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Physical Review A

Upper bound for the kinetic energy for fermion systems

Gil Zumbach

Published Mar 1, 1985• 8 citations
Publisher

Abstract

For an N-fermion system, we prove the upper bound for the kinetic energy T(\ensuremath{\psi})\ensuremath{\le}[1+C(N/q${)}^{2/3}$] F ${d}^{3}$r[\ensuremath{\nabla}\ensuremath{\rightarrow} \ensuremath{\surd}\ensuremath{\rho}(r\ensuremath{\rightarrow}) ${]}^{2}$, where \ensuremath{\rho}(r\ensuremath{\rightarrow}) is the single-particle density, q the number of spin state, and C=15(4\ensuremath{\pi}${)}^{2(}$${3/5)((2/5))}^{2}$ $^{/3}$ in three dimensions.

DOI
10.1103/physreva.31.1922

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Abstract We prove an upper bound on the energy density of the dilute spin- 12\frac{1}{2}21​ 1 2 Fermi gas capturing the leading correction to the kinetic energy 8πaρ↑ρ↓8\pi a \rho _\uparrow \rho _\downarrow 8πaρ↑​ρ↓​ 8 π a ρ ↑ ρ ↓ with an error of size smaller than aρ2(a3ρ)1/3−εa\rho ^{2}(a^3\rho )^{1/3-\varepsilon }aρ2(a3ρ)1/3−ε a ρ 2 ( a 3 ρ ) 1 / 3 - ε for any ε>0\varepsilon > 0ε>0 ε > 0 , where a denotes the scattering length of the interaction. The result is valid for a large class of interactions including interactions with a hard core. A central ingredient in the proof is a rigorous version of a fermionic cluster expansion adapted from the formal expansion of Gaudin et al. (Nucl Phys A 176(2):237–260, 1971. https://doi.org/10.1016/0375-9474(71)90267-3 ).
Any proposed approximation to the ground-state kinetic energy of a system of noninteracting fermions in terms of the particle density \ensuremath{\rho}(r) can be used to estimate the potential field v(r) that will give rise to a given \ensuremath{\rho}(r), or the \ensuremath{\rho}(r) that will result from placing a given number of particles in a given v(r). Comparison with exact quantum-mechanical calculations thus gives two possible types of tests for any proposed kinetic-energy functional. This paper reports such tests for a recently proposed nonlocal functional applicable to one-dimensional systems, comparing its predictions for several simple problems not only with the correct answers but also with the predictions of Thomas-Fermi and Thomas-Fermi-Weizs\"acker functionals, to which the new functional proves considerably superior. The comparisons yield useful insights on the virtues and defects of the new functional, and on the directions in which improvements should be sought. Particular attention is devoted to Friedel oscillations and shell structure. It is shown that functionals of the Thomas-Fermi-Weizs\"acker type can never predict multiple maxima in \ensuremath{\rho} if v has only a single minimum and no maxima; the new functional does not have this defect, though it yields density oscillations rather weaker than the exact quantum-mechanical calculations. A natural general inference from the present tests is that any satisfactory kinetic-energy functional must (as ours does) replace the entire Thomas-Fermi term by a nonlocal expression.
From the explicit solution of a two-level problem previously obtained, it is proved that the single-particle kinetic energy density functional ts{t}_{s}ts​[\ensuremath{\rho}] is the density \ensuremath{\rho} times a function solely of the ratio \ensuremath{\rho}'/\ensuremath{\rho}, with \ensuremath{\rho}' the first derivative of \ensuremath{\rho}. For an n-level system with a harmonic-oscillator potential, the form of ts{t}_{s}ts​[\ensuremath{\rho}] is explicitly obtained. Again ts{t}_{s}ts​[\ensuremath{\rho}] involves only \ensuremath{\rho} and \ensuremath{\rho}'; no higher derivatives appear.