Levy-Lieb-Based Monte Carlo Study of the Dimensionality Behaviour of the Electronic Kinetic Functional
We consider a gas of interacting electrons in the limit of nearly uniform density and treat the one dimensional (1D), two dimensional (2D) and three dimensional (3D) cases. We focus on the determination of the correlation part of the kinetic functional by employing a Monte Carlo sampling technique o...
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| Format: | Article |
| Language: | English |
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MDPI AG
2017-06-01
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| Series: | Computation |
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| Online Access: | http://www.mdpi.com/2079-3197/5/2/30 |
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| author | Seshaditya A. Luca M. Ghiringhelli Luigi Delle Site |
| author_facet | Seshaditya A. Luca M. Ghiringhelli Luigi Delle Site |
| author_sort | Seshaditya A. |
| collection | DOAJ |
| description | We consider a gas of interacting electrons in the limit of nearly uniform density and treat the one dimensional (1D), two dimensional (2D) and three dimensional (3D) cases. We focus on the determination of the correlation part of the kinetic functional by employing a Monte Carlo sampling technique of electrons in space based on an analytic derivation via the Levy-Lieb constrained search principle. Of particular interest is the question of the behaviour of the functional as one passes from 1D to 3D; according to the basic principles of Density Functional Theory (DFT) the form of the universal functional should be independent of the dimensionality. However, in practice the straightforward use of current approximate functionals in different dimensions is problematic. Here, we show that going from the 3D to the 2D case the functional form is consistent (concave function) but in 1D becomes convex; such a drastic difference is peculiar of 1D electron systems as it is for other quantities. Given the interesting behaviour of the functional, this study represents a basic first-principle approach to the problem and suggests further investigations using highly accurate (though expensive) many-electron computational techniques, such as Quantum Monte Carlo. |
| first_indexed | 2024-12-23T11:56:17Z |
| format | Article |
| id | doaj.art-edb1b616183a453cac58e61007035ea5 |
| institution | Directory Open Access Journal |
| issn | 2079-3197 |
| language | English |
| last_indexed | 2024-12-23T11:56:17Z |
| publishDate | 2017-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Computation |
| spelling | doaj.art-edb1b616183a453cac58e61007035ea52022-12-21T17:48:05ZengMDPI AGComputation2079-31972017-06-01523010.3390/computation5020030computation5020030Levy-Lieb-Based Monte Carlo Study of the Dimensionality Behaviour of the Electronic Kinetic FunctionalSeshaditya A.0Luca M. Ghiringhelli1Luigi Delle Site2Institute for Mathematics, Freie Universität Berlin, D-14195 Berlin, GermanyFritz-Haber Institute, Faradayweg 4-6, D-14195 Berlin, GermanyInstitute for Mathematics, Freie Universität Berlin, D-14195 Berlin, GermanyWe consider a gas of interacting electrons in the limit of nearly uniform density and treat the one dimensional (1D), two dimensional (2D) and three dimensional (3D) cases. We focus on the determination of the correlation part of the kinetic functional by employing a Monte Carlo sampling technique of electrons in space based on an analytic derivation via the Levy-Lieb constrained search principle. Of particular interest is the question of the behaviour of the functional as one passes from 1D to 3D; according to the basic principles of Density Functional Theory (DFT) the form of the universal functional should be independent of the dimensionality. However, in practice the straightforward use of current approximate functionals in different dimensions is problematic. Here, we show that going from the 3D to the 2D case the functional form is consistent (concave function) but in 1D becomes convex; such a drastic difference is peculiar of 1D electron systems as it is for other quantities. Given the interesting behaviour of the functional, this study represents a basic first-principle approach to the problem and suggests further investigations using highly accurate (though expensive) many-electron computational techniques, such as Quantum Monte Carlo.http://www.mdpi.com/2079-3197/5/2/30Levy-Lieb principleMonte Carlo sampling of electronskinetic-energy functionalsdimensionality |
| spellingShingle | Seshaditya A. Luca M. Ghiringhelli Luigi Delle Site Levy-Lieb-Based Monte Carlo Study of the Dimensionality Behaviour of the Electronic Kinetic Functional Computation Levy-Lieb principle Monte Carlo sampling of electrons kinetic-energy functionals dimensionality |
| title | Levy-Lieb-Based Monte Carlo Study of the Dimensionality Behaviour of the Electronic Kinetic Functional |
| title_full | Levy-Lieb-Based Monte Carlo Study of the Dimensionality Behaviour of the Electronic Kinetic Functional |
| title_fullStr | Levy-Lieb-Based Monte Carlo Study of the Dimensionality Behaviour of the Electronic Kinetic Functional |
| title_full_unstemmed | Levy-Lieb-Based Monte Carlo Study of the Dimensionality Behaviour of the Electronic Kinetic Functional |
| title_short | Levy-Lieb-Based Monte Carlo Study of the Dimensionality Behaviour of the Electronic Kinetic Functional |
| title_sort | levy lieb based monte carlo study of the dimensionality behaviour of the electronic kinetic functional |
| topic | Levy-Lieb principle Monte Carlo sampling of electrons kinetic-energy functionals dimensionality |
| url | http://www.mdpi.com/2079-3197/5/2/30 |
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