We propose an unconditionally energy-stable, orthonormality-preserving, component-wise splitting iterative scheme for the Kohn-Sham gradient flow based model in the electronic structure calculation. We first study the scheme discretized in time but still continuous in space. The component-wise splitting iterative scheme changes one wave function at a time, similar to the Gauss-Seidel iteration for solving a linear equation system. Rigorous mathematical derivations are presented to show our proposed scheme indeed satisfies the desired properties. We then study the fully-discretized scheme, where the space is further approximated by a conforming finite element subspace. For the fully-discretized scheme, not only the preservation of orthogonality and normalization (together we called orthonormalization) can be quickly shown using the same idea as for the semi-discretized scheme, but also the highlight property of the scheme, i.e., the unconditional energy stability can be rigorously proven. The scheme allows us to use large time step sizes and deal with small systems involving only a single wave function during each iteration step. Several numerical experiments are performed to verify the theoretical analysis, where the number of iterations is indeed greatly reduced as compared to similar examples solved by the Kohn-Sham gradient flow based model in the literature.

翻译：我们针对电子结构计算中基于Kohn-Sham梯度流模型，提出了一种无条件能量稳定、保持正交归一性、分量分裂的迭代格式。首先研究时间离散但空间连续的格式。该分量分裂迭代格式每次只改变一个波函数，类似于求解线性方程组的高斯-赛德尔迭代。通过严格的数学推导证明了所提格式确实满足期望性质。随后研究全离散格式，其中空间进一步用协调有限元子空间逼近。对于全离散格式，不仅正交性和归一化（统称正交归一化）的保持性可沿用半离散格式的思路快速证明，而且格式的核心特性——即无条件能量稳定性——能得到严格证明。该格式允许使用大时间步长，并在每次迭代步骤中仅处理含单个波函数的小型系统。通过多个数值实验验证了理论分析，结果显示与文献中基于Kohn-Sham梯度流模型的同类算例相比，迭代次数显著减少。