  By combining variational Monte Carlo (VMC) and complete-basis-set limit
Hartree-Fock (HF) calculations, we have obtained near-exact correlation
energies for low-density same-spin electrons on a three-dimensional sphere
(3-sphere), i.e.~the surface of a four-dimensional ball. In the VMC
calculations, we compare the efficacies of two types of one-electron basis
functions for these strongly correlated systems, and analyze the energy
convergence with respect to the quality of the Jastrow factor. The HF
calculations employ spherical Gaussian functions (SGFs) which are the
curved-space analogs of cartesian Gaussian functions. At low densities, the
electrons become relatively localized into Wigner crystals, and the natural SGF
centers are found by solving the Thomson problem (i.e. the minimum-energy
arrangement of $n$ point charges) on the 3-sphere for various values of $n$. We
have found 11 special values of $n$ whose Thomson sites are equivalent. Three
of these are the vertices of four-dimensional Platonic solids --- the
hyper-tetrahedron ($n=5$), the hyper-octahedron ($n=8$) and the 24-cell
($n=24$) --- and a fourth is a highly symmetric structure ($n=13$) which has
not previously been reported. By calculating the harmonic frequencies of the
electrons around their equilibrium positions, we also find the first-order
vibrational corrections to the Thomson energy.
