A spectrum slicing method for the Kohn–Sham problem

Solving the Kohn–Sham equation, which arises in density functional theory, is a standard procedure to determine the electronic structure of atoms, molecules, and condensed matter systems. The solution of this nonlinear eigenproblem is used to predict the spatial and energetic distribution of electronic states. However, obtaining a solution for large systems is computationally intensive because the problem scales super-linearly with the number of atoms. Here we demonstrate a divide and conquer method that partitions the necessary eigenvalue spectrum into slices and computes each partial spectrum on an independent group of processors in parallel. We focus on the elements of the spectrum slicing method that are essential to its correctness and robustness such as the choice of filter polynomial, the stopping criterion for a vector iteration, and the detection of duplicate eigenpairs computed in adjacent spectral slices. Some of the more prominent aspects of developing an optimized implementation are discussed.

► Divide and conquer method for the Kohn–Sham equation.
► New approach for scaling problem in large number of states.
► Heuristic for stopping iteration efficiently near convergence.
► Chebyshev–Jackson filtering for the Hermitian eigenproblem.
► Computational test demonstrates correctness of algorithm.