Density functional and post Hartree-Fock ab initio computations were carried out on the lowest singlet, triplet, and quintet states of 1,4-phenylenedinitrene, biphenyl-4,4'-dinitrene, (E)-stilbene-4,4'-dinitrene, and (E,E)-1,3-bis(4-nitrenophenyl)-1,3-butadiene, and (E,E,E)-1,6-bis(4-nitrenophenyl)-1,3,5-hexatriene. Near-degenerate singlet and triplet quinonoidal ground states were found for all systems using CASSCF methodology, with a slight favoring of the singlet, in accord with experimental results. The aromatic quintet dinitrene states lie much higher in energy. Restricted B3LYP hybrid density functional theory (DFT) methods give artifactually high biradical singlet state energies relative to the triplet biradical states, but unrestricted (mixed-state) B3LYP methods correctly give singlet energies that lie somewhat below the triplet state energies, as well as giving geometric results that compare well to the best CASSCF results we could achieve for these biradical states. Appropriate guidelines for selecting CASSCF versus DFT procedures in such cases are suggested in light of comparisons of computed to experimental results.