A density-functional theory for many-body lattice models is considered in which the single-particle density matrix ${\gamma }_{\mathrm{ij}}$ is the basic variable. Eigenvalue equations are derived for solving Levy’s constrained search of the interaction energy functional $W\left[{\gamma }_{\mathrm{ij}}\right].$ $W\left[{\gamma }_{\mathrm{ij}}\right]$ is expressed as the sum of Hartree-Fock energy $E_{\mathrm{HF}}\left[{\gamma }_{\mathrm{ij}}\right]$ and the correlation energy $E_C\left[{\gamma }_{\mathrm{ij}}\right].$ Exact results are obtained for $E_C\left({\gamma }_{12}\right)$ of the Hubbard model on various periodic lattices, where ${\gamma }_{\mathrm{ij}}={\gamma }_{12}$ for all nearest neighbors i and j. The functional dependence of $E_C\left({\gamma }_{12}\right)$ is analyzed by varying the number of sites $N_a,$ band filling $N_e,$ and lattice structure. The infinite one-dimensional chain and one-, two-, or three-dimensional finite clusters with periodic boundary conditions are considered. The properties of $E_C\left({\gamma }_{12}\right)$ 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, and in the crossover region $({\gamma }_{12}^{\infty }<~{\gamma }_{12}<~{\gamma }_{12}^0).\mathrm{}$ Using an appropriate scaling we observe that ${\varepsilon }_C{(g}_{12}{)=E}_C{/E}_{\mathrm{HF}}$ has a pseudo-universal behavior as a function of $g_{12}=\left({\gamma }_{12}-{\gamma }_{12}^{\infty }\right)/\left({\gamma }_{12}^0-{\gamma }_{12}^{\infty }\right).\mathrm{}$ The fact that ${\varepsilon }_C{(g}_{12})$ depends weakly on $N_a,$ $N_e,$ and lattice structure suggests that the correlation energy of extended systems could be obtained quite accurately from finite-cluster calculations. Finally, the behaviors of $E_C\left({\gamma }_{12}\right)$ for repulsive $\mathrm{}(U>\mathrm{}0\mathrm{})\mathrm{}$ and attractive $\mathrm{}(U<\mathrm{}0\mathrm{})\mathrm{}$ interactions are contrasted.

• Received 19 July 1999

DOI:https://doi.org/10.1103/PhysRevB.61.1764