Scaling of the Hamiltonian and momentum in semiconductors

We compute self-consistently the matrix elements of the Hamiltonian $H$ and the momentum $\overrightarrow{\mathrm{p}}$ in a localized basis for covalent semiconductors. This basis is connected with the Wannier functions of the system obtained by means of a variational method. The calculation has been performed for different lattice constants in order to look for a $d^{-\alpha }$ dependence law for $H$ and $\overrightarrow{\mathrm{p}}$. The results are more satisfactory in Si, where $H$ roughly scales as $d^{-2\mathrm{}}$. The scaling of the Cartesian components of $\overrightarrow{\mathrm{p}}$ is rather more complicated, but a simple behavior is obtained for the modulus $\left|\overrightarrow{\mathrm{p}}\right|$.