Fast Generation of Pipek-Mezey Wannier Functions via the Co-Iterative Augmented Hessian Method.

We report a -point extension of the second-order co-iterative augmented Hessian (CIAH) algorithm, termed -CIAH, for Pipek-Mezey (PM) localization of Wannier functions (WFs). By exploiting an efficient evaluation of the Hessian-vector product, -CIAH achieves () scaling in both CPU time and memory, matching that of previously reported first-order -space approaches while improving upon the () scaling of Γ-point CIAH, where denotes the number of -points sampling the first Brillouin zone and 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 -CIAH-based PMWF optimization, which yields an overall computational efficiency approximately 2-3-fold higher than first-order -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.