from $i,i',j,j'$. Bounds on the $p_{1111}$ that are the generalizations of [[**\[bound1\]**]{}]{}–[[**\[p11sum\]**]{}]{} for 3-OPs and 4-OPs can be formulated.

The OP-NOFT energy functional
-----------------------------

 The trial energy, $E[\Psi] = \left( \Psi, \hat{H}\Psi \right)$, the expectation value of the Hamiltonian $\hat{H}$, is an explicit functional of the 1- and 2-DM: $$E[\Psi] = E[\rho, \pi] = \operatorname{tr}\left\{\rho \hat{h}\right\} +
\operatorname{tr}\left\{ \pi \hat{w} \right\}.$$ Here $\hat{h}$ is the single-particle kinetic-energy operator plus the external potential, and $\hat{w}$ is the 2-electron Coulomb interaction. $E[\rho, \pi]$ splits into two parts, $E^d$ diagonal and $E^{od}$ off-diagonal in the SD: $$\begin{aligned}
E &=& E^d + E^{od} \nonumber \\
E^d &=& \operatorname{tr}\left\{\rho \hat{h}\right\} + \operatorname{tr}\left\{\pi^{d}
\hat{w}\right\} \nonumber \\
E^{od} &=& \operatorname{tr}\left\{\pi^{od} \hat{w}\right\}.
\label{Esum}\end{aligned}$$ The HF wave function minimizes $E^d$; $\pi^{od}$ introduces electron correlation into $E^{od}$. The explicit forms of $E^d$ and $E^{od}$ follow from [[**\[rho1diag\]**]{}]{}, [[**\[pi-d\]**]{}]{}, and [[**\[final\_pi-od\]**]{}]{}: $$E^d = \sum_i p_1(i) h_{ii} + \sum_{i<j} p_{11}(ij) \left[ {\cal
    J}_{ij} - {\cal K}_{ij}\right],
\label{E-d}$$ where $h_{ii} = (\psi_i, \hat{h}\psi_i)$, and ${\cal J}_{ij} = 
(\psi_i\psi_i,\hat{w}\psi_j\psi_j)$ and ${\cal K}_{ij} = 
(\psi_i\psi_j,\hat{w}\psi_j\psi_i)$ are the usual Coulomb and exchange integrals, respectively. $$\begin{aligned}
E^{od} &=& \sum_{\scriptscriptstyle i<i' \ne j<j'} s(ii') s(jj')
p^{1/2}_{1100}(ii'jj') p^{1/2}_{0011}(ii'jj')
\nonumber \\
&& \xi(ii'jj') \left[ {\cal K}_{ii'jj'} - {\cal K}_{ii'j'j} \right],
