The embedding potential and the corrective operator - A comparison of techniques for embedding defect cluster calculations

Cluster calculations have proved one of the most fruitful means of extracting useful information about solid-state defect properties at finite computational expense, but large clusters are necessary to obtain reliable results and uncertainty always remains concerning the effects of edge states, with ''ad hoc'' boundary conditions such as the saturation of the surface with hydrogen atoms being frequently employed. Recently methods have been developed for ''embedding'' the cluster in the perfect crystal so that the solution obtained from the cluster calculation is the same as would have been obtained from a full calculation on the whole defective solid. Here the relationships between these methods are explored and their implementation in a semi-empirical Hartree-Fock calculation is discussed. It is shown that the form of the ''embedding potential'' recently derived by Baraff and Schluter is a simple consequence of the fact that this potential is independent of the Hamiltonian and overlap matrices within the cluster. It is also shown that the ''corrective operator'' of Pisani contains in principle exactly the same information but that it is not independent of the cluster matrices, so approximation is inevitable if the same operator is to be used to embed clusters which differ internally. Simple examples are given of the use of the embedding potential to solve defect problems in the tight-binding linear chain model