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In this paper we establish a connection between density functional theory(DFT) for lattice models and common real-space DFT. We consider the lattice DFTdescription of a two-level model subject to generic interactions in Mermin'sDFT formulation in the grand canonical ensemble at finite temperature. The caseof only density-density and Hund's rule interaction studied in earlier work isshown to be equivalent to an exact-exchange description of DFT in thereal-space picture. In addition, we also include the so-called pair-hoppinginteraction which can be treated analytically and, crucially, leads tonon-integer occupations of the Kohn-Sham levels even in the limit of zerotemperature. Treating the hydrogen molecule in a minimal basis is shown to beequivalent to our two-level lattice DFT model. By means of the fractionaloccupations of the KS orbitals (which, in this case, are identical to themany-body ones) we reproduce the results of full configuration interaction,even in the dissociation limit and without breaking the spin symmetry. Beyondthe minimal basis, we embed our HOMO-LUMO model into a standard DFT calculationand, again, obtain results in overall good agreement with exact ones withoutthe need of breaking the spin symmetry.

What can lattice DFT teach us about real-space DFT?