Direct Unconstrained Variable-Metric Localization of One-Electron Orbitals.

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.