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-{"documenter":{"julia_version":"1.10.4","generation_timestamp":"2024-06-20T16:37:41","documenter_version":"1.4.1"}}
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 -\sqrt{\frac{4\pi}{2l+1}}\cdot r^{\ell}\bar{P}_{\ell}^{|m|}(\cos \theta)\sin(|m|\varphi) & m < 0\\
 \sqrt{\frac{2\pi}{2l+1}}\bar{P}_{\ell}^0(\cos \theta) & m = 0 \\
 \sqrt{\frac{4\pi}{2l+1}}\cdot r^{\ell}\bar{P}_{\ell}^{m}(\cos \theta)\cos(m\varphi)  & m > 0
-\end{cases}\]</p><p>Schmidt semi-normalized real spherical harmonics that employ the Condon-Shortley phase up to degree <span>$\ell = 3$</span> are</p><table><tr><th style="text-align: right"><span>$m\backslash\ell$</span></th><th style="text-align: right">0</th><th style="text-align: right">1</th><th style="text-align: right">2</th><th style="text-align: right">3</th></tr><tr><td style="text-align: right">3</td><td style="text-align: right"></td><td style="text-align: right"></td><td style="text-align: right"></td><td style="text-align: right"><span>$\frac{1}{2}\sqrt{\frac{5}{2}}(x^2-3y^2)x$</span></td></tr><tr><td style="text-align: right">2</td><td style="text-align: right"></td><td style="text-align: right"></td><td style="text-align: right"><span>$\frac{1}{2}\sqrt{3}(x^2-y^2)$</span></td><td style="text-align: right"><span>$\frac{1}{2}\sqrt{15}(x^2-y^2)z$</span></td></tr><tr><td style="text-align: right">1</td><td style="text-align: right"></td><td style="text-align: right"><span>$x$</span></td><td style="text-align: right"><span>$\sqrt{3}xz$</span></td><td style="text-align: right"><span>$\frac{1}{2}\sqrt{\frac{3}{2}}(5z^2-r^2)x$</span></td></tr><tr><td style="text-align: right">0</td><td style="text-align: right"><span>$1$</span></td><td style="text-align: right"><span>$z$</span></td><td style="text-align: right"><span>$\frac{1}{2}(3z^2-r^2)$</span></td><td style="text-align: right"><span>$\frac{1}{2}(5z^2-3r^2)z$</span></td></tr><tr><td style="text-align: right">-1</td><td style="text-align: right"></td><td style="text-align: right"><span>$y$</span></td><td style="text-align: right"><span>$\sqrt{3}yz$</span></td><td style="text-align: right"><span>$\frac{1}{2}\sqrt{\frac{3}{2}}(5z^2-r^2)y$</span></td></tr><tr><td style="text-align: right">-2</td><td style="text-align: right"></td><td style="text-align: right"></td><td style="text-align: right"><span>$\sqrt{3}xy$</span></td><td style="text-align: right"><span>$\sqrt{15}xyz$</span></td></tr><tr><td style="text-align: right">-3</td><td style="text-align: right"></td><td style="text-align: right"></td><td style="text-align: right"></td><td style="text-align: right"><span>$\frac{1}{2}\sqrt{\frac{5}{2}}(3x^2-y^2)y$</span></td></tr></table><h3 id="References"><a class="docs-heading-anchor" href="#References">References</a><a id="References-1"></a><a class="docs-heading-anchor-permalink" href="#References" title="Permalink"></a></h3><ol><li>Dusson, G., Bachmayr, M., Csányi, G., Drautz, R., Etter, S., van der Oord, C., &amp; Ortner, C. (2022). <a href="https://arxiv.org/pdf/1911.03550.pdf">Atomic cluster expansion: Completeness, efficiency and stability</a>. Journal of Computational Physics, 454, 110946.</li><li>Helgaker, T., Jorgensen, P., &amp; Olsen, J. (2013). <a href="https://www.wiley.com/en-us/Molecular+Electronic-Structure+Theory-p-9780471967552">Molecular electronic-structure theory</a>. John Wiley &amp; Sons.</li><li>Limpanuparb, T., &amp; Milthorpe, J. (2014). <a href="https://arxiv.org/pdf/1410.1748.pdf">Associated Legendre polynomials and spherical harmonics computation for chemistry applications</a>. arXiv preprint arXiv:1410.1748.  </li><li>Wieczorek, M. A., &amp; Meschede, M. (2018). <a href="https://agupubs.onlinelibrary.wiley.com/doi/epdf/10.1029/2018GC007529">SHTools: Tools for working with spherical harmonics.</a> Geochemistry, Geophysics, Geosystems, 19(8), 2574-2592.</li></ol></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../background/">« Background Index</a><a class="docs-footer-nextpage" href="../ace/">Cluster Expansion »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.4.1 on <span class="colophon-date" title="Thursday 20 June 2024 16:37">Thursday 20 June 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+\end{cases}\]</p><p>Schmidt semi-normalized real spherical harmonics that employ the Condon-Shortley phase up to degree <span>$\ell = 3$</span> are</p><table><tr><th style="text-align: right"><span>$m\backslash\ell$</span></th><th style="text-align: right">0</th><th style="text-align: right">1</th><th style="text-align: right">2</th><th style="text-align: right">3</th></tr><tr><td style="text-align: right">3</td><td style="text-align: right"></td><td style="text-align: right"></td><td style="text-align: right"></td><td style="text-align: right"><span>$\frac{1}{2}\sqrt{\frac{5}{2}}(x^2-3y^2)x$</span></td></tr><tr><td style="text-align: right">2</td><td style="text-align: right"></td><td style="text-align: right"></td><td style="text-align: right"><span>$\frac{1}{2}\sqrt{3}(x^2-y^2)$</span></td><td style="text-align: right"><span>$\frac{1}{2}\sqrt{15}(x^2-y^2)z$</span></td></tr><tr><td style="text-align: right">1</td><td style="text-align: right"></td><td style="text-align: right"><span>$x$</span></td><td style="text-align: right"><span>$\sqrt{3}xz$</span></td><td style="text-align: right"><span>$\frac{1}{2}\sqrt{\frac{3}{2}}(5z^2-r^2)x$</span></td></tr><tr><td style="text-align: right">0</td><td style="text-align: right"><span>$1$</span></td><td style="text-align: right"><span>$z$</span></td><td style="text-align: right"><span>$\frac{1}{2}(3z^2-r^2)$</span></td><td style="text-align: right"><span>$\frac{1}{2}(5z^2-3r^2)z$</span></td></tr><tr><td style="text-align: right">-1</td><td style="text-align: right"></td><td style="text-align: right"><span>$y$</span></td><td style="text-align: right"><span>$\sqrt{3}yz$</span></td><td style="text-align: right"><span>$\frac{1}{2}\sqrt{\frac{3}{2}}(5z^2-r^2)y$</span></td></tr><tr><td style="text-align: right">-2</td><td style="text-align: right"></td><td style="text-align: right"></td><td style="text-align: right"><span>$\sqrt{3}xy$</span></td><td style="text-align: right"><span>$\sqrt{15}xyz$</span></td></tr><tr><td style="text-align: right">-3</td><td style="text-align: right"></td><td style="text-align: right"></td><td style="text-align: right"></td><td style="text-align: right"><span>$\frac{1}{2}\sqrt{\frac{5}{2}}(3x^2-y^2)y$</span></td></tr></table><h3 id="References"><a class="docs-heading-anchor" href="#References">References</a><a id="References-1"></a><a class="docs-heading-anchor-permalink" href="#References" title="Permalink"></a></h3><ol><li>Dusson, G., Bachmayr, M., Csányi, G., Drautz, R., Etter, S., van der Oord, C., &amp; Ortner, C. (2022). <a href="https://arxiv.org/pdf/1911.03550.pdf">Atomic cluster expansion: Completeness, efficiency and stability</a>. Journal of Computational Physics, 454, 110946.</li><li>Helgaker, T., Jorgensen, P., &amp; Olsen, J. (2013). <a href="https://www.wiley.com/en-us/Molecular+Electronic-Structure+Theory-p-9780471967552">Molecular electronic-structure theory</a>. John Wiley &amp; Sons.</li><li>Limpanuparb, T., &amp; Milthorpe, J. (2014). <a href="https://arxiv.org/pdf/1410.1748.pdf">Associated Legendre polynomials and spherical harmonics computation for chemistry applications</a>. arXiv preprint arXiv:1410.1748.  </li><li>Wieczorek, M. A., &amp; Meschede, M. (2018). <a href="https://agupubs.onlinelibrary.wiley.com/doi/epdf/10.1029/2018GC007529">SHTools: Tools for working with spherical harmonics.</a> Geochemistry, Geophysics, Geosystems, 19(8), 2574-2592.</li></ol></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../background/">« Background Index</a><a class="docs-footer-nextpage" href="../ace/">Cluster Expansion »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.4.1 on <span class="colophon-date" title="Saturday 22 June 2024 16:22">Saturday 22 June 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>