Spatially localized one-electron orbitals, orthogonal
and non-orthogonal, 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 non-orthogonal 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 one to determine optimal positions of the centers
of non-orthogonal orbitals in an unconstrained and straightforward minimization procedure. It is demonstrated to produce well-localized
orthogonal and non-orthogonal orbitals with the Berghold and Pipek-Mezey localization functions for a variety of
molecules and periodic materials including large systems with nontrivial bonding.
Direct unconstrained variable-metric localization of one-electron orbitals
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