We report the new structures of aluminum hydrides derived from the Al4 tetrahedral cages. We perform ab initio quantum chemical calculation for these new aluminum hydrides. Our calculation of binding energies of the new aluminum hydrides reveal that stability of these hydrides increases as more hydrogen atoms are adsorbed, while stability of Al-H bonds decreases. We also calculate electronic stress tensor to evaluate the chemical bonds of these hydrides. As a result, we find that the bonds of the Al4 tetrahedral cage are strengthened as more hydrogen atoms are adsorbed on the aluminum hydrides. Our calculation of the potential energy surfaces and the regional chemical potential show that hydrogen atoms are likely to adsorb on bridge site at first.