Quadruply bonded dimolybdenum complexes with both symmetric and asymmetric ligand configurations were studied by ab initio quantum theory. The ligands studied were chloride, phosphine, and hydroxy groups. The effects of the different ligands, with their different pi bonding properties, on the known metal-metal quadruple bond were examined. In two types of complexes, one with weak pi-donating chloride ligands and the other with strong pi-donating hydroxy ligands, the metal-metal bonding is better described as a triple bond for some ligand configurations. Since correlation is necessary for a proper description of the quadruple bond, the complexes were studied at the multiconfigurational self-consistent field (MCSCF) level of theory. The active space of orbitals was straightforward to choose for eclipsed conformations but less so for staggered conformations. Our results agree with the experimental photoelectron spectroscopy data in that we have obtained good agreement in the separation between peaks. For comparisons with ultraviolet spectroscopy data, we performed some configuration interaction as well as MCSCF calculations on the low-lying excited states. The two different calculations provided similar results for the excitation energy of the state corresponding to the first important peak of the spectrum but did not give quantitative agreement with experimental results from a complex with trimethylphosphine rather than phosphine ligands.