  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.
