Bound states, plane waves, and orthogonalization

Abstract

Using a method of orthogonalization originally due to Des Cloizeaux, we orthonormalize an arbitrarily large finite subset of a given set of orthogonalized plane waves (OPW's). The construction is democratic in that it does not require an iterative selection of the first, second, etc., vector of the set to be orthonormalized, nor does it depend upon the use of previous vectors already orthogonalized, as is required in the Schmidt orthogonalization process. The orthonormalized vectors [completely orthogonalized plane waves (COPW'S)] depend upon a cutoff ${\overrightarrow{\mathrm{k}}}_C$ in momentum space. Two results are established concerning the behavior of the COPW's as $k_C\to \infty$: one is that completeness is obtained in the limit $k_C\to \infty$, and the second is that the COPW's themselves converge (weakly, however) to the usual OPW's. Finally, a linked-cluster expansion is derived which leads to a simple perturbation scheme for the desired vectors.