A lattice model of interacting fermions is studied with the principal aim of assessing the dependence of calculated mean-field ground states versus the N×N lattice size, with N=16, 32, and 48. A two band model on the two-dimensional square lattice is simulated, with on-site energies and interaction parameters chosen to represent crystal field split orbitals in the moderately correlated regime. Nearest neighbor hopping leads to the well known van Hove singularities (vHs) of the square lattice. Anomalies in the inverse participation ratio of the eigenstates are found to be associated with the vHs, with their prevalence decreasing inversely with N. For the chosen model, inhomogeneous spin densities are always obtained for the small lattice size N=16, with the degree of variation decreasing rapidly for most polarizations as N is increased. Various spin polarizations are treated, and one case in which spin density inhomogeneity persists for the largest lattice size is discussed and analyzed. Coupling of spin density inhomogeneities to charge density variation is minor but evident, and is primarily of intra-orbital origin.