Interpreting the correlation energy and the wave operator
If we limit the attention to a Hartree-Fock basis, then we have that
\( \langle\Phi_0\vert \hat{H}_I \vert 2p-2h\rangle \) is the only contribution and the contribution to the energy reduces to
$$
\Delta E^{(2)}=\frac{1}{4}\sum_{abij}\langle ij\vert \hat{v}\vert ab\rangle \frac{\langle ab\vert \hat{v}\vert ij\rangle}{\epsilon_i+\epsilon_j-\epsilon_a-\epsilon_b}.
$$