Leading gradient correction to the kinetic energy for two-dimensional fermion gases

Abstract

Density functional theory (DFT) is notorious for the absence of gradient corrections to the two-dimensional (2D) Thomas-Fermi kinetic-energy functional; it is widely accepted that the 2D analog of the 3D von Weizsäcker correction vanishes, together with all higher-order corrections. Contrary to this long-held belief, we show that the leading correction to the kinetic energy does not vanish, is unambiguous, and contributes perturbatively to the total energy. This insight emerges naturally in a simple extension of standard DFT, which has the effective potential energy as a functional variable on equal footing with the single-particle density.

Gradient correction and Bohm potential for two‐ and one‐dimensional electron gases at a finite temperature

“…In this regard, it is instructive to note that the Bohm potential is not a universal potential describing the non‐locality for arbitrary geometry (parameters), and therefore, using Equation , one always must check whether γ ≠ 0 for the present physical problem. Of course, the fact that γ 2 = 0 in the high‐frequency regime and in the static ground state ( ω = 0, θ → 0) of the 2D system does not mean that there is no quantum non‐locality at all . For the 2D system in the static case, van Zyl et al consistently introduced the non‐vanishing density gradient correction applying the average density approximation, which goes well beyond the LDA, and Trappe et al derived the gradient correction in terms of the effective potential for the DFT formulated as a joint functional of both the single‐particle density and the effective potential energy .…”

“…Of course, the fact that γ 2 = 0 in the high‐frequency regime and in the static ground state ( ω = 0, θ → 0) of the 2D system does not mean that there is no quantum non‐locality at all . For the 2D system in the static case, van Zyl et al consistently introduced the non‐vanishing density gradient correction applying the average density approximation, which goes well beyond the LDA, and Trappe et al derived the gradient correction in terms of the effective potential for the DFT formulated as a joint functional of both the single‐particle density and the effective potential energy . In Moldabekov et al , the relation between the second‐order functional derivative of the non‐interacting free energy functional and the dynamic polarization function in the RPA was established in the framework of the QHD model.…”

Systematic corrections to the Thomas–Fermi approximation without a gradient expansion

“…by the so-called leapfrog algorithm [21] (see also, e.g., [22]): figure 4). Clearly, this second-order leapfrog is the analog of the ST3 factorization in equation (9). There is also a leapfrog analog of ST3'.…”

“…This gradient expansion is notorious for its lack of convergence (see, e.g., [5]), the wrong sign of the von Weizsäcker term for one-dimensional systems [6], and the vanishing of all corrections for two-dimensional systems [6][7][8]-or so it seems [9]. While cures have been suggested, such as the use of a Padé approximant rather than the power series (see, e.g., [10]), or the partial resummation of the series with the aid of Airy averaging techniques [11][12][13][14][15], the situation is hardly satisfactory.…”

Julian Schwinger and the semiclassical atom

“…The puzzle disappears as soon as one recognizes that there are nonzero gradient corrections in the density-potential functional, and they can be evaluated perturbatively. 40 There is also an analog of the density-potential functionals in momentum space, where one gets an energy functional with the momentum-space density, the effective kinetic energy, and the chemical potential as independent variables. The hierarchy is repeated: We have the momentum-space version of the TF model 36 (which has succumbed to the power of mathematics 41 ), of the Scott-corrected TF model, 30 and of a model with exchange energy and gradient corrections included as well.…”