Model calculation of the static macroscopic dielectric function and the optical frequency of diamond

In this paper the dielectric matrix and its inverse are calculated for diamond using electron energies and wave functions obtained by diagonalizing a simple molecular-bonding Hamiltonian. Results of the static longitudinal macroscopic dielectric function and of the phonon frequencies at the $\Gamma$ point are presented. This macroscopic dielectric function is derived from the linear response of the system to an external perturbation in the Hartree approximation, i.e., neglecting the explicit influence of exchange and correlation in the linear-response formulas. The parameters in the Hamiltonian matrix are fitted to the experimental values of the indirect band gap and valence-band width. Results of the macroscopic dielectric function are given in the $\Delta$, $\Sigma$, and $\Lambda$ direction. The effect of the off-diagonal elements, accounting for the variations in the local electric field, on the macroscopic dielectric constant at zero wave vector, is 14%. A comparison is made with other calculations. The double summation over reciprocal-lattice vectors in the electron-nuclear part of the dynamical matrix is performed through a factorization procedure. Convergence is achieved by summing over 1185 shells. The calculated value of the optical frequencies at the $\Gamma$ point is 4.45 × ${10}^{14}$ rad/sec, compared with the experimental value of 2.51 × ${10}^{14}$ rad/sec. The results show that in this model a realistic macroscopic dielectric function does not necessarily guarantee agreement between the calculated and experimental phonon frequencies.