类Hartree-Fock方程的数值方法

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.

NUMERICAL METHODS FOR HARTREE-FOCK-LIKE EQUATIONS

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. HartreeFock-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 HartreeFock-like equations provides new insights for the iterative solution of one type of eigenvalue problems.