Electromagnetic multipole moments and transitions

These operators commute meaning that $$ \hat{Q}\frac{1}{\left(\omega-\hat{H}_0\right)}\hat{Q}=\hat{Q}\frac{1}{\left(\omega-\hat{H}_0\right)}=\frac{\hat{Q}}{\left(\omega-\hat{H}_0\right)}. $$ With these definitions we can in turn define the wave function as $$ \vert \Psi_0\rangle=\vert \Phi_0\rangle+\frac{\hat{Q}}{\omega-\hat{H}_0}\left(\omega-E+\hat{H}_I\right)\vert \Psi_0\rangle. $$ This equation is again nothing but a formal rewrite of Schr\"odinger's equation and does not represent a practical calculational scheme. It is a non-linear equation in two unknown quantities, the energy \( E \) and the exact wave function \( \vert \Psi_0\rangle \). We can however start with a guess for \( \vert \Psi_0\rangle \) on the right hand side of the last equation.