Robust Pipek–Mezey Orbital Localization in Periodic Solids

We describe a robust method for determining Pipek–Mezey (PM) Wannier functions (WF), recently introduced by Jónsson et al. (J. Chem. Theor. Chem. 2017, 13, 460), which provide some formal advantages over the more common Boys (also known as maximally-localized) Wannier functions. The Broyden–Fletcher–Goldfarb–Shanno-based PMWF solver is demonstrated to yield dramatically faster convergence compared to the alternatives (steepest ascent and conjugate gradient) in a variety of one-, two-, and three-dimensional solids (including some with vanishing gaps) and can be used to obtain Wannier functions robustly in supercells with thousands of atoms. Evaluation of the PM functional and its gradient in periodic linear combination of atomic orbital representation used a particularly simple definition of atomic charges obtained by Moore–Penrose pseudoinverse projection onto the minimal atomic orbital basis. An automated “canonicalize phase then randomize” method for generating the initial guess for WFs contributes significantly to the robustness of the solver.