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Computation 2017, 5(2), 30; doi:10.3390/computation5020030

Levy-Lieb-Based Monte Carlo Study of the Dimensionality Behaviour of the Electronic Kinetic Functional

1
Institute for Mathematics, Freie Universität Berlin, D-14195 Berlin, Germany
2
Fritz-Haber Institute, Faradayweg 4-6, D-14195 Berlin, Germany
These authors contributed equally to this work.
*
Author to whom correspondence should be addressed.
Academic Editor: Jianmin Tao
Received: 4 May 2017 / Revised: 5 June 2017 / Accepted: 8 June 2017 / Published: 10 June 2017
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Abstract

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. View Full-Text
Keywords: Levy-Lieb principle; Monte Carlo sampling of electrons; kinetic-energy functionals; dimensionality Levy-Lieb principle; Monte Carlo sampling of electrons; kinetic-energy functionals; dimensionality
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A., S.; Ghiringhelli, L.M.; Delle Site, L. Levy-Lieb-Based Monte Carlo Study of the Dimensionality Behaviour of the Electronic Kinetic Functional. Computation 2017, 5, 30.

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