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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 3P to 5S. The N = 3 and 5 systems remain 2P and 4S, 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 |Ec/E|, where Ec 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.