Sparse representations and fast algorithms for Kohn-Sham orbitals

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In this work we explore the question of how to build localized, i.e. sparse after truncation with spatially localized support, basis functions for a subspace arising in Kohn-Sham Density Functional Theory (KSDFT) applied to insulators and semiconductors with Gamma point sampling. This begins with the development of a new framework and approach to the problem based on selecting columns of the density matrix, the spectral projector onto the relevant subspace. The algorithm is a simple and robust, direct method that avoids the non-convex optimization problem—and the inherent dependence on an initial guess—typically employed by existing methods. Furthermore, our algorithm does not depend on any adjustable parameters. In fact, the only adjustable parameters present pertain to the use of the SCDM after computation, e.g. at what value the basis functions should be truncated. Importantly, exponential decay of the density matrix in a real space representation directly translates to exponentially localized basis functions. The performance and behavior of the SCDM algorithm is demonstrated for silicon crystals and collections of water molecules. Building off of our new framework, we extend the SCDM algorithm in two directions. First, we develop a two stage approximate column selection strategy to find the important columns at much lower computational cost. The constructed basis qualitatively and quantitatively matches the behavior of the SCDM and the new two stage localization method can be more than 20 times faster than the original SCDM algorithm. Second, we generalize the SCDM method to KSDFT calculations with k-point sampling of the Brillouin zone, which is needed for more general electronic structure calculations for solids. Our new method, called SCDM-k, is by construction gauge independent and a natural way to describe localized orbitals. We also explore its performance on model problems in two and three dimensions.


Type of resource text
Form electronic; electronic resource; remote
Extent 1 online resource.
Publication date 2016
Issuance monographic
Language English


Associated with Damle, Anil
Associated with Stanford University, Institute for Computational and Mathematical Engineering.
Primary advisor Ying, Lexing
Thesis advisor Ying, Lexing
Thesis advisor Candès, Emmanuel J. (Emmanuel Jean)
Thesis advisor Lin, Lin
Advisor Candès, Emmanuel J. (Emmanuel Jean)
Advisor Lin, Lin


Genre Theses

Bibliographic information

Statement of responsibility Anil Damle.
Note Submitted to the Institute for Computational and Mathematical Engineering.
Thesis Thesis (Ph.D.)--Stanford University, 2016.
Location electronic resource

Access conditions

© 2016 by Anil Damle
This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).

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