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Corrections to Thomas-Fermi Densities at Turning Points and Beyond

Raphael F. Ribeiro, Donghyung Lee, Attila Cangi, Peter Elliott, and Kieron Burke
Phys. Rev. Lett. 114, 050401 – Published 4 February 2015
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    Abstract

    Uniform semiclassical approximations for the number and kinetic-energy densities are derived for many noninteracting fermions in one-dimensional potentials with two turning points. The resulting simple, closed-form expressions contain the leading corrections to Thomas-Fermi theory, involve neither sums nor derivatives, are spatially uniform approximations, and are exceedingly accurate.

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    • Received 23 September 2014

    DOI:https://doi.org/10.1103/PhysRevLett.114.050401

    © 2015 American Physical Society

    Authors & Affiliations

    Raphael F. Ribeiro1, Donghyung Lee2, Attila Cangi3, Peter Elliott3, and Kieron Burke1

    • 1Department of Chemistry, University of California, Irvine, California 92697, USA
    • 2Samsung SDI Inc., SMRC, Samsung-ro 130, Yeongtong-gu, Suwon-si, Gyeonggi-do 443-803, Republic of Korea
    • 3Max Planck Institute of Microstructure Physics, Weinberg 2, 06120 Halle (Saale), Germany

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    Vol. 114, Iss. 5 — 6 February 2015

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    Images

    • Figure 1
      Figure 1

      Thomas-Fermi (dashed) and semiclassical (dotted) approximations to the density (solid) of 2 particles in a Morse potential, v(x)=15(e−x/2−2e−x/4).

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    • Figure 2
      Figure 2

      Error in semiclassical density for N=2 (solid), and N=8 (dashed) in the Morse potential of Fig. 1.

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    • Figure 3
      Figure 3

      Integrated measure of error [Eq. (26)] in TF density multiplied by 0.1 (top) and semiclassical uniform approximation (bottom) for the Morse potential of Fig. 1.

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    • Figure 4
      Figure 4

      Thomas-Fermi (dashed), uniform semiclassical (dotted), and exact (solid) kinetic energy density for 2 particles in the Morse potential of Fig. 1. The value of π2[nsc(x)]3/6 is also shown (dot-dashed).

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    • Figure 5
      Figure 5

      Error (see text) in kinetic energy densities in the Morse potential of Fig. 1 with the semiclassical uniform approximation (squares), Thomas-Fermi theory (dots), and tloc[nsc] (rhombs).

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