  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 nonlinear eigenvalue
problem and 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
Basic Linear Algebra Subprograms (BLAS) operations and thus can be naturally
parallelized. The global convergence of the proposed algorithm is established.
We also present a precondition technique which can dramatically accelerate the
convergence of the algorithm. The numerical experiments on all-electron
calculations show the efficiency and high scalability of the proposed
algorithm.
