Discretization error cancellation in the plane-wave approximation of periodic Hamiltonians with Coulomb singularities

TL;DR

This paper investigates how discretization errors cancel out in plane-wave approximations of periodic Hamiltonians with Coulomb singularities, explaining the phenomenon through the Kato cusp condition and providing explicit error decay formulas.

Contribution

It provides a theoretical explanation for discretization error cancellation in energy differences, using weighted Sobolev spaces and eigenfunction Fourier analysis.

Findings

Error cancellation improves accuracy of energy differences

Eigenvalue difference errors decay at the same rate as eigenvalues

Explicit formulas for eigenfunction Fourier coefficient decay

Abstract

In solid-state physics, energies of molecular systems are usually computed with a plane-wave discretization of Kohn-Sham equations. A priori estimates of plane-wave convergence for periodic Kohn-Sham calculations with pseudopotentials have been proved , however in most computations in practice, plane-wave cut-offs are not tight enough to target the desired accuracy. It is often advocated that the real quantity of interest is not the value of the energy but of energy differences for different configurations. The computed energy difference is believed to be much more accurate because of `discretization error cancellation', since the sources of numerical errors are essentially the same for different configurations. For periodic linear Hamiltonians with Coulomb potentials, error cancellation can be explained by the universality of the Kato cusp condition. Using weighted Sobolev spaces, Taylor-type expansions of the eigenfunctions are available yielding a precise characterization of this singularity. This then gives an explicit formula of the first order term of the decay of the Fourier coefficients of the eigenfunctions. It enables one to prove that errors on eigenvalue differences are reduced but converge at the same rate as the error on the eigenvalue.