具有凸密度函数的周期性科恩-沙姆方程能量的完全保证和可计算误差边界

Andrea BordignonCERMICS, Geneviève DussonLMB, Éric CancèsCERMICS, MATHERIALS, Gaspard KemlinLAMFA, Rafael Antonio Lainez ReyesIANS, Benjamin StammIANS
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引用次数: 0

摘要

本文推导了凸非线性均场模型能量的完全保证误差边界。这些结果尤其适用于具有凸密度函数的 Kohn-Sham 方程,包括还原哈特里-福克(rHF)模型,以及具有精确交换密度函数的 Kohn-Sham 模型(不幸的是,该模型并不明确,因此在实践中无法使用)。然后,我们将获得的边界分解为两部分,一部分取决于所选的离散化,另一部分取决于用于求解非线性特征值问题的自洽算法中执行的迭代次数,从而为自适应细化策略铺平了道路。我们在一系列测试案例中证明了边界的准确性,包括用 rHF 模型模拟的硅晶体和用平面波离散的氟化氢分子。我们还表明,虽然误差边界不再有保证,但对于使用 Kohn-Sham 模型模拟的硅晶体,误差边界仍然非常精确,并使用了具有实际意义的非凸交换相关函数。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Fully guaranteed and computable error bounds on the energy for periodic Kohn-Sham equations with convex density functionals
In this article, we derive fully guaranteed error bounds for the energy of convex nonlinear mean-field models. These results apply in particular to Kohn-Sham equations with convex density functionals, which includes the reduced Hartree-Fock (rHF) model, as well as the Kohn-Sham model with exact exchange-density functional (which is unfortunately not explicit and therefore not usable in practice). We then decompose the obtained bounds into two parts, one depending on the chosen discretization and one depending on the number of iterations performed in the self-consistent algorithm used to solve the nonlinear eigenvalue problem, paving the way for adaptive refinement strategies. The accuracy of the bounds is demonstrated on a series of test cases, including a Silicon crystal and an Hydrogen Fluoride molecule simulated with the rHF model and discretized with planewaves. We also show that, although not anymore guaranteed, the error bounds remain very accurate for a Silicon crystal simulated with the Kohn-Sham model using nonconvex exchangecorrelation functionals of practical interest.
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