The relative stabilities of hydrogen at (or near) the tetrahedral interstitial and bond-centered sites as well as that of hydrogen dimers in the molecular and bond-centered-antibonding configurations are calculated at the ab initio level in molecular clusters for c-C, Si, Ge, and α-Sn. The trends show that the lowest-energy configurations change as one goes down the Periodic Table. The relative stability of the possible equilibrium sites affects which charge states of H are likely to be realized in a given host. This in turn affects the diffusion properties of H and its interactions with dopants and other defect centers. The trends in equilibrium geometries and relative stabilities show that silicon is a particular case among group IV hosts in which both isolated interstitials and both dimer states are close to each other in energy. We also examine some properties of two charge states of molecular hydrogen in Si in order to determine the key features of their electron paramagnetic resonance spectra.