Figure 1

Total energy calculated from BA-LDA (open squares) and exact diagonalization (full circles) versus system size, for a 1DHM with open boundary conditions and $U=6$. For even $L$ we take $N=L/2$ (so that $n=N/L=1/2$, corresponding to quarter filling). For odd $L$ we take $N=(L-1)/2$. The error bar at $L=14$ represents the intrinsic error of the parametrization (2), as estimated from the inset. For $L=14$ the BA-LDA data agree with the exact ones within this error. Inset: Exchange-correlation energy of the infinite 1DHM as a function of $n$ for several values of $U$. The full curves represent our parametrization (2) with (3), and the symbols represent values obtained from numerical solution of the Lieb-Wu BA integral equation for $U=3,6$, and $9$. For $U=0$ and $U=\infty$ the parametrization is exact.Reuse & Permissions

Figure 2

Total energy of the 1DHM with a binary site potential and periodic boundary conditions, as described in the main text, for $U=6$ and $n=1/2$. Open squares: BA-LDA; full circles: exact diagonalization. Inset: exact $xc$ energy and LDA $xc$ energy, evaluated at exact and LDA densities, respectively. The absolute errors in the LDA $xc$ energy and total energy are different because in the LDA the Hartree, external potential, and noninteracting kinetic components of the total energy are also evaluated at the LDA density, not at the exact one. However, the $xc$ energy is the dominating source of error in the total energy and the relative error in the total energy is considerably smaller than that in its $xc$ component.Reuse & Permissions

Figure 3

Density distribution for a system with $L=201$ sites, $N=101$ electrons, $U=6$, and a localized impurity with $v_I=-1$. Main curve: periodic boundary conditions. Inset: open boundary conditions. The density of the same system without impurity has been subtracted to display clearly the impurity-added contribution.Reuse & Permissions

Figure 4

Total energy calculated from BA-LDA for the impurity model of Fig. 3, with open (full symbols) and periodic (empty symbols) boundary conditions, plotted as a function of the inverse system size. Impurity strength $v_I=-1$ (upper curve), and $v_I=-10$ (lower curve). The lines are guides for the eye.Reuse & Permissions