With these notations we rewrite the energy functional as $$ \begin{equation} \int \Phi^*\hat{H_I}\Phi d\mathbf{\tau} = \frac{1}{2}\sum_{\mu=1}^A\sum_{\nu=1}^A \langle \mu\nu|\hat{v}|\mu\nu\rangle_{AS}. \tag{11} \end{equation} $$

Adding the contribution from the one-body operator \( \hat{H}_0 \) to (11) we obtain the energy functional $$ \begin{equation} E[\Phi] = \sum_{\mu=1}^A \langle \mu | h | \mu \rangle + \frac{1}{2}\sum_{{\mu}=1}^A\sum_{{\nu}=1}^A \langle \mu\nu|\hat{v}|\mu\nu\rangle_{AS}. \tag{12} \end{equation} $$ In our coordinate space derivations below we will spell out the Hartree-Fock equations in terms of their integrals.