Generalized local-density approximation and one-dimensional finite uniform electron gases

We explicitly build a generalized local-density approximation (GLDA) correlation functional based on one-dimensional uniform electron gases (UEGs). The fundamental parameters of the GLDA—a generalization of the widely known local-density approximation used in density-functional theory—are the electronic density $\rho$ and a two-electron local parameter called the hole curvature $\eta$. The UEGs considered in this study are finite versions of the conventional infinite homogeneous electron gas and consist of $n$ electrons on an infinitely thin wire with periodic boundary conditions. We perform a comprehensive study of these finite UEGs at high, intermediate, and low densities using perturbation theory and quantum Monte Carlo calculations. We show that the present GLDA functional yields accurate estimates of the correlation energy for both weakly and strongly correlated one-dimensional systems and can be easily generalized to higher-dimensional systems.

Figure 1

Top left: $E_{\mathrm{c}}$ (in $\text{m}{E}_{\mathrm{h}}$) of 2-boxium as a function of $L$. Top right: $E_{\mathrm{c}}$ (in $\text{m}{E}_{\mathrm{h}}$) of 2-hookium as a function of $k$. Bottom left: $\Delta E_{\mathrm{c}}=E_{\mathrm{c}}-E_{\mathrm{c}}^{\mathrm{Hy}}$ (in $\text{m}{E}_{\mathrm{h}}$) of 2-boxium as a function of $L$. Bottom right: $\Delta E_{\mathrm{c}}=E_{\mathrm{c}}-E_{\mathrm{c}}^{\mathrm{Hy}}$ (in $\text{m}{E}_{\mathrm{h}}$) of 2-hookium as a function of $k$.