  We present numerically exact solutions to the full-dimensional Schrodinger
Equation for the few-electron gas (few-EG) model of electronic structure
theory. Our core methodology uses a Sum-of-Products (SOP) representation of
singular potentials to make the fully correlated multi-particle quantum
mechanical problem computationally tractable. With full access to periodic
wave-functions, we analyze the exchange/correlation, spin symmetry, and
thermodynamic ($T = 0$K) phases for few-EG systems with up to four explicit
electrons per unit cell. We also present pure-spin exchange/correlation
functionals for the same few-EG solutions. Finally, we consider the family of
few-EG calculations in juxtaposition with the closely related uniform electron
gas (UEG) system, for which previous calculations have been performed using
density functional theory and quantum Monte Carlo methods. The exact few-EG
solutions all display a similar oscillatory behavior at lower densities, which
is not predicted by previous UEG simulations.
