Harmonic analysis on isometry groups of objective structures and its applications to objective density functional theory.
2011-11
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Harmonic analysis on isometry groups of objective structures and its applications to objective density functional theory.
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2011-11
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Abstract
Objective structures (defined in James [2006]) generalize the notion of crystals and are
atomic/molecular structures in which all the constituent atoms/molecules of the structure
“see” the same environment up to orthogonal transformations and translations. It has been
conjectured [James, 2006] that the high degree of symmetry associated with these structures
can lead to interesting material properties such as ferromagnetism, ferroelectricity
and superconductivity. This provides a motivation to systematically study the electronic
properties of these structures and to formulate Density Functional Theory methods specifically
designed for objective structures. We term density functional methods/algorithms
designed for studying Objective Structures as Objective Density Functional Theory.
The purpose of this work is to serve as the first important step toward the formulation
and implementation of objective density functional theory. Keeping in mind, that density
functional theory methods designed for studying crystals obtain leverage out of the translational
symmetry of the underlying periodic system, the primary theoretical issue in formulating
objective density functional theory methods, becomes quantifying the effects of
(non-translational) symmetry on electronic structure computation. In this work, we borrow
ideas from abstract harmonic analysis/group representation theory, in order to understand
how the symmetry of objective structures generated by finite groups of isometries interacts
with the boundary value problems of Kohn-Sham density functional theory. To achieve
our goal, we first work through the formulation of a suitable group representation theory.
We then apply this representation theory to simplified versions of the boundary value problems associated with electronic structure calculation and we demonstrate how this results
in simplifications of those problems. Finally, we formulate symmetry adapted finite difference
and spectral schemes for numerical solution of the boundary value problems.
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University of Minnesota M.S. thesis. November 2011. Major: Aerospace engineering and mechanics. Advisor: Richard D. James. 1 computer file (PDF); vii, 97 pages, appendix A.
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Banerjee, Amartya Sankar. (2011). Harmonic analysis on isometry groups of objective structures and its applications to objective density functional theory.. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/119003.
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