## A general Hamiltonian

This part of the Hamiltonian is commonly defined in terms of some external potential like the three-dimensional harmonic oscillator or a particular mean-field basis. Similarly, the two-body part of the Hamiltonian is given by $$\begin{equation*} \hat{V} = \frac{1}{4}\sum_{pqrs} \langle pq|\hat{v}|rs\rangle_{\mathrm{AS}} a_p^\dagger a_q^\dagger a_s a_r \end{equation*}$$ where we have employed antisymmetric matrix elements defined as $$\begin{equation*} \langle pq|\hat{v}|rs\rangle_{\mathrm{AS}}=\langle pq|\hat{v}|rs\rangle-\langle pq|\hat{v}|sr\rangle. \end{equation*}$$ We will assume that the two-body operator $$\hat{v}$$ is given by a nucleon-nucleon interaction.