chosen a priori by a sign rule and are not variables. $p_{10}$ is related to $p_{11}$ and $p_1$ by $p_1(i) = p_{11}(ij) +
p_{10}(ij)$ and can be eliminated from the functional. Each sum in the denominator of $\xi(ij)$ in [[**\[xi\_DOCI\]**]{}]{} can be simplified by use of the sum rule of [[**\[p11sum\_DOCI\]**]{}]{} to, e.g., $$\sum_{k(\ne i,j)} p_{11}(ik) = \frac1{2} (N-2) p_1(i) - p_{11}(ij).$$

As stated in the general section on the 2-DM, we assume that the $(2,2)$ and $(2,3)$ positivity conditions are sufficient in practice. Were this not the case, it would be straightforward to add a few more positivity conditions to achieve practical sufficiency. Under this circumstance, the infimum of $E$ with respect to the NOs and the OPs, subject to the constraints [[**\[rho1cond\]**]{}]{} and [[**\[bound1\]**]{}]{}, [[**\[bound2\]**]{}]{}, [[**\[p11sum\_DOCI\]**]{}]{}, yields a variational approximation to the ground-state energy, apart from the $\xi$-approximation for $N>4$.

The last term on the rhs of [[**\[energy\]**]{}]{} originates from $\pi^{od}$ and drives electron correlation; without it the infimum is the HF energy. The second term on the rhs of [[**\[energy\]**]{}]{} originates from $\pi^d$, the form of which is represented exactly in our theory. It contains only positive contributions and is essential; the integral relation connecting $\pi$ and 