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

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 limit an infinitely dense grid, causing severe errors when neglected. The question are those respect to lattice constant crystal or helical angle chain screw-axis symmetry. This contrast atomic gradients, which can be made arbitrarily small by extension. disparate behaviour traced whether points depend on coordinate derivative energy taken. Whereas nonvanishing lattice-constant gradient identified as surface integral arising from expanding integration domain, origin helical-angle remains unknown.