Laplacian-level density functionals for the exchange-correlation energy of low-dimensional nanostructures

In modeling low-dimensional electronic nanostructures, the evaluation of the electron-electron interaction is a challenging task. Here we present an accurate and practical density-functional approach to the two-dimensional many-electron problem. In particular, we show that spin-density functionals in the class of meta-generalized-gradient approximations can be greatly simplified by reducing the explicit dependence on the Kohn-Sham orbitals to the dependence on the electron spin density and its spatial derivatives. Tests on various quantum-dot systems show that the overall accuracy is well preserved, if not even improved, by the modifications.

Figure 1

(Color online) (a) Comparison of the original (solid line) and modified (dashed line) kinetic-energy density of a spin-polarized three-electron parabolic $\left(\omega =1/4\right)$ quantum dot. (b) Local curvature of the exchange hole calculated from the original (solid line) and modified (dashed line) kinetic-energy density.

Figure 2

(Color online) Exchange-hole potentials of a spin-polarized three-electron parabolic $\left(\omega =1/4\right)$ quantum dot. (a) Result of the functionals in Ref. 9 without (solid line) and with (dashed line) the modification (see Sec. 2A). (b) The same as (a) for the functional in Ref. 11 (see Sec. 2B). The dotted line corresponds to the EXX-KLI result.

Figure 3

(Color online) (a) Original (solid lines) and modified (dashed lines) kinetic-energy densities for spin-up and spin-down electrons in a 48-electron parabolic $\left(\omega =0.3373\right)$ quantum dot at $B=3.05\text{T}$. The dotted lines show the spin densities. (b) Resulting spin-up and spin-down exchange-hole potentials using the functionals obtained from the modeling of the exchange hole (see Sec. 2A).

Figure 4

(Color online) Exchange-hole potentials of a 16-electron rectangular quantum dot calculated with the original (a) and modified (b) functionals obtained from the one-body spin-density matrix (see Sec. 2B) in comparison with the exact-exchange (Slater) potential (c).

Figure 5

(Color online) Spin-pair components of the correlation-hole potential for a spin-unpolarized six-electron parabolic $\left(\omega =1/4\right)$ quantum dot calculated using the functional obtained from the correlation-hole modeling (Ref. 13) (solid line), its modification described in Sec. 2C with $U_{x,\text{EXX}}^{\sigma }$ (obtained within the KLI approximation) as an input (dashed line), and with ${\tilde{U}}_{x,\text{model}}^{\sigma }$ as an input (dotted line).

Figure 6

(Color online) Error in the correlation energy of spin-polarized parabolic $\left(\omega =1/4\right)$ quantum dots with $N$ electrons obtained using different approximations. The circles correspond to the functional of Ref. 13 and the squares show the performance of the modified functional introduced in Sec. 2C, when the exact exchange-energy potential has been used in the expression. The triangles correspond to the case when also the exchange-energy potential has been used in a (similar) modified form.