Reduced scaling of optimal regional orbital localization via sequential exhaustion of the single-particle space

TL;DR

This paper introduces an efficient sequential algorithm for regional orbital localization within large systems, significantly reducing computational costs while maintaining localization quality, enabling practical analysis of nanoscale and extended materials.

Contribution

The paper presents a novel sequential exhaustion approach for orbital localization that reduces computational overhead for large systems without sacrificing localization accuracy.

Findings

Can handle systems with ~10,000 electrons in 0.5 hours

Maintains localization quality comparable to traditional methods

Extends easily to stochastic methodologies for non-eigenstate data

Abstract

Wannier functions have become a powerful tool in the electronic structure calculations of extended systems. The generalized Pipek-Mezey Wannier functions exhibit appealing characteristics (e.g., reaching an optimal localization and the separation of the $\sigma$-$\pi$ orbitals) when compared with other schemes. However, when applied to giant nanoscale systems, the orbital localization suffers from a large computational cost overhead when one is interested in localized states in a small fragment of the system. Herein we present a swift, efficient, and robust approach for obtaining regionally localized orbitals of a subsystem within the generalized Pipek-Mezey scheme. The proposed algorithm introduces a reduced workspace and sequentially exhausts the entire orbital space until the convergence of the localization functional. It tackles systems with $\sim$10000 electrons within 0.5 hours with no loss in localization quality compared to the traditional approach. Regionally localized orbitals with a higher extent of localization are obtained via judiciously extending the subsystem's size. Exemplifying on large bulk and a 4-nm wide slab of diamond with NV$^-$ center, we demonstrate the methodology and discuss how the choice of the localization region affects the excitation energy of the defect. Furthermore, we show how the sequential algorithm is easily extended to stochastic methodologies that do not provide individual single-particle eigenstates. It is thus a promising tool to obtain regionally localized states for solving the electronic structure problems of a subsystem embedded in giant condensed systems.