The electronic structure, lattice parameters, phonon frequencies at the Brillouin zone center, and first-order elastic-constants (C-ij) have been calculated at the ab initio Hartree-Fock level using either all-atomic-electron basis sets or basis sets associated with core pseudopotentials. Band structure, electronic density maps, and density of states have been plotted. The all-electron calculations point out a mixed covalent-ionic nature for the Mg-Si bonding. The need for flexibility and optimization of the atomic basis set is quantitatively assessed from a careful comparison,of the results obtained using several different bases. This is necessary to describe correctly the variation of the potential energy surface of the crystal associated with atomic displacements due to lattice deformations. In the present calculations, the pseudopotential approximation is. found to provide parameter values of practically similar quality as those of all-electron computations. The polarization orbitals are shown to have a fairly selective role for the calculation of the elastic constants, depending on the kind of strains involved (for instance, polarization on Si for C-44, polarization on Mg for the bulk modulus and C-11).