Large two-dimensional electronic systems: Self-consistent energies and densities at low cost

We derive a self-consistent local variant of the Thomas-Fermi approximation for (quasi-) two-dimensional (2D) systems by localizing the Hartree term. The scheme results in an explicit orbital-free representation of the electron density and energy in terms of the external potential, the number of electrons, and the chemical potential determined upon normalization. We test the method over a variety 2D nanostructures by comparing to the Kohn-Sham 2D local-density approximation (LDA) calculations up to 600 electrons. Accurate results are obtained in view of the negligible computational cost. We also assess a local upper bound for the Hartree energy.

Figure 1

Hartree energy $E_H$ from a self-consistent 2D-LDA calculation with respect to our estimate for the upper bound $E_{\mathrm{bound}}$, i.e., the right-hand side of Eq. (4) with $C=\frac{16}{3\sqrt{\pi }}$. Results as a function of $N$ are shown for three different 2D systems.

Figure 2

(a) Electron densities in harmonic quantum dots containing $N=6$, 56, and 600 electrons, respectively. The dashed lines denote the DFT results within the 2D-LDA. The solid lines correspond to the results of the present orbital-free functional. (b) The same as in (a) but for two quantum rings containing $N=12$ and 38 electrons, respectively.

Figure 3

Relative error in the total energies of harmonic quantum dots calculated with our self-consistent scheme with respect to 2D-LDA results. The inset shows the obtained chemical potentials $\mu \left(N\right)=E_{\mathrm{tot}}\left(N\right)-E_{\mathrm{tot}}(N-1)$ for $N=55–59$ in comparison with the 2D-LDA.

Figure 4

Relative error in the total energies for circular and rectangular hard-wall quantum dots calculated with our self-consistent scheme with respect to 2D-LDA results.