On the use of corresponding orbitals for the construction of mutually orthogonal orbital sets

Consider two orbital sets χ_k, k = 1…m and η₁, 1 = 1…n, which are mutually nonorthogonal. Provided that n > m, at least n − m orbitals of the set {η} can be orthogonalized to the set {χ} by a transformation within the set {η}. The orthogonalization of the remaining orbitals of {η} to the set {χ} requires a transformation in which the χk appear explicitly. The orthogonalization of one orbital set to another is relevant for SCF optimizations in a truncated basis set, in the presence of frozen occupied orbitals. Examples are frozen core calculations, ECP calculations, and embedded cluster calculations, where the cluster is embedded in a frozen environment. A simple orthogonalization scheme, which makes use of a corresponding orbital transformation, is presented. It is suggested that with a small, well-defined extension of the set {η} the complete orthogonalization can be done with a transformation in which the {χ} do not appear explicitly. © 1993 John Wiley & Sons, Inc.