Theory of the multicenter bond

Abstract

Tight-binding theory, as applied to tetrahedral semiconductors by Harrison, has been extended to crystal structures of more general coordination. Following Harrison’s application to ${\mathrm{SiO}}_2$, the total energy is formulated in terms of multicenter bonding units, the analog of the two-center bond in tetrahedral structures. The universal matrix elements, which are obtained from fits to free-electron bands and depend only on the internuclear distance, correspond to matrix elements between orthogonal Wannier functions in the zinc-blende structure, and do not apply directly to other structures. Matrix elements of nonorthogonal atomic orbitals, which are applicable to other structures, are formulated in terms of the universal matrix elements in the tetrahedral structure. These matrix elements are then applicable to new structures, and permit an extension of the theory without introducing new parameters, or modifying the theory in any fundamental way. Several properties, including the total energy, minimum bond length, and force constant, are calculated for units of ${\mathrm{Zn}}_3$ ${\mathrm{P}}_2$ and elemental boron, and are found to agree with experiment much as did the theory’s original application. The tight-binding bands of ${\mathrm{Zn}}_3$ ${\mathrm{P}}_2$, using a fluorite structure as an approximation to the true crystal structure, are compared to a pseudopotential calculation of Lin-Chung. It is found that second neighbors are necessary to obtain the correct band structure, but the total energy is not sensitive to the choice of neighbors.