Leading corrections to local approximations

For the kinetic energy of one-dimensional model finite systems the leading corrections to local approximations as a functional of the potential are derived using semiclassical methods. The corrections are simple, nonlocal functionals of the potential. Turning points produce quantum oscillations leading to energy corrections, which are completely different from the gradient corrections that occur in bulk systems with slowly varying densities. Approximations that include quantum corrections are typically much more accurate than their local analogs. The consequences for density functional theory are discussed.

Figure 1

(Color online) Exact, TF, our semiclassical (sc) and WKB density for a single particle in $v\left(x\right)=-12{\text{sin}}^2\left(\pi x\right)$ with hard walls at $x=0$ and $x=1$ $\left(\hslash =m=1\right)$.

Figure 2

(Color online) TF and renormalized exact densities for $N^{\prime }=1$, 4, and 16 particles in $v\left(x\right)=-12{\text{sin}}^2\left(\pi x\right),0\le x\le 1$, showing approach to continuum limit.

Figure 3

(Color online) Exact and approximate ground-state KEDs for $v\left(x\right)=-12{\text{sin}}^2\left(\pi x\right)$, where $0\le x\le 1$. The lowest eigenvalue is ${ϵ}_0=-4.27$ and ${\mu }_{\text{sc}}=5.52$.

Figure 4

(Color online) LDA kinetic energy multiplied by the scale factor $\gamma$ for different $\gamma$ evaluated on $n_{\gamma }^{\text{TF}}\left(x\right)$, our semiclassical density $n_{\text{sc},\gamma }\left(x\right)$, and the exact density for a single-well potential $v\left(x\right)=-10{\text{sin}}^2\left(\pi x\right)$.

Figure 5

(Color online) Fractional error in density for $v\left(x\right)=-5{\text{sin}}^2\left(\pi x\right)$; only shown in interior (${\theta }_{\text{F}}/\pi \ge 0.7$ in left half).

Figure 6

(Color online) Fractional error in density close to the edge for $v\left(x\right)=-5{\text{sin}}^2\left(\pi x\right)$, ${\theta }_{\text{F}}/\pi \le 0.7$.

Figure 7

(Color online) Same as Fig. 5, for KED.

Figure 8

(Color online) Same as Fig. 6, for KED.

Figure 9

(Color online) Leading correction to the KED for $\gamma =1$, 0.5, and 0.1 and $v\left(x\right)=-10{\text{sin}}^2\left(\pi x\right)$.

Figure 10

(Color online) Exact and approximate ground-state densities for $v\left(x\right)=-27{\text{sin}}^2\left(\pi x\right)$, where $0\le x\le 1$. The lowest eigenvalue is ${ϵ}_0=-16.3$ and ${\mu }_{\text{sc}}=0.08$. The position of the turning points is indicated by dashed lines.

Figure 11

(Color online) Exact and approximate ground-state KEDs for $v\left(x\right)=-27{\text{sin}}^2\left(\pi x\right)$, where $0\le x\le 1$. The lowest eigenvalue is ${ϵ}_0=-16.3$ and ${\mu }_{\text{sc}}=0.08$. The position of the turning points is indicated by dashed lines.

Figure 12

Contour of integration in the complex $ϵ$ plane.