Three, four and five interacting electrons in a spherical box: an exact diagonalization study extended

We study the problem of N electrons confined to a hard three-dimensional spherical box, for N = 3, 4 and 5. Full-configuration interaction and Hartree-Fock calculations have been performed. We report energies, densities, second-order density matrices and exchange-correlation holes. In the N = 4 system, we observe a spin transition as the radius R of the sphere is increased, changing the ground state from P-3 to S-5. The N = 3 and 5 systems remain P-2 and S-4, respectively, for all R investigated. Pair-correlation functions and physical exchange-correlation holes have been computed over a regime in which the systems transform from liquid droplets to Wigner molecules. These functions show marked, yet characteristic, changes in this regime. By computing \E-c/E\, where E-c is the correlation energy defined with respect to the unrestricted Hartree-Fock solution, we show that the maximum relative error in the UHF energies occurs in this crossover regime. These systems provide challenging benchmarks for post-Hartree-Fock and density functional theory methods.