We illustrate the importance of restrictions in improving ground state energy lower bounds of a model of correlated electrons on a lattice. A reduced density matrix (RDM) formalism is employed. The restrictions are derived from closely related and exactly solved models. Such conditions raise the estimates without resorting to increasing the size of the physical space, thus improving computational efficiency. Our main motivation for this work is the problematic picture of Hohenberg-Kohn-Sham density functional theory for strongly correlated fermions. We find that using small cluster representations, errors can be reduced by more than 50% depending on the nature of the model and parameter regime studied. We obtain results for one- and two-dimensional lattices at half filling in the thermodynamic limit, although the method could be easily adapted to finite molecular structures as well.