Exchange-correlation potentials for inhomogeneous electron systems in two dimensions from exact diagonalization: Comparison with the local-spin-density approximation

We consider electronic exchange and correlation effects in density-functional calculations of two-dimensional systems. Starting from wave-function calculations of total energies and electron densities of inhomogeneous model systems, we derive corresponding exchange-correlation potentials and energies. We compare these with predictions of the local-spin-density approximation and discuss its accuracy. Our data will be useful as reference data in testing, comparing, and parametrizing exchange and correlation functionals for two-dimensional electronic systems.

Figure 1

Relative errors in the LSDA energy terms (total, exchange-correlation, exchange, and correlation energies) for (a) two-electron systems in harmonic confinement and (b) two electrons in a hard-wall confinement as a function of the characteristic length scale of the system $1/\sqrt{\omega }$ or $R$.

Figure 2

Densities (a), exchange-correlation potentials (b), exchange potentials (c), and correlation potentials (d) as a function of the distance $r$ for two electrons in a spin singlet in harmonic confinement $\omega$, and (e)–(h) the same quantities for two electrons in the hard-wall confinement with radius $R$. For the latter, the results for $R=2,...,10$ are shown. Solid lines are the accurate results and dashed lines the LSDA ones. Thin dotted lines in (b), (c), (f), and (g) show the exact asymptotic $-1/r$ behavior of the exchange potential. Natural units determined by the external potential are used throughout.

Figure 3

The majority spin density (a) and majority spin exchange-correlation potential (b) as a function of the radius for four electrons with $(L,S)=(0,1)$ in harmonic confinement $\omega$, and (c), (d) the same quantities for the minority spin. Solid lines are the accurate results and dashed lines the LSDA ones. The thin dotted line shows the exact asymptotic $-1/r$ behavior of the exchange potential. HO units are used.

Figure 4

The electron density (a) and potential (b) as a function of the radius for six electrons with $(L,S)=(0,0)$ in harmonic confinement $\omega$. Solid lines are the accurate results and dashed lines the LSDA ones. The thin dotted line shows the exact asymptotic $-1/r$ behavior of the exchange potential. HO units are used.

Figure 5

The exchange-correlation potential (a) and the correlation potential (b) as a function of the local electron density for two electrons in spin singlet in harmonic confinement $\omega$ [see Figs. 2a, 2b, 2c, 2d], and (c), (d) the same quantities for the hard-wall confinement with radius $R$ [see Figs. 2e, 2f, 2g, 2h]. Panel (e) shows the same exchange-correlation potentials for six electrons with $(L,S)=(0,0)$ in harmonic confinement (see Fig. 4). Solid lines are the accurate results and dashed lines the LSDA ones. Natural units determined by the external potential are used throughout.