Here,
$$ \begin{equation} \tag{4} f^p_q = \varepsilon^p_q + \sum_i \langle pi\vert V\vert qi\rangle \end{equation} $$is the Fock matrix. We note that the Fock matrix is diagonal in the Hartree-Fock basis. The brackets \( \{\cdots\} \) in Eq. (3) denote normal ordering, i.e. all operators that annihilate the nontrivial vaccum (1) are to the right of those operators that create with respect to that vaccum. Normal ordering implies that \( \langle \Phi_0\vert H_N\vert \Phi_0\rangle = 0 \).
Normal order the expression \( \sum\limits_{pq}\varepsilon_q^p a^\dagger_p a_q \).
$$ \begin{align} \sum_{pq}\varepsilon_q^p a^\dagger_p a_q =\sum_{ab}\varepsilon_b^a a^\dagger_a a_b +\sum_{ai}\varepsilon_i^a a^\dagger_a a_i +\sum_{ai}\varepsilon_a^i a^\dagger_i a_a +\sum_{ij}\varepsilon_j^i a^\dagger_i a_j \tag{5} \end{align} $$