Solving the Hartree-Fock eigenvalue problem yields then new eigenvectors \( C_{i\alpha}^{(1)} \) and eigenvalues
\( \epsilon_i^{HF(1)} \).

- With the new eigenvalues we can set up a new Hartree-Fock potential

$$
\sum_{j = 1}^A\sum_{\gamma\delta} C_{j\gamma}^{(1)}C_{j\delta}^{(1)}\langle \alpha\gamma|\hat{v}|\beta\delta\rangle_{AS}.
$$
The diagonalization with the new Hartree-Fock potential yields new eigenvectors and eigenvalues.
This process is continued till for example
$$
\frac{\sum_{p} |\epsilon_i^{(n)}-\epsilon_i^{(n-1)}|}{m} \le \lambda,
$$
where \( \lambda \) is a user prefixed quantity (\( \lambda \sim 10^{-8} \) or smaller) and \( p \) runs over all calculated single-particle
energies and \( m \) is the number of single-particle states.