The inhomogeneous Hubbard model is investigated in the framework of lattice density-functional theory (LDFT) by considering the single-particle density matrix ${\gamma }_{ij}$ with respect to the lattice sites as the basic variable of the many-body problem. The domain of representability of ${\gamma }_{ij}$ is determined for charge-density wave states on finite bipartite lattices. Levy’s constrained search of the interaction-energy functional $W\left[{\gamma }_{ij}\right]$ is numerically solved by applying the Lanczos method to an effective Hubbard-type model. The exact functional dependence of $W\left[{\gamma }_{ij}\right]$ is analyzed by varying systematically the charge transfer $\Delta n={\gamma }_{22}-{\gamma }_{11}$, the degree of electron delocalization $g_{12}$ between the sublattices, the number of sites $N_a$, and the band filling $n=\left({\gamma }_{11}+{\gamma }_{22}\right)/2=N_e/N_a$. For each $\Delta n$ the properties of $W$ are discussed in the limits of weak $\left({\gamma }_{12}\simeq {\gamma }_{12}^0\right)$ and strong $\left({\gamma }_{12}\simeq {\gamma }_{12}^{\infty }\right)$ electronic correlations, as well as in the crossover region $\left({\gamma }_{12}^{\infty }\le {\gamma }_{12}\le {\gamma }_{12}^0\right)$. It is shown that $W$ follows quite closely a simple scaling behavior as a function of $\Delta n$ and $g_{12}=\left({\gamma }_{12}-{\gamma }_{12}^{\infty }\right)/\left({\gamma }_{12}^0-{\gamma }_{12}^{\infty }\right)$. The very good transferability of $W\left(\Delta n,g_{12}\right)$ for different $N_a$, $n$ and lattice structure opens new possibilities of applying LDFT to inhomogeneous many-body models.

• Received 6 February 2009

DOI:https://doi.org/10.1103/PhysRevB.79.235101