In the classic implementation of the LOBPCG method, orthogonalization and the R-R (Rayleigh-Ritz) procedure cost nonignorable CPU time. Especially this consumption could be very expensive to deal with situations with large block sizes. In this paper, we propose an orthogonalization-free framework of implementing the LOBPCG method for SCF (self-consistent field) iterations in solving the Kohn-Sham equation. In this framework, orthogonalization is avoided in calculations, which can decrease the computational complexity. And the R-R procedure is implemented parallelly through OpenMP, which can further reduce computational time. During numerical experiments, an effective preconditioning strategy is designed, which can accelerate the LOBPCG method remarkably. Consequently, the efficiency of the LOBPCG method can be significantly improved. Based on this, the SCF iteration can solve the Kohn-Sham equation efficiently. A series of numerical experiments are inducted to demonstrate the effectiveness of our implementation, in which significant improvements in computational time can be observed.

翻译：在LOBPCG方法的经典实现中，正交化及R-R（Rayleigh-Ritz）过程占用了不可忽略的CPU时间。当处理大规模块矩阵时，这一计算开销尤为昂贵。本文提出了一种基于避免正交化框架的LOBPCG方法实现，用于求解Kohn-Sham方程中的SCF（自洽场）迭代。在该框架中，计算过程中避免了正交化操作，从而降低了计算复杂度。同时，通过OpenMP并行实现R-R过程，进一步减少了计算时间。在数值实验中，我们设计了一种有效的预处理策略，能够显著加速LOBPCG方法，从而大幅提升其计算效率。基于此，SCF迭代可高效求解Kohn-Sham方程。通过一系列数值实验验证了该实现的有效性，实验结果显示出计算时间得到显著改善。