A sequence of diffuse functions to be used in connections with the previously defined polarization consistent basis sets are proposed based on energetic criteria and results for molecular properties. At the Hartree-Fock level the addition of a single set of diffuse s- and p-functions significantly improves the convergence of calculated electron affinities. A corresponding analysis at the density functional level indicates that only systems with high electron affinities have well-defined basis set limits with common exchange-correlation functionals that have electron self-interaction errors. The majority of reported density functional calculations of electron affinities appear to be artifacts of the limited basis set used. The good agreement with experiments for such calculations is most likely due to a reasonable modeling of the physics of the anionic species, rather than being a theoretically sound procedure. For molecular properties like dipole and quadrupole moments, and static polarizabilities, the addition of diffuse functions up to d-functions is required to reach the basis set limit in a consistent fashion, but higher order angular momentum functions are significantly less important.