Adding a three-body interaction
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.