Efficiency of Generalized Regular k-point grids

Abstract

Most DFT practitioners use regular grids (Monkhorst-Pack, MP) for integrations in the Brillioun zone. Although regular grids are the natural choice and easy to generate, more general grids whose generating vectors are not merely integer divisions of the reciprocal lattice vectors, are usually more efficient. 1 We demonstrate the efficiency of generalized regular (GR) grids compared to Monkhorst-Pack (MP) and simultaneously commensurate (SC) grids. In the case of metals, for total energy accuracies of one meV/atom, GR grids are 60% faster on average than MP grids and 20% faster than SC grids. GR grids also have greater freedom in choosing the k-point density, enabling the practitioner to achieve a target accuracy with the minimum computational cost. arXiv:1804.04741v2 [cond-mat.mtrl-sci]

The Basics of Electronic Structure Theory for Periodic Systems

“…If the total-energy calculations are intended to be used as the basis of a thermodynamics analysis, highly precise values, with an accuracy of 1 meV per atom or better (Grabowksi et al, 2007 ; Grabowski et al, 2011 ), are required. Test calculations performed in an automated way for k -point grids of various shapes and densities (Morgan et al, 2018 ) suggest that an accurate sampling of the Fermi surface, and thus a dense k -point grid in the range of 30 3 irreducible k- points per atom, is necessary to achieve this goal. Here, we present case studies carried out with the VASP code (Kresse and Furthmüller, 1996b ) for four representative materials, aluminum (Al), vanadium (V), silicon (Si), and titanium (Ti).…”

“…For this reason, one uses automatically generated, very dense k -point sets that allow one to reach an accuracy of the total energy better than 1 meV per atom. As has been shown in a recent study (Morgan et al, 2018 ), in order to guarantee this accuracy level for all phases (with differently sized and shaped unit cells), a k -point density as high as 5,000 k -points /Å −3 is typically required. Moreover, methods based on machine learning attempt to select k -point grids that are most suitable for the problem at hand (Choudhary and Tavazza, 2019 ).…”

Numerical quadrature in the Brillouin zone for periodic Schrödinger operators

“…The most famous method is the linear tetrahedron method and its improved version by Blöchl [2] (the Blöchl scheme is not covered by the results in this paper, and we plan to investigate it in a forthcoming paper). Other numerical quadratures have been proposed [21,22] (see also [12] for an adaptive numerical scheme). In this paper, we study these quadrature rules, and prove that an interpolation of quantities of interest to order p coupled to a reconstruction of the Fermi surface with a method of order q leads, in general, to a total error of order L −(min(p+1,q+1)) : this is the content of Theorem 4.5.…”

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

“…The poor convergence of the electronic energy means that DFT codes must use extremely dense grids [2,3] to achieve an accuracy of several meV/atom. To reduce computation time, it is common practice to evaluate eigenvalues at symmetrically equivalent k-points only once.…”

“…In most DFT codes, even for very dense grids, the setup and symmetry reduction of the grid takes a few seconds at most. Our motivation for an improved algorithm (despite the speed of current routines) is two-fold: 1) enable an automatic grid-generation technique that allows us to scan over thousands of candidate grids, in a few seconds, to find one with the best possible symmetry reduction [4,5] (in other words, enable a k-point generation method in the same spirit as that of [2] but have the grid generation done on-the-fly [6]), and 2) eliminate (or at least greatly reduce) the probability of incorrect symmetry reduction 3 as the result of finite precision errors (the danger of these increases as the density of the integration grid increases).…”