Let us go back to the case of two-particles in the single-particle states \( a \) and \( b \) outside \( {}^{16}\mbox{O} \) as a closed shell core, say \( {}^{18}\mbox{O} \). The representation of the Slater determinant is $$ |^{18}\mathrm{O}\rangle =|(ab)M\rangle = a^{\dagger}_aa^{\dagger}_b|^{16}\mathrm{O}\rangle=|\Phi^{ab}\rangle. $$ The anti-symmetrized matrix element is detailed as $$ \langle (ab) M | \hat{V} | (cd) M \rangle = \langle (j_am_aj_bm_b)M=m_a+m_b | \hat{V} | (j_cm_cj_dm_d)M=m_a+m_b \rangle, $$ and note that anti-symmetrization means $$ \langle (ab) M | \hat{V} | (cd) M \rangle =-\langle (ba) M | \hat{V} | (cd) M \rangle =\langle (ba) M | \hat{V} | (dc) M \rangle, $$ $$ \langle (ab) M | \hat{V} | (cd) M \rangle =-\langle (ab) M | \hat{V} | (dc) M \rangle. $$