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Post-processing of the planewave approximation of Schrödinger equations. Part I: linear operators
IMA Journal of Numerical Analysis ( IF 2.1 ) Pub Date : 2020-09-18 , DOI: 10.1093/imanum/draa044
Eric Cancès 1 , Geneviève Dusson 2 , Yvon Maday 3 , Benjamin Stamm 4 , Martin Vohralík 5
Affiliation  

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 (Cancès, Dusson, Maday, Stamm and Vohralík, (2014), A perturbation-method-based a posteriori estimator for the planewave discretization of nonlinear Schrödinger equations. C. R. Math., 352, 941--946; Cancès, Dusson, Maday, Stamm and Vohralík, (2016), A perturbation-method-based postprocessing for the planewave discretization of Kohn–Sham models. J. Comput. Phys., 307, 446--459.) We consider here a Schrödinger operator |${{\mathscr{H}} \,}= -\frac{1}{2}\varDelta +{\mathscr{V}}$| on |$L^2(\varOmega )$|⁠, where |$\varOmega $| is a cubic box with periodic boundary conditions and where |${\mathscr{V}}$| is a multiplicative operator by a regular-enough function |${\mathscr{V}}$|⁠. The quantities of interest are, on the one hand, the ground-state energy defined as the sum of the lowest |$N$| eigenvalues of |${{\mathscr{H}} \,}$|⁠, and, on the other hand, the ground-state density matrix that is the spectral projector on the vector space spanned by the associated eigenvectors. Such a problem is central in first-principle molecular simulation, since it corresponds to the so-called linear subproblem in Kohn–Sham density functional theory. Interpreting the exact eigenpairs of |${{\mathscr{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_{\textrm{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 plane-wave basis and two fast Fourier transforms per eigenvalue.

中文翻译:

Schrödinger方程的平面波逼近的后处理。第一部分:线性算子

在本文中,我们证明对先前工作中引入和测试的Schrödinger方程的平面波近似进行基于扰动的后处理的先验误差估计(Cancès,Dusson,Maday,Stamm和Vohralík,(2014年),《扰动-method基于后验为的非线性薛定谔方程的离散化平面波估计。CR数学式352,941--946;Cancès,Dusson,Maday,斯塔姆和Vohralík,(2016),A扰动方法为基础的后处理科恩深水车型的平面波离散。J. COMPUT。物理学。307,446--459)。我们这里薛定谔运营商考虑| $ {{\ mathscr {H}} \,} =-\ frac {1} {2} \ varDelta + {\ mathscr {V}} $ | | $ L ^ 2(\ varOmega)$ |⁠上,其中| $ \ varOmega $ | 是具有周期性边界条件的立方箱,其中| $ {\ mathscr {V}} $ | 是一个正则函数| $ {\ mathscr {V}} $ |⁠的乘法运算符。一方面,感兴趣的数量是基态能量,定义为最低| $ N $ |的总和。的特征值| $ {{\ mathscr {H}} \} $ |⁠另一方面,是基态密度矩阵,它是由相关特征向量跨越的向量空间上的频谱投影仪。这个问题在第一原理分子模拟中很重要,因为它对应于Kohn-Sham密度泛函理论中的所谓线性子问题。解释| $ {{\ mathscr {H}} \,} $ |的确切特征对。作为在平面波(即傅立叶)基础上通过变分逼近获得的数值本征对的扰动,我们计算了本征函数的一阶校正,然后将其转换为基态密度矩阵的校正。这使我们可以将精度提高一个与动能截止|| {{E _ {\ textrm {c}}} ^ {-1} $ |的倒数成比例的因子Hilbert-Schmidt范数中的基态能量和基态密度矩阵的计算量较低。实际上,校正的计算仅需要在较大的平面波基础上计算解的残差,并且每个特征值需要两个快速傅立叶变换。
更新日期:2020-09-20
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