approximation, this method does not suffer from double-counting the second-order contributions [@Row1968b; @VKC2000]. For the OBDM in a generic basis one finds [@CGP1998; @VKC2000] $$\begin{aligned}
\lefteqn{\langle 0 | a_{pm_p}^{\dagger} a_{p'm_{p'}} | 0 \rangle  
= } \nonumber \\ 
&&\sum_{hm_h \nu \nu'} 
(\delta_{\nu\nu'} 
 -\frac{1}{2} \! \sum_{p_1h_1} 
\langle 0   | a_{h_1m_{h_1}}^{\dagger} a_{p_1m_{p_1}} | \nu' \rangle  
\nonumber \\ && 
\mbox{} \hspace{3cm} \times 
\langle \nu | a_{p_1m_{p_1}}^{\dagger} a_{h_1m_{h_1}} | 0    \rangle 
) 
\nonumber 
\\ &&  
\mbox{}\hspace{8mm}\times 
\langle 0   | a_{p  m_p    }^{\dagger} a_{h  m_h    } | \nu  \rangle 
\langle \nu'| a_{h  m_h    }^{\dagger} a_{p' m_{p' }} | 0    \rangle 
, 
\label{E:gOBDMp} 
\\ 
\lefteqn{\langle 0 | a_{hm_h}^{\dagger} a_{h'm_{h'}} | 0 \rangle 
=  
\delta_{h,h'}\delta_{m_hm_{h'}} }
\nonumber \\ 
&-& 
\sum_{pm_p \nu \nu'} 
(\delta_{\nu\nu'} 
 -\frac{1}{2} \! \sum_{p_1h_1} 
\langle 0   | a_{h_1m_{h_1}}^{\dagger} a_{p_1m_{p_1}} | \nu' \rangle 
\nonumber \\ && 
\mbox{} \hspace{3cm} \times 
\langle \nu | a_{p_1m_{p_1}}^{\dagger} a_{h_1m_{h_1}} | 0    \rangle 
) 
\nonumber 
\\ 
&& 
\mbox{}\hspace{8mm}\times 
\langle 0   | a_{p  m_p    }^{\dagger} a_{h  m_h    } | \nu  \rangle 
\langle \nu'| a_{h  m_h    }^{\dagger} a_{p' m_{p' }} | 0    \rangle 
. 
\label{E:gOBDMh} \end{aligned}$$ In the natural-orbital basis, and using angular momentum coupling, we have

$$\begin{aligned}
\langle 0 | a_{p}^{\dagger} a_{p'} | 0 \rangle 
&=& 
\hat{j}_p^{-2}  \sum_{{J}^{\pi} \nu \nu' h }
\hat{J}^2       [ 
