This paper explores the possibility of combining projected Hartree-Fock and density functional theories for treating static and dynamic correlations in molecular systems with mean-field computational cost. The combination of spin-projected unrestricted Hartree-Fock (SUHF) with the TPSS correlation functional (SUHF+TPSS) yields excellent results for non-metallic molecular dissociations and singlet-triplet splittings. However, SUHF+TPSS fails to provide the qualitatively correct dissociation curve for the notoriously difficult case of the chromium dimer. By tuning the TPSS correlation parameters and adding complex conjugation symmetry breaking and restoration to SUHF, the right curve shape for Cr2 can be obtained; unfortunately, such a combination is found to lead to overcorrelation in the general case.