Comparison of a Hartree, a Hartree-Fock, and an exact treatment of quantum-dot helium

We compare energies, pair correlation functions, and particle densities of the ground state of quantum-dot helium in a magnetic field obtained by a Hartree, a Hartree-Fock (HF), and an exact treatment. The exact and HF results for the triplet state agree well, which illustrates the importance of the exchange interaction for systems of few electrons. The results for the singlet state differ significantly and we show that this is caused by a lack of correlation between the angular momenta of the electrons in the HF approximation.