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documentation:standard_operators:crystal_field [2016/10/10 09:40] – external edit 127.0.0.1documentation:standard_operators:crystal_field [2025/11/20 02:45] (current) – external edit 127.0.0.1
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 ====== Crystal field operator ====== ====== Crystal field operator ======
  
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 In order to evaluate $\big\langle \varphi_{\tau_1}(\vec{r}) \big| V(\vec{r}) \big| \varphi_{\tau_2}(\vec{r}) \big\rangle$ we expand $V(\vec{r})$ on renormalized spherical Harmonics: In order to evaluate $\big\langle \varphi_{\tau_1}(\vec{r}) \big| V(\vec{r}) \big| \varphi_{\tau_2}(\vec{r}) \big\rangle$ we expand $V(\vec{r})$ on renormalized spherical Harmonics:
 \begin{equation} \begin{equation}
-C_{k,m}(\theta,\phi) = \sqrt{\frac{4 \pi}{2l+1}} Y_{k,m}(\theta,\phi).+C_{k,m}(\theta,\phi) = \sqrt{\frac{4 \pi}{2k+1}} Y_{k,m}(\theta,\phi).
 \end{equation} \end{equation}
 Using a tailor series in $r^k$ we find: Using a tailor series in $r^k$ we find:
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 with: with:
 \begin{align} \begin{align}
-A_{k,m}        &= \frac{1}{\sqrt{(l-m)!}}\frac{1}{\sqrt{(l+m)!}} \\ +A_{k,m}        &= \frac{1}{\sqrt{(k-m)!}}\frac{1}{\sqrt{(k+m)!}} \\ 
 \nonumber &\times \big\langle R_{n_1,l_1}(r) \big| \\ \nonumber &\times \big\langle R_{n_1,l_1}(r) \big| \\
-\nonumber &\quad \partial_z^{l-|m|}(-\rm{Sign}[m]\partial_x+\imath\partial_y)^{|m|}V(r,\theta,\phi)\big|_{r=0} \\+\nonumber &\quad \partial_z^{k-|m|}(-\rm{Sign}[m]\partial_x+\imath\partial_y)^{|m|}V(r,\theta,\phi)\big|_{r=0} \\
 \nonumber & \quad\quad\quad \big| R_{n_2,l_2}(r) \big\rangle. \nonumber & \quad\quad\quad \big| R_{n_2,l_2}(r) \big\rangle.
 \end{align} \end{align}
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 ===== Table of contents ===== ===== Table of contents =====
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