The equation $$ \vert \psi_n \rangle =\frac{1}{\left( E_n -\hat{H}_0\right)}\hat{V}\vert \psi_n \rangle, $$ is normally solved in an iterative fashion. We assume first that $$ \vert\psi_n \rangle = \vert\phi_n \rangle, $$ where \( \vert\phi_n \rangle \) are the eigenfunctions of $$ \hat{H}_0\vert \phi_n \rangle=\omega_n\vert \phi_n \rangle $$ the so-called unperturbed problem. In our case, these will simply be the kinetic energies of the relative motion.