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.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Price includes VAT (USA)
Tax calculation will be finalised during checkout.
Babuška, I., Osborn, J.E.: Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems. Math. Comput. 52(186), 275–297 (1989)
Babuška, I., Rosenzweig, M.B.: A finite element scheme for domains with corners. Numer. Math. 20, 1–21 (1972/73)
Cancès, E., Chakir, R., Maday, Y.: Numerical analysis of the planewave discretization of some orbital-free and Kohn–Sham models. ESAIM Math. Model. Numer. Anal. 46(2), 341–388 (2012)
Cancès, E., Dusson, G.: Discretization error cancellation in electronic structure calculation: toward a quantitative study. ESAIM Math. Model. Numer. Anal. 51(5), 1617–1636 (2017)
Cancès, E., Dusson, G., Maday, Y., Stamm, B., Vohralík, M.: A perturbation-method-based post-processing for the planewave discretization of Kohn–Sham models. J. Comput. Phys. 307, 446–459 (2016)
Chen, H., Gong, X., He, L., Yang, Z., Zhou, A.: Numerical analysis of finite dimensional approximations of Kohn–Sham models. Adv. Comput. Math. 38(2), 225–256 (2013)
Chen, H., Schneider, R.: Numerical analysis of augmented plane wave methods for full-potential electronic structure calculations. ESAIM Math. Model. Numer. Anal. 49(3), 755–785 (2015)
Dupuy, M.-S.: The variational projector augmented-wave method for the plane-wave discretization of linear Schrödinger operators (2018). (in preparation)
Egorov, Y.V., Schulze, B.-W.: Pseudo-Differential Operators, Singularities, Applications, Volume 93 of Operator Theory: Advances and Applications. Birkhäuser Verlag, Basel (1997)
Flad, H.-J., Schneider, R., Schulze, B.-W.: Asymptotic regularity of solutions to Hartree–Fock equations with Coulomb potential. Math. Methods Appl. Sci. 31(18), 2172–2201 (2008)
Grisvard, P.: Singularities in Boundary Value Problems, Volume 22 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics]. Springer, Berlin (1992)
Hunsicker, E., Nistor, V., Sofo, J.O.: Analysis of periodic Schrödinger operators: regularity and approximation of eigenfunctions. J. Math. Phys. 49(8), 083501 (2008)
Kato, T.: On the eigenfunctions of many-particle systems in quantum mechanics. Commun. Pure Appl. Math. 10, 151–177 (1957)
Kozlov, V.A., Maz’ya, V.G., Rossmann, J.: Elliptic Boundary Value Problems in Domains with Point Singularities, Volume 52 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (1997)
Melrose, R.B.: The Atiyah–Patodi–Singer Index Theorem, Volume 4 of Research Notes in Mathematics. A K Peters Ltd., Wellesley (1993)
Pieniazek, S.N., Clemente, F.R., Houk, K.N.: Sources of error in DFT computations of C–C bond formation thermochemistries: \(\pi \rightarrow \sigma \) transformations and error cancellation by DFT methods. Angew. Chem. Int. Edition 47(40), 7746–7749 (2008)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press [Harcourt Brace Jovanovich, Publishers], New York (1978)
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
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
- Eigenvalue problems
- Spectral method
- Error analysis
Mathematics Subject Classification