The most common choice is to start with the function which is expected to exhibit the largest overlap with the wave function we are searching after, namely \( \vert \Phi_0\rangle \). This can again be inserted in the solution for \( \vert \Psi_0\rangle \) in an iterative fashion and if we continue along these lines we end up with $$ \vert \Psi_0\rangle=\sum_{i=0}^{\infty}\left\{\frac{\hat{Q}}{\omega-\hat{H}_0}\left(\omega-E+\hat{H}_I\right)\right\}^i\vert \Phi_0\rangle, $$ for the wave function and $$ \Delta E=\sum_{i=0}^{\infty}\langle \Phi_0\vert \hat{H}_I\left\{\frac{\hat{Q}}{\omega-\hat{H}_0}\left(\omega-E+\hat{H}_I\right)\right\}^i\vert \Phi_0\rangle, $$ which is now a perturbative expansion of the exact energy in terms of the interaction \( \hat{H}_I \) and the unperturbed wave function \( \vert \Psi_0\rangle \).