We report ab initio investigations of the adsorption of atomic hydrogen on the (100) surfaces of Rh and Pd in the local-density-functional and generalized-gradient approximations. Our calculations have been performed using a plane-wave basis, using optimized ultrasoft pseudopotentials for describing the electron - ion interactions. Detailed results are reported for the adsorption energies, the stabilities of various adsorption geometries, and the adsorption-induced changes in the surface relaxations and in the work-functions. We find that the adsorption of a monolayer of hydrogen changes the inward relaxation of the top layer of the substrate into an outward relaxation. However, the change of the substrate relaxation has only a very small influence on the adsorption energy and geometry. For both metals the stable adsorption sites are the fourfold hollows. The site preference has its origin in a maximum gain of covalent bonding energy resulting from the overlap of the hydrogen s and the metal orbitals and from a minimal Pauli repulsion. Non-local exchange - correlation corrections have only a small influence on the atomic adsorption process and on the relaxation of the substrate, but influence the adsorption energy through corrections to the binding energy of the hydrogen molecule. Relativistic effects, however, turn out to be quite important.