Ab initio Hartree-Fock calculations of a hydrogen molecule in group IV element crystals (diamond, silicon and germanium) have been performed with cluster models The total energy has a minimum value when the hydrogen molecule is at a tetrahedral site. The charge of the hydrogen molecule in silicon and germanium crystals is neutral but positive in diamond. The calculated vibrational frequencies of H2 are 5533, 4423 and 4402 cm-1 in diamond, silicon and germanium crystals, respectively.