A variational framework for spectral discretization of the density matrix in Kohn Sham density functional theory

摘要

Kohn-Sham density functional theory (KSDFT) is currently the main work-horse of quantumudmechanical calculations in physics, chemistry, and materials science. From a mechanicaludengineering perspective, we are interested in studying the role of defects in theudmechanical properties in materials. In real materials, defects are typically found at udvery small concentrations e.g., vacancies occur at parts per million,uddislocation density in metals ranges from $10^{10} m^{-2}$ to $10^{15} m^{-2}$,udand grain sizes vary from nanometers to micrometers in polycrystalline materials, etc. In order to model materials atudrealistic defect concentrations using DFT, we would needudto work with system sizes beyond millions of atoms. Due to the cubic-scalingudcomputational cost with respect to the number of atoms in conventional DFT implementations, such system sizes areudunreachable. Since the early 1990s, there has been a huge interest in developing DFTudimplementations that have linear-scaling computational cost. A promisingudapproach to achieving linear-scaling cost is to approximate the density matrix inudKSDFT. The focus of thisudthesis is to provide a firm mathematical framework to study the convergence ofudthese approximations. We reformulate the Kohn-Sham densityudfunctional theory as a nested variational problem in the density matrix,udthe electrostatic potential, and a field dual to the electron density. Theudcorresponding functional is linear in the density matrix and thus amenable toudspectral representation. Based on this reformulation, we introduce a newudapproximation scheme, called spectral binning, which does not require smoothingudof the occupancy function and thus applies at arbitrarily low temperatures. Weudproof convergence of the approximate solutions with respect to spectral binningudand with respect to an additional spatial discretization of the domain. For audstandard one-dimensional benchmark problem, we present numerical experiments forudwhich spectral binning exhibits excellent convergence characteristics andudoutperforms other linear-scaling methods.