Citation
Wang, Xin C. (2015) A Variational Framework for Spectral Discretization of the Density Matrix in KohnSham Density Functional Theory. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/Z99021QK. https://resolver.caltech.edu/CaltechTHESIS:04132015160812309
Abstract
KohnSham density functional theory (KSDFT) is currently the main workhorse 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 cubicscaling 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 linearscaling computational cost. A promising approach to achieving linearscaling 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 KohnSham 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 onedimensional benchmark problem, we present numerical experiments for which spectral binning exhibits excellent convergence characteristics and outperforms other linearscaling methods.
Item Type:  Thesis (Dissertation (Ph.D.))  

Subject Keywords:  Density functional theory; KohnSham; 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): 
 
Thesis Committee: 
 
Defense Date:  5 January 2015  
Record Number:  CaltechTHESIS:04132015160812309  
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:04132015160812309  
DOI:  10.7907/Z99021QK  
ORCID: 
 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  8819  
Collection:  CaltechTHESIS  
Deposited By:  Xin Wang  
Deposited On:  16 Apr 2015 00:05  
Last Modified:  04 Oct 2019 00:07 
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