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### First-principle calculation of SP^{3} hybridization and bonding in diamond-structure semiconductor crystals

#### Abstract

electrons in the diamond-structure semiconductor crystals has been calculated for the k = 0 state. This charge density represents the electronic distribution in the direct lattice of the crystals. Normally, the charge density in a condensed matter such as crystals is obtained from extremely complicated functions. However, in this work, the charge density is calculated from firstprinciples. The basic inputs are the lattice constant of the crystal and the atomic number of the atom. To obtain the required expression for the hybrids, we must choose some orientation for the tetrahedral bonds with respect to the Cartesian axes. The most convenient orientation chosen is to inscribe the tetrahedron in a cube whose edges are parallel to the Cartesian axes as shown in Figure 3.1. The tetrahedral bonds point towards the corners of a unit cube. The centre of the cube is the origin of the coordinates. The charge density is calculated along the four tetrahedral bonds and the [100], [010], and [001] directions. The density along the bonds directions is found to be the same. Also, the density along the [100], [010], and [001] directions are the same at equal distances from the center of the Wigner-Seitz cell (which is the origin of coordinate axes employed). In general, the density rises from zero at the center of a cell to a certain maximum value and then drops down as the distance is further increased. The results obtained also explain the known hardness of carbon (diamond).

*Journal of the Nigerian Association of Mathematical Physics*, Volume 15 (November, 2009), pp 155 - 162