Quantum oscillations in the kinetic energy density: Gradient corrections from the Airy gas

We derive a closed-form expression for the quantum corrections to the kinetic energy density in the Thomas-Fermi limit of a linear potential model system in three dimensions (the Airy gas). The universality of the expression is tested numerically in a number of three-dimensional model systems: (i) jellium surfaces, (ii) confinement in a hydrogenlike potential (the Bohr atom), (iii) particles confined by a harmonic potential in one and (iv) all three dimensions, and (v) a system with a cosine potential (the Mathieu gas). Our results confirm that the usual gradient expansion of extended Thomas-Fermi theory does not describe the quantum oscillations for systems that incorporate surface regions where the electron density drops off to zero. We find that the correction derived from the Airy gas is universally applicable to relevant spatial regions of systems of types (i), (ii), and (iv), but somewhat surprisingly not (iii). We discuss possible implications of our findings to the development of functionals for the kinetic energy density.

Figure 1

The ETF GE and the AG-GE in Eq. (31) compared to the exact KED for (a), (b) the AG and (c), (d) a jellium surface with $r_s=2$. The AG system shown in (a) and (b) is the system used to derive the AG-GE. These panels are placed side by side with the corresponding application of the expressions in (c), (d) the jellium system for comparison. (a) The behavior far inside the AG surface $\left(\zeta \to -\infty \right)$ as a function of the scaled coordinate $\zeta$. The GE derived from the AG unsurprisingly describes the oscillations in the KED well, whereas the ETF GE does not. (b) The behavior in the surface region of the AG. The vertical line shows the position of the classical turning point. The inset shows the difference between the two expansions and the exact KED. (c) The regime of slowly varying electron density at large negative $z$ of the jellium model. Also for the jellium system, the AG-GE describes the oscillations in the KED well, whereas the ETF GE fails to do so. This closely mimics the behavior seen for the AG in (a). (d) The surface region of the jellium model. In both (b) and (d), the ETF GE deviates less than the AG-GE from the exact KED over the surface region.

Figure 2

The ETF GE and the AG-GE in Eq. (31) compared to the exact KED for the (a), (b) isotropic harmonic oscillator and (c), (d) Bohr atom. Both systems show the same characteristics as were observed in Fig. 1. The AG-GE accurately describes the oscillations where the density is slowly varying, i.e., $s$ and $q$ are small [shown in (a) and (c)], whereas ETF fails to do so. However, across the edge [shown in (b) and (d)], the error in ETF GE is generally smaller than for the AG-GE.

Figure 3

Convergence of the ETF GE and the AG-GE in Eq. (31) in an arbitrarily chosen single point for the (a) isotropic HO and (b) Bohr atom. (a) The isotropic HO at $r_0=0.2$ for the curvature parameter $w\to 0$, i.e., as the system fills up with more particles. Even at moderate values of $w$, the AG-GE reproduces the exact KED with great accuracy, whereas the ETF GE does not. (b) The Bohr atom at the arbitrarily chosen point $r_0=0.5$ as the number of filled shells $N_{\mathrm{shell}}$ increases. Similar to (a), the AG-GE accurately reproduces the KED in the limit of large $N_{\mathrm{shell}}$, but the ETF GE does not.

Figure 4

The parameter space of the Mathieu gas. The shaded area is defined by values of the chemical potential $\mu$ that belong to one of the possible energy bands of the energy spectrum in the $z$ dimension. The light area is defined by values of the chemical potential $\mu$ in the FE continuum in $x$ and $y$ between the bands. Yellow lines correspond to values of the chemical potential situated precisely on the band edges of the energy spectrum in the $z$ dimension. The straight lines in the parameter space correspond to different paths along which one can approach the limit of slowly varying electron density in the MG. The black line splits the parameter space into two distinct regions. The blue dashed line indicates a path to the limit of slowly varying density for which $\overline{\lambda }<1/2$, and the red dashed line indicates a path to the limit of slowly varying density for which $\overline{\lambda }>1/2$.

Figure 5

The ETF GE and the AG-GE in Eq. (31) compared to the exact KED for three different MG systems as a function of scaled spatial coordinate $\overline{z}$. (a), (b) The MG with $\overline{\lambda }=0.1,\overline{p}=0.02$, which has no forbidden regions and thus is FE-like. The ETF GE accurately describes the KED, whereas the AG-GE fails to do so. (c), (d) The MG with $\overline{\lambda }=0.5,\overline{p}=0.015$. In this MG, the chemical potential tangents the top of the potential. The ETF GE describes the system fairly well, but there appears to be a minor discrepancy in the amplitude of the oscillations. The AG-GE gives a description that is severely downshifted as compared to the exact KED. (e), (f) The MG with $\overline{\lambda }=0.9,\overline{p}=0.02$, where the chemical potential is far below the maximum values of the cosine potential, i.e., a system with an infinite number of classically forbidden regions. Both the ETF GE and AG-GE have oscillations that are similar to the exact KED. However, the oscillations in ETF appear to be of too small amplitude, whereas the AG-GE is shifted down relative to the exact KED.

Figure 6

Convergence of the ETF GE and the AG-GE in Eq. (31) in one single arbitrarily chosen spatial point for MG systems $\left(\overline{z}=0.01\right)$ with (a)–(c) different parameters and (d) in the HG for $\overline{z}=0$. (a) The convergence as the scaled wave vector $\overline{p}$ approaches zero for a MG with $\overline{\lambda }=0.1$. In this MG, there are no classical turning points and the ETF GE appears to reproduce the exact KED well, whereas the AG-GE generally fails. (b) A MG with $\overline{\lambda }=0.5$ for which the chemical potential tangents the top of the potential. (c) A MG with $\overline{\lambda }=0.9$ where the chemical potential is far below the maximum values of the cosine potential, i.e., a system with an infinite number of classical turning points. Both the ETF GE and the AG-GE have oscillations that are similar to the exact KED in (b) and (c). However, the oscillations in ETF are of too small amplitude, whereas the AG-GE appears to be shifted down. (d) The HG in the limit of a curvature parameter $w\to 0$. The general behavior in the MG with large $\overline{\lambda }$ in (c) and the HG in (d) are very similar.

Figure 7

The HG filled up to the 30th energy level in the $z$-dimension energy spectra. (a) The exact KED for this system compared to the ETF GE and the AG-GE in Eq. (31) as a function of the scaled coordinate $\overline{z}$. The ETF GE underestimates the amplitude of the exact oscillation, whereas the AG-GE has a relative offset compared to the amplitude. (b) The surface region of the same system, where the vertical line is the classical turning point. The AG-GE has larger errors than the ETF GE in this surface region.

Figure 8

Exact KED for a jellium surface $\left(r_s=1\right)$ compared to the KEDs of other AG-based functionals (Baltin [25, 26], Vitos et al. [23], and the $A_{1/6}$ functional of Constantin and Ruzsinszky [24]). (a) The limit of a slowly varying density. The expression by Baltin lacks a term proportional to the Laplacian and has oscillations of much less amplitude than the other functionals. The functional by Vitos et al. and the $A_{1/6}$ both somewhat underestimate the amplitude of the oscillations, whereas the AG-GE of Eq. (31) reproduces the amplitude correctly. (b) The surface region of the same system. The KED is well described by the functionals by Vitos et al. and Constantin and Ruzsinszky. The AG-GE and the expression by Baltin both significantly deviate from the exact KED in this region.

Figure 9

Contour plot that shows $s^2$ vs $q$ over all values of $r$ for the Bohr atoms filled up to and including the third shell. As the shell variable $N_{\mathrm{shell}}$ is increased, the number of loops increases and the curve approaches the origin, which demonstrates how the limit of slowly varying density is reached in this system within an intermediate region of $r$ values.

Figure 10

Contour plot of the scaled gradient $s^2$ vs the scaled Laplacian $q$ in the limit of slowly varying density for the case of a jellium surface $\left(r_s=1\right)$ as $z\to -\infty$ and a Mathieu gas $\left(\overline{\lambda }=0.1\right)$ as $\overline{p}\to 0$.