Geometry of Degeneracy in Potential and Density Space
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866910319179202560 |
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| author | Penz, Markus van Leeuwen, Robert |
| author_facet | Penz, Markus van Leeuwen, Robert |
| contents | In a previous work [J. Chem. Phys. 155, 244111 (2021)], we found counterexamples to the fundamental Hohenberg-Kohn theorem from density-functional theory in finite-lattice systems represented by graphs. Here, we demonstrate that this only occurs at very peculiar and rare densities, those where density sets arising from degenerate ground states, called degeneracy regions, touch each other or the boundary of the whole density domain. Degeneracy regions are shown to generally be in the shape of the convex hull of an algebraic variety, even in the continuum setting. The geometry arising between density regions and the potentials that create them is analyzed and explained with examples that, among other shapes, feature the Roman surface. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_12366 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Geometry of Degeneracy in Potential and Density Space Penz, Markus van Leeuwen, Robert Quantum Physics Mathematical Physics Chemical Physics In a previous work [J. Chem. Phys. 155, 244111 (2021)], we found counterexamples to the fundamental Hohenberg-Kohn theorem from density-functional theory in finite-lattice systems represented by graphs. Here, we demonstrate that this only occurs at very peculiar and rare densities, those where density sets arising from degenerate ground states, called degeneracy regions, touch each other or the boundary of the whole density domain. Degeneracy regions are shown to generally be in the shape of the convex hull of an algebraic variety, even in the continuum setting. The geometry arising between density regions and the potentials that create them is analyzed and explained with examples that, among other shapes, feature the Roman surface. |
| title | Geometry of Degeneracy in Potential and Density Space |
| topic | Quantum Physics Mathematical Physics Chemical Physics |
| url | https://arxiv.org/abs/2206.12366 |