The correlation is from the off-diagonal terms of the residual interaction, which is the TBME of the shell model. Consider following Hamiltonian of these nuclei
the correlation energy between the two neutrons is the term . Be aware that this is a residual interaction, not the nucleon-nucleon interaction .
where is mean field.
The separation energies is proportional to the terms
Thus, the neutron-neutron correlation energy is
For 18O, MeV, and MeV, thus, MeV.
In the shell model calculation, the single particle energy of the 1d5/2 and 2s1/2 neutron are -4.143 MeV and -3.27 MeV respectively. The residual interaction is
The eigenenergy of the ground state of 18O is -10.539 on top of 16O.
The non-correlated binding energy of 18O is 2*-4.143 = -8.286 MeV.
Therefore, the theoretical correlated energy is 2.258 MeV.