Two electrons in an external oscillator potential: exact solution versus one-particle approximations

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Abstract
We use the two-electron oscillator (with Coulomb interaction between the electrons) as a test system for a selection of the most common one-particle approximations: Hartree-Fock (HF), local spin density approximation (LSDA), with self-interaction correction (SIC), generalized gradient correction (GGA) and corrections for excitation energies. Moreover we compared excitation energies from total energy differences with the concept of the Slater transition state (STS) and with the difference of approximate and exact Kohn-Sham energies. By tuning the external oscillator frequency, one can realize the strong, intermediate and weak correlation regime within one system. The results are compared with exact charge densities, Kohn-Sham potentials, ground-state energies and excitation energies. Unlike previous papers on this model, we used self-consistent (and not the exact) charge densities as input for the density functional theory, which makes it possible to check the accuracy of the approximated density itself. We found that the LSDA describes the charge density even in the Wigner crystal limit qualitatively correct, although only SIC provides quantitatively satisfactory results. For all oscillator frequencies, the spurious wiggles in the GGA exchange potential are located near the classical turning point suggesting that they are a consequence of the divergence of the underlying Kirshnitz expansion in this region. It is also observed that the well known large error in the LSDA and GGA exchange potential is not present in self-interaction free methods such as HF and SIC giving rise to the assumption that self-interaction is responsible for this defect. KS eigenvalue differences (as zeroth approximations for excitation energies) calculated from the exact and LSDA effective potential are very close. Therefore, their common large difference with the exact excitation energies cannot be fixed by nonlocal corrections to the LSDA.