All-electron calculations play an important role in density functional theory, in which improving computational efficiency is one of the most needed and challenging tasks. In the model formulations, both the nonlinear eigenvalue problem and the total energy minimization problem pursue orthogonal solutions. Most existing algorithms for solving these two models invoke orthogonalization process either explicitly or implicitly in each iteration. Their efficiency suffers from this process in view of its cubic complexity and low parallel scalability in terms of the number of electrons for large scale systems. To break through this bottleneck, we propose an orthogonalization-free algorithm framework based on the total energy minimization problem. It is shown that the desired orthogonality can be gradually achieved without invoking orthogonalization in each iteration. Moreover, this framework fully consists of BLAS operations and thus can be naturally parallelized. The global convergence of the proposed algorithm is established. We also present a preconditioning technique which can dramatically accelerate the convergence of the algorithm. The numerical experiments on all-electron calculations show the effectiveness and high scalability of the proposed algorithm. 

　　Publication: 

　　SIAM Journal on Scientific Computing, 2022, Volume 44, Issue 3, pp: B723-B745. 

　　Author: 

　　Bin Gao 

　　ICTEAM Institute, UCLouvain, Louvain-la-Neuve, Belgium 

　　Email: gaobin@lsec.cc.ac.cn 

　　Guanghui Hu 

　　Department of Mathematics, University of Macau, Macao SAR, China; Zhuhai UM Science & Technology Research Institute, Zhuhai, Guangdong, China; Guangdong-Hong Kong-Macao Joint Laboratory for Data-Driven Fluid Mechanics and Engineering Applications, University of Macau, Macao SAR, China 

　　Yang Kuang 

　　School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou, Guangdong, China; Department of Mathematics, National University of Singapore, Singapore 

　　Xin Liu 

　　State Key Laboratory of Scientific and Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and University of Chinese Academy of Sciences, Beijing, China 

　　Email: liuxin@lsec.cc.ac.cn