If we compare this to the correlation energy obtained from full configuration interaction theory with a Hartree-Fock basis, we found that
$$ E-E_0 =\Delta E= \sum_{abij}\langle ij | \hat{v}| ab \rangle C_{ij}^{ab}, $$where the energy \( E_0 \) is the reference energy and \( \Delta E \) defines the so-called correlation energy.