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

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