Reminder on definitions
The action of the creation and annihilation operators with respect to a reference state \( \Phi_0 \) are then given by
\( a_i|\Phi_0\rangle = |\Phi_i\rangle \) where a state labeled by \( |\Phi_i\rangle \) means that a particle in a
single-particle state \( i \) has been removed. Similarly, we have \( a_a^\dagger|\Phi_0\rangle = |\Phi^a\rangle \), \( a_i^\dagger|\Phi_0\rangle = 0 \)
and \( a_a|\Phi_0\rangle = 0 \). With the above definitions, we write our Hamiltonian as
$$
\begin{equation*}
\hat{H}=\hat{H}_0+\hat{V}+\hat{W},
\end{equation*}
$$
where
the single-particle part is given by
$$
\begin{equation*}
\hat{H}_0 = \sum_{pq} \langle p|\hat{h}_0|q\rangle a_p^\dagger a_q.
\end{equation*}
$$