We study a system of two electrons interacting with a Coulomb potential in a sphere of radius R, bounded by an infinite wall using exact diagonalization. We have also investigated the influence of an additional parabolic potential (of strength $k)$ arising from a uniform background smeared throughout the sphere. The convergence of the ground state energy of the singlet spin state of the system is investigated as a function of sphere size (essentially $r_s,$ the Wigner–Seitz density parameter) for cases where there is no background potential $\mathrm{}(k=0\mathrm{})\mathrm{}$ and for when $k\ne 0.$ With $k=0\mathrm{}$ and small $r_s,$ we observe a maximum in the ground state density at the origin of the sphere. At $r_s\approx 8\mathrm{a}.\mathrm{u}.\mathrm{},$ the ground state density acquires a minimum at the origin. For this and larger systems we identify the formation of a “Wigner” molecule state. We further investigate the ground state density as a function of k and also the correlation hole density as a function of $r_s$ and k. We invert the Kohn–Sham equation for a two electron system and calculate the local effective potential and correlation potential (to within an additive constant) as functions of the radial coordinate for a number of values of $r_s$ and k.

• Received 24 June 2002

DOI:https://doi.org/10.1103/PhysRevB.66.235118