Tight-binding theory and elastic constants

Harrison has developed a tight-binding theory of semiconductors that admits simple calculation of the total energy without recourse to integration over the Brillouin zone. The elastic constants, which can be calculated from changes in the total energy under certain distortions, provide a critical test of many assumptions of the theory. The validity of several independent assumptions is examined using the elastic constants as a means for testing them. It is found that the principal error in the theory lies in the assumed form of the two-body repulsion, $V_0$. It is argued that the form of $V_0$ which Harrison assumes is in error mainly because of his neglect of the orthogonalization of valence wave functions to neighboring cores. An accurate ab initio treatment of $V_0$ contains significant self-consistency corrections; this seems to preclude a simple parameter-free formulation of $V_0$ analogous to the universal matrix elements. The tight-binding theory is shown to predict a universal relation between the three independent elastic constants. Such a relation is well borne out by experiment, providing an independent test of the theory.