for $i \le N/2$, $R(i+,i-)=0$, and $s(i)=
+1$; for $i > N/2$, $R(i+,i-)=2$, and $s(i)= -1$. They were kept fixed during optimization.

At convergence, a subset of NOs, the active NOs, had $p_1 \ge
10^{-3}$. The remaining NOs contributed negligibly to the energy. The same active NOs and signs were also found for several test cases starting from a sufficiently large set of random NOs, half with positive and half with negative signs[^1]. It is significant for the rule of [[**\[signprod\]**]{}]{} for arbitrary matrix-element signs that for the systems tested, the PWBT-based rule of the Table and the alternative of random initial assignment of signs to pairs yield the same results for positive matrix elements. The procedure of [[**\[signprod\]**]{}]{} also yields half positive and half negative signs for the pairs when signs are assigned randomly to the individual NSOs with no reference to the matrix-element signs.

We also performed restricted HF, DFT (PBE [@Perdew1996] and/or PBE0 [@Adamo1999]), and CCSD(T) calculations, with the same basis. The relatively small 6-31G$^{**}$ set is adequate for the comparisons of interest. We report the dissociation energy curves of the dimers H$_2$, LiH and HF in Figs. S1, S2, and S3 in Section S3 of the 