satisfies two important sum rules $$\begin{aligned}
\int d{{\bf{r}}}_2 \, \, \pi({{\bf{r}}}{{\bf{r}}}_2; {{\bf{r}}}' {{\bf{r}}}_2) &=& (N-1) \rho({{\bf{r}}},{{\bf{r}}}') \\
\int d{{\bf{r}}}_1 \, d{{\bf{r}}}_2 \, \, \pi({{\bf{r}}}_1 {{\bf{r}}}_2;{{\bf{r}}}_1 {{\bf{r}}}_2) w(r_{12})
&\ge& 0. \end{aligned}$$

The OP-NOFT-0 form for $\pi$, [[**\[pisum\]**]{}]{}–[[**\[xi\_DOCI\]**]{}]{}, is exact when $N=2$ with $\xi=1$. It is equivalent to DOCI for $N=4$. When $N>4$, the $\xi$-approximation of [[**\[xi\_DOCI\]**]{}]{} and the assumption on the $N$-representability condition break the equivalence to DOCI. Our numerical results presented below and our detailed examinations of its formal structure at dissociation suggest that it is a very good approximation to DOCI, with algebraic instead of exponential complexity.

The expectation value $E = \langle \Psi | H |
\Psi \rangle$ of the Hamiltonian $H = h + \frac1{2} \sum_{i\ne j}
w(r_{ij})$, where $h$ is a sum of one-body terms, becomes: $$\begin{aligned}
E &=& 2 \sum_i p_1(i) \langle\phi_i | h | \phi_i\rangle + \nonumber \\
&&\sum_{ij} p_{11}(ij) \left(2 J_{ij} - K_{ij}\right) + \nonumber \\
&&\sum_{i\ne j} s(i) s(j) p_{10}^{1/2}(ij)\,p_{01}^{1/2}(ij)\,
\xi(ij) \, K_{ij}, \label{energy}\end{aligned}$$ where $J_{ij}$ and $K_{ij}$ are Hartree- and exchange integrals defined in terms of the NOs $\{\phi_i\}$. With [[**\[xi\_DOCI\]**]{}]{} for $\xi(ij)$, $E$ in [[**\[energy\]**]{}]{} is a functional of the NOs, and the 1- and 2-state OPs. The signs are 