The equations for $r$ and $R$ can be separated via the ansatz for the wave function $u(r,R) = \psi(r)\phi(R)$ and the energy is given by the sum of the relative energy $E_r$ and the center-of-mass energy $E_R$, that is $$E^{(2)}=E_r+E_R.$$ We add then the repulsive Coulomb interaction between two electrons, namely a term $$V(r_1,r_2) = \frac{\beta e^2}{|\mathbf{r}_1-\mathbf{r}_2|}=\frac{\beta e^2}{r},$$ with $\beta e^2=1.44$ eVnm.