{"id":243,"date":"2023-04-06T10:34:43","date_gmt":"2023-04-06T02:34:43","guid":{"rendered":"http:\/\/47.93.1.71\/?p=243"},"modified":"2023-04-06T10:39:58","modified_gmt":"2023-04-06T02:39:58","slug":"243","status":"publish","type":"post","link":"https:\/\/blog.changelife.top\/index.php\/2023\/04\/06\/243\/","title":{"rendered":"Spin-orbit"},"content":{"rendered":"\n<!DOCTYPE html>\n<html  lang=\"en\">\n<head prefix=\"og: http:\/\/ogp.me\/ns# article: http:\/\/ogp.me\/ns\/article#\">\n\n        <meta name=\"viewport\" content=\"width=device-width, initial-scale=1\"\/>\n        <meta name=\"generator\" content=\"MindTouch\"\/>\n        <meta name=\"robots\" content=\"index,follow\"\/>\n        <meta property=\"og:title\" content=\"Spin-orbit Coupling\"\/>\n        <meta property=\"og:site_name\" content=\"Chemistry LibreTexts\"\/>\n        <meta property=\"og:type\" content=\"article\"\/>\n        <meta property=\"og:locale\" content=\"en_US\"\/>\n        <meta property=\"og:url\" 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the background color of the dropdown button when the dropdown content is shown *\/\n.LTdropdown:hover .dropbtn {\n    filter: brightness(85%);\n}\n\nnav.elm-header-user-nav > div.elm-nav-container > ol {\n\tdisplay: flex;\n}\n\n\nnav.elm-header-user-nav > div.elm-nav-container > ol > li {\n\tdisplay: flex;\n}\n\nli.elm-global-search {\n\tflex: 1\n}\n\n#mt-site-search-input {\n\twidth: auto;\n}\n\n\n.mt-breadcrumbs a span.nb-comment-highlight {\n   font-size: unset;\n   margin-right: unset;\n   vertical-align: unset;\n}\n\n@font-face {\n    font-family: 'moreicons';\n    src: url(https:\/\/cdn.libretexts.net\/github\/LibreTextsMain\/Miscellaneous\/icosso.woff) format(\"woff\");\n    font-weight: normal;\n    font-style: normal\n}\n.icon-SSO:before {\n    font-family: 'moreicons' !important;\n    content: \"\\e95d\";\n    vertical-align: middle;\n    cursor: pointer;\n    margin-right: 0.25em;\n}\n.sso-user {\n    border: 2px solid white;\n    border-radius: 5px; \n    vertical-align: middle;  \n}\n.sso-user:hover:after {\n    content: \" (Logout)\";\n}\n\n.authenticated-sso.elm-header-user-nav {\n    background-color: #0f67a6;\n        \n}\n.authenticated-sso.elm-header-user-nav .mt-user-menu-user  {\n    background: #323232;\n    border-radius: 5px;\n}\n.authenticated-sso.elm-header-user-nav .mt-user-menu > li > a {\n    color: white;\n}\n\n\/*Media Queries*\/\n@media (max-width: 600px) {\n\t.mobileOnly {\n\t\tdisplay: block;\n\t}\n\n\t.mobileHide {\n\t\tdisplay: none;\n\t}\n\n\t.LTBranding {\n\t\tflex-direction: column;\n\t}\n\n}\n\n@media (max-width: 875px) {\n\t#RemixerIcon, #DownloadIcon, #SSOInstructorIcon {\n\t\twidth: 1.95em;\n\t}\n}\n\n@media (min-width: 1000px) {\n\tnav.elm-header-global-nav ol.mt-breadcrumbs {\n\t\tmax-width: 85% !important;\n\t}\n\n\t#TOCbutton {\n\t\twidth: auto;\n\t}\n}\n\n@media (max-width: 1001px) {\n\n\t#TOCbutton {\n\t\twidth: 20px;\n\t\tjustify-content: flex-start;\n\t}\n}\n\/*\n@media (min-width: 83em) {\n   article.elm-content-container {\n      min-width: unset;\n   }\n}\n*\/\n\/*\n@media (min-width: 37.5em){\n    .elm-header-user-nav {\n        z-index: 1;\n    }\n}\n*\/\n\n.mt-content-container img {\n    height: auto!important;\n    object-fit: contain;\n}\n.mt-content-container .mt-sortable-listing-link img {\n    height: 150px!important;\n}\n\n.mt-video-widget > p {\n    text-align: center;\n}\n\noption[value=\"article:topic-portfolio\"] {\n    display: none;\n}\noption[value=\"article:howto\"] {\n    display: none;\n}\noption[value=\"article:reference\"] {\n    display: none;\n}\n\n.live-tag-link {\n    white-space: normal;\n    user-select: text;\n    -webkit-user-drag: none;\n}\n\nspan.mt-font-ma-shan-zheng{\n    font-family: 'Ma Shan Zheng';\n        \n}\n\n.cke_librebox .cke_combo_open {\n    margin-left: 0;\n}\n\n.cke_librebox .cke_combo_inlinelabel {\n    padding-left: 2px;\n    padding-right: 0;\n}\n\n.cke_librequery .cke_combo_open {\n    margin-left: 0;\n}\n\n.cke_librequery .cke_combo_inlinelabel {\n    padding-left: 2px;\n    padding-right: 0;\n}\n\n.cke_button_icon.cke_button__librequeryoptions_icon {\n  background-image: url('https:\/\/cdn.libretexts.net\/Icons\/query.ico') !important;\n  background-position: 0px !important;\n  background-size: 16px !important;\n}\n\n\n\/** Heading 7 **\/\n.lt-heading-7 {\n    text-align: left!important;\n    font: normal normal 700 112.5%\/1.2 Tahoma,Arial,serif;\n    color: #000;\n    margin: 2rem 0 0.5rem;\n    padding: 0;\n}\n\n@media (min-width: 37.5em) {\n    .lt-heading-7 {\n        font-size: 112.5% !important;\n    }\n}\n\n@media (min-width: 18.75em) {\n    .lt-heading-7 {\n        font-size: 105.5% !important;\n    }\n}\n\n\n\/** AutoAttribution **\/ \n#librelens-attribution-list li {\n    font-size: 1rem !important;\n}\/*]]>*\/<\/style><style type=\"text\/css\">\/*<![CDATA[*\/\n.printOnly {\n\tdisplay: none;\n}\n\n.hideprint {\n\tvisibility: visible !important;\n\tdisplay: block !important;\n}\n\n.showprint {\n\tvisibility: hidden !important;\n\tdisplay: none !important;\n}\n\n@media print {\n\th1#title {\n\t\tmargin-top: 0!important\n\t}\n\t.box-interactive,\n\t.box-query,\n\t.box-adapt {\n\t\tvisibility: hidden !important;\n\t\tdisplay: none !important;\n\t}\n\ttable.mt-responsive-table {\n\t\t\/*table-layout: auto;\n        *\/\n\t}\n\t.elm-meta-top .mt-article-pagination * {\n\t\tdisplay: none;\n\t}\n\t.mt-author-container {\n\t\tbackground: #f0f0f0;\n\t\tborder-radius: .5em;\n\t\tvisibility: hidden !important;\n\t\tdisplay: none !important;\n\t}\n\t#elm-main-content {\n\t\tpadding-bottom: 0;\n\t}\n\tmain.elm-skin-container {\n\t\tdisplay: inline;\n\t}\n\t.mt-video-widget:not(.cke_widget_element) {\n\t\tmargin: 0 auto !important;\n\t\twidth: 60% !important;\n\t\tpage-break-inside: avoid;\n\t}\n\t.elm-skin-container {\n\t\tbackground-color: #ffffff !important;\n\t}\n\t.mt-sortable-listing {\n\t\tbackground-color: #ffffff !important;\n\t}\n\t.twocolumn {\n\t\t-webkit-column-count: 2;\n\t\t\/* Chrome, Safari, Opera *\/\n\t\t-moz-column-count: 2;\n\t\t\/* Firefox *\/\n\t\tcolumn-count: 2;\n\t}\n\t.threecolumn {\n\t\t-webkit-column-count: 3;\n\t\t\/* Chrome, Safari, Opera *\/\n\t\t-moz-column-count: 3;\n\t\t\/* Firefox *\/\n\t\tcolumn-count: 3;\n\t}\n\t.hideprint {\n\t\tvisibility: hidden !important;\n\t\tdisplay: none !important;\n\t}\n\t.showprint {\n\t\tvisibility: visible !important;\n\t\tdisplay: block !important;\n\t}\n\t.printOnly {\n\t\tdisplay: block;\n\t}\n\t.printHide {\n\t\tdisplay: none;\n\t}\n\timg {\n\t\tzoom: 0.60;\n\t\tpage-break-inside: avoid;\n\t}\n\t.mt-author-container img {\n\t\tzoom: 1;\n\t}\n\t.MathJax_Display {\n\t\tmargin-top: 0.7em !important;\n\t\tmargin-bottom: 0.7em !important;\n\t}\n\t#flash-messages {\n\t\tvisibility: hidden !important;\n\t\tdisplay: none !important;\n\t}\n\tblockquote {\n\t\tdisplay: block;\n\t\tfont-size: 120%;\n\t\tcolor: black;\n\t\tmargin-top: 0px;\n\t\tmargin-bottom: 0px;\n\t\tmargin-left: 40px;\n\t\tpadding-left: 15px;\n\t\tpadding-top: 0px;\n\t\tpadding-bottom: 0px;\n\t\tborder-left: 3px solid #0f67a6;\n\t\tborder-top: 0px;\n\t\tborder-bottom: 0px;\n\t\tborder-right: 0px;\n\t}\n\tp {\n\t\ttext-align: justify;\n\t\torphans: 2;\n\t\twidows: 2;\n\t\tmargin-block-start: 0.5em;\n\t\tmargin-block-end: 0.5em;\n\t\tmargin-inline-start: 0px;\n\t\tmargin-inline-end: 0px;\n\t}\n\t.definition,\n\t.note1,\n\t.example,\n\t.exercise,\n\t.skills {\n\t\tborder: solid 1px hsla(0, 0%, 0%, 0.5);\n\t\t-moz-border-radius: 5px;\n\t\t-webkit-border-radius: 5px;\n\t\tborder-radius: 5px;\n\t\tpadding-left: 7px;\n\t\tpadding-right: 7px;\n\t\tpadding-bottom: 0px;\n\t\tpadding-top: 0px;\n\t\tmargin-bottom: 5px;\n\t\tmargin-top: 5px;\n\t}\n\t.definition {\n\t\tbackground-color: hsla(7, 93%, 41%, 0.03);\n\t}\n\t.example {\n\t\tbackground-color: hsla(215, 93%, 39%, 0.03);\n\t}\n\t.note1 {\n\t\tbackground-color: hsla(78, 93%, 39%, 0.03);\n\t}\n\t.exercise {\n\t\tbackground-color: hsla(283, 93%, 24%, 0.03);\n\t}\n\t.skills {\n\t\tbackground-color: hsla(7, 93%, 41%, 0.03);\n\t}\n\tol.elm-meta-top {\n\t\tdisplay: none;\n\t}\n\tbody:not(.cke_editable) .mt-responsive-table th,\n\tbody:not(.cke_editable) .mt-responsive-table td,\n\tbody:not(.cke_editable) table th[data-th] {\n\t\tdisplay: table-cell !important;\n\t\tbackground-image: none !important;\n\t\tfont-size: 95%;\n\t}\n\t#mt-page-health-container,\n\theader.elm-header,\n\t#mt-toc-container,\n\t#mt-content-side,\n\tdiv#mt-summary,\n\tli.elm-pdf-export,\n\tli.elm-social-share,\n\tfooter.elm-footer,\n\tfooter.elm-content-footer,\n\tli.elm-meta-icons,\n\tli.elm-last-modified,\n\tli.elm-page-restriction,\n\tbody:not(.cke_editable) .mt-responsive-table td:before {\n\t\tdisplay: none !important;\n\t}\n\tbody {\n\t\tfont-size: 0.80em !important;\n\t\tfont-family: 'Big Caslon', 'Book Antiqua', 'Palatino Linotype', Georgia, serif !important;\n\t}\n\t.mt-content-container ol,\n\t.mt-content-container p,\n\t.mt-content-container ul {\n\t\tfont-family: 'Big Caslon', 'Book Antiqua', 'Palatino Linotype', Georgia, serif !important;\n\t\tfont-size: 100%;\n\t\tline-height: 1.4;\n\t}\n\th1,\n\th2,\n\th3,\n\th4,\n\th5 {\n\t\tcolor: hsl(205, 82%, 42%) !important;\n\t\talign-content: left;\n\t\tfont-family: \"Tahoma\", Arial, serif !important;\n\t\ttext-align: left !important;\n\t\tmargin: 0.8em 0 0 0 !important;\n\t\tpage-break-after: avoid !important;\n\t\tpage-break-inside: avoid !important;\n\t\tbreak-after: avoid !important;\n\t\tbreak-after: avoid-page !important;\n\t\torphans: 1;\n\t\tmargin-block-end: -3px !important;\n\t}\n\th1 {\n\t\tfont-size: 18px !important;\n\t}\n\th2 {\n\t\tfont-size: 15px !important;\n\t}\n\th3 {\n\t\tfont-size: 13px !important;\n\t}\n\th4 {\n\t\tfont-size: 11px !important;\n\t}\n\th5 {\n\t\tfont-size: 10px !important;\n\t}\n\t.mt-new-features-container {\n\t\tdisplay: none;\n\t}\n\t\/*https:\/\/stackoverflow.com\/questions\/9238868\/how-do-i-avoid-a-page-break-immediately-after-a-heading\/53742871#53742871*\/\n\th2,\n\th3,\n\th4,\n\th5,\n\th6 {\n\t\tpage-break-inside: avoid;\n\t}\n\th2::after {\n\t\tcontent: \"\";\n\t\tdisplay: block;\n\t\theight: 2.5rem;\n\t\t\/* pretend that the header is at least 8rem high so this header cannot fit near the end of page *\/\n\t\tmargin-bottom: -2.5rem;\n\t\t\/* however, reduce the margin the same amount so that the following content can stay in its usual place *\/\n\t}\n\th3::after {\n\t\tcontent: \"\";\n\t\tdisplay: block;\n\t\theight: 2.5rem;\n\t\t\/* pretend that the header is at least 8rem high so this header cannot fit near the end of page *\/\n\t\tmargin-bottom: -2.5rem;\n\t\t\/* however, reduce the margin the same amount so that the following content can stay in its usual place *\/\n\t}\n\th4::after {\n\t\tcontent: \"\";\n\t\tdisplay: block;\n\t\theight: 2.5rem;\n\t\t\/* pretend that the header is at least 8rem high so this header cannot fit near the end of page *\/\n\t\tmargin-bottom: -2.5rem;\n\t\t\/* however, reduce the margin the same amount so that the following content can stay in its usual place *\/\n\t}\n\t.box-legend::before {\n\t\tcontent: \"\";\n\t\tdisplay: block;\n\t\theight: 5rem;\n\t\t\/* pretend that the header is at least 8rem high so this header cannot fit near the end of page *\/\n\t\tmargin-bottom: -5rem;\n\t\t\/* however, reduce the margin the same amount so that the following content can stay in its usual place *\/\n\t\tpage-break-inside: avoid;\n\t\tbreak-inside: avoid;\n\t}\n\tp.boxtitle {\n\t\tfont-style: normal;\n\t\tfont-size: 120%;\n\t\tcolor: DarkSlateGray;\n\t\tline-height: 80% !important;\n\t\tbackground: none;\n\t\tmargin-top: 0 !important;\n\t\tmargin-bottom: 0 !important;\n\t\tpadding: 0 !important;\n\t\ttext-align: left !important;\n\t\tborder-bottom: 1px solid DarkSlateGray;\n\t\tfont-family: \"Tahoma\", Arial, serif !important;\n\t\tpage-break-after: avoid;\n\t\tpage-break-inside: avoid;\n\t\tbreak-after: avoid-page;\n\t\torphans: 1;\n\t}\n\tdt {\n\t\tcolor: initial;\n\t}\n\tdl,\n\tdt {\n\t\ttext-align: left !important;\n\t}\n\t.row {\n\t\tpadding: 0;\n\t\tmargin: 0;\n\t\tlist-style: none;\n\t\tdisplay: -webkit-box;\n\t\tdisplay: -moz-box;\n\t\tdisplay: -ms-flexbox;\n\t\tdisplay: -webkit-flex;\n\t\tdisplay: flex;\n\t\t-webkit-flex-flow: row wrap;\n\t\tjustify-content: center;\n\t\tbreak-inside: avoid;\n\t}\n\t#elm-main-content figcaption {\n\t\tcaption-side: bottom;\n\t\ttext-align: justify;\n\t\tfont-size: 90%;\n\t\tmargin-left: 5%;\n\t\tmargin-right: 5%;\n\t\tmargin-top: 0.3em;\n\t\tbreak-inside: avoid-page;\n\t}\n\t.rightfig {\n\t\tdisplay: table;\n\t\tfloat: right;\n\t\tmargin: 0em 0 0em 1em;\n\t}\n\t.rightfigcaption {\n\t\tdisplay: table-caption;\n\t\tcaption-side: bottom;\n\t\tmargin: 0.3em 0 0 0;\n\t\ttext-align: justify;\n\t\tfont-size: 90%;\n\t}\n\t\/**************************** table bar ***************************\/\n\tcaption {\n\t\tdisplay: table-caption;\n\t\ttext-align: center;\n\t\tfont-family: 'Big Caslon', 'Book Antiqua', 'Palatino Linotype', Georgia, serif !important;\n\t}\n\tbody:not(.cke_editable) .mt-responsive-table td,\n\tbody:not(.cke_editable) .mt-responsive-table th {\n\t\tpadding: 0.4em !important;\n\t\tpadding-top: 0.3em !important;\n\t\tpadding-right: 0.4em !important;\n\t\tpadding-bottom: 0.3em !important;\n\t\tpadding-left: 0.4em !important;\n\t\tvertical-align: middle;\n\t}\n\tbody:not(.cke_editable) .mt-responsive-table p {\n\t\tmargin: 0em !important;\n\t}\n\tbody:not(.cke_editable) .mt-responsive-table tr td:last-child {\n\t\tborder: none !important;\n\t}\n\tbody:not(.cke_editable) .mt-responsive-table tr th:last-child {\n\t\tborder-bottom-width: 0.15em !important;\n\t}\n\ttable {\n\t\tmargin-top: 0.5em !important;\n\t\tmargin-bottom: 1em !important;\n\t\t-webkit-print-color-adjust: exact;\n\t\tpage-break-inside: none;\n\t}\n\t.mt-content-container table p {\n\t\tmargin: 0em !important;\n\t}\n\tth,\n\ttd {\n\t\tfont-family: 'Big Caslon', 'Book Antiqua', 'Palatino Linotype', Georgia, serif !important;\n\t\t\/*text-align: center;\n        *\/\n\t\t\/*vertical-align: middle;\n        *\/\n\t}\n\t\/**************************** lists ***************************\/\n\tul,\n\tol {\n\t\ttext-align: left !important;\n\t\tmargin: 0.3em 0 0.5em 0;\n\t}\n\tol li,\n\tul li {\n\t\ttext-align: left !important;\n\t\tfont-family: 'Big Caslon', 'Book Antiqua', 'Palatino Linotype', Georgia, serif !important;\n\t\tmargin-top: 0em !important;\n\t\tmargin-bottom: 0em !important;\n\t}\n\t.attributions li,\n\t.attributions p {\n\t\tline-height: 1.25 !important;\n\t\tmargin-block-start: 0em !important;\n\t\tmargin-block-end: 0em !important;\n  \t\tfont-size: 80% !important;\n\n\t}\n\t\/**************************** author bar ***************************\/\n\t.mt-author-container {\n\t\t\/*background: @secondary-container-background-color;*\/\n\t\t\/*border: 1px solid @secondary-container-border-color;*\/\n\t\t\/*border-radius: @border-radius-default;*\/\n\t\tclear: both;\n\t\t\/*color: @tertiary-font-color;*\/\n\t\tfont-size: 85%;\n\t\tmargin: 0 0 1em;\n\t\toverflow: hidden;\n\t\tpadding: .5em 1em;\n\t}\n\t.mt-author-container ul {\n\t\tlist-style: none;\n\t\tmargin: 0;\n\t\tpadding: 0;\n\t}\n\t.mt-author-container .mt-author-image {\n\t\tfloat: left;\n\t\tmargin: 0 1em 0 0;\n\t}\n\t.mt-author-container .mt-author-image-crop {\n\t\t\/*border: 1px solid @primary-container-border-color;*\/\n\t\tborder-radius: 50%;\n\t\theight: 3em;\n\t\toverflow: hidden;\n\t\tposition: relative;\n\t\twidth: 3em;\n\t}\n\t.mt-author-container .mt-author-image-crop img {\n\t\tdisplay: inline;\n\t\theight: 100%;\n\t\tmargin: 0 auto;\n\t\twidth: auto;\n\t}\n\t.mt-author-container img {\n\t\tmax-height: 3em;\n\t\tmax-width: 3em;\n\t\tvertical-align: middle;\n\t}\n\telm-skin-container {\n\t\tdisplay: block !important;\n\t}\n\thypothesis-sidebar,\n\thypothesis-adder,\n\t.h-sidebar-iframe,\n\t.annotator-frame {\n\t\tdisplay: none;\n\t}\n\t\/* canvas, img, object, video {\n         max-width: 100%;\n         max-height: 500px;\n    }\n     *\/\n\t\/**************** Box container styles *******************\/\n\t.mt-content-container .boxobjectives,\n\t.boxexample,\n\t.boxexercise,\n\t.boxquery,\n\t.boxnotewithlegend,\n\t.boxnotewithoutlegend,\n\t.boxprocedure,\n\t.boxdefinition,\n\t.boxtheorem,\n\t.boxlemma,\n\t.boxnotation,\n\t.boxproposition {\n\t\t-moz-border-radius: 8px;\n\t\t-webkit-border-radius: 8px;\n\t\tborder-radius: 8px;\n\t\tpadding-top: 0em;\n\t\tpadding-bottom: 0em;\n\t\tpadding-left: 1em;\n\t\tpadding-right: 1em;\n\t\tmargin-bottom: 1em;\n\t\toverflow: visible;\n\t\tborder: 0px;\n\t\torphans: 2;\n\t}\n\t.mt-content-container .boxnotewithlegend,\n\t.boxprocedure,\n\t.notation {\n\t\tbackground-color: hsla(20, 0%, 93%, 1);\n\t}\n\t.mt-content-container .boxexample,\n\t.boxexercise {\n\t\tbackground-color: hsla(220, 100%, 97%, 1);\n\t}\n\t.mt-content-container .boxwarning {\n\t\tbackground-color: hsla(0, 100%, 96%, 1);\n\t}\n\t.mt-content-container .boxdefinition,\n\t.boxtheorem,\n\t.boxlemma,\n\t.boxnotation,\n\t.boxproposition {\n\t\tbackground-color: hsla(28, 80%, 95%, 1);\n\t}\n\t.mt-content-container .boxobjectives {\n\t\tbackground-color: hsla(0, 0%, 90%, 1);\n\t}\n\t.mt-content-container .boxnotewithoutlegend {\n\t\tbackground-color: hsla(20, 0%, 93%, 1);\n\t\tpadding-right: 1em;\n\t\tpadding-left: 1em;\n\t\tpadding-top: 0.5em !important;\n\t\tpadding-bottom: 0.5em !important;\n\t\tmargin-right: 1em;\n\t\tmargin-left: 1em;\n\t}\n\t.mt-content-container .boxquery {\n\t\tborder: 2px solid green;\n\t\tpage-break-inside: avoid;\n\t\tvisibility: hidden !important;\n\t\tdisplay: none !important;\n\t}\n\t\/**************** Box legend container styles *******************\/\n\t.mt-content-container .boxlegend {\n\t\tfont-style: normal !important;\n\t\tfont-size: 1.1em !important;\n\t\tfont-weight: normal !important;\n\t\tfont-family: \"Tahoma\", Arial, serif !important;\n\t\tpadding-top: 0.2em !important;\n\t\tpadding-bottom: 0.2em !important;\n\t\tpadding-left: 0.3em !important;\n\t\tpadding-right: 0.3em !important;\n\t\tmargin-left: 0.35em;\n\t\tcolor: white;\n\t\tborder-radius: 3px;\n\t\toverflow: visible;\n\t\tbreak-inside: avoid !important;\n\t}\n\t.boxnotewithlegend .boxlegend {\n\t\tbackground-color: hsla(20, 0%, 45%, 1);\n\t}\n\t.boxnotewithoutlegend .boxlegend {\n\t\tbackground-color: hsla(20, 0%, 45%, 1);\n\t\tfont-size: 0em !important;\n\t\tvisibility: hidden !important;\n\t}\n\t.boxprocedure .boxlegend {\n\t\tbackground-color: hsla(20, 0%, 45%, 1);\n\t}\n\t.boxnotation .boxlegend {\n\t\tbackground-color: hsla(20, 0%, 45%, 1);\n\t}\n\t.boxexample .boxlegend {\n\t\tbackground-color: hsla(220, 100%, 60%, 1);\n\t}\n\t.boxexercise .boxlegend {\n\t\tbackground-color: hsla(0, 100%, 56%, 1);\n\t}\n\t.boxquery .boxlegend {\n\t\tbackground-color: hsl(120, 100%, 25%, 1);\n\t}\n\t.boxobjectives .boxlegend {\n\t\tbackground-color: hsla(0, 0%, 10%, 1);\n\t}\n\t.boxdefinition .boxlegend {\n\t\tbackground-color: hsla(28, 80%, 52%, 1);\n\t}\n\t.boxtheorem .boxlegend {\n\t\tbackground-color: hsla(28, 80%, 52%, 1);\n\t}\n\t.boxlemma .boxlegend {\n\t\tbackground-color: hsla(28, 80%, 52%, 1);\n\t}\n\t.boxnotation .boxlegend {\n\t\tbackground-color: hsla(28, 80%, 52%, 1);\n\t}\n\t.boxproposition .boxlegend {\n\t\tbackground-color: hsla(28, 80%, 52%, 1);\n\t}\n\t\/****************************************** Current New boxes **********************************************************************\/\n\t.box-objectives,\n\t.box-emphasis,\n\t.box-structure,\n\t.box-example,\n\t.box-exercise,\n\t.box-warning,\n\t.box-query,\n\t.box-note,\n\t.box-procedure,\n\t.box-definition,\n\t.box-theorem,\n\t.box-lemma,\n\t.box-notation,\n\t.box-proposition,\n\t.box-interactive {\n\t\tdisplay: block;\n\t\tborder-top: none;\n\t\tborder-bottom: none;\n\t\tborder-right: none;\n\t\tpadding: 0px 10px 1px 8px;\n\t\tmargin-bottom: 0.8em !important;\n\t\tmargin-top: 0.8em !important;\n\t\torphans: 3;\n\t\tborder-radius: 4px 8px 8px 4px;\n\t\t-moz-border-radius: 4px 8px 8px 4px;\n\t\t-webkit-border-radius: 4px 8px 8px 4px;\n\t}\n\t.box-emphasis {\n\t\tpadding: 5px 10px 5px 10px;\n\t}\n\t\/**************** Tuning the box colors ***************\/\n\t.box-objectives {\n\t\tbackground-color: hsla(204, 83%, 97%, 1);\n\t\tborder-left: 4px solid hsla(204, 83%, 41%, 1);\n\t}\n\t.box-example {\n\t\tbackground-color: hsla(204, 83%, 97%, 1);\n\t\tborder-left: 4px solid hsla(204, 83%, 41%, 1);\n\t}\n\t.box-exercise {\n\t\tbackground-color: hsla(204, 83%, 97%, 1);\n\t\tborder-left: 4px solid hsla(204, 83%, 41%, 1);\n\t}\n\t.box-warning {\n\t\tbackground-color: hsl(0, 100%, 96%);\n\t\tborder-left: 4px solid hsl(0, 100%, 31%);\n\t}\n\t.box-note {\n\t\tbackground-color: hsl(180, 8%, 94%);\n\t\tborder-left: 4px solid hsl(180, 8%, 40%);\n\t}\n\t.box-emphasis {\n\t\tbackground-color: hsl(180, 8%, 94%);\n\t\tborder-left: 4px solid hsl(180, 8%, 40%);\n\t}\n\t.box-structure {\n\t\tbackground-color: hsl(240, 67%, 97%);\n\t\tborder-left: 4px solid hsl(240, 67%, 62%);\n\t}\n\t.box-query {\n\t\tborder: 3px solid hsla(220, 100%, 60%, 1);\n\t}\n\t.box-interactive {\n\t\tborder: 3px solid black;\n\t}\n\t.box-definition,\n\t.box-theorem,\n\t.box-lemma,\n\t.box-notation,\n\t.box-proposition {\n\t\tbackground-color: hsl(124, 33%, 94%);\n\t\tborder-left: 4px solid hsl(132, 36%, 27%);\n\t}\n\t.boxprocedure,\n\t.notation {\n\t\tbackground-color: hsla(20, 0%, 93%, 1);\n\t\tborder-left: 4px solid hsl(43, 81%, 36%, 1)\n\t}\n\t\/************** Defining the Legends ******************\/\n\t.box-legend {\n\t\tdisplay: block;\n\t\toverflow: hidden;\n\t\tmargin: 0 0 5px 0;\n\t\ttransform: translate(-11px, 0px);\n\t\tbreak-inside: avoid-page !important;\n\t\tpage-break-inside: avoid !important;\n\t}\n\t.box-legend>span {\n\t\tfloat: left;\n\t\tfont-family: \"Tahoma\", Arial, serif !important;\n\t\tfont-size: 13px !important;\n\t\tfont-weight: normal !important;\n\t\tcolor: white;\n\t\tpadding: 3px 5px 3px 5px !important;\n\t\tmargin: 0 0 3px 0;\n\t\tline-height: 0.8em;\n\t\tborder-radius: 1px;\n\t}\n\t.box-legend p {\n\t\tfloat: left;\n\t\tfont-family: \"Tahoma\", Arial, serif !important;\n\t\tfont-size: 14px !important;\n\t\tfont-weight: normal !important;\n\t\tcolor: white;\n\t\tfont-size: 14px !important;\n\t\tpadding: 4px 5px 4px 5px !important;\n\t\tmargin: 0 3px 3px 0;\n\t\tline-height: 0.8em;\n\t\tborder-radius: 1px;\n\t}\n\t\/************** Tuning the Legends ******************\/\n\t.box-objectives .box-legend>span {\n\t\tbackground-color: hsl(212, 77%, 47%, 1);\n\t}\n\t.box-example .box-legend>span {\n\t\tbackground-color: hsl(212, 77%, 47%, 1);\n\t}\n\t.box-exercise .box-legend>span {\n\t\tbackground-color: hsl(212, 77%, 47%, 1);\n\t}\n\t.box-query .box-legend>span {\n\t\tbackground-color: hsl(212, 77%, 47%, 1);\n\t}\n\t.box-interactive .box-legend>span {\n\t\tbackground-color: black;\n\t}\n\t.box-note .box-legend>span {\n\t\tbackground-color: hsl(180, 8%, 40%);\n\t}\n\t.box-warning .box-legend>span {\n\t\tbackground-color: hsl(0, 100%, 31%);\n\t}\n\t.box-structure .box-legend>span {\n\t\tbackground-color: hsl(240, 67%, 62%);\n\t}\n\t.box-procedure .box-legend>span {\n\t\tbackground-color: hsla(39, 100%, 58%, 1);\n\t}\n\t.box-definition .box-legend>span {\n\t\tbackground-color: hsl(132, 36%, 27%);\n\t}\n\t.box-theorem .box-legend>span {\n\t\tbackground-color: hsl(132, 36%, 27%);\n\t}\n\t.box-lemma .box-legend>span {\n\t\tbackground-color: hsl(132, 36%, 27%);\n\t}\n\t.box-notation .box-legend>span {\n\t\tbackground-color: hsl(132, 36%, 27%);\n\t}\n\t.box-proposition .box-legend>span {\n\t\tbackground-color: hsl(132, 36%, 27%);\n\t}\n\t.box-definition>header,\n\t.box-structure>header,\n\t.box-note>header,\n\t.box-theorem>header,\n\t.box-example>header,\n\t.box-exercise>header,\n\t.box-query>header,\n\t.box-adapt>header,\n\t.box-objectives>header,\n\t.box-warning>header {\n\t\tdisplay: inline-flex;\n\t\toverflow: hidden;\n\t\tbreak-inside: avoid-page;\n\t\tpage-break-inside: avoid;\n\t\ttext-align: left;\n\t\twidth: auto;\n\t\tmargin: 0 0px 2px 0px !important;\n\t\tpadding: 2px 6px 2px 4px !important;\n\t\ttransform: translate(-11px, 0px) !important;\n\t\tborder-radius: 3px 5px 5px 0;\n\t\t-moz-border-radius: 3px 5px 5px 0;\n\t\t-webkit-border-radius: 3px 5px 5px 0;\n\t}\n\t.box-definition>header *,\n\t.box-structure>header *,\n\t.box-note>header *,\n\t.box-theorem>header *,\n\t.box-example>header *,\n\t.box-exercise>header *,\n\t.box-query>header *,\n\t.box-adapt>header *,\n\t.box-objectives>header *,\n\t.box-warning>header * {\n\t\tfont-family: \"Tahoma\", Arial, serif !important;\n\t\tfont-size: 14px !important;\n\t\tfont-weight: 400 !important;\n\t\tcolor: white !important;\n\t\t;\n\t\twidth: auto !important;\n\t\t;\n\t\tfloat: left !important;\n\t\t;\n\t\tline-height: 1 !important;\n\t\t;\n\t\tmargin-block-start: 0 !important;\n\t\tmargin-block-end: 0 !important;\n\t}\n\t.box-definition>header,\n\t.box-theorem>header {\n\t\tbackground-color: hsl(132, 36%, 27%);\n\t}\n\t.box-definition>header * {\n\t\tbackground-color: hsl(132, 36%, 27%);\n\t}\n\t.box-example>header,\n\t.box-exercise>header,\n\t.box-query>header,\n\t.box-objectives>header,\n\t.box-adapt>header {\n\t\tbackground-color: hsla(204, 83%, 41%, 1);\n\t}\n\t.box-note>header {\n\t\tbackground-color: hsl(180, 8%, 40%);\n\t}\n\t.box-example>header *,\n\t.box-objectives>header *,\n\t.box-exercise>header * {\n\t\tbackground-color: hsla(204, 83%, 41%, 1);\n\t}\n\t.box-note>header * {\n\t\tbackground-color: hsl(180, 8%, 40%);\n\t}\n\t.box-structure>header {\n\t\tbackground-color: hsl(240, 67%, 62%);\n\t}\n\t.box-structure>header 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(max-width: 1350px) {\n\t#topHeader > ol {\n\t\tfont-size: 0px;\n\t}\n}\n\n@media only screen and (min-width: 480px) and (max-width: 767px) {\n\t.partners-logo a {\n\t\tmax-width: 150px;\n\t}\n}\ndiv.elm-nav-container > a > img {\n\tmax-width: unset;\n}\n\nnav.elm-nav.my-custom-class {\n\tpadding: .25rem 0;\n}\n\n#bottomFooter p.mt-align-center {\n    margin: 5px;\n    line-height: initial;\n}\n\n.LTBranding {\n\tdisplay: flex;\n\tjustify-content: space-between;\n\talign-items: center;\n\theight: min-content;\n}\n\n.LTBranding img{\n\tobject-fit: contain;\n}\/*]]>*\/<\/style><style type=\"text\/css\">\/*<![CDATA[*\/\nul.jmolPopupMenu {\noverflow: initial;\nfloat: initial;\nmax-height: none;\nopacity: initial;\n}\nul.jmolPopupMenu .ui-menu, .ui-selectmenu-button {\noverflow: initial;\n}\n\nul.jmolPopupMenu .ui-menu, ul.jmolPopupMenu .ui-menu-item {\nfloat: initial;\nmax-height: none;\n}\n\nul.jmolPopupMenu .ui-state-disabled {\nopacity: initial;\n}\n\ndiv.iCn3D .icn3d-mn.ui-menu {\n    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\\newcommand{\\kernel}{\\mathrm{null}\\,}\\) \\( \\newcommand{\\range}{\\mathrm{range}\\,}\\) \\( \\) \\( \\) \\( \\) \\( \\newcommand{\\norm}[1]{\\| #1 \\|}\\) \\( \\newcommand{\\inner}[2]{\\langle #1, #2 \\rangle}\\) \\( \\)\\(\\newcommand{\\AA}{\\unicode[.8,0]{x212B}}\\)<\/div>\n<\/p>\n<link href=\"https:\/\/pro.fontawesome.com\/releases\/v5.10.1\/css\/all.css\" rel=\"stylesheet\" \/>\n \n<\/header>\n                <aside class=\"mt-content-side\"><\/aside>\n            <aside id=\"mt-toc-container\" data-title=\"Table of contents\" data-collapsed=\"true\">\n                <nav class=\"mt-toc-content mt-collapsible-section\">\n                    <ol><li><a href=\"#The_Spin-orbit_coupling_Hamiltonian\" rel=\"internal\">The Spin-orbit coupling Hamiltonian<\/a><\/li><\/ol>\n                <\/nav>\n            <\/aside>\n            <section class=\"mt-content-container\">\n                <p class=\"lt-chem-5558\">Spin-orbit coupling refers to the interaction of a particle&#8217;s &quot;spin&quot; motion with its &quot;orbital&quot; motion.<\/p>\n\n<div mt-section-origin=\"Bookshelves\/Physical_and_Theoretical_Chemistry_Textbook_Maps\/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)\/Spectroscopy\/Electronic_Spectroscopy\/Spin-orbit_Coupling\" class=\"mt-section\" id=\"section_1\"><span id=\"The_Spin-orbit_coupling_Hamiltonian\"><\/span><h2 class=\"lt-chem-5558 editable\">The Spin-orbit coupling Hamiltonian<\/h2>\n\n<p class=\"lt-chem-5558\">The magnitude of spin-orbit coupling splitting is measured spectroscopically as<\/p>\n\n<p class=\"mt-align-center   lt-chem-5558\">\\[\\begin{align*} H_{so} &amp;=\\dfrac{1}{2} hcA \\left( (l+s)(l+s+1)-l(l+1)-s(s+1)\\right) <br> &amp;= \\dfrac{1}{2} hcA \\left(l^2 +s^2 + ls + sl + l +s -l^2 -l -s^2 -s) \\right) <br> &amp;= hcA \\textbf{l} \\cdot \\textbf{s} \\end{align*}\\]<\/p>\n\n<p class=\"lt-chem-5558\">The expression can be modified by realizing that \\(j = l + s\\).<\/p>\n\n<p class=\"mt-align-center   lt-chem-5558\">\\[H_{so}=\\dfrac{1}{2} hcA \\biggr(j(j+1)-l(l+1)-s(s+1)\\biggr)\\]&#8203;<\/p>\n\n<p class=\"lt-chem-5558\">where \\(A\\) is the magnitude of the spin-orbit coupling in wave numbers. The magnitude of the spin orbit coupling can be calculated in terms of molecule parameters by the substitution<\/p>\n\n<p class=\"mt-align-center   lt-chem-5558\">\\[hcA\\,\\widehat{L}\\cdot\\widehat{S}=\\dfrac{Z\\alpha^2}{2}\\dfrac{1}{r^3}\\widehat{L}\\cdot\\widehat{S}\\]<\/p>\n\n<p class=\"lt-chem-5558\">where \\(a\\) is the fine structure constant (\\(a = 1\/137.037\\)) and the carrots indicate that \\(L\\) and \\(S\\) are operators. The fine structure constant is a dimensionless constant, \\(a = \\dfrac{e^2}{&aacute;c}\\). \\(Z\\) is an effective atomic number. The spin orbit coupling splitting can be calculated from<\/p>\n\n<p class=\"mt-align-center   lt-chem-5558\">\\[E_{so}=\\int \\Psi^*H_{SO}\\Psi\\,d\\tau=\\dfrac{Z}{2(137)^2}\\int\\Psi^*\\dfrac{\\widehat{L}\\cdot\\widehat{S}}{r^3}\\Psi\\,d\\tau\\]<\/p>\n\n<p class=\"lt-chem-5558\">This expression can be recast to give an spin-orbit coupling energy in terms of molecular parameters<\/p>\n\n<p class=\"mt-align-center   lt-chem-5558\">\\[E_{so}=\\dfrac{1}{2} \\biggr(j(j+1)-l(l+1)-s(s+1)\\biggr)=\\dfrac{Z}{2(137)^2}\\biggr\\langle\\dfrac{1}{r^3}\\biggr\\rangle\\]<\/p>\n\n<p class=\"lt-chem-5558\">where<\/p>\n\n<p class=\"mt-align-center   lt-chem-5558\">\\[\\biggr\\langle\\dfrac{1}{r^3}\\biggr\\rangle=\\int\\Psi^*\\biggr(\\dfrac{1}{r^3}\\biggr)\\Psi\\,d\\tau\\]<\/p>\n\n<p class=\"lt-chem-5558\">We can evaluate this integral explicitly for a given atomic orbital.<\/p>\n\n<p class=\"lt-chem-5558\">For example for Y210 we have<\/p>\n\n<p class=\"mt-align-center   lt-chem-5558\">\\[\\Psi_{210}=\\dfrac{1}{4\\sqrt{2\\pi}}\\biggr(\\dfrac{Z}{a_0}\\biggr)^\\dfrac{3}{2}\\dfrac{Zr}{a_0}e^{-Zr\/2a_0}\\cos\\theta\\]<\/p>\n\n<p class=\"lt-chem-5558\">so that the integral is<\/p>\n\n<p class=\"mt-align-center   lt-chem-5558\">\\[\\biggr\\langle\\dfrac{1}{r^3}\\biggr\\rangle=\\dfrac{1}{32\\pi}\\biggr<br \/>\n(\\dfrac{Z}{a_0}\\biggr)^5\\int_{0}^{2z}d\\phi\\int_{0}^{z}\\cos^2\\theta\\sin\\theta\\,d\\theta cos\\theta\\int_{0}^{\\infty}r^2e^{Zr\/a_0}\\biggr(\\dfrac{1}{r^3}\\biggr)r^2\\,dr\\]<\/p>\n\n<p class=\"lt-chem-5558\">which integrates to<\/p>\n\n<p class=\"mt-align-center   lt-chem-5558\">\\[\\biggr\\langle\\dfrac{1}{r^3}\\biggr\\rangle=\\dfrac{1}{32\\pi}\\biggr(\\dfrac{Z}{a_0}\\biggr)^52\\pi\\biggr(\\dfrac{2}{3}\\biggr)\\biggr(\\dfrac{a_0\\,^2}{Z^2}\\biggr)=\\dfrac{1}{24}\\biggr(\\dfrac{Z}{a_0}\\biggr)^3\\]<\/p>\n\n<p class=\"lt-chem-5558\">Or \\(Z^3\/24\\) in atomic units.<\/p>\n\n<p class=\"lt-chem-5558\">Therefore in atomic units we have<\/p>\n\n<p class=\"mt-align-center   lt-chem-5558\">\\[\\biggr\\langle\\dfrac{1}{r^3}\\biggr\\rangle=\\dfrac{Z^3}{n^3l(l+1\/2)(l+1)}\\]<\/p>\n\n<p class=\"lt-chem-5558\">Therefore, in general the spin-orbit splitting is given by<\/p>\n\n<p class=\"mt-align-center   lt-chem-5558\">\\[E_{so}=\\dfrac{Z^4}{2(137)^2n^3}\\Biggr(\\dfrac{j(j+1)-l(l+1)-s(s+1)}{2l(l+1\/2)(l+1)}\\Biggr)\\]<\/p>\n\n<p class=\"lt-chem-5558\">Note that the spin-orbit coupling increases as the fourth power of the effective nuclear charge Z, but only as the third power of the principal quantum number n. 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