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Discretization Error Cancellation in the Plane-Wave Approximation of Periodic Hamiltonians with Coulomb Singularities

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. In the present work, we focused on periodic linear Hamiltonians with Coulomb potentials where 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. As a consequence, the error on the difference of discretized eigenvalues for different configurations is indeed reduced by an explicit factor. However, this error converges at the same rate as the error on the eigenvalue. Plane-wave methods for periodic Hamiltonians with Coulomb potentials are thus still inefficient to compute energy differences.

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Correspondence to Mi-Song Dupuy.

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Dupuy, MS. Discretization Error Cancellation in the Plane-Wave Approximation of Periodic Hamiltonians with Coulomb Singularities. J Sci Comput 80, 859–877 (2019). https://doi.org/10.1007/s10915-019-00959-6

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Keywords

  • Eigenvalue problems
  • Spectral method
  • Error analysis

Mathematics Subject Classification

  • 65N15
  • 65G99
  • 35P15
  • 65N35