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.

翻译：本文针对凸非线性平均场模型的能量推导了完全可保证的误差界。这些结果特别适用于具有凸密度泛函的Kohn-Sham方程，包括简化Hartree-Fock（rHF）模型，以及采用精确交换-密度泛函的Kohn-Sham模型（遗憾的是该泛函非显式表达，因而无法实际应用）。随后我们将所得误差界分解为两部分：一部分取决于所选离散化方案，另一部分取决于求解非线性特征值问题的自洽算法迭代次数，从而为自适应细化策略奠定基础。通过一系列测试案例验证了误差界的准确性，包括采用平面波离散化、通过rHF模型模拟的硅晶体和氟化氢分子。我们还证明，对于采用具有实际意义的非凸交换关联泛函的Kohn-Sham模型所模拟的硅晶体，虽然误差界不再具有完全可保证性，但仍保持极高的精确度。