Before we continue with the expressions for the two-body operator, we introduce a nomenclature we will use for the rest of this text. It is inspired by the notation used in quantum chemistry. We reserve the labels \( i,j,k,\dots \) for hole states and \( a,b,c,\dots \) for states above \( F \), viz. particle states. This means also that we will skip the constraint \( \leq F \) or \( > F \) in the summation symbols. Our operator \( \hat{H}_0 \) reads now $$ \begin{eqnarray} \hat{H}_0 &=& \sum_{ab} \langle a|\hat{h}|b\rangle b_a^\dagger b_b + \sum_{ai} \left[ \langle a|\hat{h}|i\rangle b_a^\dagger b_i^\dagger + \langle i|\hat{h}|a\rangle b_i b_a \right] \nonumber \\ &+& \sum_{i} \langle i|\hat{h}|i\rangle - \sum_{ij} \langle j|\hat{h}|i\rangle b_i^\dagger b_j \tag{78} \end{eqnarray} $$