## Rewriting the energy functional

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

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