Extending the Scale with Real-Space Methods for the Electronic Structure Problem
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In principle, the electronic structure of a material can be determined by a solution of the many-body Schrödinger equation. This was first noted by Dirac shortly after the invention of quantum mechanics in 1929. However, Dirac also noted that the solution of the many-body quantum mechanical equations was much too difficult to be solved. He challenged his colleagues to develop “practical methods of applying quantum mechanics,” which can lead to an explanation of the main features of complex atomic systems.” In this chapter, we explore concepts and algorithms targeting “Dirac’s challenge.” Two key physical concepts will be employed: pseudopotential theory and density functional theory. For many weakly correlated systems, this formalism works well for ground-state properties such as phase stability, structural properties, and vibrational modes. However, applying this approach to large systems, e.g., systems with thousands of atoms, remains a challenge even with contemporary computational platforms. The goal of this chapter is to show how new algorithms can be used to extend computations to these systems. The approach centers on solving the nonlinear Kohn-Sham equation by a nonlinear form of the subspace iteration technique. This approach results in a significant speedup, often by more than an order of magnitude with no loss of accuracy. Numerical results are presented for nanoscale systems with tens of thousands of atoms and new methods are proposed to extend our work to even larger systems.
This work is supported by a subaward from the Center for Computational Study of Excited-State Phenomena in Energy Materials at the Lawrence Berkeley National Laboratory, which is funded by the US Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division under Contract No. DE-AC02-05CH11231, as part of the Computational Materials Sciences Program.
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