A robust algorithm for $k$-point grid generation and symmetry reduction

TL;DR

This paper introduces a new, efficient algorithm for generating and reducing $k$-point grids in reciprocal space, optimizing computational speed and symmetry reduction for electronic structure calculations.

Contribution

The authors present a novel algorithm that efficiently computes and reduces generalized regular $k$-point grids using integer matrices and finite group theory, improving over traditional methods.

Findings

Faster generation and reduction of $k$-point grids.

More symmetry reduction leading to fewer irreducible $k$-points.

Open source implementation available.

Abstract

We develop an algorithm for i) computing generalized regular $k$-point grids, ii) reducing the grids to their symmetrically distinct points, and iii) mapping the reduced grid points into the Brillouin zone. The algorithm exploits the connection between integer matrices and finite groups to achieve a computational complexity that is linear with the number of $k$-points. The favorable scaling means that, at a given $k$-point density, all possible commensurate grids can be generated (as suggested by Moreno and Soler) and quickly reduced to identify the grid with the fewest symmetrically unique $k$-points. These optimal grids provide significant speed-up compared to Monkhorst-Pack $k$-point grids; they have better symmetry reduction resulting in fewer irreducible $k$-points at a given grid density. The integer nature of this new reduction algorithm also simplifies issues with finite precision in current implementations. The algorithm is available as open source software.