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Journal Articles IMA Journal of Numerical Analysis Year : 2021

Post-processing of the planewave approximation of Schrödinger equations. Part I: linear operators

Abstract

In this article, we prove a priori error estimates for the perturbation-based post-processing of the plane-wave approximation of Schrödinger equations introduced and tested numerically in previous works [6, 7]. We consider here a Schrödinger operator $H = − 1/2 ∆ + V$ on $L^2 (Ω$), where $Ω$ is a cubic box with periodic boundary conditions, and where V is a multiplicative operator by a regular-enough function V. The quantities of interest are, on the one hand, the ground-state energy defined as the sum of the lowest N eigenvalues of H , and, on the other hand, the ground-state density matrix, that is the spectral projector on the vector space spanned by the associated eigenvectors. Such a problem is central in first-principle molecular simulation, since it corresponds to the so-called linear subproblem in Kohn-Sham density functional theory (DFT). Interpreting the exact eigenpairs of H as perturbations of the numerical eigenpairs obtained by a variational approximation in a plane-wave (i.e. Fourier) basis, we compute first-order corrections for the eigenfunctions, which are turned into corrections on the ground-state density matrix. This allows us to increase the accuracy by a factor proportional to the inverse of the kinetic energy cutoff $E_c^{-1}$ of both the ground-state energy and the ground-state density matrix in Hilbert--Schmidt norm at a low computational extra-cost. Indeed, the computation of the corrections only requires the computation of the residual of the solution in a larger planewave basis and two Fast Fourier Transforms per eigenvalue. Schrödinger operator, perturbation method, planewave approximation, eigenvalue problem, post-processing.
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Dates and versions

hal-01908039 , version 1 (29-10-2018)
hal-01908039 , version 2 (20-11-2018)
hal-01908039 , version 3 (22-01-2020)
hal-01908039 , version 4 (17-03-2020)

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Eric Cancès, Geneviève Dusson, Yvon Maday, Benjamin Stamm, Martin Vohralík. Post-processing of the planewave approximation of Schrödinger equations. Part I: linear operators. IMA Journal of Numerical Analysis, 2021, 41 (4), pp.2423-2455. ⟨10.1093/imanum/draa044⟩. ⟨hal-01908039v4⟩
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