Exchange-correlation potential with a proper long-range behavior for harmonically confined electron droplets

The exchange-correlation potentials stemming from the local-density approximation and several generalized-gradient approximations are known to have incorrect asymptotic decay. This failure is independent of the dimensionality but so far the problem has been corrected—within the mentioned approximations—only in three dimensions. Here we provide a cured exchange-correlation potential for two-dimensional harmonically confined systems that cover a wide range of applications in quantum Hall and semiconductor physics, especially in quantum-dot modeling. The given potential is a generalized-gradient approximation and we demonstrate that it agrees very well with the analytic result of a two-electron quantum dot in the asymptotic regime, and yields plausible results for larger two-dimensional systems.

Figure 1

(Color online) Exchange-correlation potentials of a parabolic $\left(\omega =1\right)$ two-electron quantum dot calculated from the analytic density. The result from the present model (solid line) has been calculated with $\beta =0.01$ in Eq. (10).

Figure 2

(Color online) (a) Same as Fig. 1 but here calculated with several values for the parameter $\beta$ in Eq. (10). (b) Relative deviation of the integrated model potential from the exact result [see Eq. (16)] as a function of $\beta$ (note the logarithmic scale).

Figure 3

(Color online) (a) Exchange-correlation potentials of a 42-electron parabolic $\left(\omega =0.5\right)$ quantum dot calculated with the LDA (dashed line) and the present model (solid line). The dotted line shows the desired asymptotic behavior $\left(-1/r\right)$. (b) Quantities $x$ (solid line) and $F\left(x\right)$ (dashed line) for the same system. Note the different range in the $x$ axis in comparison with (a), and that the values for $F\left(x\right)$ have been multiplied by a factor ${10}^3$.

Figure 4

(Color online) (a) Exchange-correlation potentials of a two-electron parabolic $\left(\omega =0.5,r_0=3\right)$ quantum ring calculated with the LDA (dashed line) and the present model (solid line). The dotted line shows the desired asymptotic behavior $\left(-1/r\right)$. (b) The same as in figure (a) but for ten electrons.