correlation. Moreover, only in the case of H$_8$, H$_{16}$ and N$_2$ at large separations did inclusion of the $(2,3)$ positivity condition turn out to be essential to enforce numerically the variational character of the ground-state solution.

Discussion
==========

We have introduced a new method for correlated electronic-structure calculations, OP-NOFT, that scales algebraically. Its DOCI-like simplification, OP-NOFT-0 scales favorably with system size, with HF energy-minimization scaling. The variational character of the energies calculated via OP-NOFT-0 supports the accuracy of the $\xi$ approximation [[**\[xi\_DOCI\]**]{}]{} and of limiting the positivity conditions to [[**\[bound1\]**]{}]{}–[[**\[bound2\]**]{}]{}. OP-NOFT-0 is restricted to the $A=0$ sector of the Hilbert space. While providing an accurate description of single-bond breaking and achieving a considerable improvement over single-reference methods in all cases studied, it overestimates the dissociation energy of a triple-bond, missing the correlations between same-spin electrons in the open-shell fragments. That error should be eliminated by including the contribution of the $A=2,4$ sectors as shown in Ref. [@Bytautas2011]. Including those sectors would require use of the full theory and its functional, [[**\[E-od\]**]{}]{}. While 4-state NSO OPs and 4-index integrals would be required, the theory would still scale polynomially.

It will be straightforward to add the computation of interatomic forces to the 