  Based on the numerical method proposed in [G. Hu, X. Xie, F. Xu, J. Comput.
Phys., 355 (2018), 436-449.] for Kohn-Sham equation, further improvement on the
efficiency is obtained in this paper by i). designing a numerical method with
the strategy of separately handling the nonlinear Hartree potential and
exchange-correlation potential, and ii).parallelizing the algorithm in an
eigenpairwise approach. The feasibility of two approaches are analyzed in
detail, and the new algorithm is described completely. Compared with previous
results, a significant improvement of numerical efficiency can be observed from
plenty of numerical experiments, which make the new method more suitable for
the practical problems.
