The electronic structures and molecular properties of square-planar 6pi-electron ring molecules and ions E2N2 and E-4(2+) (E = S, Se, Te) were studied using various ab initio methods and density functionals. All species were found to contain singlet diradical character in their electronic structures. Detailed analysis of the CAS wave function of S2N2 in terms of different valence bond structures gives the largest weight for a Lewis-type singlet diradical VB structure in which the two unpaired electrons reside on nitrogen atoms, though the relative importance of the different VB structures is highly dependent on the level of theory. The diradical character in both E2N2 and E-4(2+) was found to increase in the series S < Se < Te. The diradical nature of the chemical species is manifested in the prediction of molecular properties, in which the coupled cluster and multiconfigurational approaches, as well as the BPW91 functional show consistent performance. (77) Se NMR chemical shifts of chalcogen cations SxSe4-x2+ (x = 0-3) were calculated with CAS, BPW91, and B3PW91 methods using the GIAO formalism. The hybrid functional B3PW91 shows inferior performance, but both CAS and BPW91 unquestionably confirm the experimental assignment and are able to predict the NMR chemical shifts of these computationally difficult cases with excellent accuracy.