Real-space methods for electronic structure calculations of 100,000 atoms and beyond

With density functional theory and the use of pseudopotentials, the electronic structure problem can be effectively solved for many weakly coupled systems. However, computational cost of the Kohn–Sham equation is still a problem, frequently restricting the systems of interest to just a few thousand or fewer atoms. Here, we discuss novel methods that let us solve systems that contain more than 100,000 atoms. We concentrate on new computational algorithms based on real-space solutions of the Kohn-Sham equation. Our strategy has several benefits. First, the global communication required for fast Fourier transforms is avoided by real-space formalisms, such as finite differences and finite elements, which also provide superior scalability for big calculations across hundreds or thousands of computer nodes. Second, finite-difference techniques with a uniform real-space grid offer simple implementation; for instance, the grid spacing alone determines how quickly a Kohn-Sham solution converges. Based on a Chebyshev-filtered subspace iteration method, we developed a promising approach for solving the Kohn–Sham equations in real space [1–3]. We show with computations of confined systems with over 100,000 atoms or 400,000 electrons, that this method effectively reduces the communication overhead and improves the utilization of the vector processing capabilities provided by most modern parallel computers. We also describe our progress on even larger systems, surpassing 1,000,000 electrons.