Uniform electron gases. I. Electrons on a ring.

We introduce a new paradigm for one-dimensional uniform electron gases (UEGs). In this model, n electrons are confined to a ring and interact via a bare Coulomb operator. We use Rayleigh-Schrödinger perturbation theory to show that, in the high-density regime, the ground-state reduced (i.e., per electron) energy can be expanded as ε(r(s),n)=ε0(n)r(s)(-2)+ε1(n)r(s)(-1)+ε2(n)+ε3(n)r(s+)⋯ , where r(s) is the Seitz radius. We use strong-coupling perturbation theory and show that, in the low-density regime, the reduced energy can be expanded as ε(r(s),n)=η0(n)r(s)(-1)+η1(n)r(s)(-3/2)+η2(n)r(s)(-2)+⋯ . We report explicit expressions for ε0(n), ε1(n), ε2(n), ε3(n), η0(n), and η1(n) and derive the thermodynamic (large-n) limits of each of these. Finally, we perform numerical studies of UEGs with n = 2, 3, [ellipsis (horizontal)], 10, using Hylleraas-type and quantum Monte Carlo methods, and combine these with the perturbative results to obtain a picture of the behavior of the new model over the full range of n and r(s) values.