Clean and oxygen covered 6H-SiC(0001) surfaces with a (root3 x root3)R30 degrees reconstruction are investigated using the local spin-density approximation (LSDA) of density-functional theory. For the Si(T-4) model of the clean surface, the ground state is found to be anti-ferromagnetic whereby a dangling bond derived band splits into a fully occupied and an empty band separated by a gap of 0.7 eV. The widths of the occupied and empty surface-state bands are in good agreement with photoemission and inverse photoemission experiments. For the oxygen covered (0001) surface, our calculations confirm a structural model suggested on the basis of a recent LEED analysis. Within the LSDA a semiconducting surface with a band gap of 1.0 eV is obtained for this system.