Interpreting the correlation energy and the wave operator

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.

We see that if we set $$ C_{ij}^{ab} =\frac{1}{4}\frac{\langle ab \vert \hat{v} \vert ij \rangle}{\epsilon_i+\epsilon_j-\epsilon_a-\epsilon_b}, $$ we have a perfect agreement between FCI and MBPT. However, FCI includes such \( 2p-2h \) correlations to infinite order. In order to make a meaningful comparison we would at least need to sum such correlations to infinite order in perturbation theory.

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