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documentation:language_reference:functions:shellaverage [2026/11/12 18:01] – created Maurits W. Haverkortdocumentation:language_reference:functions:shellaverage [2026/11/12 18:05] (current) – Maurits W. Haverkort
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-\begin{equation} \nonumber 
 P[O] = \int \mathrm{d}U \; U \, O \, U^{\dagger} . P[O] = \int \mathrm{d}U \; U \, O \, U^{\dagger} .
-\end{equation} 
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-\begin{equation} \nonumber 
 \hat{1}_{\Lambda^p} = \sum_{\tau_1 < \tau_2 < \cdots < \tau_p} a^{\dagger}_{\tau_1} \cdots a^{\dagger}_{\tau_p} a^{\phantom{\dagger}}_{\tau_p} \cdots a^{\phantom{\dagger}}_{\tau_1} = \frac{1}{p!} \hat{N} (\hat{N}-1) \cdots (\hat{N}-p+1) . \hat{1}_{\Lambda^p} = \sum_{\tau_1 < \tau_2 < \cdots < \tau_p} a^{\dagger}_{\tau_1} \cdots a^{\dagger}_{\tau_p} a^{\phantom{\dagger}}_{\tau_p} \cdots a^{\phantom{\dagger}}_{\tau_1} = \frac{1}{p!} \hat{N} (\hat{N}-1) \cdots (\hat{N}-p+1) .
-\end{equation} 
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-\begin{eqnarray} \nonumber +U_{\mathrm{av}}(p) = F^0 - \frac{2}{25} F^2  
-U_{\mathrm{av}}(p) &=& F^0 - \frac{2}{25} F^2 \\ \nonumber +$$ 
-U_{\mathrm{av}}(d) &=& F^0 - \frac{2}{63} \left( F^2 + F^4 \right) \\ \nonumber +$$ 
-U_{\mathrm{av}}(f) &=& F^0 - \frac{4}{195} F^2 - \frac{2}{143} F^4 - \frac{100}{5577} F^6 , +U_{\mathrm{av}}(d) = F^0 - \frac{2}{63} \left( F^2 + F^4 \right) 
-\end{eqnarray}+$$ 
 +$$ 
 +U_{\mathrm{av}}(f) = F^0 - \frac{4}{195} F^2 - \frac{2}{143} F^4 - \frac{100}{5577} F^6,
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-\begin{equation} \nonumber 
 U_{\mathrm{av}}(pd) = F^0_{pd} - \frac{1}{15} G^1_{pd} - \frac{3}{70} G^3_{pd} . U_{\mathrm{av}}(pd) = F^0_{pd} - \frac{1}{15} G^1_{pd} - \frac{3}{70} G^3_{pd} .
-\end{equation} 
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-\begin{equation} \nonumber 
 \epsilon_{\uparrow} \hat{N}_{\uparrow} + \epsilon_{\downarrow} \hat{N}_{\downarrow} \epsilon_{\uparrow} \hat{N}_{\uparrow} + \epsilon_{\downarrow} \hat{N}_{\downarrow}
 + \tfrac{1}{2} U_{\uparrow\uparrow} \hat{N}_{\uparrow} (\hat{N}_{\uparrow}-1) + \tfrac{1}{2} U_{\uparrow\uparrow} \hat{N}_{\uparrow} (\hat{N}_{\uparrow}-1)
 + \tfrac{1}{2} U_{\downarrow\downarrow} \hat{N}_{\downarrow} (\hat{N}_{\downarrow}-1) + \tfrac{1}{2} U_{\downarrow\downarrow} \hat{N}_{\downarrow} (\hat{N}_{\downarrow}-1)
 + U_{\uparrow\downarrow} \hat{N}_{\uparrow} \hat{N}_{\downarrow} . + U_{\uparrow\downarrow} \hat{N}_{\uparrow} \hat{N}_{\downarrow} .
-\end{equation} 
 $$ $$
  

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