Correlation energy of the one-dimensional Coulomb gas

We introduce a new paradigm for finite and infinite strict-one-dimensional uniform electron gases. In this model, $n$ electrons are confined to a ring and interact via a bare Coulomb operator. In the high-density limit (small-$r_s$, where $r_s$ is the Seitz radius), we find that the reduced correlation energy is $\text{Ec}(r_s,n)={\text{eps}}^{(2)}(n)+O(r_s)$, and we report explicit expressions for ${\text{eps}}^{(2)}(n)$. In the thermodynamic (large-$n$) limit of this, we show that $\text{Ec}(r_s)=-{\pi }^2\mathrm{/}360+O(r_s)$. In the low-density (large-$r_s$) limit, the system forms a Wigner crystal and we find that $\Ec(r_s) = -[\ln(\sqrt{2\pi})-3/4] r_s^{-1} + 0.359933 r_s^{-3/2} + O(r_s^{-2})$. Using these results, we propose a correlation functional that interpolates between the high- and low-density limits. The accuracy of the functional for intermediate densities is established by comparison with diffusion Monte Carlo results. Application to a non-uniform system is also reported.