$\pi$ separates into a part $\pi^d$ diagonal in the indices, i.e. with $ii' = jj'$, and an off-diagonal part, $\pi^{od}$, with $ii' \neq jj'$: $$\begin{aligned}
\pi^d(x'_1x'_2;x_1x_2) &=& \frac1{2}\sum_{i\ne j} p_{11}(ij) \nonumber\\
&&\left(
\psi_i(x'_1)\psi_j(x'_2)-\psi_j(x'_1)\psi_i(x'_2)\right)\nonumber\\
&&\left(
\psi_i(x_1)\psi_j(x_2)-\psi_j(x_1)\psi_i(x_2)\right)
\label{pi-d}\\
\pi^{od}(x'_1x'_2;x_1x_2) &=& 
\sum_{\scriptscriptstyle
\overset{i<i' \ne j<j',{{\bf{k}}}}
{i,i',j,j' \not\in {{\bf{k}}}}}
C_{ii'{{\bf{k}}}} C_{jj'{{\bf{k}}}}\nonumber\\
&&\left(
\psi_i(x'_1)\psi_{i'}(x'_2)-\psi_{i'}(x'_1)\psi_i(x'_2)\right)\nonumber\\
&&\left(
\psi_j(x_1)\psi_{j'}(x_2)-\psi_{j'}(x_1)\psi_j(x_2)\right)
\label{pi-od}\end{aligned}$$

Electron correlation is expressed through $\pi^{od}$. The analogous off-diagonal part of $\rho(x',x)$ is suppressed by the OC, an advantage of the NSO basis. Note also that the PDC has eliminated 3-index terms from $\pi^{od}$ in [[**\[pi-od\]**]{}]{}.

The $p_{11}(ij) = \sum_{{{\bf{n}}}} C_{{{\bf{n}}}}^2 \, \nu_{i,{{\bf{n}}}} \nu_{j,{{\bf{n}}}}$ in $\pi^d$ are joint 2-state occupation probabilities (2-OPs). Establishing the $N$-representability conditions for the 2-OPs is a hard problem because new conditions constrain the 2-OPs that arise from the positivity conditions for the $q$-OPs, i.e. the $p_{11\cdots 1}
(i_1,i_2,\cdots,i_q) = \sum_{{{\bf{n}}}} C_{{{\bf{n}}}}^2 \, \nu_{i_1,{{\bf{n}}}}
\nu_{i_2,{{\bf{n}}}} \cdots \nu_{i_q,{{\bf{n}}}}$, at any order $q$. These conditions derive from the so-called $(2,q)$ positivity conditions [@Mazziotti2012a] restricted to the diagonal elements of the 2-DM [@Ayers2007]. Limiting ourselves to the $(2,2)$ and $(2,3)$ conditions, the following conditions for the 2-OPs hold: $$\begin{aligned}
\mbox{sup}(p_1(i)+p_1(j)-1,0) &\le& p_{11}(ij) \,\le\, p_1(<) \label{bound1}\\
\mbox{sup}(p_1(i)+p_1(j)+p_1(k)-1,0) &\le&
p_{11}(ij)+\nonumber\\
&&p_{11}(ik)+p_{11}(jk) \label{bound2}\end{aligned}$$ $p_1(<)$ is the lesser of $p_1(i)$ and 