Robust Pipek-Mezey Orbital Localization in Periodic Solids

TL;DR

This paper introduces a robust and fast BFGS-based method for Pipek-Mezey Wannier function localization in periodic solids, outperforming previous algorithms in convergence speed and robustness for large supercells.

Contribution

The paper presents a new BFGS-based solver for Pipek-Mezey Wannier functions that converges faster and is more robust than existing methods, especially in large and complex solids.

Findings

BFGS solver achieves faster convergence than steepest ascent and conjugate gradient methods.

Method is effective in 1D, 2D, and 3D solids, including systems with vanishing gaps.

Robust initial guess generation improves solver reliability.

Abstract

We describe a robust method for determining Pipek-Mezey (PM) Wannier functions (WF), recently introduced by J\'onsson 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 (BFGS) based PMWF solver is demonstrated to yield dramatically faster convergence compared to the alternatives (steepest ascent and conjugate gradient) in a variety of 1-, 2-, and 3-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 LCAO 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" (CPR) method for generating the initial guess for WFs contributes significantly to the robustness of the solver.