The reference state (1) is a non-trivial vacuum of our theory. We normal order this Hamiltonian with respect to the nontrivial vacuum state given by the Hartree-Fock reference and obtain the normal-ordered Hamiltonian
$$ \begin{equation} \tag{3} H_N = \sum_{pq} f_{pq} \left\{a^\dagger_p a_q\right\} + \frac{1}{4}\sum_{pqrs}\langle pq\vert V\vert rs\rangle \left\{a^\dagger_pa^\dagger_q a_sa_r\right\}. \end{equation} $$