  Spatially localized one-electron orbitals, orthogonal and nonorthogonal, are
widely used in electronic structure theory to describe chemical bonding and
speed up calculations. In order to avoid linear dependencies of localized
orbitals, the existing localization methods either constrain orbital
transformations to be unitary, that is, metric preserving, or, in the case of
variable-metric methods, fix the centers of nonorthogonal localized orbitals.
Here, we developed a different approach to orbital localization, in which these
constraints are replaced with a single restriction that specifies the maximum
allowed deviation from the orthogonality for the final set of localized
orbitals. This reformulation, which can be viewed as a generalization of
existing localization methods, enables one to choose the desired balance
between the orthogonality and locality of the orbitals. Furthermore, the
approach is conceptually and practically simple as it obviates the necessity in
unitary transformations and allows to determine optimal positions of the
centers of nonorthogonal orbitals in a unconstrained and straightforward
minimization procedure. It is demonstrated to produce well-localized orthogonal
and nonorthogonal orbitals with the Berghold and Pipek-Mezey localization
functions for a variety of molecules and periodic materials including large
systems with non-trivial bonding.
