We apply density functional theory (DFT) and the DFT+U technique to study the adsorption of transition metal porphine molecules on atomistically flat Au(111) surfaces. DFT calculations using the Perdew-Burke-Ernzerhof exchange correlation functional correctly predict the palladium porphine (PdP) low-spin ground state. PdP is found to adsorb preferentially on gold in a flat geometry, not in an edgewise geometry, in qualitative agreement with experiments on substituted porphyrins. It exhibits no covalent bonding to Au(111), and the binding energy is a small fraction of an electronvolt. The DFT+U technique, parametrized to B3LYP-predicted spin state ordering of the Mn cl-electrons, is found to be crucial for reproducing the correct magnetic moment and geometry of the isolated manganese porphine (MnP) molecule. Adsorption of Mn(II)P on Au(111) substantially alters the Mn ion spin state. Its interaction with the gold substrate is stronger and more site-specific than that of PdP. The binding can be partially reversed by applying an electric potential, which leads to significant changes in the electronic and magnetic properties of adsorbed MnP and similar to 0.1 angstrom changes in the Mn-nitrogen distances within the porphine macrocycle. We conjecture that this DFT+U approach may be a useful general method for modeling first-row transition metal ion complexes in a condensed-matter setting.