corresponding spin independent 2-NSO OPs, e.g. $p_{1111}(i+i- k+k-) = p_{11}(ik)$. Correspondingly, the signs in [[**\[final\_pi-od\]**]{}]{} depend only on a single orbital index, $s(i+i-) = s(i)$, and the $\xi$ depend only on two-orbital indices, $\xi(i+i- j+j-) = \xi(ij)$. With these simplifications, the 2-DM of [[**\[pi-d\]**]{}]{} and [[**\[final\_pi-od\]**]{}]{} becomes $$\pi({{\bf{r}}}'_1 {{\bf{r}}}'_2;{{\bf{r}}}_1 {{\bf{r}}}_2) =
\pi^d({{\bf{r}}}'_1 {{\bf{r}}}'_2;{{\bf{r}}}_1 {{\bf{r}}}_2) +
\pi^{od}({{\bf{r}}}'_1 {{\bf{r}}}'_2;{{\bf{r}}}_1 {{\bf{r}}}_2),
\label{pisum}$$ after tracing out the spins, with $$\begin{aligned}
\pi^d({{\bf{r}}}'_1 {{\bf{r}}}'_2;{{\bf{r}}}_1 {{\bf{r}}}_2) &=& 
2 \sum_{ij} p_{11}(ij) \nonumber\\
&&\big(2 \phi_i({{\bf{r}}}'_1) \phi_j({{\bf{r}}}'_2) \phi_i({{\bf{r}}}_1) \phi_j({{\bf{r}}}_2) - \nonumber \\
&&\phi_i({{\bf{r}}}'_1) \phi_j({{\bf{r}}}'_2) \phi_j({{\bf{r}}}_1) \phi_i({{\bf{r}}}_2) \big) 
\label{pi-d_DOCI}\\
\pi^{od}({{\bf{r}}}'_1 {{\bf{r}}}'_2;{{\bf{r}}}_1 {{\bf{r}}}_2) &=& 2 \sum_{i\ne j} s(i) s(j) \nonumber \\
&&\left[ p_{10}(ij) p_{01}(ij)\right]^{1/2} \xi(ij) \nonumber \\
&& \phi_i({{\bf{r}}}'_1) \phi_i({{\bf{r}}}'_2) \phi_j({{\bf{r}}}_1) \phi_j({{\bf{r}}}_2).
\label{pi-od_DOCI}\end{aligned}$$ The sum in [[**\[pi-d\_DOCI\]**]{}]{} includes the term $i=j$, for which $p_{11}(ii) =
p_1(i)$ because of double occupancy, and $\xi(ij)$ in [[**\[pi-od\_DOCI\]**]{}]{} is now $$\xi(ij) \approx \frac{
\sum_{k(\ne i,j)} p^{1/2}_{11}(ik) p^{1/2}_{11}(jk)
}{\left[
\sum_{k(\ne i,j)} p_{11}(ik) \, \, \sum_{k(\ne i,j)} p_{11}(jk)
\right]^{1/2}}.
\label{xi_DOCI}$$

The one- and two-orbital OPs of OP-NOFT-0 lie within the same bounds as in the general case, [[**\[bound1\]**]{}]{}–[[**\[bound2\]**]{}]{}, if we impose only the $(2,2)$ and $(2,3)$ positivity conditions, and their sum rules become, respectively, [[**\[rho1cond\]**]{}]{} and $$2 \sum_{j(\ne i)} p_{11}(ij) = (N-2) p_1(i).
\label{p11sum_DOCI}$$

The $\pi$ of [[**\[pisum\]**]{}]{} 