The energetics and band structures of the elemental semiconductor Si in two fourfold-coordinated clathrate structures have been studied. For the Si(34) and Si(46) structures we predict indirect band gaps near 1.9 eV, or 0.7 eV wider than that of Si in the well-known sp3 diamond structure. We find that the energy differences between the clathrate structures and the diamond structure are about 0.07 eV, which is nearly three times smaller than those for more compact high-pressure structures of Si. The calculations are done using an ab initio tight-binding-like molecular-dynamics method, and further checked by the fully self-consistent plane-wave method. The possible origin of the small energy differences between Si in the clathrate and diamond structures is discussed in terms of strain energy. We compare the results for Si with those for similar structures involving carbon.