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）过程会消耗不可忽略的CPU时间。尤其在处理大块尺寸问题时，这一消耗可能非常昂贵。本文针对求解Kohn-Sham方程的自洽场（SCF）迭代，提出了一种无正交化框架的LOBPCG方法实现。该框架避免了计算中的正交化步骤，从而降低了计算复杂度。同时，R-R过程通过OpenMP实现并行化，可进一步减少计算时间。在数值实验中，设计了一种有效的预处理策略，能显著加速LOBPCG方法，进而大幅提升其效率。基于此，SCF迭代能够高效求解Kohn-Sham方程。一系列数值实验验证了本实现的有效性，实验中可观察到计算时间的显著改善。