Defining $$ h_{\alpha\beta}^{HF}=\langle \alpha | h | \beta \rangle+ \sum_{j=1}^A\sum_{\gamma\delta} C^*_{j\gamma}C_{j\delta}\langle \alpha\gamma|\hat{v}|\beta\delta\rangle_{AS}, $$ we can rewrite the new equations as $$ \begin{equation} \sum_{\gamma}h_{\alpha\beta}^{HF}C_{i\beta}=\epsilon_i^{HF}C_{i\alpha}. \tag{17} \end{equation} $$ The latter is nothing but a standard eigenvalue problem.

It suffices to tabulate the matrix elements \( \langle \alpha | h | \beta \rangle \) and \( \langle \alpha\gamma|\hat{v}|\beta\delta\rangle_{AS} \) once and for all. Successive iterations require thus only a look-up in tables over one-body and two-body matrix elements. These details will be discussed below when we solve the Hartree-Fock equations numerically.