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We report a k-point extension of the second-order co-iterative augmented Hessian (CIAH) algorithm, termed k-CIAH, for Pipek–Mezey (PM) localization of Wannier functions (WFs). By exploiting an efficient evaluation of the Hessian–vector product, k-CIAH achieves O(Nk2n3) scaling in both CPU time and memory, matching that of previously reported first-order k-space approaches while improving upon the O(Nk3n3) scaling of Γ-point CIAH, where Nk denotes the number of k-points sampling the first Brillouin zone and n characterizes the unit-cell size. Benchmark calculations on a diverse set of solids─including insulators, semiconductors, metals, and surfaces─demonstrate the fast and robust convergence of k-CIAH-based PMWF optimization, which yields an overall computational efficiency approximately 2–3-fold higher than first-order k-space methods and orders of magnitude higher than Γ-point CIAH for localizing 1000–5000 orbitals. The quality of the resulting PMWFs is further validated by accurate electronic band structures obtained via PMWF-based Wannier interpolation.
Yang et al. (Wed,) studied this question.