We are going to study two electrons in a harmonic oscillator well which also interact via a repulsive Coulomb interaction. Let us start with the single-electron equation written as $$-\frac{\hbar^2}{2 m} \frac{d^2}{dr^2} u(r) + \frac{1}{2}k r^2u(r) = E^{(1)} u(r),$$ where $E^{(1)}$ stands for the energy with one electron only. For two electrons with no repulsive Coulomb interaction, we have the following Schroedinger equation $$\left( -\frac{\hbar^2}{2 m} \frac{d^2}{dr_1^2} -\frac{\hbar^2}{2 m} \frac{d^2}{dr_2^2}+ \frac{1}{2}k r_1^2+ \frac{1}{2}k r_2^2\right)u(r_1,r_2) = E^{(2)} u(r_1,r_2) .$$