## 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 $$\begin{equation*} \hat{H}_0 = \sum_{pq} \langle p|\hat{h}_0|q\rangle a_p^\dagger a_q. \end{equation*}$$