Variable-Metric Localization of Occupied and Virtual Orbitals.

The key idea of the variable-metric approach to orbital localization is to allow nonorthogonality between orbitals while, at the same time, preventing them from becoming linearly dependent. The variable-metric localization has been shown to improve the locality of occupied nonorthogonal orbitals relative to their orthogonal counterparts. In this work, numerous localization algorithms are designed and tested to exploit the conceptual simplicity of the variable-metric approach with the goal of creating a straightforward and reliable localization procedure for virtual orbitals. The implemented algorithms include the steepest descent, conjugate gradient (CG), limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS), and hybrid procedures as well as trust-region (TR) methods based on the CG and Cauchy-point subproblem solvers. Comparative analysis shows that the CG-based TR algorithm is the best overall method to obtain nonorthogonal localized molecular orbitals (NLMOs), occupied or virtual. The L-BFGS and CG algorithms can also be used to obtain NLMOs reliably but often at higher computational cost. Extensive tests demonstrate that the implemented methods allow us to obtain well-localized Boys-Foster (i.e., maximally localized Wannier functions) and Pipek-Mezey, orthogonal and nonorthogonal, and occupied and virtual orbitals for a variety of gas-phase molecules and periodic materials. The tests also show that virtual NLMOs, which have not been described before, are, on average, 13% (Boys-Foster) and 18% (Pipek-Mezey) more localized than their orthogonal counterparts.