We have till now seen the following definitions of a two-body matrix elements with quantum numbers \( p=j_pm_p \) etc we have a two-body state defined as $$ |(pq)M\rangle = a^{\dagger}_pa^{\dagger}_q|\Phi_0\rangle, $$ where \( |\Phi_0\rangle \) is a chosen reference state, say for example the Slater determinant which approximates \( {}^{16}\mbox{O} \) with the \( 0s \) and the \( 0p \) shells being filled, and \( M=m_p+m_q \). Recall that we label single-particle states above the Fermi level as \( abcd\dots \) and states below the Fermi level for \( ijkl\dots \). In case of two-particles in the single-particle states \( a \) and \( b \) outside \( {}^{16}\mbox{O} \) as a closed shell core, say \( {}^{18}\mbox{O} \), we would write the representation of the Slater determinant as $$ |^{18}\mathrm{O}\rangle =|(ab)M\rangle = a^{\dagger}_aa^{\dagger}_b|^{16}\mathrm{O}\rangle=|\Phi^{ab}\rangle. $$ In case of two-particles removed from say \( {}^{16}\mbox{O} \), for example two neutrons in the single-particle states \( i \) and \( j \), we would write this as $$ |^{14}\mathrm{O}\rangle =|(ij)M\rangle = a_ja_i|^{16}\mathrm{O}\rangle=|\Phi_{ij}\rangle. $$