$p_1(j)$. In addition the sum rule $$\begin{aligned}
\sum_{j (\ne i)} p_{11}(ij) &=& (N-1) p_1(i) \label{p11sum}\end{aligned}$$ must be satisfied. Conditions [[**\[bound1\]**]{}]{}–[[**\[p11sum\]**]{}]{} were first established by Weinhold and Bright Wilson [@Weinhold1967a]. They are necessary but not sufficient conditions for $N$-representability [@Ayers2007; @Davidson1969]. Establishing a complete set of conditions is a QMA-hard problem because the number of $(2,q)$ positivity conditions increases combinatorially with increasing $q$. Fortunately numerical calculations on atoms and molecules indicate that sufficiently accurate lower-bound ground-state energies often result by imposing $(2,q)$-positivity conditions with $q\le 3$ [@Zhao2004; @Mazziotti2012b]. This suggests that even in the most difficult situations, fermionic problems in atoms and molecules should require only a finite and small set of positivity conditions. Here we shall limit ourselves to conditions [[**\[bound1\]**]{}]{}–[[**\[p11sum\]**]{}]{}, as we found in our numerical calculations that they are sufficient to produce accurate lower-bounds. If higher order conditions were found to be necessary, it would not be hard to add a few more of them in the numerical scheme presented later.

The $\pi^d$ of [[**\[pi-d\]**]{}]{} contains only 2-OPs and products of 2 distinct NSOs; it has at most algebraic complexity $\sim M^3$ deriving from condition [[**\[bound2\]**]{}]{}. Thus when only conditions [[**\[bound1\]**]{}]{}–[[**\[p11sum\]**]{}]{} are imposed, the exponential complexity of the 