Many-body perturbation theory
Defining \( e=E-\hat{H}_0 \) and recalling that \( \hat{H}_0 \) commutes with
\( \hat{Q} \) by construction and that \( \hat{Q} \) is an idempotent operator
\( \hat{Q}^2=\hat{Q} \).
Using this equation in the above expansion for \( \Delta E \) we can write the denominator
$$
\hat{Q}\frac{1}{\hat{e}-\hat{Q}\hat{H}_I\hat{Q}}=
$$
$$
\hat{Q}\left[\frac{1}{\hat{e}}+\frac{1}{\hat{e}}\hat{Q}\hat{H}_I\hat{Q}
\frac{1}{\hat{e}}+\frac{1}{\hat{e}}\hat{Q}\hat{H}_I\hat{Q}
\frac{1}{\hat{e}}\hat{Q}\hat{H}_I\hat{Q}\frac{1}{\hat{e}}+\dots\right]\hat{Q}.
$$