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 (r_s,n)=ε₀(n) r_s^-2+ε₁(n)rs-¹+ε ₂(n)+₃(n)r_s+, where r_s+⋯ where r_sis 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)rs- 1+η₁(n)rs^-3/2+η₂(n)rs^-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,⋯, 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.