$\rho$ depends only on $\pi^d$ and guarantees that the functional $E$ is self-interaction free. [[**\[energy\]**]{}]{} is a generalization of the NOFT formulations of 1-DM functional theories, which require only 1-state OPs. The extra complexity from 2-state OPs and implicit 4-state OPs is more than compensated by the substantial gain in accuracy it makes possible. The computational cost of calculating $E$ from [[**\[energy\]**]{}]{} scales like HF energy-functional minimization with a greater prefactor ($M^3$ vs $(N/2)^3$) due to fractional occupation of NOs.

Numerical results for simple molecular systems
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To test this $A=0$ version, we studied several diatomic molecules and linear chains of H atoms with open boundary conditions. We included all electrons (core and valence) and expanded the NOs in a contracted Gaussian 6-31G$^{**}$ basis, unless otherwise specified. The constrained minimization of the functional was performed by damped Car-Parrinello dynamics [@Car1985], as detailed in Section S2 of the SI.

We started the minimization from NOs and OPs obeying the constraints but otherwise random. The matrix element $K_{ij}$ is positive in [[**\[energy\]**]{}]{}. The signs were therefore taken from the sign rule of the Table S1 in Sec. S1 of the SI, except that the $(1,1)$ case does not occur in zero seniority: 