Methods of embedding for defect and surface problems

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Abstract
In the theory of defects and surfaces, the solution for a small part of the solid is often all that is of interest. This paper considers the problem of finding within such a small subdomain or 'cluster' the correct solution to the equation of motion of the whole system. It is first shown how the 'embedding-potential' method of 'perturbed-cluster' theory is equivalent to the usual Green function methods of 'perturbed-crystal' theory. Then other 'perturbed-cluster' methods are considered and it is shown that, because of the further assumptions about the solution that are involved, they fail to remove properly some of the spurious features introduced by performing a cluster calculation. This is demonstrated by the solution of simple model systems. It is shown how the matrix elements of certain operators and the correct normalisation of the static function can be calculated from a solution in the cluster region only. Next it is shown that the real-space embedding procedure introduced by Inglesfield (1981) corresponds in a discrete basis to a fundamentally different way of calculating the embedding potential, which is especially suitable for use in surface problems. This too is illustrated with a simple example, and some implicit equations for the embedding potential that do not involve the Green function of the host crystal are derived for the case where different possible cluster boundaries are related by lattice symmetry operations. Finally the implementation of these schemes and the application of the ideas to other quantum problems involving subdomains are briefly considered.

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