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Ab initio quantum mechanical methods have been applied to the S6 and O6 molecules at their respective D3d hexagonal chair equilibrium geometries. Double zeta plus polarization (DZ + P) and triple zeta plus double polarization (TZ + 2P) basis sets have been used in conjunction with the self-consistent field (SCF) method and second-order perturbation theory. Equilibrium geometries, harmonic vibrational frequencies, infrared intensities, and Raman intensities have been predicted for the two cyclic molecules. Two previous vibrational difficulties between theory and experiment for S6 have been resolved. The O6 molecule appears to be similar to the well-characterized S6 in several respects. However, its dissociation energy and vibrational frequencies reveal a much flatter potential energy surface for O6 in the region of the equilibrium geometry. While still predicted to correspond to a genuine potential minimum, O6 nevertheless lies about 100 kcal mol-1 above three separated O2 molecules. From a methodological viewpoint, the single configuration Hartree-Fock approach is found to be qualitatively satisfactory for S6 but very poor for O6.