In this paper, we propose a novel eigenpair-splitting method, inspired by the divide-and-conquer strategy, for solving the generalized eigenvalue problem arising from the Kohn-Sham equation. Unlike the commonly used domain decomposition approach in divide-and-conquer, which solves the problem on a series of subdomains, our eigenpair-splitting method focuses on solving a series of subequations defined on the entire domain. This method is realized through the integration of two key techniques: a multi-mesh technique for generating approximate spaces for the subequations, and a soft-locking technique that allows for the independent solution of eigenpairs. Numerical experiments show that the proposed eigenpair-splitting method can dramatically enhance simulation efficiency, and its potential towards practical applications is also demonstrated well through an example of the HOMO-LUMO gap calculation. Furthermore, the optimal strategy for grouping eigenpairs is discussed, and the possible improvements to the proposed method are also outlined.

翻译：本文受分治策略启发，提出了一种新颖的特征对分裂方法，用于求解Kohn-Sham方程导出的广义特征值问题。与分治法中常用的、在一系列子域上求解问题的区域分解方法不同，我们的特征对分裂方法聚焦于求解定义在整个区域上的一系列子方程。该方法通过整合两项关键技术得以实现：用于为子方程生成近似空间的多网格技术，以及允许特征对独立求解的软锁定技术。数值实验表明，所提出的特征对分裂方法能显著提升模拟效率，并通过HOMO-LUMO能隙计算的实例充分展示了其在实际应用中的潜力。此外，本文讨论了特征对分组的最优策略，并概述了该方法可能的改进方向。