An additional reminder

Introducing a reference state \( |\Phi_0\rangle \) as our new vacuum state leads to a redefinition of the Hamiltonian in terms of a constant reference energy \( E_0 \) defined as $$ \begin{equation*} E_0 = \sum_{i\le F}\langle i | \hat{h}_0|i\rangle+\frac{1}{2}\sum_{ij\le F} \langle ij|\hat{v}|ij\rangle +\frac{1}{6}\sum_{ijk\le F} \langle ijk|\hat{w}|ijk\rangle, \end{equation*} $$ and a normal-ordered Hamiltonian $$ \begin{equation*} \hat{H}_N=\sum_{pq}\langle p|\tilde{f}|q\rangle a^\dagger_p a_q+\frac{1}{4} \sum_{pqrs} \langle pq|\tilde{v}|rs\rangle a^\dagger_p a^\dagger_q a_s a_r +\frac{1}{36} \sum_{\substack{pqr \\stu}} \langle pqr|\hat{w}|stu\rangle a^\dagger_p a^\dagger_q a^\dagger_r a_u a_t a_s \end{equation*} $$ where $$ \begin{equation*} \langle p| \tilde{f}|q\rangle = \langle p|\hat{h}_0|q\rangle +\sum_{i\le F} \langle pi|\hat{v}|qi\rangle+\frac{1}{2}\sum_{ij\le\alpha_F} \langle pij|\hat{w}|qij\rangle, \end{equation*} $$ represents a correction to the single-particle operator \( \hat{h}_0 \) due to contributions from the nucleons below the Fermi level.