Finally, the three-body part of our Hamiltonian operator is defined by $$\begin{equation*} \hat{W} =\frac{1}{36} \sum_{pqrstu} \langle pqr|\hat{w}|stu\rangle_{\mathrm{AS}}a_p^\dagger a_q^\dagger a_r^\dagger a_u a_t a_s, \end{equation*}$$ where we have defined the antisymmetric matrix elements $$\begin{equation*} \langle pqr|\hat{w}|stu\rangle_{\mathrm{AS}}= \langle pqr|\hat{w}|stu\rangle + \langle pqr|\hat{w}|tus\rangle + \langle pqr|\hat{w}|ust\rangle- \langle pqr|\hat{w}|sut\rangle - \langle pqr|\hat{w}|tsu\rangle - \langle pqr|\hat{w}|uts\rangle. \end{equation*}$$ We will in the discussions to come drop the $$\mathrm{AS}$$ subscript, assuming thereby that all matrix elements are antisymmetrized.