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A Variational Framework for Spectral Discretization of the Density Matrix in Kohn-Sham Density Functional Theory


Wang, Xin C. (2015) A Variational Framework for Spectral Discretization of the Density Matrix in Kohn-Sham Density Functional Theory. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/Z99021QK.


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.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Density functional theory; Kohn-Sham; Linear scaling methods; density matrix; spectral binning;
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Mechanical Engineering
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Bhattacharya, Kaushik (co-advisor)
  • Ortiz, Michael (co-advisor)
Thesis Committee:
  • Ravichandran, Guruswami (chair)
  • Bhattacharya, Kaushik
  • Ortiz, Michael
  • Minnich, Austin J.
Defense Date:5 January 2015
Record Number:CaltechTHESIS:04132015-160812309
Persistent URL:
Wang, Xin C.0000-0003-3854-4831
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:8819
Deposited By: Xin Wang
Deposited On:16 Apr 2015 00:05
Last Modified:04 Oct 2019 00:07

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