We present the results of local density band structure calculations on the adsorption of atomic oxygen on Ge(001). As a reference for calculating adsorption energies, also the (2 X 1) symmetric dimer reconstruction of the clean Ge(001) is investigated. The surface states that form on this clean surface closely resemble those for silicon, although the calculated reconstruction energy is only 1.059 eV per dimer, half of the value for Si(001). For half a monolayer coverage of oxygen atoms the adsorption energy is 2.13 eV per adsorbing atom, also much less than the corresponding value for silicon. Electronic structure and optimized geometry for germanium and silicon are very similar, however. For one monolayer adsorption of oxygen the inability of small clusters to describe the systems is demonstrated. The optimum geometry is derived using simple dangling bond concepts, and is confirmed by slab calculations which find an adsorption energy of 2.112 eV per adsorbing atom. A comparison is made for the optimized structure, electronic surface states and adsorption energies with the isoelectronic systems of O, S and Se on Si(001).