SI. For H$_2$, our functional depends only on 1-state OPs and reduces to the exact expression of Löwdin and Shull [@Lowdin1956]. The OP-NOFT-0 dissociation energy curve thus coincides with CASSCF at all interatomic separations. Even in a system as simple as H$_2$, (spin-restricted) HF and DFT fail badly at dissociation because these single reference theories cannot recover the Heitler-London form of the wavefunction. The 4-electron case of LiH provides the first test of 2-state OPs. The conditions [[**\[bound1\]**]{}]{},[[**\[bound2\]**]{}]{}, and [[**\[p11sum\_DOCI\]**]{}]{} simplify in this case as discussed in Section S4 of the SI. Expression [[**\[xi\_DOCI\]**]{}]{} for $\xi$ is exact here, but the restriction to $A=0$ is not. That the OP-NOFT-0 dissociation curve of LiH almost coincides with CASSCF indicates that higher seniorities contribute negligibly to its ground-state energy. Finally, HF, a 10-electron system, provides the first complete test of the theory. Here the OP-NOFT-0 dissociation energy curve follows that of CASSCF with a positive energy shift over the entire separation range. This indicates that the $\xi$ and $A=0$ approximations work well. We cannot exclude compensation of errors between the two approximations, but the results below suggest that it is not the main factor behind their quality.

The results in Section S3 