Interpreting the correlation energy and the wave operator

In a shell-model calculation, the unknown coefficients in \( \hat{C} \) are the eigenvectors which result from the diagonalization of the Hamiltonian matrix.

How can we use perturbation theory to determine the same coefficients? Let us study the contributions to second order in the interaction, namely $$ \Delta E^{(2)}=\langle\Phi_0\vert \hat{H}_I\frac{\hat{Q}}{W_0-\hat{H}_0}\hat{H}_I\vert \Phi_0\rangle. $$

The intermediate states given by \( \hat{Q} \) can at most be of a \( 2p-2h \) nature if we have a two-body Hamiltonian. This means that second order in the perturbation theory can have \( 1p-1h \) and \( 2p-2h \) at most as intermediate states. When we diagonalize, these contributions are included to infinite order. This means that higher-orders in perturbation theory bring in more complicated correlations.