In an FCI 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
$$ \Delta E^{(2)}=\langle\Phi_0\vert \hat{H}_I\frac{\hat{Q}}{W_0-\hat{H}_0}\hat{H}_I\vert \Phi_0\rangle. $$