Nonrelativistic and relativistic DFT calculations are performed on four-coordinate metal porphyrins MP and their six-coordinate adducts MP(py)2 and MP(py)(CO) (py=pyridine) with M=Fe, Ru, and Os. The electronic structures of the MPs are investigated by considering all possible low-lying states with different configurations of nd-electrons. FeP and OsP have a 3A2g ground state, while this state is nearly degenerate with 3Eg for RuP. Without relativistic corrections, the ground states of both RuP and OsP would be 3Eg. For the six-coordinate adducts with py and CO, the strong-field axial ligands raise the energy of the M dz2-orbital, thereby making the MII ion diamagnetic. The calculated redox properties of MP(py)2 and MP(py)(CO) are in agreement with experiment. The difference between RuP(py)(CO) and OsP(py)(CO), in terms of site of oxidation, is due to relativistic effects.