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类Hartree-Fock方程的数值方法
NUMERICAL METHODS FOR HARTREE-FOCK-LIKE EQUATIONS

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林霖  
文摘 本文的主要目的是介绍近年来大基组下的类Hartree-Fock方程数值求解的一些进展.类Hartree-Fock方程出现在Hartree-Fock理论和含杂化泛函的Kohn-Sham密度泛函理论中,是电子结构理论中一类重要的方程.该方程在复杂的化学和材料体系的电子结构计算中有广泛地应用.由于计算代价的原因,类Hartree-Fock方程一般只被用在较小规模的量子体系(含几十到几百个电子)的计算.从数学角度上讲,类Hartree-Fock方程是一个非线性积分-微分方程组,其计算代价主要来自于积分算子的部分,也就是Fock交换算子.通过发展和结合自适应压缩交换算子方法(ACE),投影的C-DIIS方法(PC-DIIS)方法,以及插值可分密度近似方法(ISDF),我们大大降低了杂化泛函密度泛函理论的计算代价.以含1000个硅原子的体系为例,我们将平面波基组下的杂化泛函的计算代价降至接近不含Fock交换算子的半局域泛函计算的水平.同时,我们发现类Hartree-Fock方程的数学结构也为一类特征值问题的迭代求解提供了新的思路.
其他语种文摘 The main goal of this paper is to introduce some recent developments of numerical methods for solving Hartree-Fock-like equations in the context of large basis sets. Hartree-Fock-like equations are an important type of equations in electronic structure theory. They appear in the Hartree-Fock theory, as well as the Kohn-Sham density functional theory with hybrid exchange-correlation functionals, and are widely used in electronic structure calculations of complex chemical and materials systems. Because of its high computational cost, Hartree-Fock-like equations are typically only used in systems consisting of tens to hundreds of electrons. From a mathematical perspective, Hartree-Fock-like equations are a system of nonlinear integro-differential equations. The computational cost is mainly due to the integral operator part, namely the Fock exchange operator. Through the development of the adaptive compression method (ACE), the projected commutator-direct inversion in the iterative subspace (PC-DIIS) method, and the interpolative separable density fitting (ISDF) method, we demonstrate that the cost of Kohn-Sham density functional theory calculations with hybrid functionals can be significantly reduced. Using a silicon system with 1000 atoms for example, we have reduced the cost of hybrid functional calculations with a planewave basis set to a level that is close to the cost of semi-local functional calculations, which do not involve the Fock exchange operator. Meanwhile, we find that the structure of Hartree-Fock-like equations provides new insights for the iterative solution of one type of eigenvalue problems.
来源 计算数学 ,2019,41(2):113-125 【核心库】
关键词 类Hartree-Fock方程 ; 非线性特征值问题 ; 积分-微分算子 ; 量子化学 ; 电子结构理论 ; 密度泛函理论
地址

加州大学伯克利分校数学系, 劳伦斯伯克利国家实验室, 美国, 伯克利, 94720

语种 中文
文献类型 研究性论文
ISSN 0254-7791
学科 数学
基金 美国国家科学基金 ;  美国能源部（DOE）项目
文献收藏号 CSCD:6488852

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1 戴小英 电子结构计算的数值方法与理论 计算数学,2020,42(2):131-158
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