Born–Oppenheimer approximation

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class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Aproximaci%C3%B3_de_Born-Oppenheimer" title="Aproximació de Born-Oppenheimer – Catalan" lang="ca" hreflang="ca" data-title="Aproximació de Born-Oppenheimer" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Bornova%E2%80%93Oppenheimerova_aproximace" title="Bornova–Oppenheimerova aproximace – Czech" lang="cs" hreflang="cs" data-title="Bornova–Oppenheimerova aproximace" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Born-Oppenheimer-approksimationen" title="Born-Oppenheimer-approksimationen – Danish" lang="da" hreflang="da" data-title="Born-Oppenheimer-approksimationen" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Born-Oppenheimer-N%C3%A4herung" title="Born-Oppenheimer-Näherung – German" lang="de" hreflang="de" data-title="Born-Oppenheimer-Näherung" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Aproximaci%C3%B3n_de_Born-Oppenheimer" title="Aproximación de Born-Oppenheimer – Spanish" lang="es" hreflang="es" data-title="Aproximación de Born-Oppenheimer" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Born-Oppenheimerren_hurbilketa" title="Born-Oppenheimerren hurbilketa – Basque" lang="eu" hreflang="eu" data-title="Born-Oppenheimerren hurbilketa" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D9%82%D8%B1%DB%8C%D8%A8_%D8%A8%D9%88%D8%B1%D9%86%E2%80%93%D8%A7%D9%88%D9%BE%D9%86%D9%87%D8%A7%DB%8C%D9%85%D8%B1" title="تقریب بورن–اوپنهایمر – Persian" lang="fa" hreflang="fa" data-title="تقریب بورن–اوپنهایمر" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Approximation_de_Born-Oppenheimer" title="Approximation de Born-Oppenheimer – French" lang="fr" hreflang="fr" data-title="Approximation de Born-Oppenheimer" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B3%B4%EB%A5%B8-%EC%98%A4%ED%8E%9C%ED%95%98%EC%9D%B4%EB%A8%B8_%EA%B7%BC%EC%82%AC" title="보른-오펜하이머 근사 – Korean" lang="ko" hreflang="ko" data-title="보른-오펜하이머 근사" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Hampiran_Born%E2%80%93Oppenheimer" title="Hampiran Born–Oppenheimer – Indonesian" lang="id" hreflang="id" data-title="Hampiran Born–Oppenheimer" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Approssimazione_di_Born-Oppenheimer" title="Approssimazione di Born-Oppenheimer – Italian" lang="it" hreflang="it" data-title="Approssimazione di Born-Oppenheimer" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A7%D7%99%D7%A8%D7%95%D7%91_%D7%91%D7%95%D7%A8%D7%9F-%D7%90%D7%95%D7%A4%D7%A0%D7%94%D7%99%D7%99%D7%9E%D7%A8" title="קירוב בורן-אופנהיימר – Hebrew" lang="he" hreflang="he" data-title="קירוב בורן-אופנהיימר" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Born%E2%80%93Oppenheimer-k%C3%B6zel%C3%ADt%C3%A9s" title="Born–Oppenheimer-közelítés – Hungarian" lang="hu" hreflang="hu" data-title="Born–Oppenheimer-közelítés" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Born-Oppenheimerbenadering" title="Born-Oppenheimerbenadering – Dutch" lang="nl" hreflang="nl" data-title="Born-Oppenheimerbenadering" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%9C%E3%83%AB%E3%83%B3%E2%80%93%E3%82%AA%E3%83%83%E3%83%9A%E3%83%B3%E3%83%8F%E3%82%A4%E3%83%9E%E3%83%BC%E8%BF%91%E4%BC%BC" title="ボルン–オッペンハイマー近似 – Japanese" lang="ja" hreflang="ja" data-title="ボルン–オッペンハイマー近似" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Przybli%C5%BCenie_Borna-Oppenheimera" title="Przybliżenie Borna-Oppenheimera – Polish" lang="pl" hreflang="pl" data-title="Przybliżenie Borna-Oppenheimera" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Aproxima%C3%A7%C3%A3o_de_Born-Oppenheimer" title="Aproximação de Born-Oppenheimer – Portuguese" lang="pt" hreflang="pt" data-title="Aproximação de Born-Oppenheimer" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a 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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">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">The notion that the motion of atomic nuclei and electrons can be separated</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with the <a href="/wiki/Born_approximation" title="Born approximation">Born approximation</a>.</div><p>In <a href="/wiki/Quantum_chemistry" title="Quantum chemistry">quantum chemistry</a> and <a href="/wiki/Molecular_physics" title="Molecular physics">molecular physics</a>, the <b>Born–Oppenheimer</b> (<b>BO</b>) <b>approximation</b> is the best-known mathematical approximation in molecular dynamics. Specifically, it is the assumption that the <a href="/wiki/Wave_function" title="Wave function">wave functions</a> of <a href="/wiki/Atomic_nucleus" title="Atomic nucleus">atomic nuclei</a> and <a href="/wiki/Electron" title="Electron">electrons</a> in a molecule can be treated separately, based on the fact that the nuclei are much heavier than the electrons. Due to the larger relative mass of a nucleus compared to an electron, the coordinates of the nuclei in a system are approximated as fixed, while the coordinates of the electrons are dynamic.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The approach is named after <a href="/wiki/Max_Born" title="Max Born">Max Born</a> and his 23-year-old graduate student <a href="/wiki/J._Robert_Oppenheimer" title="J. Robert Oppenheimer">J. Robert Oppenheimer</a>, the latter of whom proposed it in 1927 during a period of intense ferment in the development of quantum mechanics.<sup id="cite_ref-BornOppie_2-0" class="reference"><a href="#cite_note-BornOppie-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>The approximation is widely used in quantum chemistry to speed up the computation of molecular wavefunctions and other properties for large molecules. There are cases where the assumption of separable motion no longer holds, which make the approximation lose validity (it is said to "break down"), but even then the approximation is usually used as a starting point for more refined methods. </p><p>In molecular <a href="/wiki/Spectroscopy" title="Spectroscopy">spectroscopy</a>, using the <b>BO</b> approximation means considering molecular energy as a sum of independent terms, e.g.: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\text{total}}=E_{\text{electronic}}+E_{\text{vibrational}}+E_{\text{rotational}}+E_{\text{nuclear spin}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>total</mtext> </mrow> </msub> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>electronic</mtext> </mrow> </msub> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>vibrational</mtext> </mrow> </msub> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>rotational</mtext> </mrow> </msub> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>nuclear spin</mtext> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\text{total}}=E_{\text{electronic}}+E_{\text{vibrational}}+E_{\text{rotational}}+E_{\text{nuclear spin}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24d60711530895a71e0afbbb53474d6941b3e348" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:55.824ex; height:2.843ex;" alt="{\displaystyle E_{\text{total}}=E_{\text{electronic}}+E_{\text{vibrational}}+E_{\text{rotational}}+E_{\text{nuclear spin}}.}" /></span> These terms are of different orders of magnitude and the nuclear spin energy is so small that it is often omitted. The electronic energies <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\text{electronic}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>electronic</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\text{electronic}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf71db642d18cffaa5705ae8f6a1081e1d5548de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.803ex; height:2.509ex;" alt="{\displaystyle E_{\text{electronic}}}" /></span> consist of kinetic energies, interelectronic repulsions, internuclear repulsions, and electron–nuclear attractions, which are the terms typically included when computing the electronic structure of molecules. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Example">Example</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Born%E2%80%93Oppenheimer_approximation&action=edit&section=1" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Benzene" title="Benzene">benzene</a> molecule consists of 12 nuclei and 42 electrons. The <a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a>, which must be solved to obtain the <a href="/wiki/Energy_level" title="Energy level">energy levels</a> and wavefunction of this molecule, is a <a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential eigenvalue equation</a> in the three-dimensional coordinates of the nuclei and electrons, giving 3 × 12 = 36 nuclear plus 3 × 42 = 126 electronic, totalling 162 variables for the wave function. The <a href="/wiki/Computational_complexity_of_mathematical_operations" title="Computational complexity of mathematical operations">computational complexity</a>, i.e., the computational power required to solve an eigenvalue equation, increases faster than the square of the number of coordinates.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>When applying the BO approximation, two smaller, consecutive steps can be used: For a given position of the nuclei, the <i>electronic</i> Schrödinger equation is solved, while treating the nuclei as stationary (not "coupled" with the dynamics of the electrons). This corresponding <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalue</a> problem then consists only of the 126 electronic coordinates. This electronic computation is then repeated for other possible positions of the nuclei, i.e. deformations of the molecule. For benzene, this could be done using a grid of 36 possible nuclear position coordinates. The electronic energies on this grid are then connected to give a <a href="/wiki/Potential_energy_surface" title="Potential energy surface">potential energy surface</a> for the nuclei. This potential is then used for a second Schrödinger equation containing only the 36 coordinates of the nuclei. </p><p>So, taking the most optimistic estimate for the complexity, instead of a large equation requiring at least <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 162^{2}=26\,244}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>162</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>26</mn> <mspace width="thinmathspace"></mspace> <mn>244</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 162^{2}=26\,244}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8c98561d0aaea379d9a74e14508f8df28fffa98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.839ex; height:2.676ex;" alt="{\displaystyle 162^{2}=26\,244}" /></span> hypothetical calculation steps, a series of smaller calculations requiring <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 126^{2}N=15\,876\,N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>126</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>N</mi> <mo>=</mo> <mn>15</mn> <mspace width="thinmathspace"></mspace> <mn>876</mn> <mspace width="thinmathspace"></mspace> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 126^{2}N=15\,876\,N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16048cee5f8e5aa0b16f678a0c9311e2e2491e56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:18.354ex; height:2.676ex;" alt="{\displaystyle 126^{2}N=15\,876\,N}" /></span> (with <i>N</i> being the number of grid points for the potential) and a very small calculation requiring <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 36^{2}=1296}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>36</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1296</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 36^{2}=1296}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbec3dbd93d2a33d01b0a3beb2ddb4dda55a144a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.127ex; height:2.676ex;" alt="{\displaystyle 36^{2}=1296}" /></span> steps can be performed. In practice, the scaling of the problem is larger than <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac9810bbdafe4a6a8061338db0f74e25b7952620" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.449ex; height:2.676ex;" alt="{\displaystyle n^{2}}" /></span>, and more approximations are applied in <a href="/wiki/Computational_chemistry" title="Computational chemistry">computational chemistry</a> to further reduce the number of variables and dimensions. </p><p>The slope of the potential energy surface can be used to simulate <a href="/wiki/Molecular_dynamics" title="Molecular dynamics">molecular dynamics</a>, using it to express the mean force on the nuclei caused by the electrons and thereby skipping the calculation of the nuclear Schrödinger equation. </p> <div class="mw-heading mw-heading2"><h2 id="Detailed_description">Detailed description</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Born%E2%80%93Oppenheimer_approximation&action=edit&section=2" title="Edit section: Detailed description"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The BO approximation recognizes the large difference between the <a href="/wiki/Electron" title="Electron">electron</a> mass and the masses of atomic nuclei, and correspondingly the time scales of their motion. Given the same amount of momentum, the nuclei move much more slowly than the electrons. In mathematical terms, the BO approximation consists of expressing the <a href="/wiki/Wavefunction" class="mw-redirect" title="Wavefunction">wavefunction</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi _{\mathrm {total} }}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi _{\mathrm {total} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28bc274d6d976b12856348eb7c403db53c9be146" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.421ex; height:2.509ex;" alt="{\displaystyle \Psi _{\mathrm {total} }}" /></span>) of a molecule as the product of an electronic wavefunction and a nuclear (<a href="/wiki/Molecular_vibration" title="Molecular vibration">vibrational</a>, <a href="/wiki/Rotational_spectroscopy" title="Rotational spectroscopy">rotational</a>) wavefunction. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi _{\mathrm {total} }=\psi _{\mathrm {electronic} }\psi _{\mathrm {nuclear} }}"> <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">e</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi _{\mathrm {total} }=\psi _{\mathrm {electronic} }\psi _{\mathrm {nuclear} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f58071229cf08ef79589a8c99d0e590eaea43c69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.077ex; height:2.509ex;" alt="{\displaystyle \Psi _{\mathrm {total} }=\psi _{\mathrm {electronic} }\psi _{\mathrm {nuclear} }}" /></span>. This enables a separation of the <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian operator</a> into electronic and nuclear terms, where cross-terms between electrons and nuclei are neglected, so that the two smaller and decoupled systems can be solved more efficiently. </p><p>In the first step, the nuclear <a href="/wiki/Kinetic_energy" title="Kinetic energy">kinetic energy</a> is neglected,<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup> that is, the corresponding operator <i>T</i><sub>n</sub> is subtracted from the total <a href="/wiki/Molecular_Hamiltonian" title="Molecular Hamiltonian">molecular Hamiltonian</a>. In the remaining electronic Hamiltonian <i>H</i><sub>e</sub> the nuclear positions are no longer variable, but are constant parameters (they enter the equation "parametrically"). The electron–nucleus interactions are <i>not</i> removed, i.e., the electrons still "feel" the <a href="/wiki/Coulomb_potential" class="mw-redirect" title="Coulomb potential">Coulomb potential</a> of the nuclei clamped at certain positions in space. (This first step of the BO approximation is therefore often referred to as the <i>clamped-nuclei</i> approximation.) </p><p>The electronic <a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{\text{e}}(\mathbf {r} ,\mathbf {R} )\chi (\mathbf {r} ,\mathbf {R} )=E_{\text{e}}\chi (\mathbf {r} ,\mathbf {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </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> <mi>χ</mi> <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"> <mtext>e</mtext> </mrow> </msub> <mi>χ</mi> <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 H_{\text{e}}(\mathbf {r} ,\mathbf {R} )\chi (\mathbf {r} ,\mathbf {R} )=E_{\text{e}}\chi (\mathbf {r} ,\mathbf {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/522ac7ee37376228aa3c0467fe8ca5a960537cb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.425ex; height:2.843ex;" alt="{\displaystyle H_{\text{e}}(\mathbf {r} ,\mathbf {R} )\chi (\mathbf {r} ,\mathbf {R} )=E_{\text{e}}\chi (\mathbf {r} ,\mathbf {R} )}" /></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi (\mathbf {r} ,\mathbf {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>χ</mi> <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 \chi (\mathbf {r} ,\mathbf {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0a3b7809b458f0a5a4cb91e6a3b42da88bb3a32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.404ex; height:2.843ex;" alt="{\displaystyle \chi (\mathbf {r} ,\mathbf {R} )}" /></span>, the electronic wavefunction for given positions of nuclei (fixed <b>R</b>), is solved approximately.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>note 2<span class="cite-bracket">]</span></a></sup> The quantity <b>r</b> stands for all electronic coordinates and <b>R</b> for all nuclear coordinates. The electronic energy <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalue</a> <i>E</i><sub>e</sub> depends on the chosen positions <b>R</b> of the nuclei. Varying these positions <b>R</b> in small steps and repeatedly solving the electronic <a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a>, one obtains <i>E</i><sub>e</sub> as a function of <b>R</b>. This is the <a href="/wiki/Potential_energy_surface" title="Potential energy surface">potential energy surface</a> (PES): <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{e}(\mathbf {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </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_{e}(\mathbf {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff1c56f5f1d88f11cab494108e8b4fee4f312958" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.526ex; height:2.843ex;" alt="{\displaystyle E_{e}(\mathbf {R} )}" /></span>. Because this procedure of recomputing the electronic wave functions as a function of an infinitesimally changing nuclear geometry is reminiscent of the conditions for the <a href="/wiki/Adiabatic_theorem" title="Adiabatic theorem">adiabatic theorem</a>, this manner of obtaining a PES is often referred to as the <i>adiabatic approximation</i> and the PES itself is called an <i>adiabatic surface</i>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>note 3<span class="cite-bracket">]</span></a></sup> </p><p>In the second step of the BO approximation, the nuclear kinetic energy <i>T</i><sub>n</sub> (containing partial derivatives with respect to the components of <b>R</b>) is reintroduced, and the Schrödinger equation for the nuclear motion<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>note 4<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [T_{\text{n}}+E_{\text{e}}(\mathbf {R} )]\phi (\mathbf {R} )=E\phi (\mathbf {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>n</mtext> </mrow> </msub> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>E</mi> <mi>ϕ</mi> <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 [T_{\text{n}}+E_{\text{e}}(\mathbf {R} )]\phi (\mathbf {R} )=E\phi (\mathbf {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbe27bb3cf9414ccc6d468886bd7053ddb69b5f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.398ex; height:2.843ex;" alt="{\displaystyle [T_{\text{n}}+E_{\text{e}}(\mathbf {R} )]\phi (\mathbf {R} )=E\phi (\mathbf {R} )}" /></span></dd></dl> <p>is solved. This second step of the BO approximation involves separation of vibrational, translational, and rotational motions. This can be achieved by application of the <a href="/wiki/Eckart_conditions" title="Eckart conditions">Eckart conditions</a>. The eigenvalue <i>E</i> is the total energy of the molecule, including contributions from electrons, nuclear vibrations, and overall rotation and translation of the molecule.<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="This text was moved from introduction where context didn't seem sufficient. Averaging over electronic configurations may perhaps be relevant for defining a molecular equilibrium, but seems wrong to define a potential energy surface associated with a particular electronic state (July 2019)">clarification needed</span></a></i>]</sup> In accord with the <a href="/wiki/Hellmann%E2%80%93Feynman_theorem" title="Hellmann–Feynman theorem">Hellmann–Feynman theorem</a>, the nuclear potential is taken to be an average over electron configurations of the sum of the electron–nuclear and internuclear electric potentials. </p> <div class="mw-heading mw-heading2"><h2 id="Derivation">Derivation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Born%E2%80%93Oppenheimer_approximation&action=edit&section=3" title="Edit section: Derivation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It will be discussed how the BO approximation may be derived and under which conditions it is applicable. At the same time we will show how the BO approximation may be improved by including <a href="/wiki/Vibronic_coupling" title="Vibronic coupling">vibronic coupling</a>. To that end the second step of the BO approximation is generalized to a set of coupled eigenvalue equations depending on nuclear coordinates only. Off-diagonal elements in these equations are shown to be nuclear kinetic energy terms. </p><p>It will be shown that the BO approximation can be trusted whenever the PESs, obtained from the solution of the electronic Schrödinger equation, are well separated: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{0}(\mathbf {R} )\ll E_{1}(\mathbf {R} )\ll E_{2}(\mathbf {R} )\ll \cdots {\text{ for all }}\mathbf {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </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"> <mn>1</mn> </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"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>≪</mo> <mo>⋯</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> for all </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{0}(\mathbf {R} )\ll E_{1}(\mathbf {R} )\ll E_{2}(\mathbf {R} )\ll \cdots {\text{ for all }}\mathbf {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cea42d94b5683553b180754d817bca1fd0db557d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.686ex; height:2.843ex;" alt="{\displaystyle E_{0}(\mathbf {R} )\ll E_{1}(\mathbf {R} )\ll E_{2}(\mathbf {R} )\ll \cdots {\text{ for all }}\mathbf {R} }" /></span>.</dd></dl> <p>We start from the <i>exact</i> non-relativistic, time-independent molecular Hamiltonian: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=H_{\text{e}}+T_{\text{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> </msub> <mo>+</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>n</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H=H_{\text{e}}+T_{\text{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/432aa886aa92d42f2819f78a01b1362a606b4233" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.4ex; height:2.509ex;" alt="{\displaystyle H=H_{\text{e}}+T_{\text{n}}}" /></span></dd></dl> <p>with </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{\text{e}}=-\sum _{i}{{\frac {1}{2}}\nabla _{i}^{2}}-\sum _{i,A}{\frac {Z_{A}}{r_{iA}}}+\sum _{i>j}{\frac {1}{r_{ij}}}+\sum _{B>A}{\frac {Z_{A}Z_{B}}{R_{AB}}}\quad {\text{and}}\quad T_{\text{n}}=-\sum _{A}{{\frac {1}{2M_{A}}}\nabla _{A}^{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> </msub> <mo>=</mo> <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> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msubsup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>A</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>A</mi> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>></mo> <mi>j</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> <mo>></mo> <mi>A</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> </mfrac> </mrow> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="1em"></mspace> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>n</mtext> </mrow> </msub> <mo>=</mo> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msubsup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{\text{e}}=-\sum _{i}{{\frac {1}{2}}\nabla _{i}^{2}}-\sum _{i,A}{\frac {Z_{A}}{r_{iA}}}+\sum _{i>j}{\frac {1}{r_{ij}}}+\sum _{B>A}{\frac {Z_{A}Z_{B}}{R_{AB}}}\quad {\text{and}}\quad T_{\text{n}}=-\sum _{A}{{\frac {1}{2M_{A}}}\nabla _{A}^{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2e73f2ac1e8075f6a38435566604cf238f73392" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:81.001ex; height:6.843ex;" alt="{\displaystyle H_{\text{e}}=-\sum _{i}{{\frac {1}{2}}\nabla _{i}^{2}}-\sum _{i,A}{\frac {Z_{A}}{r_{iA}}}+\sum _{i>j}{\frac {1}{r_{ij}}}+\sum _{B>A}{\frac {Z_{A}Z_{B}}{R_{AB}}}\quad {\text{and}}\quad T_{\text{n}}=-\sum _{A}{{\frac {1}{2M_{A}}}\nabla _{A}^{2}}.}" /></span></dd></dl> <p>The position vectors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} \equiv \{\mathbf {r} _{i}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>≡</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} \equiv \{\mathbf {r} _{i}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f39adbe5f79e48aa50649fc593e7a3e34227623" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.427ex; height:2.843ex;" alt="{\displaystyle \mathbf {r} \equiv \{\mathbf {r} _{i}\}}" /></span> of the electrons and the position vectors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} \equiv \{\mathbf {R} _{A}=(R_{Ax},R_{Ay},R_{Az})\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo>≡</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>z</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} \equiv \{\mathbf {R} _{A}=(R_{Ax},R_{Ay},R_{Az})\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ca22431ff6b33456d6054535f53241874ffe775" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.083ex; height:3.009ex;" alt="{\displaystyle \mathbf {R} \equiv \{\mathbf {R} _{A}=(R_{Ax},R_{Ay},R_{Az})\}}" /></span> of the nuclei are with respect to a Cartesian <a href="/wiki/Inertial_frame" class="mw-redirect" title="Inertial frame">inertial frame</a>. Distances between particles are written as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{iA}\equiv |\mathbf {r} _{i}-\mathbf {R} _{A}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>A</mi> </mrow> </msub> <mo>≡</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>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{iA}\equiv |\mathbf {r} _{i}-\mathbf {R} _{A}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a978c25c0d3b01d3b82edea3f7005b3e4da0870e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.683ex; height:2.843ex;" alt="{\displaystyle r_{iA}\equiv |\mathbf {r} _{i}-\mathbf {R} _{A}|}" /></span> (distance between electron <i>i</i> and nucleus <i>A</i>) and similar definitions hold for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/857845aef8b93395ad10279211c6c49180bb8791" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.526ex; height:2.343ex;" alt="{\displaystyle r_{ij}}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{AB}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{AB}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baa22b9617fe3558b77fc9e242bb5e851f495c55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.476ex; height:2.509ex;" alt="{\displaystyle R_{AB}}" /></span>. </p><p>We assume that the molecule is in a homogeneous (no external force) and isotropic (no external torque) space. The only interactions are the two-body Coulomb interactions among the electrons and nuclei. The Hamiltonian is expressed in <a href="/wiki/Atomic_units" title="Atomic units">atomic units</a>, so that we do not see the Planck constant, the dielectric constant of the vacuum, electronic charge, or electronic mass in this formula. The only constants explicitly entering the formula are <i>Z<sub>A</sub></i> and <i>M<sub>A</sub></i> – the atomic number and mass of nucleus <i>A</i>. </p><p>It is useful to introduce the total nuclear momentum and to rewrite the nuclear kinetic energy operator as follows: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\text{n}}=\sum _{A}\sum _{\alpha =x,y,z}{\frac {P_{A\alpha }P_{A\alpha }}{2M_{A}}}\quad {\text{with}}\quad P_{A\alpha }=-i{\frac {\partial }{\partial R_{A\alpha }}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>n</mtext> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </munder> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>=</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>α</mi> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>α</mi> </mrow> </msub> </mrow> <mrow> <mn>2</mn> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext>with</mtext> </mrow> <mspace width="1em"></mspace> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>α</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>α</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\text{n}}=\sum _{A}\sum _{\alpha =x,y,z}{\frac {P_{A\alpha }P_{A\alpha }}{2M_{A}}}\quad {\text{with}}\quad P_{A\alpha }=-i{\frac {\partial }{\partial R_{A\alpha }}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/946075c53e2222cd4574f4f8a7ad6dce5f72ccdd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:50.325ex; height:6.843ex;" alt="{\displaystyle T_{\text{n}}=\sum _{A}\sum _{\alpha =x,y,z}{\frac {P_{A\alpha }P_{A\alpha }}{2M_{A}}}\quad {\text{with}}\quad P_{A\alpha }=-i{\frac {\partial }{\partial R_{A\alpha }}}.}" /></span></dd></dl> <p>Suppose we have <i>K</i> electronic eigenfunctions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{k}(\mathbf {r} ;\mathbf {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</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> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi _{k}(\mathbf {r} ;\mathbf {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83655a629cc686e2e43c83ea44c623ad0b8bfe29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.492ex; height:2.843ex;" alt="{\displaystyle \chi _{k}(\mathbf {r} ;\mathbf {R} )}" /></span> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{\text{e}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{\text{e}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3f30e7735ab482a29b661f31cce5148e3e9bd86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.894ex; height:2.509ex;" alt="{\displaystyle H_{\text{e}}}" /></span>; that is, we have solved </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{\text{e}}\chi _{k}(\mathbf {r} ;\mathbf {R} )=E_{k}(\mathbf {R} )\chi _{k}(\mathbf {r} ;\mathbf {R} )\quad {\text{for}}\quad k=1,\ldots ,K.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> </msub> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</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> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</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> <mo stretchy="false">)</mo> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext>for</mtext> </mrow> <mspace width="1em"></mspace> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>K</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{\text{e}}\chi _{k}(\mathbf {r} ;\mathbf {R} )=E_{k}(\mathbf {R} )\chi _{k}(\mathbf {r} ;\mathbf {R} )\quad {\text{for}}\quad k=1,\ldots ,K.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6c7f65f4ec57d05a7d1f65d8cd62f3e977dcd4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.387ex; height:2.843ex;" alt="{\displaystyle H_{\text{e}}\chi _{k}(\mathbf {r} ;\mathbf {R} )=E_{k}(\mathbf {R} )\chi _{k}(\mathbf {r} ;\mathbf {R} )\quad {\text{for}}\quad k=1,\ldots ,K.}" /></span></dd></dl> <p>The electronic wave functions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e95cb03b1819c3a4fbb19e9fc9607b7869e4cb60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.544ex; height:2.009ex;" alt="{\displaystyle \chi _{k}}" /></span> will be taken to be real, which is possible when there are no magnetic or spin interactions. The <i>parametric dependence</i> of the functions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e95cb03b1819c3a4fbb19e9fc9607b7869e4cb60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.544ex; height:2.009ex;" alt="{\displaystyle \chi _{k}}" /></span> on the nuclear coordinates is indicated by the symbol after the semicolon. This indicates that, although <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e95cb03b1819c3a4fbb19e9fc9607b7869e4cb60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.544ex; height:2.009ex;" alt="{\displaystyle \chi _{k}}" /></span> is a real-valued function of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><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} }" /></span>, its functional form depends on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><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} }" /></span>. </p><p>For example, in the molecular-orbital-linear-combination-of-atomic-orbitals <a href="/wiki/Molecular_orbital#Qualitative_discussion" title="Molecular orbital">(LCAO-MO)</a> approximation, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e95cb03b1819c3a4fbb19e9fc9607b7869e4cb60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.544ex; height:2.009ex;" alt="{\displaystyle \chi _{k}}" /></span> is a molecular orbital (MO) given as a linear expansion of atomic orbitals (AOs). An AO depends visibly on the coordinates of an electron, but the nuclear coordinates are not explicit in the MO. However, upon change of geometry, i.e., change of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><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} }" /></span>, the LCAO coefficients obtain different values and we see corresponding changes in the functional form of the MO <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e95cb03b1819c3a4fbb19e9fc9607b7869e4cb60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.544ex; height:2.009ex;" alt="{\displaystyle \chi _{k}}" /></span>. </p><p>We will assume that the parametric dependence is continuous and differentiable, so that it is meaningful to consider </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{A\alpha }\chi _{k}(\mathbf {r} ;\mathbf {R} )=-i{\frac {\partial \chi _{k}(\mathbf {r} ;\mathbf {R} )}{\partial R_{A\alpha }}}\quad {\text{for}}\quad \alpha =x,y,z,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>α</mi> </mrow> </msub> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</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> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</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> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>α</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext>for</mtext> </mrow> <mspace width="1em"></mspace> <mi>α</mi> <mo>=</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{A\alpha }\chi _{k}(\mathbf {r} ;\mathbf {R} )=-i{\frac {\partial \chi _{k}(\mathbf {r} ;\mathbf {R} )}{\partial R_{A\alpha }}}\quad {\text{for}}\quad \alpha =x,y,z,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c72ece83af9be9c6a892cf5c6d5553a860fa0c87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:47.162ex; height:6.176ex;" alt="{\displaystyle P_{A\alpha }\chi _{k}(\mathbf {r} ;\mathbf {R} )=-i{\frac {\partial \chi _{k}(\mathbf {r} ;\mathbf {R} )}{\partial R_{A\alpha }}}\quad {\text{for}}\quad \alpha =x,y,z,}" /></span></dd></dl> <p>which in general will not be zero. </p><p>The total wave function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi (\mathbf {R} ,\mathbf {r} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <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 (\mathbf {R} ,\mathbf {r} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d8e1a738ba1fd02c5bdb285f4812d9c80681db8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.757ex; height:2.843ex;" alt="{\displaystyle \Psi (\mathbf {R} ,\mathbf {r} )}" /></span> is expanded in terms of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{k}(\mathbf {r} ;\mathbf {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</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> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi _{k}(\mathbf {r} ;\mathbf {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83655a629cc686e2e43c83ea44c623ad0b8bfe29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.492ex; height:2.843ex;" alt="{\displaystyle \chi _{k}(\mathbf {r} ;\mathbf {R} )}" /></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi (\mathbf {R} ,\mathbf {r} )=\sum _{k=1}^{K}\chi _{k}(\mathbf {r} ;\mathbf {R} )\phi _{k}(\mathbf {R} ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <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> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </munderover> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</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> <mo stretchy="false">)</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi (\mathbf {R} ,\mathbf {r} )=\sum _{k=1}^{K}\chi _{k}(\mathbf {r} ;\mathbf {R} )\phi _{k}(\mathbf {R} ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73d798a8b58251572f223570d158de67c5fbd1b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.023ex; height:7.343ex;" alt="{\displaystyle \Psi (\mathbf {R} ,\mathbf {r} )=\sum _{k=1}^{K}\chi _{k}(\mathbf {r} ;\mathbf {R} )\phi _{k}(\mathbf {R} ),}" /></span></dd></dl> <p>with </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \chi _{k'}(\mathbf {r} ;\mathbf {R} )|\chi _{k}(\mathbf {r} ;\mathbf {R} )\rangle _{(\mathbf {r} )}=\delta _{k'k},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>k</mi> <mo>′</mo> </msup> </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 class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</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> <mo stretchy="false">)</mo> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mrow> </msub> <mo>=</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>k</mi> <mo>′</mo> </msup> <mi>k</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \chi _{k'}(\mathbf {r} ;\mathbf {R} )|\chi _{k}(\mathbf {r} ;\mathbf {R} )\rangle _{(\mathbf {r} )}=\delta _{k'k},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e88552f7631552c7b0a005cb738395d6cf7d617" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:29.518ex; height:3.176ex;" alt="{\displaystyle \langle \chi _{k'}(\mathbf {r} ;\mathbf {R} )|\chi _{k}(\mathbf {r} ;\mathbf {R} )\rangle _{(\mathbf {r} )}=\delta _{k'k},}" /></span></dd></dl> <p>and where the subscript <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {r} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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 (\mathbf {r} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7e96acdcfa10ddcd93e4c06c02b663a8b747270" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.911ex; height:2.843ex;" alt="{\displaystyle (\mathbf {r} )}" /></span> indicates that the integration, implied by the <a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">bra–ket notation</a>, is over electronic coordinates only. By definition, the matrix with general element </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\big (}\mathbb {H} _{\text{e}}(\mathbf {R} ){\big )}_{k'k}\equiv \langle \chi _{k'}(\mathbf {r} ;\mathbf {R} )|H_{\text{e}}|\chi _{k}(\mathbf {r} ;\mathbf {R} )\rangle _{(\mathbf {r} )}=\delta _{k'k}E_{k}(\mathbf {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>k</mi> <mo>′</mo> </msup> <mi>k</mi> </mrow> </msub> <mo>≡</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>k</mi> <mo>′</mo> </msup> </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 class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</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> <mo stretchy="false">)</mo> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mrow> </msub> <mo>=</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>k</mi> <mo>′</mo> </msup> <mi>k</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </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 {\big (}\mathbb {H} _{\text{e}}(\mathbf {R} ){\big )}_{k'k}\equiv \langle \chi _{k'}(\mathbf {r} ;\mathbf {R} )|H_{\text{e}}|\chi _{k}(\mathbf {r} ;\mathbf {R} )\rangle _{(\mathbf {r} )}=\delta _{k'k}E_{k}(\mathbf {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bf29fd4703f16579d2df7e96cd53466ed48e804" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:53.316ex; height:3.343ex;" alt="{\displaystyle {\big (}\mathbb {H} _{\text{e}}(\mathbf {R} ){\big )}_{k'k}\equiv \langle \chi _{k'}(\mathbf {r} ;\mathbf {R} )|H_{\text{e}}|\chi _{k}(\mathbf {r} ;\mathbf {R} )\rangle _{(\mathbf {r} )}=\delta _{k'k}E_{k}(\mathbf {R} )}" /></span></dd></dl> <p>is diagonal. After multiplication by the real function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{k'}(\mathbf {r} ;\mathbf {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>k</mi> <mo>′</mo> </msup> </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 \chi _{k'}(\mathbf {r} ;\mathbf {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6fee05501c232a18585156a1431d9f5b68d6a7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.024ex; height:2.843ex;" alt="{\displaystyle \chi _{k'}(\mathbf {r} ;\mathbf {R} )}" /></span> from the left and integration over the electronic coordinates <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><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} }" /></span> the total Schrödinger equation </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\Psi (\mathbf {R} ,\mathbf {r} )=E\Psi (\mathbf {R} ,\mathbf {r} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <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> <mi mathvariant="normal">Ψ</mi> <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 H\Psi (\mathbf {R} ,\mathbf {r} )=E\Psi (\mathbf {R} ,\mathbf {r} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29240276fdd232712e05f633503c02307ec99fa5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.451ex; height:2.843ex;" alt="{\displaystyle H\Psi (\mathbf {R} ,\mathbf {r} )=E\Psi (\mathbf {R} ,\mathbf {r} )}" /></span></dd></dl> <p>is turned into a set of <i>K</i> coupled eigenvalue equations depending on nuclear coordinates only </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\mathbb {H} _{\text{n}}(\mathbf {R} )+\mathbb {H} _{\text{e}}(\mathbf {R} )]{\boldsymbol {\phi }}(\mathbf {R} )=E{\boldsymbol {\phi }}(\mathbf {R} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>n</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ϕ</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ϕ</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [\mathbb {H} _{\text{n}}(\mathbf {R} )+\mathbb {H} _{\text{e}}(\mathbf {R} )]{\boldsymbol {\phi }}(\mathbf {R} )=E{\boldsymbol {\phi }}(\mathbf {R} ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfc268c44531893ccfa51898d111e668a41237de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.94ex; height:2.843ex;" alt="{\displaystyle [\mathbb {H} _{\text{n}}(\mathbf {R} )+\mathbb {H} _{\text{e}}(\mathbf {R} )]{\boldsymbol {\phi }}(\mathbf {R} )=E{\boldsymbol {\phi }}(\mathbf {R} ).}" /></span></dd></dl> <p>The column vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\phi }}(\mathbf {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ϕ</mi> </mrow> <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 {\boldsymbol {\phi }}(\mathbf {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/501782f98e712fbee8b6a0267060025f296175ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.467ex; height:2.843ex;" alt="{\displaystyle {\boldsymbol {\phi }}(\mathbf {R} )}" /></span> has elements <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{k}(\mathbf {R} ),\ k=1,\ldots ,K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{k}(\mathbf {R} ),\ k=1,\ldots ,K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d47c70674f9af4059a1cd6dca4f5cb057774c981" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.618ex; height:2.843ex;" alt="{\displaystyle \phi _{k}(\mathbf {R} ),\ k=1,\ldots ,K}" /></span>. The matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {H} _{\text{e}}(\mathbf {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </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 \mathbb {H} _{\text{e}}(\mathbf {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d07a81467b26f539d248fc5f35c5ab4add05f560" 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 \mathbb {H} _{\text{e}}(\mathbf {R} )}" /></span> is diagonal, and the nuclear Hamilton matrix is non-diagonal; its off-diagonal (<i>vibronic coupling</i>) terms <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\big (}\mathbb {H} _{\text{n}}(\mathbf {R} ){\big )}_{k'k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>n</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>k</mi> <mo>′</mo> </msup> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\big (}\mathbb {H} _{\text{n}}(\mathbf {R} ){\big )}_{k'k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b448a98fad59bbdf836b4aef987b8728ca648cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.373ex; height:3.343ex;" alt="{\displaystyle {\big (}\mathbb {H} _{\text{n}}(\mathbf {R} ){\big )}_{k'k}}" /></span> are further discussed below. The vibronic coupling in this approach is through nuclear kinetic energy terms. </p><p>Solution of these coupled equations gives an approximation for energy and wavefunction that goes beyond the Born–Oppenheimer approximation. Unfortunately, the off-diagonal kinetic energy terms are usually difficult to handle. This is why often a <a href="/wiki/Diabatic_representation" title="Diabatic representation">diabatic</a> transformation is applied, which retains part of the nuclear kinetic energy terms on the diagonal, removes the kinetic energy terms from the off-diagonal and creates coupling terms between the adiabatic PESs on the off-diagonal. </p><p>If we can neglect the off-diagonal elements the equations will uncouple and simplify drastically. In order to show when this neglect is justified, we suppress the coordinates in the notation and write, by applying the <a href="/wiki/Leibniz_rule_(generalized_product_rule)" class="mw-redirect" title="Leibniz rule (generalized product rule)">Leibniz rule</a> for differentiation, the matrix elements of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\text{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>n</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\text{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37f3235aa9ecfb6c191beb6ba0f65fe704bcd062" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.504ex; height:2.509ex;" alt="{\displaystyle T_{\text{n}}}" /></span> as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\text{n}}(\mathbf {R} )_{k'k}\equiv {\big (}\mathbb {H} _{\text{n}}(\mathbf {R} ){\big )}_{k'k}=\delta _{k'k}T_{\text{n}}-\sum _{A,\alpha }{\frac {1}{M_{A}}}\langle \chi _{k'}|P_{A\alpha }|\chi _{k}\rangle _{(\mathbf {r} )}P_{A\alpha }+\langle \chi _{k'}|T_{\text{n}}|\chi _{k}\rangle _{(\mathbf {r} )}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>n</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>k</mi> <mo>′</mo> </msup> <mi>k</mi> </mrow> </msub> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>n</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>k</mi> <mo>′</mo> </msup> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>k</mi> <mo>′</mo> </msup> <mi>k</mi> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>n</mtext> </mrow> </msub> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mo>,</mo> <mi>α</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mfrac> </mrow> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>k</mi> <mo>′</mo> </msup> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>α</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>α</mi> </mrow> </msub> <mo>+</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>k</mi> <mo>′</mo> </msup> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>n</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\text{n}}(\mathbf {R} )_{k'k}\equiv {\big (}\mathbb {H} _{\text{n}}(\mathbf {R} ){\big )}_{k'k}=\delta _{k'k}T_{\text{n}}-\sum _{A,\alpha }{\frac {1}{M_{A}}}\langle \chi _{k'}|P_{A\alpha }|\chi _{k}\rangle _{(\mathbf {r} )}P_{A\alpha }+\langle \chi _{k'}|T_{\text{n}}|\chi _{k}\rangle _{(\mathbf {r} )}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/635f0bd28b0597d943e499720706783255fe4ccb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:79.549ex; height:6.676ex;" alt="{\displaystyle T_{\text{n}}(\mathbf {R} )_{k'k}\equiv {\big (}\mathbb {H} _{\text{n}}(\mathbf {R} ){\big )}_{k'k}=\delta _{k'k}T_{\text{n}}-\sum _{A,\alpha }{\frac {1}{M_{A}}}\langle \chi _{k'}|P_{A\alpha }|\chi _{k}\rangle _{(\mathbf {r} )}P_{A\alpha }+\langle \chi _{k'}|T_{\text{n}}|\chi _{k}\rangle _{(\mathbf {r} )}.}" /></span></dd></dl> <p>The diagonal (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k'=k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>k</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k'=k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87d3996fb3876f66262f58737fa1471fd40fabe6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.206ex; height:2.509ex;" alt="{\displaystyle k'=k}" /></span>) matrix elements <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \chi _{k}|P_{A\alpha }|\chi _{k}\rangle _{(\mathbf {r} )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>α</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \chi _{k}|P_{A\alpha }|\chi _{k}\rangle _{(\mathbf {r} )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00a70ec17f016a04c0e991311fcaa47bbea4ad7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.491ex; height:3.176ex;" alt="{\displaystyle \langle \chi _{k}|P_{A\alpha }|\chi _{k}\rangle _{(\mathbf {r} )}}" /></span> of the operator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{A\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{A\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90fcc56427f6c97215ae945b7226a73dcc1baa75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.009ex; height:2.509ex;" alt="{\displaystyle P_{A\alpha }}" /></span> vanish, because we assume time-reversal invariant, so <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e95cb03b1819c3a4fbb19e9fc9607b7869e4cb60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.544ex; height:2.009ex;" alt="{\displaystyle \chi _{k}}" /></span> can be chosen to be always real. The off-diagonal matrix elements satisfy </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \chi _{k'}|P_{A\alpha }|\chi _{k}\rangle _{(\mathbf {r} )}={\frac {\langle \chi _{k'}|[P_{A\alpha },H_{\text{e}}]|\chi _{k}\rangle _{(\mathbf {r} )}}{E_{k}(\mathbf {R} )-E_{k'}(\mathbf {R} )}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>k</mi> <mo>′</mo> </msup> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>α</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>k</mi> <mo>′</mo> </msup> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">[</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>α</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> </msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mrow> </msub> </mrow> <mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </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"> <msup> <mi>k</mi> <mo>′</mo> </msup> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \chi _{k'}|P_{A\alpha }|\chi _{k}\rangle _{(\mathbf {r} )}={\frac {\langle \chi _{k'}|[P_{A\alpha },H_{\text{e}}]|\chi _{k}\rangle _{(\mathbf {r} )}}{E_{k}(\mathbf {R} )-E_{k'}(\mathbf {R} )}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e7f587131133e6a1d1716ac4981b3dd78038cc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:39.847ex; height:6.676ex;" alt="{\displaystyle \langle \chi _{k'}|P_{A\alpha }|\chi _{k}\rangle _{(\mathbf {r} )}={\frac {\langle \chi _{k'}|[P_{A\alpha },H_{\text{e}}]|\chi _{k}\rangle _{(\mathbf {r} )}}{E_{k}(\mathbf {R} )-E_{k'}(\mathbf {R} )}}.}" /></span></dd></dl> <p>The matrix element in the numerator is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \chi _{k'}|[P_{A\alpha },H_{\mathrm {e} }]|\chi _{k}\rangle _{(\mathbf {r} )}=iZ_{A}\sum _{i}\left\langle \chi _{k'}\left|{\frac {(\mathbf {r} _{iA})_{\alpha }}{r_{iA}^{3}}}\right|\chi _{k}\right\rangle _{(\mathbf {r} )}\quad {\text{with}}\quad \mathbf {r} _{iA}\equiv \mathbf {r} _{i}-\mathbf {R} _{A}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>k</mi> <mo>′</mo> </msup> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">[</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>α</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>H</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"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mrow> </msub> <mo>=</mo> <mi>i</mi> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mrow> <mo>⟨</mo> <mrow> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>k</mi> <mo>′</mo> </msup> </mrow> </msub> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>A</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mrow> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>|</mo> </mrow> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mo>⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mrow> </msub> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext>with</mtext> </mrow> <mspace width="1em"></mspace> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>A</mi> </mrow> </msub> <mo>≡</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \chi _{k'}|[P_{A\alpha },H_{\mathrm {e} }]|\chi _{k}\rangle _{(\mathbf {r} )}=iZ_{A}\sum _{i}\left\langle \chi _{k'}\left|{\frac {(\mathbf {r} _{iA})_{\alpha }}{r_{iA}^{3}}}\right|\chi _{k}\right\rangle _{(\mathbf {r} )}\quad {\text{with}}\quad \mathbf {r} _{iA}\equiv \mathbf {r} _{i}-\mathbf {R} _{A}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c75e0c451a3487401904583a3a16eab3999de0a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:76.371ex; height:8.009ex;" alt="{\displaystyle \langle \chi _{k'}|[P_{A\alpha },H_{\mathrm {e} }]|\chi _{k}\rangle _{(\mathbf {r} )}=iZ_{A}\sum _{i}\left\langle \chi _{k'}\left|{\frac {(\mathbf {r} _{iA})_{\alpha }}{r_{iA}^{3}}}\right|\chi _{k}\right\rangle _{(\mathbf {r} )}\quad {\text{with}}\quad \mathbf {r} _{iA}\equiv \mathbf {r} _{i}-\mathbf {R} _{A}.}" /></span></dd></dl> <p>The matrix element of the one-electron operator appearing on the right side is finite. </p><p>When the two surfaces come close, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{k}(\mathbf {R} )\approx E_{k'}(\mathbf {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </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"> <msup> <mi>k</mi> <mo>′</mo> </msup> </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_{k}(\mathbf {R} )\approx E_{k'}(\mathbf {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f2399038873cb82dedb33224a380f342066509b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.863ex; height:2.843ex;" alt="{\displaystyle E_{k}(\mathbf {R} )\approx E_{k'}(\mathbf {R} )}" /></span>, the nuclear momentum coupling term becomes large and is no longer negligible. This is the case where the BO approximation breaks down, and a coupled set of nuclear motion equations must be considered instead of the one equation appearing in the second step of the BO approximation. </p><p>Conversely, if all surfaces are well separated, all off-diagonal terms can be neglected, and hence the whole matrix of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\alpha }^{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\alpha }^{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b64915f647552e4facb1c116bfc6a9479ee27461" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.286ex; height:2.843ex;" alt="{\displaystyle P_{\alpha }^{A}}" /></span> is effectively zero. The third term on the right side of the expression for the matrix element of <i>T</i><sub>n</sub> (the <i>Born–Oppenheimer diagonal correction</i>) can approximately be written as the matrix of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\alpha }^{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\alpha }^{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b64915f647552e4facb1c116bfc6a9479ee27461" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.286ex; height:2.843ex;" alt="{\displaystyle P_{\alpha }^{A}}" /></span> squared and, accordingly, is then negligible also. Only the first (diagonal) kinetic energy term in this equation survives in the case of well separated surfaces, and a diagonal, uncoupled, set of nuclear motion equations results: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [T_{\text{n}}+E_{k}(\mathbf {R} )]\phi _{k}(\mathbf {R} )=E\phi _{k}(\mathbf {R} )\quad {\text{for}}\quad k=1,\ldots ,K,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>n</mtext> </mrow> </msub> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </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> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext>for</mtext> </mrow> <mspace width="1em"></mspace> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>K</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [T_{\text{n}}+E_{k}(\mathbf {R} )]\phi _{k}(\mathbf {R} )=E\phi _{k}(\mathbf {R} )\quad {\text{for}}\quad k=1,\ldots ,K,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67b4d3d6f6517ae29b1d50f82bc3b374875e2917" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:51.496ex; height:2.843ex;" alt="{\displaystyle [T_{\text{n}}+E_{k}(\mathbf {R} )]\phi _{k}(\mathbf {R} )=E\phi _{k}(\mathbf {R} )\quad {\text{for}}\quad k=1,\ldots ,K,}" /></span></dd></dl> <p>which are the normal second step of the BO equations discussed above. </p><p>We reiterate that when two or more potential energy surfaces approach each other, or even cross, the Born–Oppenheimer approximation breaks down, and one must fall back on the coupled equations. Usually one invokes then the <a href="/wiki/Diabatic_representation" title="Diabatic representation">diabatic</a> approximation. </p> <div class="mw-heading mw-heading2"><h2 id="Born–Oppenheimer_approximation_with_correct_symmetry"><span id="Born.E2.80.93Oppenheimer_approximation_with_correct_symmetry"></span>Born–Oppenheimer approximation with correct symmetry</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Born%E2%80%93Oppenheimer_approximation&action=edit&section=4" title="Edit section: Born–Oppenheimer approximation with correct symmetry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To include the correct symmetry within the Born–Oppenheimer (BO) approximation,<sup id="cite_ref-BornOppie_2-1" class="reference"><a href="#cite_note-BornOppie-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> a molecular system presented in terms of (mass-dependent) nuclear coordinates <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><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} }" /></span> and formed by the two lowest BO adiabatic potential energy surfaces (PES) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{1}(\mathbf {q} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{1}(\mathbf {q} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ba91d1e020f09536435a989952d5f224b364b20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.609ex; height:2.843ex;" alt="{\displaystyle u_{1}(\mathbf {q} )}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{2}(\mathbf {q} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{2}(\mathbf {q} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b30a020a4b81415788f31fb3c9677d8be4d0483" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.609ex; height:2.843ex;" alt="{\displaystyle u_{2}(\mathbf {q} )}" /></span> is considered. To ensure the validity of the BO approximation, the energy <i>E</i> of the system is assumed to be low enough so that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{2}(\mathbf {q} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{2}(\mathbf {q} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b30a020a4b81415788f31fb3c9677d8be4d0483" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.609ex; height:2.843ex;" alt="{\displaystyle u_{2}(\mathbf {q} )}" /></span> becomes a closed PES in the region of interest, with the exception of sporadic infinitesimal sites surrounding degeneracy points formed by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{1}(\mathbf {q} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{1}(\mathbf {q} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ba91d1e020f09536435a989952d5f224b364b20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.609ex; height:2.843ex;" alt="{\displaystyle u_{1}(\mathbf {q} )}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{2}(\mathbf {q} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{2}(\mathbf {q} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b30a020a4b81415788f31fb3c9677d8be4d0483" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.609ex; height:2.843ex;" alt="{\displaystyle u_{2}(\mathbf {q} )}" /></span> (designated as (1, 2) degeneracy points). </p><p>The starting point is the nuclear adiabatic BO (matrix) equation written in the form<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {\hbar ^{2}}{2m}}(\nabla +\tau )^{2}\Psi +(\mathbf {u} -E)\Psi =0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>m</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mo>+</mo> <mi>τ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Ψ</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>−</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">Ψ</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {\hbar ^{2}}{2m}}(\nabla +\tau )^{2}\Psi +(\mathbf {u} -E)\Psi =0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d538b22d279a7e9b42c3e97004137bbcc91370cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:33.964ex; height:5.676ex;" alt="{\displaystyle -{\frac {\hbar ^{2}}{2m}}(\nabla +\tau )^{2}\Psi +(\mathbf {u} -E)\Psi =0,}" /></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi (\mathbf {q} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi (\mathbf {q} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a47da525b2dcde37f46a96bff99d8e222f9f1007" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.033ex; height:2.843ex;" alt="{\displaystyle \Psi (\mathbf {q} )}" /></span> is a column vector containing the unknown nuclear wave functions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{k}(\mathbf {q} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{k}(\mathbf {q} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf781b2b83acfa1f6f5add023b4fcbb3447b85a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.827ex; height:2.843ex;" alt="{\displaystyle \psi _{k}(\mathbf {q} )}" /></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} (\mathbf {q} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} (\mathbf {q} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46dd20a4842dac69271790368f375c1ca5c4f587" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.71ex; height:2.843ex;" alt="{\displaystyle \mathbf {u} (\mathbf {q} )}" /></span> is a diagonal matrix containing the corresponding adiabatic potential energy surfaces <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{k}(\mathbf {q} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{k}(\mathbf {q} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cc51859416c8e229e04ea1c01b184984bfe3b81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.643ex; height:2.843ex;" alt="{\displaystyle u_{k}(\mathbf {q} )}" /></span>, <i>m</i> is the reduced mass of the nuclei, <i>E</i> is the total energy of the system, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3d0e93b78c50237f9ea83d027e4ebbdaef354b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \nabla }" /></span> is the <a href="/wiki/Gradient" title="Gradient">gradient</a> operator with respect to the nuclear coordinates <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><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} }" /></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\tau } (\mathbf {q} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\tau } (\mathbf {q} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b6c9971bd011dc39b96b91e0570d52775245b38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.427ex; height:2.843ex;" alt="{\displaystyle \mathbf {\tau } (\mathbf {q} )}" /></span> is a matrix containing the vectorial non-adiabatic coupling terms (NACT): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\tau } _{jk}=\langle \zeta _{j}|\nabla \zeta _{k}\rangle .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">∇</mi> <msub> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\tau } _{jk}=\langle \zeta _{j}|\nabla \zeta _{k}\rangle .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c72bc3a77e735291dfb0eabe81d154fdf38c2b93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.955ex; height:3.009ex;" alt="{\displaystyle \mathbf {\tau } _{jk}=\langle \zeta _{j}|\nabla \zeta _{k}\rangle .}" /></span></dd></dl> <p>Here <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\zeta _{n}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\zeta _{n}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d77148dcf213d40c55578f20b16daca54694bcf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.788ex; height:2.843ex;" alt="{\displaystyle |\zeta _{n}\rangle }" /></span> are eigenfunctions of the <a href="/wiki/Electronic_Hamiltonian" class="mw-redirect" title="Electronic Hamiltonian">electronic Hamiltonian</a> assumed to form a complete <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a> in the given region in <a href="/wiki/Configuration_space_(physics)" title="Configuration space (physics)">configuration space</a>. </p><p>To study the scattering process taking place on the two lowest surfaces, one extracts from the above BO equation the two corresponding equations: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi _{1}+({\tilde {u}}_{1}-E)\psi _{1}-{\frac {\hbar ^{2}}{2m}}[2\mathbf {\tau } _{12}\nabla +\nabla \mathbf {\tau } _{12}]\psi _{2}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>m</mi> </mrow> </mfrac> </mrow> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mi>E</mi> <mo stretchy="false">)</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <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>m</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">[</mo> <mn>2</mn> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mi mathvariant="normal">∇</mi> <mo>+</mo> <mi mathvariant="normal">∇</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo stretchy="false">]</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi _{1}+({\tilde {u}}_{1}-E)\psi _{1}-{\frac {\hbar ^{2}}{2m}}[2\mathbf {\tau } _{12}\nabla +\nabla \mathbf {\tau } _{12}]\psi _{2}=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3aff6f9682c2f4f90475850ddacdb069a6dad75c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:54.929ex; height:5.676ex;" alt="{\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi _{1}+({\tilde {u}}_{1}-E)\psi _{1}-{\frac {\hbar ^{2}}{2m}}[2\mathbf {\tau } _{12}\nabla +\nabla \mathbf {\tau } _{12}]\psi _{2}=0,}" /></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi _{2}+({\tilde {u}}_{2}-E)\psi _{2}+{\frac {\hbar ^{2}}{2m}}[2\mathbf {\tau } _{12}\nabla +\nabla \mathbf {\tau } _{12}]\psi _{1}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>m</mi> </mrow> </mfrac> </mrow> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mi>E</mi> <mo stretchy="false">)</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <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>m</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">[</mo> <mn>2</mn> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mi mathvariant="normal">∇</mi> <mo>+</mo> <mi mathvariant="normal">∇</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo stretchy="false">]</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi _{2}+({\tilde {u}}_{2}-E)\psi _{2}+{\frac {\hbar ^{2}}{2m}}[2\mathbf {\tau } _{12}\nabla +\nabla \mathbf {\tau } _{12}]\psi _{1}=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3299f944b4522f4ba4cdfe50bf397750c44c46f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:54.929ex; height:5.676ex;" alt="{\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi _{2}+({\tilde {u}}_{2}-E)\psi _{2}+{\frac {\hbar ^{2}}{2m}}[2\mathbf {\tau } _{12}\nabla +\nabla \mathbf {\tau } _{12}]\psi _{1}=0,}" /></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {u}}_{k}(\mathbf {q} )=u_{k}(\mathbf {q} )+(\hbar ^{2}/2m)\tau _{12}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <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>m</mi> <mo stretchy="false">)</mo> <msubsup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {u}}_{k}(\mathbf {q} )=u_{k}(\mathbf {q} )+(\hbar ^{2}/2m)\tau _{12}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0ef4a71dccc6d0cbaa289cfccc8522ce248a837" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.669ex; height:3.343ex;" alt="{\displaystyle {\tilde {u}}_{k}(\mathbf {q} )=u_{k}(\mathbf {q} )+(\hbar ^{2}/2m)\tau _{12}^{2}}" /></span> (<i>k</i> = 1, 2), and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\tau } _{12}=\mathbf {\tau } _{12}(\mathbf {q} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\tau } _{12}=\mathbf {\tau } _{12}(\mathbf {q} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02ab6bd1b35a54deb02f6ca158ef9ed0cb76b9d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.108ex; height:2.843ex;" alt="{\displaystyle \mathbf {\tau } _{12}=\mathbf {\tau } _{12}(\mathbf {q} )}" /></span> is the (vectorial) NACT responsible for the coupling between <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{1}(\mathbf {q} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{1}(\mathbf {q} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ba91d1e020f09536435a989952d5f224b364b20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.609ex; height:2.843ex;" alt="{\displaystyle u_{1}(\mathbf {q} )}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{2}(\mathbf {q} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{2}(\mathbf {q} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b30a020a4b81415788f31fb3c9677d8be4d0483" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.609ex; height:2.843ex;" alt="{\displaystyle u_{2}(\mathbf {q} )}" /></span>. </p><p>Next a new function is introduced:<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi =\psi _{1}+i\psi _{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>χ</mi> <mo>=</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>i</mi> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi =\psi _{1}+i\psi _{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6206f9ce643f823733a5c49690025609afd9806" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.978ex; height:2.509ex;" alt="{\displaystyle \chi =\psi _{1}+i\psi _{2},}" /></span></dd></dl> <p>and the corresponding rearrangements are made: </p> <ol><li>Multiplying the second equation by <i>i</i> and combining it with the first equation yields the (complex) equation <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\chi +({\tilde {u}}_{1}-E)\chi +i{\frac {\hbar ^{2}}{2m}}[2\mathbf {\tau } _{12}\nabla +\nabla \mathbf {\tau } _{12}]\chi +i(u_{1}-u_{2})\psi _{2}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>m</mi> </mrow> </mfrac> </mrow> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>χ</mi> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mi>χ</mi> <mo>+</mo> <mi>i</mi> <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>m</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">[</mo> <mn>2</mn> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mi mathvariant="normal">∇</mi> <mo>+</mo> <mi mathvariant="normal">∇</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mi>χ</mi> <mo>+</mo> <mi>i</mi> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\chi +({\tilde {u}}_{1}-E)\chi +i{\frac {\hbar ^{2}}{2m}}[2\mathbf {\tau } _{12}\nabla +\nabla \mathbf {\tau } _{12}]\chi +i(u_{1}-u_{2})\psi _{2}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66756d95e7adec66fedb13f852f35405eadc7a07" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:68.023ex; height:5.676ex;" alt="{\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\chi +({\tilde {u}}_{1}-E)\chi +i{\frac {\hbar ^{2}}{2m}}[2\mathbf {\tau } _{12}\nabla +\nabla \mathbf {\tau } _{12}]\chi +i(u_{1}-u_{2})\psi _{2}=0.}" /></span></li> <li>The last term in this equation can be deleted for the following reasons: At those points where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{2}(\mathbf {q} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{2}(\mathbf {q} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b30a020a4b81415788f31fb3c9677d8be4d0483" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.609ex; height:2.843ex;" alt="{\displaystyle u_{2}(\mathbf {q} )}" /></span> is classically closed, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{2}(\mathbf {q} )\sim 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">)</mo> <mo>∼</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{2}(\mathbf {q} )\sim 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4989a86d5b7898efdf85649c1017b0643ad93419" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.053ex; height:2.843ex;" alt="{\displaystyle \psi _{2}(\mathbf {q} )\sim 0}" /></span> by definition, and at those points where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{2}(\mathbf {q} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{2}(\mathbf {q} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b30a020a4b81415788f31fb3c9677d8be4d0483" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.609ex; height:2.843ex;" alt="{\displaystyle u_{2}(\mathbf {q} )}" /></span> becomes classically allowed (which happens at the vicinity of the (1, 2) degeneracy points) this implies that: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{1}(\mathbf {q} )\sim u_{2}(\mathbf {q} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">)</mo> <mo>∼</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{1}(\mathbf {q} )\sim u_{2}(\mathbf {q} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a94b87bb8c7782b3a0ad2dbbaecd5401d39ff125" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.316ex; height:2.843ex;" alt="{\displaystyle u_{1}(\mathbf {q} )\sim u_{2}(\mathbf {q} )}" /></span>, or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{1}(\mathbf {q} )-u_{2}(\mathbf {q} )\sim 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">)</mo> <mo>∼</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{1}(\mathbf {q} )-u_{2}(\mathbf {q} )\sim 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa591a5dc3c63264884b86367f6ba36a3ffc2803" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.319ex; height:2.843ex;" alt="{\displaystyle u_{1}(\mathbf {q} )-u_{2}(\mathbf {q} )\sim 0}" /></span>. Consequently, the last term is, indeed, negligibly small at every point in the region of interest, and the equation simplifies to become <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\chi +({\tilde {u}}_{1}-E)\chi +i{\frac {\hbar ^{2}}{2m}}[2\mathbf {\tau } _{12}\nabla +\nabla \mathbf {\tau } _{12}]\chi =0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>m</mi> </mrow> </mfrac> </mrow> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>χ</mi> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mi>χ</mi> <mo>+</mo> <mi>i</mi> <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>m</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">[</mo> <mn>2</mn> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mi mathvariant="normal">∇</mi> <mo>+</mo> <mi mathvariant="normal">∇</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mi>χ</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\chi +({\tilde {u}}_{1}-E)\chi +i{\frac {\hbar ^{2}}{2m}}[2\mathbf {\tau } _{12}\nabla +\nabla \mathbf {\tau } _{12}]\chi =0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f44a50feb7b2cc32399613d465fa961f5db4bf6d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:52.395ex; height:5.676ex;" alt="{\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\chi +({\tilde {u}}_{1}-E)\chi +i{\frac {\hbar ^{2}}{2m}}[2\mathbf {\tau } _{12}\nabla +\nabla \mathbf {\tau } _{12}]\chi =0.}" /></span></li></ol> <p>In order for this equation to yield a solution with the correct symmetry, it is suggested to apply a perturbation approach based on an elastic potential <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{0}(\mathbf {q} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{0}(\mathbf {q} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2da43424debf0e00750e3a5375d26e4f9d518c09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.609ex; height:2.843ex;" alt="{\displaystyle u_{0}(\mathbf {q} )}" /></span>, which coincides with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{1}(\mathbf {q} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{1}(\mathbf {q} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ba91d1e020f09536435a989952d5f224b364b20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.609ex; height:2.843ex;" alt="{\displaystyle u_{1}(\mathbf {q} )}" /></span> at the asymptotic region. </p><p>The equation with an elastic potential can be solved, in a straightforward manner, by substitution. Thus, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab65ab05ef061a305d53a395741e8dec6d267986" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.509ex; height:2.009ex;" alt="{\displaystyle \chi _{0}}" /></span> is the solution of this equation, it is presented as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{0}(\mathbf {q} |\Gamma )=\xi _{0}(\mathbf {q} )\exp \left[-i\int _{\Gamma }d\mathbf {q} '\cdot \mathbf {\tau } (\mathbf {q} '|\Gamma )\right],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">)</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mi>i</mi> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Γ</mi> </mrow> </msub> <mi>d</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>′</mo> </msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi _{0}(\mathbf {q} |\Gamma )=\xi _{0}(\mathbf {q} )\exp \left[-i\int _{\Gamma }d\mathbf {q} '\cdot \mathbf {\tau } (\mathbf {q} '|\Gamma )\right],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/171e32ba7ce0feaa6fea3ab79bc601b5d41d0892" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:41.418ex; height:6.176ex;" alt="{\displaystyle \chi _{0}(\mathbf {q} |\Gamma )=\xi _{0}(\mathbf {q} )\exp \left[-i\int _{\Gamma }d\mathbf {q} '\cdot \mathbf {\tau } (\mathbf {q} '|\Gamma )\right],}" /></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" /></span> is an arbitrary contour, and the exponential function contains the relevant symmetry as created while moving along <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" /></span>. </p><p>The function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi _{0}(\mathbf {q} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi _{0}(\mathbf {q} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f8a7a107ee1d6200c9b4fd2c913a8dbdbf660cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.298ex; height:2.843ex;" alt="{\displaystyle \xi _{0}(\mathbf {q} )}" /></span> can be shown to be a solution of the (unperturbed/elastic) equation </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\xi _{0}+(u_{0}-E)\xi _{0}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>m</mi> </mrow> </mfrac> </mrow> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mi>E</mi> <mo stretchy="false">)</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\xi _{0}+(u_{0}-E)\xi _{0}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1f1447b706fd77a8ce7f68221d900ee8fd9b3b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.54ex; height:5.676ex;" alt="{\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\xi _{0}+(u_{0}-E)\xi _{0}=0.}" /></span></dd></dl> <p>Having <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{0}(\mathbf {q} |\Gamma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi _{0}(\mathbf {q} |\Gamma )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/187d1e521391a643fb2176311d06f3aef39b9697" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.834ex; height:2.843ex;" alt="{\displaystyle \chi _{0}(\mathbf {q} |\Gamma )}" /></span>, the full solution of the above decoupled equation takes the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi (\mathbf {q} |\Gamma )=\chi _{0}(\mathbf {q} |\Gamma )+\eta (\mathbf {q} |\Gamma ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>χ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>η</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi (\mathbf {q} |\Gamma )=\chi _{0}(\mathbf {q} |\Gamma )+\eta (\mathbf {q} |\Gamma ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5a4f94b4f8b9aae90c9a04e01ee6fb32f115fc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.693ex; height:2.843ex;" alt="{\displaystyle \chi (\mathbf {q} |\Gamma )=\chi _{0}(\mathbf {q} |\Gamma )+\eta (\mathbf {q} |\Gamma ),}" /></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta (\mathbf {q} |\Gamma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta (\mathbf {q} |\Gamma )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/651486349106509ca72f74a725943943ab2499f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.494ex; height:2.843ex;" alt="{\displaystyle \eta (\mathbf {q} |\Gamma )}" /></span> satisfies the resulting inhomogeneous equation: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\eta +({\tilde {u}}_{1}-E)\eta +i{\frac {\hbar ^{2}}{2m}}[2\mathbf {\tau } _{12}\nabla +\nabla \mathbf {\tau } _{12}]\eta =(u_{1}-u_{0})\chi _{0}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>m</mi> </mrow> </mfrac> </mrow> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>η</mi> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mi>η</mi> <mo>+</mo> <mi>i</mi> <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>m</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">[</mo> <mn>2</mn> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mi mathvariant="normal">∇</mi> <mo>+</mo> <mi mathvariant="normal">∇</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mi>η</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\eta +({\tilde {u}}_{1}-E)\eta +i{\frac {\hbar ^{2}}{2m}}[2\mathbf {\tau } _{12}\nabla +\nabla \mathbf {\tau } _{12}]\eta =(u_{1}-u_{0})\chi _{0}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e2c0f96b61584603d397af830f85ad360862e9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:62.302ex; height:5.676ex;" alt="{\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\eta +({\tilde {u}}_{1}-E)\eta +i{\frac {\hbar ^{2}}{2m}}[2\mathbf {\tau } _{12}\nabla +\nabla \mathbf {\tau } _{12}]\eta =(u_{1}-u_{0})\chi _{0}.}" /></span></dd></dl> <p>In this equation the inhomogeneity ensures the symmetry for the perturbed part of the solution along any contour and therefore for the solution in the required region in configuration space. </p><p>The relevance of the present approach was demonstrated while studying a two-arrangement-channel model (containing one inelastic channel and one reactive channel) for which the two adiabatic states were coupled by a <a href="/wiki/Jahn%E2%80%93Teller_effect" title="Jahn–Teller effect">Jahn–Teller</a> <a href="/wiki/Conical_intersection" title="Conical intersection">conical intersection</a>.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> A nice fit between the symmetry-preserved single-state treatment and the corresponding two-state treatment was obtained. This applies in particular to the reactive state-to-state probabilities (see Table III in Ref. 5a and Table III in Ref. 5b), for which the ordinary BO approximation led to erroneous results, whereas the symmetry-preserving BO approximation produced the accurate results, as they followed from solving the two coupled equations. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Born%E2%80%93Oppenheimer_approximation&action=edit&section=5" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Adiabatic_ionization" class="mw-redirect" title="Adiabatic ionization">Adiabatic ionization</a></li> <li><a href="/wiki/Adiabatic_process_(quantum_mechanics)" class="mw-redirect" title="Adiabatic process (quantum mechanics)">Adiabatic process (quantum mechanics)</a></li> <li><a href="/wiki/Avoided_crossing" title="Avoided crossing">Avoided crossing</a></li> <li><a href="/wiki/Born%E2%80%93Huang_approximation" title="Born–Huang approximation">Born–Huang approximation</a></li> <li><a href="/wiki/Franck%E2%80%93Condon_principle" title="Franck–Condon principle">Franck–Condon principle</a></li> <li><a href="/wiki/Kohn_anomaly" title="Kohn anomaly">Kohn anomaly</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Born%E2%80%93Oppenheimer_approximation&action=edit&section=6" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Authors often justify this step by stating that "the heavy nuclei move more slowly than the light <a href="/wiki/Electron" title="Electron">electrons</a>". Classically, this statement makes sense only if the <a href="/wiki/Momentum" title="Momentum">momentum</a> <i>p</i> of electrons and nuclei is of the same order of magnitude. In that case <i>m</i><sub>n</sub> ≫ <i>m</i><sub>e</sub> implies <i>p</i><sup>2</sup>/(2<i>m</i><sub>n</sub>) ≪ <i>p</i><sup>2</sup>/(2<i>m</i><sub>e</sub>). It is easy to show that for two bodies in circular orbits around their center of mass (regardless of individual masses), the momenta of the two bodies are equal and opposite, and that for any collection of particles in the center-of-mass frame, the net momentum is zero. Given that the center-of-mass frame is the lab frame (where the molecule is stationary), the momentum of the nuclei must be equal and opposite to that of the electrons. A hand-waving justification can be derived from quantum mechanics as well. The corresponding operators do not contain mass and the molecule can be treated as a <a href="/wiki/Particle_in_a_box" title="Particle in a box">box containing the electrons and nuclei</a>. Since the kinetic energy is <i>p</i><sup>2</sup>/(2<i>m</i>), it follows that, indeed, the kinetic energy of the nuclei in a molecule is usually much smaller than the kinetic energy of the electrons, the mass ratio being on the order of 10<sup>4</sup>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (January 2015)">citation needed</span></a></i>]</sup></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Typically, the electronic Schrödinger equation for molecules cannot be solved exactly. Approximation methods include the <a href="/wiki/Hartree-Fock_method" class="mw-redirect" title="Hartree-Fock method">Hartree-Fock method</a></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">It is assumed, in accordance with the <a href="/wiki/Adiabatic_theorem" title="Adiabatic theorem">adiabatic theorem</a>, that the same electronic state (for instance, the electronic ground state) is obtained upon small changes of the nuclear geometry. The method would give a discontinuity (jump) in the PES if electronic state switching would occur.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (January 2015)">citation needed</span></a></i>]</sup></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">This equation is time-independent, and stationary wavefunctions for the nuclei are obtained; nevertheless, it is traditional to use the word "motion" in this context, although classically motion implies time dependence.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (January 2015)">citation needed</span></a></i>]</sup></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Born%E2%80%93Oppenheimer_approximation&action=edit&section=7" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626" /><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFHanson" class="citation web cs1">Hanson, David. <a rel="nofollow" class="external text" href="https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Book%3A_Quantum_States_of_Atoms_and_Molecules_(Zielinksi_et_al)/10%3A_Theories_of_Electronic_Molecular_Structure/10.01%3A_The_Born-Oppenheimer_Approximation">"The Born-Oppenheimer Approximation"</a>. <i>Chemistry Libretexts</i>. Chemical Education Digital Library<span class="reference-accessdate">. Retrieved <span class="nowrap">2 August</span> 2022</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Chemistry+Libretexts&rft.atitle=The+Born-Oppenheimer+Approximation&rft.aulast=Hanson&rft.aufirst=David&rft_id=https%3A%2F%2Fchem.libretexts.org%2FBookshelves%2FPhysical_and_Theoretical_Chemistry_Textbook_Maps%2FBook%253A_Quantum_States_of_Atoms_and_Molecules_%28Zielinksi_et_al%29%2F10%253A_Theories_of_Electronic_Molecular_Structure%2F10.01%253A_The_Born-Oppenheimer_Approximation&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABorn%E2%80%93Oppenheimer+approximation" class="Z3988"></span></span> </li> <li id="cite_note-BornOppie-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-BornOppie_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-BornOppie_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMax_BornJ._Robert_Oppenheimer1927" class="citation journal cs1 cs1-prop-foreign-lang-source">Max Born; J. Robert Oppenheimer (1927). <a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fandp.19273892002">"Zur Quantentheorie der Molekeln"</a> [On the Quantum Theory of Molecules]. <i>Annalen der Physik</i> (in German). <b>389</b> (20): <span class="nowrap">457–</span>484. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1927AnP...389..457B">1927AnP...389..457B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fandp.19273892002">10.1002/andp.19273892002</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annalen+der+Physik&rft.atitle=Zur+Quantentheorie+der+Molekeln&rft.volume=389&rft.issue=20&rft.pages=%3Cspan+class%3D%22nowrap%22%3E457-%3C%2Fspan%3E484&rft.date=1927&rft_id=info%3Adoi%2F10.1002%2Fandp.19273892002&rft_id=info%3Abibcode%2F1927AnP...389..457B&rft.au=Max+Born&rft.au=J.+Robert+Oppenheimer&rft_id=https%3A%2F%2Fdoi.org%2F10.1002%252Fandp.19273892002&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABorn%E2%80%93Oppenheimer+approximation" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBirdSherwin2006" class="citation book cs1">Bird, Kai; Sherwin, Martin K. (2006). <i>American Prometheus: The Triumph and Tragedy of J. Robert Oppenheimer</i> (1st ed.). Vintage Books. pp. <span class="nowrap">65–</span>66. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0375726262" title="Special:BookSources/978-0375726262"><bdi>978-0375726262</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=American+Prometheus%3A+The+Triumph+and+Tragedy+of+J.+Robert+Oppenheimer&rft.pages=%3Cspan+class%3D%22nowrap%22%3E65-%3C%2Fspan%3E66&rft.edition=1st&rft.pub=Vintage+Books&rft.date=2006&rft.isbn=978-0375726262&rft.aulast=Bird&rft.aufirst=Kai&rft.au=Sherwin%2C+Martin+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABorn%E2%80%93Oppenheimer+approximation" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein, <i>Introduction to Algorithms</i>, 3rd ed., MIT Press, Cambridge, MA, 2009, § 28.2.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBornHuang1954" class="citation book cs1"><a href="/wiki/Max_Born" title="Max Born">Born, M.</a>; <a href="/wiki/Huang_Kun" title="Huang Kun">Huang, K.</a> (1954). "IV". <i><a href="/wiki/Dynamical_Theory_of_Crystal_Lattices" title="Dynamical Theory of Crystal Lattices">Dynamical Theory of Crystal Lattices</a></i>. New York: Oxford University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=IV&rft.btitle=Dynamical+Theory+of+Crystal+Lattices&rft.place=New+York&rft.pub=Oxford+University+Press&rft.date=1954&rft.aulast=Born&rft.aufirst=M.&rft.au=Huang%2C+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABorn%E2%80%93Oppenheimer+approximation" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation book cs1">"Born-Oppenheimer Approach: Diabatization and Topological Matrix". <i>Beyond Born-Oppenheimer: Electronic Nonadiabatic Coupling Terms and Conical Intersections</i>. Hoboken, NJ, USA: John Wiley & Sons, Inc. 28 March 2006. pp. <span class="nowrap">26–</span>57. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2F0471780081.ch2">10.1002/0471780081.ch2</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-78008-3" title="Special:BookSources/978-0-471-78008-3"><bdi>978-0-471-78008-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Born-Oppenheimer+Approach%3A+Diabatization+and+Topological+Matrix&rft.btitle=Beyond+Born-Oppenheimer%3A+Electronic+Nonadiabatic+Coupling+Terms+and+Conical+Intersections&rft.place=Hoboken%2C+NJ%2C+USA&rft.pages=%3Cspan+class%3D%22nowrap%22%3E26-%3C%2Fspan%3E57&rft.pub=John+Wiley+%26+Sons%2C+Inc.&rft.date=2006-03-28&rft_id=info%3Adoi%2F10.1002%2F0471780081.ch2&rft.isbn=978-0-471-78008-3&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABorn%E2%80%93Oppenheimer+approximation" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBaerEnglman1997" class="citation journal cs1">Baer, Michael; Englman, Robert (1997). "A modified Born-Oppenheimer equation: application to conical intersections and other types of singularities". <i>Chemical Physics Letters</i>. <b>265</b> (<span class="nowrap">1–</span>2). Elsevier BV: <span class="nowrap">105–</span>108. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1997CPL...265..105B">1997CPL...265..105B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fs0009-2614%2896%2901411-x">10.1016/s0009-2614(96)01411-x</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0009-2614">0009-2614</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Chemical+Physics+Letters&rft.atitle=A+modified+Born-Oppenheimer+equation%3A+application+to+conical+intersections+and+other+types+of+singularities&rft.volume=265&rft.issue=%3Cspan+class%3D%22nowrap%22%3E1%E2%80%93%3C%2Fspan%3E2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E105-%3C%2Fspan%3E108&rft.date=1997&rft.issn=0009-2614&rft_id=info%3Adoi%2F10.1016%2Fs0009-2614%2896%2901411-x&rft_id=info%3Abibcode%2F1997CPL...265..105B&rft.aulast=Baer&rft.aufirst=Michael&rft.au=Englman%2C+Robert&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABorn%E2%80%93Oppenheimer+approximation" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBaerCharutzKosloffBaer1996" class="citation journal cs1">Baer, Roi; Charutz, David M.; Kosloff, Ronnie; Baer, Michael (22 November 1996). "A study of conical intersection effects on scattering processes: The validity of adiabatic single-surface approximations within a quasi-Jahn–Teller model". <i>The Journal of Chemical Physics</i>. <b>105</b> (20). AIP Publishing: <span class="nowrap">9141–</span>9152. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1996JChPh.105.9141B">1996JChPh.105.9141B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.472748">10.1063/1.472748</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0021-9606">0021-9606</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Journal+of+Chemical+Physics&rft.atitle=A+study+of+conical+intersection+effects+on+scattering+processes%3A+The+validity+of+adiabatic+single-surface+approximations+within+a+quasi-Jahn%E2%80%93Teller+model&rft.volume=105&rft.issue=20&rft.pages=%3Cspan+class%3D%22nowrap%22%3E9141-%3C%2Fspan%3E9152&rft.date=1996-11-22&rft.issn=0021-9606&rft_id=info%3Adoi%2F10.1063%2F1.472748&rft_id=info%3Abibcode%2F1996JChPh.105.9141B&rft.aulast=Baer&rft.aufirst=Roi&rft.au=Charutz%2C+David+M.&rft.au=Kosloff%2C+Ronnie&rft.au=Baer%2C+Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABorn%E2%80%93Oppenheimer+approximation" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAdhikariBilling1999" class="citation journal cs1">Adhikari, Satrajit; Billing, Gert D. (1999). "The conical intersection effects and adiabatic single-surface approximations on scattering processes: A time-dependent wave packet approach". <i>The Journal of Chemical Physics</i>. <b>111</b> (1). AIP Publishing: <span class="nowrap">40–</span>47. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1999JChPh.111...40A">1999JChPh.111...40A</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.479360">10.1063/1.479360</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0021-9606">0021-9606</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Journal+of+Chemical+Physics&rft.atitle=The+conical+intersection+effects+and+adiabatic+single-surface+approximations+on+scattering+processes%3A+A+time-dependent+wave+packet+approach&rft.volume=111&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E40-%3C%2Fspan%3E47&rft.date=1999&rft.issn=0021-9606&rft_id=info%3Adoi%2F10.1063%2F1.479360&rft_id=info%3Abibcode%2F1999JChPh.111...40A&rft.aulast=Adhikari&rft.aufirst=Satrajit&rft.au=Billing%2C+Gert+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABorn%E2%80%93Oppenheimer+approximation" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCharutzBaerBaer1997" class="citation journal cs1">Charutz, David M.; Baer, Roi; Baer, Michael (1997). "A study of degenerate vibronic coupling effects on scattering processes: are resonances affected by degenerate vibronic coupling?". <i>Chemical Physics Letters</i>. <b>265</b> (6). Elsevier BV: <span class="nowrap">629–</span>637. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1997CPL...265..629C">1997CPL...265..629C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fs0009-2614%2896%2901494-7">10.1016/s0009-2614(96)01494-7</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0009-2614">0009-2614</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Chemical+Physics+Letters&rft.atitle=A+study+of+degenerate+vibronic+coupling+effects+on+scattering+processes%3A+are+resonances+affected+by+degenerate+vibronic+coupling%3F&rft.volume=265&rft.issue=6&rft.pages=%3Cspan+class%3D%22nowrap%22%3E629-%3C%2Fspan%3E637&rft.date=1997&rft.issn=0009-2614&rft_id=info%3Adoi%2F10.1016%2Fs0009-2614%2896%2901494-7&rft_id=info%3Abibcode%2F1997CPL...265..629C&rft.aulast=Charutz&rft.aufirst=David+M.&rft.au=Baer%2C+Roi&rft.au=Baer%2C+Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABorn%E2%80%93Oppenheimer+approximation" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Born%E2%80%93Oppenheimer_approximation&action=edit&section=8" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Resources related to the Born–Oppenheimer approximation: </p> <ul><li><a rel="nofollow" class="external text" href="http://gallica.bnf.fr/ark:/12148/bpt6k15386r">The original article</a> (in German)</li> <li><a rel="nofollow" class="external text" href="http://elib.bsu.by/bitstream/123456789/154381/1/1927-084%20AP%20Born%20%26%20Oppenheimer%20-%20On%20the%20Quantum%20Theory%20of%20Molecules.pdf">Translation by S. M. Blinder</a></li> <li><a rel="nofollow" class="external text" href="https://www.theochem.ru.nl/files/dbase/born-oppenheimer-translated-s-m-blinder.pdf">Another version of the same translation by S. M. Blinder</a></li> <li><a rel="nofollow" class="external text" href="http://www.tcm.phy.cam.ac.uk/%7Epdh1001/thesis/node13.html">The Born–Oppenheimer approximation</a>, a section from Peter Haynes' doctoral thesis</li></ul> <p class="mw-empty-elt"> </p> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output 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Born–Oppenheimer approximation - Wikipedia