Calculation of the energy spectrum of a two-electron spherical quantum dot

We study the energy spectrum of the two-electron spherical parabolic quantum dot using the exact Schrödinger, Hartree-Fock and Kohn-Sham equations. The results obtained by applying the shifted-1/N method are compared with those obtained by using an accurate numerical technique, showing that the relative error is reasonably small, although the first method consistently underestimates the correct values. The approximate ground-state HF and local-density KS energies, estimated using the shifted-1/N method, are compared with accurate numerical self-consistent solutions. We make some perturbative analyses of the exact energy in terms of the confinement strength, and we propose some interpolation formulae. A similar analysis is performed for both mean-field approximations and interpolation formulae are also proposed for these exchange-only ground-state cases.