki/Spettroscopia" title="Spettroscopia">Spettroscopia</a>.</div> </div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Vibrationrotationenergy.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Vibrationrotationenergy.svg/310px-Vibrationrotationenergy.svg.png" decoding="async" width="310" height="404" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Vibrationrotationenergy.svg/465px-Vibrationrotationenergy.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Vibrationrotationenergy.svg/620px-Vibrationrotationenergy.svg.png 2x" data-file-width="744" data-file-height="969" /></a><figcaption>Diagramma delle transizioni energetiche roto-vibrazionali in una molecola tra due stati vibrazionali. Le transizioni del Q branch non sono permesse, in quanto <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta L=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>L</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta L=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f144590a6bf1ae1fafca7fa37551ff06c7d51d60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.78ex; height:2.176ex;" alt="{\displaystyle \Delta L=0}"> non è permesso, mentre si ha <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta L=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>L</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta L=-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fef8e0959bfaef048e5e06108400e18072aad7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.588ex; height:2.343ex;" alt="{\displaystyle \Delta L=-1}"> per il P branch e <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta L=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>L</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta L=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45be1d9ff1fe224e2c9f3eb97addbfdc7d16a827" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.78ex; height:2.176ex;" alt="{\displaystyle \Delta L=1}"> per il Q branch.</figcaption></figure> Lo spettro elettromagnetico molecolare è generato dalle transizioni tra due autostati dell'energia totale. Nel caso si studi lo <a href="/wiki/Spettro_di_emissione" title="Spettro di emissione">spettro di emissione</a> la molecola passa da uno stato eccitato allo stato fondamentale, mentre nel caso si studi lo <a href="/wiki/Spettro_di_assorbimento" title="Spettro di assorbimento">spettro di assorbimento</a> si osserva la transizione inversa. Tale passaggio comporta l'emissione o l'assorbimento di un <a href="/wiki/Fotone" title="Fotone">fotone</a>, la cui <a href="/wiki/Frequenza" title="Frequenza">frequenza</a> è data dalla <a href="/wiki/Legge_di_Planck" title="Legge di Planck">legge di Planck</a>: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu ={\frac {\Delta E}{h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>E</mi> </mrow> <mi>h</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu ={\frac {\Delta E}{h}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/160da598141e3fffd5fda89ce100ed4e84f85fab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.878ex; height:5.509ex;" alt="{\displaystyle \nu ={\frac {\Delta E}{h}}}"></dd></dl> dove <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/345f0ed5ff6ec6eddc5f908d379f032c52f119c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.711ex; height:2.176ex;" alt="{\displaystyle \Delta E}"> è la differenza di energia tra i due stati di partenza e arrivo: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta E=E'_{\mathrm {e} }-E_{\mathrm {e} }+E'_{vib}-E_{vib}+E'_{rot}-E_{rot}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>E</mi> <mo>=</mo> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mo>′</mo> </msubsup> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo>+</mo> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mi>i</mi> <mi>b</mi> </mrow> <mo>′</mo> </msubsup> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mi>i</mi> <mi>b</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> <mo>′</mo> </msubsup> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta E=E'_{\mathrm {e} }-E_{\mathrm {e} }+E'_{vib}-E_{vib}+E'_{rot}-E_{rot}\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/794779a137e28f22582da2d82fe576e229e85c5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:43.143ex; height:2.843ex;" alt="{\displaystyle \Delta E=E'_{\mathrm {e} }-E_{\mathrm {e} }+E'_{vib}-E_{vib}+E'_{rot}-E_{rot}\ }"></dd></dl> Le transizioni elettroniche dallo stato fondamentale ai primi stati eccitati sono dell'ordine di alcuni <a href="/wiki/Elettronvolt" title="Elettronvolt">eV</a>, e sono osservate nella regione del visibile e dell'<a href="/wiki/Radiazione_ultravioletta" title="Radiazione ultravioletta">ultravioletto</a> dello <a href="/wiki/Spettro_elettromagnetico" title="Spettro elettromagnetico">spettro elettromagnetico</a>, mentre le transizioni roto-vibrazionali sono osservate nella regione dell'<a href="/wiki/Infrarosso" class="mw-redirect" title="Infrarosso">infrarosso</a>.<a href="#cite_note-settantanove-36">[36]</a> Le transizioni tra due autostati dell'energia totale vengono studiate attraverso le transizioni tra autostati del <a href="/wiki/Momento_di_dipolo_elettrico" class="mw-redirect" title="Momento di dipolo elettrico">momento di dipolo elettrico</a>, definito come:<a href="#cite_note-cimiraglia-6">[6]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {d} =\mathbf {d} _{\mathrm {e} }+\mathbf {d} _{\mathrm {n} }=-e\sum _{i}{\mathbf {r_{i}} }+\sum _{\alpha }{eZ_{\alpha }\mathbf {R_{\alpha }} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>e</mi> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> </msub> </mrow> </mrow> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {d} =\mathbf {d} _{\mathrm {e} }+\mathbf {d} _{\mathrm {n} }=-e\sum _{i}{\mathbf {r_{i}} }+\sum _{\alpha }{eZ_{\alpha }\mathbf {R_{\alpha }} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/828a828fe14cad9ed5c343f37e1d9d7a5852b77b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:38.307ex; height:5.509ex;" alt="{\displaystyle \mathbf {d} =\mathbf {d} _{\mathrm {e} }+\mathbf {d} _{\mathrm {n} }=-e\sum _{i}{\mathbf {r_{i}} }+\sum _{\alpha }{eZ_{\alpha }\mathbf {R_{\alpha }} }}"></dd></dl> con e la carica dell'elettrone. Tale operatore è esplicitato dall'espressione: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {d} =\int {\psi _{vib}'^{*}\psi _{rot}'^{*}}\left[\int \psi _{el}^{*}\mathbf {d} \psi _{el}dx_{e}\right]\psi _{vib}\psi _{rot}d\tau =\langle {\psi _{vib}'\psi _{rot}'}|\mathbf {\mu } |\psi _{vib}\psi _{rot}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mi>i</mi> <mi>b</mi> </mrow> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mrow> </msubsup> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mrow> </msubsup> </mrow> <mrow> <mo>[</mo> <mrow> <mo>∫</mo> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mi>l</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mi>l</mi> </mrow> </msub> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mi>i</mi> <mi>b</mi> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> <mi>d</mi> <mi>τ</mi> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mi>i</mi> <mi>b</mi> </mrow> <mo>′</mo> </msubsup> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> <mo>′</mo> </msubsup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mi>i</mi> <mi>b</mi> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {d} =\int {\psi _{vib}'^{*}\psi _{rot}'^{*}}\left[\int \psi _{el}^{*}\mathbf {d} \psi _{el}dx_{e}\right]\psi _{vib}\psi _{rot}d\tau =\langle {\psi _{vib}'\psi _{rot}'}|\mathbf {\mu } |\psi _{vib}\psi _{rot}\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16e3a71f413c36ee9c61f3107e06dcdcd39d8a6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:64.802ex; height:6.176ex;" alt="{\displaystyle \mathbf {d} =\int {\psi _{vib}'^{*}\psi _{rot}'^{*}}\left[\int \psi _{el}^{*}\mathbf {d} \psi _{el}dx_{e}\right]\psi _{vib}\psi _{rot}d\tau =\langle {\psi _{vib}'\psi _{rot}'}|\mathbf {\mu } |\psi _{vib}\psi _{rot}\rangle }"></dd></dl> dove <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\mu } }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\mu } }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e407f1e8e83fffbed7e61f4112b7eef6b22b9e67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mathbf {\mu } }"> è l'operatore di momento dipolare elettronico della molecola: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\mu } =\int \psi _{el}^{*}\mathbf {d} \psi _{el}dx_{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mo>=</mo> <mo>∫</mo> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mi>l</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mi>l</mi> </mrow> </msub> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\mu } =\int \psi _{el}^{*}\mathbf {d} \psi _{el}dx_{e}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf934330bd49bcf5f2c9774127e76f90017b34c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.114ex; height:5.676ex;" alt="{\displaystyle \mathbf {\mu } =\int \psi _{el}^{*}\mathbf {d} \psi _{el}dx_{e}}"></dd></dl> Ognuno dei livelli vibrazionali che caratterizzano una superficie adiabatica è associato a diversi stati rotazionali. Nel diagramma spettroscopico le transizioni rotazionali costituiscono due rami: il primo è detto R Branch, e rappresenta le transizioni rotazionali tra i numeri quantici <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |l_{i}\rangle \to |l_{i}+1\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |l_{i}\rangle \to |l_{i}+1\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dde51cbd65b410666d470bcf8ddcb8c10a09183c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.706ex; height:2.843ex;" alt="{\displaystyle |l_{i}\rangle \to |l_{i}+1\rangle }">, mentre il secondo, detto P branch, rappresenta le transizioni <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |l_{i}\rangle \to |l_{i}-1\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |l_{i}\rangle \to |l_{i}-1\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65b529dfc347d0cc48b262dfc1e10438bddb370a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.706ex; height:2.843ex;" alt="{\displaystyle |l_{i}\rangle \to |l_{i}-1\rangle }">. Tra i due rami vi è un vuoto, motivato dal fatto che la transizione <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta l=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>l</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta l=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/966f254d09d15f5065fc8b3eeec2b57d89c3df87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.89ex; height:2.176ex;" alt="{\displaystyle \Delta l=0}"> è proibita dalle regole di selezione.<a href="#cite_note-37">[37]</a> Quando la transizione viene effettuata da un elettrone, essa genera anche transizioni tra autostati dell'energia roto-vibrazionale dei nuclei: tali transizioni sono dette vibroniche, e sono causate dal fatto che a due differenti superfici adiabatiche corrispondono geometrie diverse della molecola. In particolare, nelle molecole biatomiche, corrispondono a distanze internucleari differenti. <div class="mw-heading mw-heading3"><h3 id="Spettro_nucleare">Spettro nucleare</h3>[<a href="/w/index.php?title=Molecola&veaction=edit&section=17" title="Modifica la sezione Spettro nucleare" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Molecola&action=edit&section=17" title="Edit section's source code: Spettro nucleare">modifica wikitesto</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Ir_hcl_rot-vib_mrtz.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/29/Ir_hcl_rot-vib_mrtz.svg/480px-Ir_hcl_rot-vib_mrtz.svg.png" decoding="async" width="480" height="198" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/29/Ir_hcl_rot-vib_mrtz.svg/720px-Ir_hcl_rot-vib_mrtz.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/29/Ir_hcl_rot-vib_mrtz.svg/960px-Ir_hcl_rot-vib_mrtz.svg.png 2x" data-file-width="807" data-file-height="333" /></a><figcaption>Spettro di assorbimento rotovibrazionale della molecola di HCl nella transizione tra lo stato fondamentale ed il primo stato eccitato: a sinistra l'R Branch e a destra il P branch.</figcaption></figure> <div class="mw-heading mw-heading4"><h4 id="Spettro_nelle_molecole_biatomiche">Spettro nelle molecole biatomiche</h4>[<a href="/w/index.php?title=Molecola&veaction=edit&section=18" title="Modifica la sezione Spettro nelle molecole biatomiche" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Molecola&action=edit&section=18" title="Edit section's source code: Spettro nelle molecole biatomiche">modifica wikitesto</a>]</div> Nel caso di molecole biatomiche omonucleari il momento di dipolo elettrico è nullo per motivi di simmetria,<a href="#cite_note-38">[38]</a> e questo fatto spiega la trasparenza dell'<a href="/wiki/Atmosfera_terrestre" title="Atmosfera terrestre">atmosfera terrestre</a>, composta prevalentemente da O2 e N2. Nelle molecole biatomiche eteronucleari, invece, l'elemento di matrice della componente <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{z}=d(r)cos\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{z}=d(r)cos\theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9178cbc9f4cfd4f8d48d757325b035032f01fe4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.698ex; height:2.843ex;" alt="{\displaystyle d_{z}=d(r)cos\theta }"> lungo l'asse z del momento di dipolo è:<a href="#cite_note-cimiraglia-6">[6]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle v'l'm'|d_{z}|vlm\rangle =\int g_{v'}(r)d(r)g_{v}(r)dr\int Y_{l'm'}^{*}(\theta ,\phi )Y_{lm}(\theta ,\phi )\cos \theta \sin \theta d\theta d\phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msup> <mi>v</mi> <mo>′</mo> </msup> <msup> <mi>l</mi> <mo>′</mo> </msup> <msup> <mi>m</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>v</mi> <mi>l</mi> <mi>m</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo>∫</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>v</mi> <mo>′</mo> </msup> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>r</mi> <mo>∫</mo> <msubsup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>l</mi> <mo>′</mo> </msup> <msup> <mi>m</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>,</mo> <mi>ϕ</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> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mi>d</mi> <mi>θ</mi> <mi>d</mi> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle v'l'm'|d_{z}|vlm\rangle =\int g_{v'}(r)d(r)g_{v}(r)dr\int Y_{l'm'}^{*}(\theta ,\phi )Y_{lm}(\theta ,\phi )\cos \theta \sin \theta d\theta d\phi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbcb5d8025724b5586c998a8befb87ebe5b36883" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:73.937ex; height:5.676ex;" alt="{\displaystyle \langle v'l'm'|d_{z}|vlm\rangle =\int g_{v'}(r)d(r)g_{v}(r)dr\int Y_{l'm'}^{*}(\theta ,\phi )Y_{lm}(\theta ,\phi )\cos \theta \sin \theta d\theta d\phi }"></dd></dl> dove <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |vlm\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>v</mi> <mi>l</mi> <mi>m</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |vlm\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a64c307f0bce4a51967a924ab9e65afa4850dff4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.413ex; height:2.843ex;" alt="{\displaystyle |vlm\rangle }"> sono gli autostati simultanei dell'energia vibrazionale e rotazionale. Lo stesso accade per le componenti x e y. Dalle proprietà delle armoniche sferiche e dallo sviluppo di <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(r)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3bedebe43a3ade30faa9295adad0e84f710b290" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.074ex; height:2.843ex;" alt="{\displaystyle d(r)}"> attorno alla distanza di equilibrio si ottengono le regole di selezione: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta l=\pm 1\qquad \Delta m=0,\pm 1\qquad \Delta v=0,\pm 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>l</mi> <mo>=</mo> <mo>±</mo> <mn>1</mn> <mspace width="2em" /> <mi mathvariant="normal">Δ</mi> <mi>m</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mo>±</mo> <mn>1</mn> <mspace width="2em" /> <mi mathvariant="normal">Δ</mi> <mi>v</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mo>±</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta l=\pm 1\qquad \Delta m=0,\pm 1\qquad \Delta v=0,\pm 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f1d87c54f7459036df4c5f6ec339f2f47024b5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:41.559ex; height:2.509ex;" alt="{\displaystyle \Delta l=\pm 1\qquad \Delta m=0,\pm 1\qquad \Delta v=0,\pm 1}"></dd></dl> che definiscono le transizione permesse tra autostati dell'operatore associato all'<a href="/wiki/Osservabile" title="Osservabile">osservabile</a> dipolo elettrico. <div class="mw-heading mw-heading4"><h4 id="Spettro_nelle_molecole_poliatomiche">Spettro nelle molecole poliatomiche</h4>[<a href="/w/index.php?title=Molecola&veaction=edit&section=19" title="Modifica la sezione Spettro nelle molecole poliatomiche" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Molecola&action=edit&section=19" title="Edit section's source code: Spettro nelle molecole poliatomiche">modifica wikitesto</a>]</div> L'operatore di momento dipolare elettronico di una molecola poliatomica è dato da:<a href="#cite_note-cimiraglia-6">[6]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\mu } =\mu _{a}\mathbf {e} _{a}+\mu _{b}\mathbf {e} _{b}+\mu _{c}\mathbf {e} _{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\mu } =\mu _{a}\mathbf {e} _{a}+\mu _{b}\mathbf {e} _{b}+\mu _{c}\mathbf {e} _{c}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/579da271610c28cfbff11a5d1fe881426b58efe8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.029ex; height:2.509ex;" alt="{\displaystyle \mathbf {\mu } =\mu _{a}\mathbf {e} _{a}+\mu _{b}\mathbf {e} _{b}+\mu _{c}\mathbf {e} _{c}}"></dd></dl> in cui <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {e} _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {e} _{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba77c5a75ef9e230d1a36183785477a2eb3c5c1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.025ex; height:2.009ex;" alt="{\displaystyle \mathbf {e} _{i}}"> sono i <a href="/wiki/Versore" title="Versore">versori</a> degli assi d'inerzia. Il momento di dipolo elettrico diventa: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {d} =\langle {\psi _{vib}'\psi _{rot}'}|\mu _{a}\mathbf {e} _{a}+\mu _{b}\mathbf {e} _{b}+\mu _{c}\mathbf {e} _{c}|\psi _{vib}\psi _{rot}\rangle =\langle \psi _{vib}'|\mu _{a}|\psi _{vib}\rangle \langle \psi _{rot}'|\mathbf {e} _{a}|\psi _{rot}\rangle +\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mi>i</mi> <mi>b</mi> </mrow> <mo>′</mo> </msubsup> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> <mo>′</mo> </msubsup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mi>i</mi> <mi>b</mi> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mi>i</mi> <mi>b</mi> </mrow> <mo>′</mo> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mi>i</mi> <mi>b</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo fence="false" stretchy="false">⟨</mo> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> <mo>′</mo> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {d} =\langle {\psi _{vib}'\psi _{rot}'}|\mu _{a}\mathbf {e} _{a}+\mu _{b}\mathbf {e} _{b}+\mu _{c}\mathbf {e} _{c}|\psi _{vib}\psi _{rot}\rangle =\langle \psi _{vib}'|\mu _{a}|\psi _{vib}\rangle \langle \psi _{rot}'|\mathbf {e} _{a}|\psi _{rot}\rangle +\dots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e843d7c4baa25e673355cd702c861e10d518d4d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:77.688ex; height:3.009ex;" alt="{\displaystyle \mathbf {d} =\langle {\psi _{vib}'\psi _{rot}'}|\mu _{a}\mathbf {e} _{a}+\mu _{b}\mathbf {e} _{b}+\mu _{c}\mathbf {e} _{c}|\psi _{vib}\psi _{rot}\rangle =\langle \psi _{vib}'|\mu _{a}|\psi _{vib}\rangle \langle \psi _{rot}'|\mathbf {e} _{a}|\psi _{rot}\rangle +\dots }"></dd></dl> Detto <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132d0144479d6f47c30ad82a65d458966ccbe928" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.008ex; height:2.509ex;" alt="{\displaystyle \mathbf {Q} }"> il vettore delle coordinate normali, le cui componenti sono: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{i}=\sum _{\alpha =1}^{3N}R_{\alpha \beta }q_{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mi>N</mi> </mrow> </munderover> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{i}=\sum _{\alpha =1}^{3N}R_{\alpha \beta }q_{\alpha }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f5e67930a4ef646555d890389e473702e01af1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.79ex; height:7.343ex;" alt="{\displaystyle Q_{i}=\sum _{\alpha =1}^{3N}R_{\alpha \beta }q_{\alpha }}"></dd></dl> ed espandendo in serie di Taylor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (Q_{1},\dots Q_{3N-6})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo stretchy="false">(</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mi>N</mi> <mo>−</mo> <mn>6</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu (Q_{1},\dots Q_{3N-6})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ff458465bc09d8f06e2488cb7925e93a65e2370" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.7ex; height:2.843ex;" alt="{\displaystyle \mu (Q_{1},\dots Q_{3N-6})}"> attorno alla posizione di equilibrio: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\mu } =\mu _{eq}\langle \psi _{vib}'|\psi _{vib}\rangle +\sum _{i=1}^{3N}\left({\frac {\partial \mu _{a}}{\partial Q_{i}}}\right)\langle \psi _{vib}'|Q_{i}|\psi _{vib}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mi>q</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟨</mo> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mi>i</mi> <mi>b</mi> </mrow> <mo>′</mo> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mi>i</mi> <mi>b</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mi>N</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo fence="false" stretchy="false">⟨</mo> <msubsup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mi>i</mi> <mi>b</mi> </mrow> <mo>′</mo> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mi>i</mi> <mi>b</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\mu } =\mu _{eq}\langle \psi _{vib}'|\psi _{vib}\rangle +\sum _{i=1}^{3N}\left({\frac {\partial \mu _{a}}{\partial Q_{i}}}\right)\langle \psi _{vib}'|Q_{i}|\psi _{vib}\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dc31fda93859b6b81ef99b0e393356c5fa07e78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:46.299ex; height:7.343ex;" alt="{\displaystyle \mathbf {\mu } =\mu _{eq}\langle \psi _{vib}'|\psi _{vib}\rangle +\sum _{i=1}^{3N}\left({\frac {\partial \mu _{a}}{\partial Q_{i}}}\right)\langle \psi _{vib}'|Q_{i}|\psi _{vib}\rangle }"></dd></dl> si ottengono i due termini che generano le transizioni. Le transizioni dovute al primo termine del secondo membro sono nella regione delle <a href="/wiki/Microonde" title="Microonde">microonde</a> dello spettro, mentre le transizioni dovute al secondo termine nell'<a href="/wiki/Radiazione_infrarossa" title="Radiazione infrarossa">infrarosso</a>. Il secondo termine fornisce inoltre le regole di selezione relative all'oscillatore armonico corrispondente: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta v_{i}=\pm 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo>±</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta v_{i}=\pm 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd4fd4e8d6b091591704e9736f049307ff4daf7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.932ex; height:2.509ex;" alt="{\displaystyle \Delta v_{i}=\pm 1}">. Per quanto riguarda lo spettro rotazionale, si ha che gli spherical top ed i symmetrical top planari hanno dipolo nullo, e pertanto non generano transizioni di dipolo. Nel caso di symmetrical top non planari, il dipolo è diretto lungo l'asse di simmetria, e le transizioni tra autostati degli operatori <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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{z}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77a5b940110c5e1fe03782a31c5e700939ae20e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.585ex; height:2.509ex;" alt="{\displaystyle L_{z}}"> ed <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{c}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40dfe0c821ab1ddeb7ab27ddaa97764086f42789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.527ex; height:2.509ex;" alt="{\displaystyle L_{c}}"> sono rispettivamente: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta L=\pm 1\qquad \Delta M=0,\pm 1\qquad \Delta m=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>L</mi> <mo>=</mo> <mo>±</mo> <mn>1</mn> <mspace width="2em" /> <mi mathvariant="normal">Δ</mi> <mi>M</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mo>±</mo> <mn>1</mn> <mspace width="2em" /> <mi mathvariant="normal">Δ</mi> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta L=\pm 1\qquad \Delta M=0,\pm 1\qquad \Delta m=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b3ecc670c4392bcef9bcb9933e99a365b4fc35f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:39.759ex; height:2.509ex;" alt="{\displaystyle \Delta L=\pm 1\qquad \Delta M=0,\pm 1\qquad \Delta m=0}"></dd></dl> e si rilevano nella regione delle microonde dello spettro. <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Franck-Condon-diagram.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Franck-Condon-diagram.png/290px-Franck-Condon-diagram.png" decoding="async" width="290" height="358" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Franck-Condon-diagram.png/435px-Franck-Condon-diagram.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Franck-Condon-diagram.png/580px-Franck-Condon-diagram.png 2x" data-file-width="2677" data-file-height="3307" /></a><figcaption>Transizioni elettroniche tra due superfici adiabatiche approssimate dal <a href="/wiki/Potenziale_di_Morse" title="Potenziale di Morse">potenziale di Morse</a>, in cui si evidenzia la sovrapposizione delle funzioni d'onda che sta alla base del <a href="/wiki/Principio_di_Franck_Condon" class="mw-redirect" title="Principio di Franck Condon">principio di Franck Condon</a>.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Spettro_elettronico">Spettro elettronico</h3>[<a href="/w/index.php?title=Molecola&veaction=edit&section=20" title="Modifica la sezione Spettro elettronico" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Molecola&action=edit&section=20" title="Edit section's source code: Spettro elettronico">modifica wikitesto</a>]</div> Una transizione elettronica molecolare consiste in una transizione da parte dell'elettrone tra due <a href="/wiki/Superficie_di_energia_potenziale" title="Superficie di energia potenziale">superfici adiabatiche</a>. Tali transizioni sono simili a quelle atomiche, e consistono nella promozione di un elettrone da un <a href="/wiki/Orbitale_molecolare" title="Orbitale molecolare">orbitale molecolare</a> ad un altro orbitale vuoto.<a href="#cite_note-settantanove-36">[36]</a> Le regole di selezione si ricavano osservando che l'operatore di spin totale: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {S} =\sum _{i}s_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {S} =\sum _{i}s_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2541bb3fa1b5f632cd3f6fe741d4e97eee48278" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:10.216ex; height:5.509ex;" alt="{\displaystyle \mathbf {S} =\sum _{i}s_{i}}"></dd></dl> commuta con l'hamiltoniana elettronica e con <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {d} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {d} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ec3b626fc045b6ff579316e29978fccfed884c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.176ex;" alt="{\displaystyle \mathbf {d} }">, l'operatore di dipolo non agisce sullo spin, e pertanto si ha che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \mathbf {S} =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \mathbf {S} =0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2e55768f4fdcd6e115ff5c16278f3e782c5200a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.682ex; height:2.176ex;" alt="{\displaystyle \Delta \mathbf {S} =0}">.<a href="#cite_note-cimiraglia-6">[6]</a> Per l'operatore di momento angolare nelle molecole biatomiche: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{z}=\sum _{i}L_{iz}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>z</mi> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{z}=\sum _{i}L_{iz}\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3486bb630f0d8c9a3a970f833bddaae7508587f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.158ex; height:5.509ex;" alt="{\displaystyle L_{z}=\sum _{i}L_{iz}\ }"></dd></dl> solo la componente lungo l'asse z commuta con <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{z}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d31b01121be60eacac8bc444a879e1dca908afc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.211ex; height:2.509ex;" alt="{\displaystyle d_{z}}">, ottenendo che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta M=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>M</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta M=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6055d28afbbe23c47b8d230efbdbfbe0727983ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.639ex; height:2.176ex;" alt="{\displaystyle \Delta M=0}">, mentre per le altre due componenti si ricava che <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta M=\pm 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>M</mi> <mo>=</mo> <mo>±</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta M=\pm 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a9ce15af0108b08ca189013e47e8ba4692fcab3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.447ex; height:2.176ex;" alt="{\displaystyle \Delta M=\pm 1}">. In definitiva si ha: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta M=0,\pm 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>M</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mo>±</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta M=0,\pm 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb51d62ddc2f84d3c997b2f44e66092510aba26b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.644ex; height:2.509ex;" alt="{\displaystyle \Delta M=0,\pm 1}"> <div class="mw-heading mw-heading4"><h4 id="Il_principio_di_Franck_Condon">Il principio di Franck Condon</h4>[<a href="/w/index.php?title=Molecola&veaction=edit&section=21" title="Modifica la sezione Il principio di Franck Condon" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Molecola&action=edit&section=21" title="Edit section's source code: Il principio di Franck Condon">modifica wikitesto</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r130657691"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r139142988"> <div class="hatnote noprint vedi-anche"> <div class="hatnote-content"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/18px-Magnifying_glass_icon_mgx2.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/27px-Magnifying_glass_icon_mgx2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Magnifying_glass_icon_mgx2.svg/36px-Magnifying_glass_icon_mgx2.svg.png 2x" data-file-width="286" data-file-height="280" /> Lo stesso argomento in dettaglio: <a href="/wiki/Principio_di_Franck_Condon" class="mw-redirect" title="Principio di Franck Condon">Principio di Franck Condon</a>.</div> </div> Il principio di Franck Condon afferma la probabilità associata ad una transizione vibrazionale, data da: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=\left\langle \psi '\right|\mathbf {d} \left|\psi \right\rangle =\int {\psi '^{*}}\mathbf {d} \psi d\tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mrow> <mo>⟨</mo> <msup> <mi>ψ</mi> <mo>′</mo> </msup> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> <mrow> <mo>|</mo> <mi>ψ</mi> <mo>⟩</mo> </mrow> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>ψ</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> <mi>ψ</mi> <mi>d</mi> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=\left\langle \psi '\right|\mathbf {d} \left|\psi \right\rangle =\int {\psi '^{*}}\mathbf {d} \psi d\tau }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71a1c1cf92dff5c9bb103e9a3cbd56add9deb183" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.033ex; height:5.676ex;" alt="{\displaystyle P=\left\langle \psi '\right|\mathbf {d} \left|\psi \right\rangle =\int {\psi '^{*}}\mathbf {d} \psi d\tau }"></dd></dl> aumenta all'aumentare della sovrapposizione delle funzioni d'onda dei rispettivi stati iniziale e finale. Questo comporta che i livelli vibrazionali associati allo stato finale sono favoriti nel momento in cui la transizione comporta un cambiamento minimo nelle coordinate nucleari. Una conseguenza del principio è che, ad esempio, come mostrato nella figura a sinistra, se le funzioni d'onda tra lo stato fondamentale della superficie adiabatica iniziale e il secondo stato eccitato della superficie adiabatica finale si sovrappongono, tale transizione è più probabile delle altre dal momento che minimizza la variazione delle coordinate dei nuclei. <div class="mw-heading mw-heading2"><h2 id="Note">Note</h2>[<a href="/w/index.php?title=Molecola&veaction=edit&section=22" title="Modifica la sezione Note" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Molecola&action=edit&section=22" title="Edit section's source code: Note">modifica wikitesto</a>]</div> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> <cite class="citation cita" style="font-style:normal"><a href="#CITEREFPauling,_Linus">Pauling, Linus</a></cite>. </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite class="citation cita" style="font-style:normal"><a href="#CITEREFEbbin,_Darrell">Ebbin, Darrell</a></cite>. </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite class="citation web" style="font-style:normal"> A. 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Mullins, <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontost0000breh">Introduction To The Structure Of Matter: A Course In Modern Physics</a>, Greenville, NC, U.S.A., John Wiley & Sons, 1989, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/978-0-471-60531-7" title="Speciale:RicercaISBN/978-0-471-60531-7">978-0-471-60531-7</a>.</cite></li> <li><cite id="CITEREFmanini" class="citation libro" style="font-style:normal">(<abbr title="inglese">EN</abbr>) Nicola Manini, Introduction to the Physics of Matter, Milano, CUSL, 2008, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/978-88-8132-552-8" title="Speciale:RicercaISBN/978-88-8132-552-8">978-88-8132-552-8</a>.</cite></li> <li><cite id="CITEREFspinicci" class="citation libro" style="font-style:normal"> Roberto Spinicci, Elementi di Chimica, Firenze, Firenze University Press, 2009, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/978-88-6453-062-8" title="Speciale:RicercaISBN/978-88-6453-062-8">978-88-6453-062-8</a>.</cite></li> <li><cite id="CITEREFlinus" class="citation libro" style="font-style:normal">(<abbr title="inglese">EN</abbr>) Pauling, Linus, General Chemistry, New York, Dover Publications, Inc., 1970, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/0-486-65622-5" title="Speciale:RicercaISBN/0-486-65622-5">0-486-65622-5</a>.</cite></li> <li><cite id="CITEREFebbin" class="citation libro" style="font-style:normal">(<abbr title="inglese">EN</abbr>) Ebbin, Darrell, D., General Chemistry, 3rd Ed., Boston, Houghton Mifflin Co., 1990, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/0-395-43302-9" title="Speciale:RicercaISBN/0-395-43302-9">0-395-43302-9</a>.</cite></li> <li><cite class="citation libro" style="font-style:normal">(<abbr title="inglese">EN</abbr>) Brown, T.L., Chemistry – the Central Science, 9th Ed., New Jersey, Prentice Hall, 2003, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/0-13-066997-0" title="Speciale:RicercaISBN/0-13-066997-0">0-13-066997-0</a>.</cite></li> <li><cite class="citation libro" style="font-style:normal">(<abbr title="inglese">EN</abbr>) Chang, Raymond, <a rel="nofollow" class="external text" href="https://archive.org/details/chemistry0006chan">Chemistry, 6th Ed.</a>, New York, McGraw Hill, 1998, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/0-07-115221-0" title="Speciale:RicercaISBN/0-07-115221-0">0-07-115221-0</a>.</cite></li> <li><cite class="citation libro" style="font-style:normal">(<abbr title="inglese">EN</abbr>) Zumdahl, Steven S., Chemistry, 4th ed., Boston, Houghton Mifflin, 1997, <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Speciale:RicercaISBN/0-669-41794-7" title="Speciale:RicercaISBN/0-669-41794-7">0-669-41794-7</a>.</cite></li></ul> <div class="mw-heading mw-heading2"><h2 id="Voci_correlate">Voci correlate</h2>[<a href="/w/index.php?title=Molecola&veaction=edit&section=24" title="Modifica la sezione Voci correlate" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Molecola&action=edit&section=24" title="Edit section's source code: Voci correlate">modifica wikitesto</a>]</div> <ul><li><a href="/wiki/Atomo" title="Atomo">Atomo</a></li> <li><a href="/wiki/Composto_organico" title="Composto organico">Composto organico</a></li> <li><a href="/wiki/Formula_chimica" title="Formula chimica">Formula chimica</a></li> <li><a href="/wiki/Macromolecola" title="Macromolecola">Macromolecola</a></li> <li><a href="/wiki/Molecola_biatomica" title="Molecola biatomica">Molecola biatomica</a></li> <li><a href="/wiki/Interazione_debole" title="Interazione debole">Interazione debole</a></li> <li><a href="/wiki/Isomeria" title="Isomeria">Isomeria</a></li> <li><a href="/wiki/Legame_chimico" title="Legame chimico">Legame chimico</a></li> <li><a href="/wiki/Simmetria_molecolare" title="Simmetria molecolare">Simmetria molecolare</a></li> <li><a href="/wiki/Storia_della_chimica" title="Storia della chimica">Storia della chimica</a></li> <li><a href="/wiki/Orbitale_molecolare" title="Orbitale molecolare">Orbitale molecolare</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Altri_progetti">Altri progetti</h2>[<a href="/w/index.php?title=Molecola&veaction=edit&section=25" title="Modifica la sezione Altri progetti" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Molecola&action=edit&section=25" title="Edit section's source code: Altri progetti">modifica wikitesto</a>]</div> <div id="interProject" class="toccolours" style="display: none; clear: both; margin-top: 2em">Altri progetti<ul title="Collegamenti verso gli altri progetti Wikimedia"> <li class="" title=""><a href="https://it.wiktionary.org/wiki/molecola" class="extiw" title="wikt:molecola">Wikizionario</a></li> <li class="" title=""><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Molecules?uselang=it">Wikimedia Commons</a></li></ul></div> <ul><li><a href="https://it.wiktionary.org/wiki/" title="Collabora a Wikizionario"><img alt="Collabora a Wikizionario" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Wiktionary_small.svg/18px-Wiktionary_small.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Wiktionary_small.svg/27px-Wiktionary_small.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Wiktionary_small.svg/36px-Wiktionary_small.svg.png 2x" data-file-width="350" data-file-height="350" /></a> <a href="https://it.wiktionary.org/wiki/" class="extiw" title="wikt:">Wikizionario</a> contiene il lemma di dizionario «<a href="https://it.wiktionary.org/wiki/molecola" class="extiw" title="wikt:molecola">molecola</a>»</li> <li><a href="https://commons.wikimedia.org/wiki/?uselang=it" title="Collabora a Wikimedia Commons"><img alt="Collabora a Wikimedia Commons" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png" decoding="async" width="18" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/27px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/36px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a> <a class="external text" href="https://commons.wikimedia.org/wiki/?uselang=it">Wikimedia Commons</a> contiene immagini o altri file sulla <a class="external text" href="https://commons.wikimedia.org/wiki/Category:Molecules?uselang=it">molecola</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Collegamenti_esterni">Collegamenti esterni</h2>[<a href="/w/index.php?title=Molecola&veaction=edit&section=26" title="Modifica la sezione Collegamenti esterni" class="mw-editsection-visualeditor">modifica</a> | <a href="/w/index.php?title=Molecola&action=edit&section=26" title="Edit section's source code: Collegamenti esterni">modifica wikitesto</a>]</div> <ul><li class="mw-empty-elt"></li> <li><cite id="CITEREFTreccani.it" class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://www.treccani.it/enciclopedia/molecola">molecola</a>, su Treccani.it – Enciclopedie on line, <a href="/wiki/Istituto_dell%27Enciclopedia_Italiana" title="Istituto dell'Enciclopedia Italiana">Istituto dell'Enciclopedia Italiana</a>.</cite> <a href="https://www.wikidata.org/wiki/Q11369#P3365" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></li> <li><cite id="CITEREFEnciclopedia_Italiana_1934" class="citation libro" style="font-style:normal"> Edoardo Amaldi, <a rel="nofollow" class="external text" href="https://www.treccani.it/enciclopedia/molecola_(Enciclopedia-Italiana)/">MOLECOLA</a>, in <a href="/wiki/Enciclopedia_Treccani" title="Enciclopedia Treccani">Enciclopedia Italiana</a>, <a href="/wiki/Istituto_dell%27Enciclopedia_Italiana" title="Istituto dell'Enciclopedia Italiana">Istituto dell'Enciclopedia Italiana</a>, 1934.</cite> <a href="https://www.wikidata.org/wiki/Q11369#P4223" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></li> <li><cite id="CITEREFEnciclopedia_Italiana_2000" class="citation libro" style="font-style:normal"> Amilcare Carpi De Resmini, <a rel="nofollow" class="external text" href="https://www.treccani.it/enciclopedia/molecola_res-adc63305-c2b6-11e2-a3c8-00271042e8d9_(Enciclopedia-Italiana)/">Molecola</a>, in <a href="/wiki/Enciclopedia_Treccani" title="Enciclopedia Treccani">Enciclopedia Italiana</a>, VI Appendice, <a href="/wiki/Istituto_dell%27Enciclopedia_Italiana" title="Istituto dell'Enciclopedia Italiana">Istituto dell'Enciclopedia Italiana</a>, 2000.</cite> <a href="https://www.wikidata.org/wiki/Q11369#P4223" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></li> <li><cite id="CITEREFBritannica.com" class="citation web" style="font-style:normal">(<abbr title="inglese">EN</abbr>) <a rel="nofollow" class="external text" href="https://www.britannica.com/science/molecule">molecule</a>, su <a href="/wiki/Enciclopedia_Britannica" title="Enciclopedia Britannica">Enciclopedia Britannica</a>, Encyclopædia Britannica, Inc.</cite> <a href="https://www.wikidata.org/wiki/Q11369#P1417" title="Modifica su Wikidata"><img alt="Modifica su Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/10px-Blue_pencil.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/15px-Blue_pencil.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Blue_pencil.svg/20px-Blue_pencil.svg.png 2x" data-file-width="600" data-file-height="600" /></a></li> <li><cite id="CITEREFTreccani_molecola_(Dizionario-delle-Scienze-Fisiche)" class="citation testo" style="font-style:normal"> <a rel="nofollow" class="external text" href="http://www.treccani.it/enciclopedia/molecola_(Dizionario-delle-Scienze-Fisiche)">Molecola</a>, in Dizionario delle scienze fisiche, Roma, Istituto dell'Enciclopedia Italiana, 1996.</cite></li> <li><cite id="CITEREFTreccani_molecole_(Enciclopedia-della-Scienza-e-della-Tecnica)" class="citation testo" style="font-style:normal"> <a rel="nofollow" class="external text" href="http://www.treccani.it/enciclopedia/molecole_(Enciclopedia-della-Scienza-e-della-Tecnica)">Molecola</a>, in Enciclopedia della scienza e della tecnica, Roma, Istituto dell'Enciclopedia Italiana, 2007-2008.</cite></li> <li><cite id="CITEREFTreccani_molecola_(Enciclopedia-dei-ragazzi)" class="citation testo" style="font-style:normal"> <a rel="nofollow" class="external text" href="http://www.treccani.it/enciclopedia/molecola_(Enciclopedia-dei-ragazzi)">Molecola</a>, in Enciclopedia dei ragazzi, Roma, Istituto dell'Enciclopedia Italiana, 2004-2006.</cite></li></ul> <style data-mw-deduplicate="TemplateStyles:r141815314">.mw-parser-output .navbox{border:1px solid #aaa;clear:both;margin:auto;padding:2px;width:100%}.mw-parser-output .navbox th{padding-left:1em;padding-right:1em;text-align:center}.mw-parser-output .navbox>tbody>tr:first-child>th{background:#ccf;font-size:90%;width:100%;color:var(--color-base,black)}.mw-parser-output .navbox_navbar{float:left;margin:0;padding:0 10px 0 0;text-align:left;width:6em}.mw-parser-output .navbox_title{font-size:110%}.mw-parser-output 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title="Tachione">Tachione</a> · <a href="/wiki/Bosoni_X_e_Y" title="Bosoni X e Y">Bosoni X e Y</a> · <a href="/wiki/Bosoni_W%27_e_Z%27" title="Bosoni W' e Z'">bosoni W' e Z'</a> · <a href="/wiki/Neutrino_sterile" title="Neutrino sterile">Neutrino sterile</a> · <a href="/wiki/Preone_(fisica)" title="Preone (fisica)">Preone</a> · <a href="/wiki/Fermione_di_Majorana" title="Fermione di Majorana">Fermione di Majorana</a></td></tr></tbody></table></td></tr><tr><th class="subnavbox_group" style="width:5em;font-weight:normal; width:9%;background:#e6e6ff">Altre</th><td colspan="1"><a href="/wiki/Ghost_di_Faddeev-Popov" title="Ghost di Faddeev-Popov">Ghost di Faddeev-Popov</a></td></tr></tbody></table></td><td rowspan="3" class="navbox_image"><a href="/wiki/File:Modello_Standard_delle_Particelle_Elementari.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/Modello_Standard_delle_Particelle_Elementari.svg/400px-Modello_Standard_delle_Particelle_Elementari.svg.png" decoding="async" width="400" height="383" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/Modello_Standard_delle_Particelle_Elementari.svg/600px-Modello_Standard_delle_Particelle_Elementari.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/40/Modello_Standard_delle_Particelle_Elementari.svg/800px-Modello_Standard_delle_Particelle_Elementari.svg.png 2x" data-file-width="1390" data-file-height="1330" /></a></td></tr><tr><th colspan="1" class="navbox_group" style="width:5em;font-weight:normal; width:10%"><a href="/wiki/Stato_legato" class="mw-redirect" title="Stato legato">Composte</a></th><td colspan="1" class="navbox_list navbox_even" style="background:transparent"><table class="subnavbox"><tbody><tr><th class="subnavbox_group" style="width:5em;font-weight:normal; width:9%;background:#e6e6ff"><a href="/wiki/Adrone" title="Adrone">Adroni</a></th><td colspan="1"><table class="subnavbox"><tbody><tr><th class="subnavbox_group" style="width:5em;font-weight:normal; width:12%;background:#e6e6ff"><a href="/wiki/Barione" title="Barione">Barioni</a>/ <a href="/wiki/Iperone_(fisica)" title="Iperone (fisica)">iperoni</a></th><td colspan="1"><a href="/wiki/Nucleone" title="Nucleone">Nucleone</a> (<a href="/wiki/Protone" title="Protone">Protone</a> · <a href="/wiki/Neutrone" title="Neutrone">Neutrone</a>) · <a href="/wiki/Barione_delta" title="Barione delta">Barione delta</a> · <a href="/wiki/Barione_lambda" title="Barione lambda">Barione lambda</a> · <a href="/wiki/Barione_sigma" title="Barione sigma">Barione sigma</a> · <a href="/wiki/Barione_Xi" title="Barione Xi">Barione Xi</a> · <a href="/wiki/Barione_omega" title="Barione omega">Barione omega</a></td></tr><tr><th class="subnavbox_group" style="width:5em;font-weight:normal; width:12%;background:#e6e6ff"><a href="/wiki/Mesone" title="Mesone">Mesoni</a>/ <a href="/wiki/Quarkonio" title="Quarkonio">quarkoni</a></th><td colspan="1"><a href="/wiki/Pione" title="Pione">Pione</a> · <a href="/wiki/Mesone_rho" title="Mesone rho">Mesone rho</a> · <a href="/wiki/Mesone_eta" title="Mesone eta">Mesoni eta e eta primo</a> · <a href="/wiki/Mesone_phi" title="Mesone phi">Mesone phi</a> · <a href="/wiki/Mesone_omega" title="Mesone omega">Mesone omega</a> · <a href="/wiki/Mesone_J/Psi" title="Mesone J/Psi">Mesone J/Psi</a> · <a href="/wiki/Mesone_upsilon" title="Mesone upsilon">Mesone upsilon</a> · <a href="/wiki/Kaone" title="Kaone">Kaone</a> · <a href="/wiki/Mesone_B" title="Mesone B">Mesone B</a> · <a href="/wiki/Mesone_D" title="Mesone D">Mesone D</a></td></tr><tr><th class="subnavbox_group" style="width:5em;font-weight:normal; width:12%;background:#e6e6ff"><a href="/wiki/Adrone_esotico" title="Adrone esotico">Adroni esotici</a></th><td colspan="1"><a href="/wiki/Tetraquark" title="Tetraquark">Tetraquark</a> · <a href="/wiki/Pentaquark" title="Pentaquark">Pentaquark</a></td></tr></tbody></table></td></tr><tr><th class="subnavbox_group" style="width:5em;font-weight:normal; width:9%;background:#e6e6ff"><a href="/wiki/Categoria:Particelle_composte_ipotetiche" title="Categoria:Particelle composte ipotetiche">Ipotetiche</a></th><td colspan="1"><table class="subnavbox"><tbody><tr><th class="subnavbox_group" style="font-weight:normal; width:12%;background:#e6e6ff"><a href="/wiki/Barione_esotico" title="Barione esotico">Barioni esotici</a></th><td colspan="1"><a href="/wiki/Esaquark" title="Esaquark">Esaquark</a></td></tr><tr><th class="subnavbox_group" style="font-weight:normal; width:12%;background:#e6e6ff"><a href="/wiki/Mesone_esotico" title="Mesone esotico">Mesoni esotici</a></th><td colspan="1"><a href="/wiki/Glueball" title="Glueball">Glueball</a> (<a href="/w/index.php?title=Odderone&action=edit&redlink=1" class="new" title="Odderone (la pagina non esiste)">Odderone</a>) · <a href="/wiki/Mesone_theta" title="Mesone theta">Mesone theta</a> · <a href="/wiki/Mesone_T" title="Mesone T">Mesone T</a></td></tr><tr><th class="subnavbox_group" style="font-weight:normal; width:12%;background:#e6e6ff">Altre</th><td colspan="1"><a href="/wiki/Molecola_mesonica" title="Molecola mesonica">Molecola mesonica</a> · <a href="/wiki/Pomerone" title="Pomerone">Pomerone</a></td></tr></tbody></table></td></tr><tr><th class="subnavbox_group" style="width:5em;font-weight:normal; width:9%;background:#e6e6ff">Altre</th><td colspan="1"><a href="/wiki/Nucleo_atomico" title="Nucleo atomico">Nuclei atomici</a> · <a href="/wiki/Atomo" title="Atomo">Atomi</a> · <a href="/wiki/Atomo_esotico" title="Atomo esotico">Atomi esotici</a> (<a href="/wiki/Positronio" title="Positronio">Positronio</a> · <a href="/wiki/Muonio_(fisica)" title="Muonio (fisica)">Muonio</a> · <a href="/wiki/Tauonio" class="mw-redirect" title="Tauonio">Tauonio</a> · <a href="/wiki/Pionio" title="Pionio">Pionio</a>) · <a href="/wiki/Atomolecola" title="Atomolecola">Atomolecole</a> · <a class="mw-selflink selflink">Molecole</a> · <a href="/wiki/Superatomo" title="Superatomo">Superatomo</a></td></tr></tbody></table></td></tr><tr><th colspan="1" class="navbox_group" style="width:5em;font-weight:normal; width:10%"><a href="/wiki/Quasiparticella" title="Quasiparticella">Quasiparticelle</a></th><td colspan="1" class="navbox_list navbox_odd" style="background:transparent"><a href="/wiki/Droplettone" title="Droplettone">Droplettone</a> · <a href="/wiki/Eccitone" title="Eccitone">Eccitone</a> · <a href="/wiki/Fasone" title="Fasone">Fasone</a> · <a href="/w/index.php?title=Frattone_(particella)&action=edit&redlink=1" class="new" title="Frattone (particella) (la pagina non esiste)">Frattone</a> · <a href="/wiki/Fonone" title="Fonone">Fonone</a> · <a href="/wiki/Lacuna_(fisica)" title="Lacuna (fisica)">Lacuna</a> · <a href="/wiki/Levitone" title="Levitone">Levitone</a> · <a href="/wiki/Magnone_(fisica)" title="Magnone (fisica)">Magnone</a> · <a href="/w/index.php?title=Plasmarone&action=edit&redlink=1" class="new" title="Plasmarone (la pagina non esiste)">Plasmarone</a> · <a href="/wiki/Plasmone" title="Plasmone">Plasmone</a> · <a href="/wiki/Polaritone" title="Polaritone">Polaritone</a> · <a href="/wiki/Polarone" title="Polarone">Polarone</a> · <a href="/wiki/Qualunquone" title="Qualunquone">Qualunquone</a> · <a href="/wiki/Rotone" title="Rotone">Rotone</a> · <a href="/wiki/Skyrmione" title="Skyrmione">Skyrmione</a> · <a href="/wiki/Skyrmione_magnetico" title="Skyrmione magnetico">Skyrmione magnetico</a> · <a href="/wiki/Solitone" title="Solitone">Solitone</a> · <a href="/wiki/Solitone_di_Davydov" title="Solitone di Davydov">Solitone di Davydov</a> · <a href="/wiki/Spinone_(fisica)" title="Spinone (fisica)">Spinone</a> · <a href="/wiki/Topolaritone" title="Topolaritone">Topolaritone</a></td></tr><tr><th colspan="3" class="navbox_abovebelow"><a href="/wiki/Cronologia_della_scoperta_delle_particelle" title="Cronologia della scoperta delle particelle">Cronologia della scoperta delle particelle</a></th></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r140554510">.mw-parser-output .CdA{border:1px solid #aaa;width:100%;margin:auto;font-size:90%;padding:2px}.mw-parser-output .CdA th{background-color:#f2f2f2;font-weight:bold;width:20%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .CdA{border-color:#54595D}html.skin-theme-clientpref-night .mw-parser-output .CdA th{background-color:#202122}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .CdA{border-color:#54595D}html.skin-theme-clientpref-os .mw-parser-output .CdA th{background-color:#202122}}</style><table class="CdA"><tbody><tr><th><a href="/wiki/Aiuto:Controllo_di_autorit%C3%A0" title="Aiuto:Controllo di autorità">Controllo di autorità</a></th><td><a href="/wiki/Nuovo_soggettario" title="Nuovo soggettario">Thesaurus BNCF</a> <a rel="nofollow" class="external text" href="https://thes.bncf.firenze.sbn.it/termine.php?id=4215">4215</a> · <a href="/wiki/Library_of_Congress_Control_Number" title="Library of Congress Control Number">LCCN</a> (<abbr title="inglese">EN</abbr>) <a rel="nofollow" class="external text" href="http://id.loc.gov/authorities/subjects/sh85086597">sh85086597</a> · <a href="/wiki/Gemeinsame_Normdatei" title="Gemeinsame Normdatei">GND</a> (<abbr title="tedesco">DE</abbr>) <a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4039972-2">4039972-2</a> · <a href="/wiki/Biblioteca_nazionale_di_Spagna" title="Biblioteca nazionale di Spagna">BNE</a> (<abbr title="spagnolo">ES</abbr>) <a rel="nofollow" class="external text" href="http://catalogo.bne.es/uhtbin/authoritybrowse.cgi?action=display&authority_id=XX524812">XX524812</a> <a rel="nofollow" class="external text" href="http://datos.bne.es/resource/XX524812">(data)</a> · <a href="/wiki/Biblioteca_nazionale_di_Francia" title="Biblioteca nazionale di Francia">BNF</a> (<abbr title="francese">FR</abbr>) <a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb119469207">cb119469207</a> <a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb119469207">(data)</a> · <a href="/wiki/Biblioteca_nazionale_di_Israele" title="Biblioteca nazionale di Israele">J9U</a> (<abbr title="inglese">EN</abbr>, <abbr title="ebraico">HE</abbr>) <a rel="nofollow" class="external text" href="http://olduli.nli.org.il/F/?func=find-b&local_base=NLX10&find_code=UID&request=987007541144005171">987007541144005171</a> · <a href="/wiki/Biblioteca_della_Dieta_nazionale_del_Giappone" title="Biblioteca della Dieta nazionale del Giappone">NDL</a> (<abbr title="inglese">EN</abbr>, <abbr title="giapponese">JA</abbr>) <a rel="nofollow" class="external text" href="https://id.ndl.go.jp/auth/ndlna/00561032">00561032</a></td></tr></tbody></table> <div class="noprint" style="width:100%; padding: 3px 0; display: flex; flex-wrap: wrap; row-gap: 4px; column-gap: 8px; box-sizing: border-box;"><div style="flex-basis: calc( 100% / 2 - 8px / 2 );"><style data-mw-deduplicate="TemplateStyles:r140555418">.mw-parser-output .itwiki-template-occhiello{width:100%;line-height:25px;border:1px solid #CCF;background-color:#F0EEFF;box-sizing:border-box}.mw-parser-output .itwiki-template-occhiello-progetto{background-color:#FAFAFA}@media screen{html.skin-theme-clientpref-night .mw-parser-output .itwiki-template-occhiello{background-color:#202122;border-color:#54595D}html.skin-theme-clientpref-night .mw-parser-output .itwiki-template-occhiello-progetto{background-color:#282929}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .itwiki-template-occhiello{background-color:#202122;border-color:#54595D}html.skin-theme-clientpref-os .mw-parser-output .itwiki-template-occhiello-progetto{background-color:#282929}}</style><div class="itwiki-template-occhiello"><a href="/wiki/File:Gnome-applications-science.svg" class="mw-file-description" title="Chimica"><img alt=" " src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Gnome-applications-science.svg/25px-Gnome-applications-science.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Gnome-applications-science.svg/38px-Gnome-applications-science.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2e/Gnome-applications-science.svg/50px-Gnome-applications-science.svg.png 2x" data-file-width="48" data-file-height="48" /></a> <a href="/wiki/Portale:Chimica" title="Portale:Chimica">Portale Chimica</a></div></div><div style="flex-basis: calc( 100% / 2 - 8px / 2 );"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r140555418"><div class="itwiki-template-occhiello"><a href="/wiki/File:Nuvola_apps_katomic.svg" class="mw-file-description" title="Fisica"><img alt=" " src="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Nuvola_apps_katomic.svg/25px-Nuvola_apps_katomic.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Nuvola_apps_katomic.svg/38px-Nuvola_apps_katomic.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/55/Nuvola_apps_katomic.svg/50px-Nuvola_apps_katomic.svg.png 2x" data-file-width="128" data-file-height="128" /></a> <a href="/wiki/Portale:Fisica" title="Portale:Fisica">Portale Fisica</a></div></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Estratto da "<a dir="ltr" href="https://it.wikipedia.org/w/index.php?title=Molecola&oldid=142133299">https://it.wikipedia.org/w/index.php?title=Molecola&oldid=142133299</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Categoria:Categorie" title="Categoria:Categorie">Categorie</a>: <ul><li><a href="/wiki/Categoria:Entit%C3%A0_molecolari" title="Categoria:Entità molecolari">Entità molecolari</a></li><li><a href="/wiki/Categoria:Concetti_fondamentali_di_chimica" title="Categoria:Concetti fondamentali di chimica">Concetti fondamentali di chimica</a></li><li><a href="/wiki/Categoria:Fisica_molecolare" title="Categoria:Fisica molecolare">Fisica molecolare</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Categorie nascoste: <ul><li><a href="/wiki/Categoria:Voci_con_modulo_citazione_e_parametro_coautori" title="Categoria:Voci con modulo citazione e parametro coautori">Voci con modulo citazione e parametro coautori</a></li><li><a href="/wiki/Categoria:P3365_letta_da_Wikidata" title="Categoria:P3365 letta da Wikidata">P3365 letta da Wikidata</a></li><li><a href="/wiki/Categoria:P4223_letta_da_Wikidata" title="Categoria:P4223 letta da Wikidata">P4223 letta da Wikidata</a></li><li><a href="/wiki/Categoria:P1417_letta_da_Wikidata" title="Categoria:P1417 letta da Wikidata">P1417 letta da Wikidata</a></li><li><a href="/wiki/Categoria:Voci_con_codice_Thesaurus_BNCF" title="Categoria:Voci con codice Thesaurus BNCF">Voci con codice Thesaurus BNCF</a></li><li><a href="/wiki/Categoria:Voci_con_codice_LCCN" title="Categoria:Voci con codice LCCN">Voci con codice LCCN</a></li><li><a href="/wiki/Categoria:Voci_con_codice_GND" title="Categoria:Voci con codice GND">Voci con codice GND</a></li><li><a href="/wiki/Categoria:Voci_con_codice_BNE" title="Categoria:Voci con codice BNE">Voci con codice BNE</a></li><li><a href="/wiki/Categoria:Voci_con_codice_BNF" title="Categoria:Voci con codice BNF">Voci con codice BNF</a></li><li><a href="/wiki/Categoria:Voci_con_codice_J9U" title="Categoria:Voci con codice J9U">Voci con codice J9U</a></li><li><a href="/wiki/Categoria:Voci_con_codice_NDL" title="Categoria:Voci con codice NDL">Voci con codice NDL</a></li><li><a href="/wiki/Categoria:Voci_non_biografiche_con_codici_di_controllo_di_autorit%C3%A0" title="Categoria:Voci non biografiche con codici di controllo di autorità">Voci non biografiche con codici di controllo di autorità</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Questa pagina è stata modificata per l'ultima volta il 12 nov 2024 alle 09:37.</li> <li id="footer-info-copyright">Il testo è disponibile secondo la <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.it">licenza 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Molecola - Wikipedia