Application of the Method of Tight Binding to the Calculation of the Energy Band Structures of Diamond, Silicon, and Sodium Crystals

The method of tight binding is used to calculate the energy band structure of the diamond, silicon, and sodium crystals. The wave functions of the valence and conduction bands of diamond are expanded in terms of Bloch sums constructed from the 1s, 2s, and 2p Hartree-Fock atomic orbitals, and two different crystal potentials the muffin-tin and the overlapping atomic potential (OAP) were used. With the muffin-tin potential, the tight-binding and the augmented-plane-wave (APW) method yield nearly identical valence band structure, and their conduction bands show only minor differences. When the OAP is used, the results of the tight-binding scheme differ appreciably from those of Bassani and Yoshimine by the method of orthogonalized plane waves (OPW), the discrepancy being attrubuted to incomplete convergence of the latter calculations. The tight-binding structures of the valence band derived from the two different potentials agree quite well with each other, but considerable deviations are found in the conduction band. The x-ray form factors calculated by means of the tight-binding wave functions are in good agreement with experiment and represent a considerable improvement over a simple superposition of atomic charges. The convergence of the tight-binding method with respect to the higher atomic orbitals has been examined. It is found that addition of 3s, 3p, and 3d Bloch sums to the wave functions of diamond has only small effects on the energy of the valence and conduction bands. A similar tight-binding calculation has been performed for the band structure of silicon using OAP, and the results are in good agreement with those of a modified scheme of the method of OPW using as basis functions 609 OPW's as well as the Bloch sums of the core states. For the base of sodium, the method of tight binding gives conduction-band energies in good agreement with the APW-type calculations of Schlosser and Marcus. Generalization of the method of tight binding by using single-Gaussian Bloch sums is discussed, and the use of this scheme leads to substantial improvements for the case of diamond.