## 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.