Nonvanishing quadrature derivatives in the analytical gradients of density functional energies in crystals and helices

TL;DR

This paper reveals that certain quadrature derivatives in analytical energy gradients for crystals and helices do not vanish even with dense grids, leading to significant errors if ignored, due to their dependence on the coordinate of differentiation.

Contribution

It identifies the nonvanishing quadrature derivatives in lattice constant and helical angle gradients and explains their origins, highlighting their impact on computational accuracy.

Findings

Quadrature derivatives in lattice constant gradients do not vanish with dense grids.

Quadrature derivatives in atomic gradients can be minimized by grid extension.

The origin of nonvanishing derivatives in lattice gradients is a surface integral from expanding domains.

Abstract

It is shown that the quadrature derivatives in some analytical gradients of energies evaluated with a multi-centre radial-angular grid do not vanish even in the limit of an infinitely dense grid, causing severe errors when neglected. The gradients in question are those with respect to a lattice constant of a crystal or to the helical angle of a chain with screw axis symmetry. This is in contrast with the quadrature derivatives in atomic gradients, which can be made arbitrarily small by grid extension. The disparate behaviour is traced to whether the grid points depend on the coordinate with respect to which the derivative of energy is taken. Whereas the nonvanishing quadrature derivative in the lattice-constant gradient is identified as the surface integral arising from an expanding integration domain, the analytical origin of the nonvanishing quadrature derivative in the helical-angle gradient remains unknown.