Many-body perturbation theory
We can now this expression in terms of a perturbative expression in terms
of
\hat{H}_I where we iterate the last expression in terms of
\Delta E
\Delta E=\sum_{i=1}^{\infty}\Delta E^{(i)}.
We get the following expression for
\Delta E^{(i)}
\Delta E^{(1)}=\langle \Phi_0\vert \hat{H}_I\vert \Phi_0\rangle,
which is just the contribution to first order in perturbation theory,
\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,
which is the contribution to second order.