Large-scale electronic structure calculations

First-principles electronic structure calculations determine material properties using the fundamental equations of quantum mechanics. The computational bottleneck of these calculations is typically the self-consistent solution of Schrödinger (or Kohn-Sham) equations, which in turn require the repeated solution of large eigenvalue problems of dimension n = 10⁵-10⁶ for the lowest m = 10²-10³ eigenvalues and eigenvectors. Recently, a new method to solve large eigenvalue problems has been pioneered by A. Knyazev, the locally-optimal conjugate gradient method. Variants of this method will be discussed in the context of electronic structure calculations, focusing specifically on a real-space formulation of density-functional theory using a finite-element basis set.