A Variational Framework for Spectral Discretization of the Density Matrix in Kohn-Sham Density Functional Theory
Kohn-Sham density functional theory (KSDFT) is currently the main work-horse of quantum mechanical calculations in physics, chemistry, and materials science. From a mechanical engineering perspective, we are interested in studying the role of defects in the mechanical properties in materials. In...
Summary: | Kohn-Sham density functional theory (KSDFT) is currently the main work-horse of quantum
mechanical calculations in physics, chemistry, and materials science. From a mechanical
engineering perspective, we are interested in studying the role of defects in the
mechanical properties in materials. In real materials, defects are typically found at
very small concentrations e.g., vacancies occur at parts per million,
dislocation density in metals ranges from $10^{10} m^{-2}$ to $10^{15} m^{-2}$,
and grain sizes vary from nanometers to micrometers in polycrystalline materials, etc. In order to model materials at
realistic defect concentrations using DFT, we would need
to work with system sizes beyond millions of atoms. Due to the cubic-scaling
computational cost with respect to the number of atoms in conventional DFT implementations, such system sizes are
unreachable. Since the early 1990s, there has been a huge interest in developing DFT
implementations that have linear-scaling computational cost. A promising
approach to achieving linear-scaling cost is to approximate the density matrix in
KSDFT. The focus of this
thesis is to provide a firm mathematical framework to study the convergence of
these approximations. We reformulate the Kohn-Sham density
functional theory as a nested variational problem in the density matrix,
the electrostatic potential, and a field dual to the electron density. The
corresponding functional is linear in the density matrix and thus amenable to
spectral representation. Based on this reformulation, we introduce a new
approximation scheme, called spectral binning, which does not require smoothing
of the occupancy function and thus applies at arbitrarily low temperatures. We
proof convergence of the approximate solutions with respect to spectral binning
and with respect to an additional spatial discretization of the domain. For a
standard one-dimensional benchmark problem, we present numerical experiments for
which spectral binning exhibits excellent convergence characteristics and
outperforms other linear-scaling methods. |
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