Exchange-energy functionals for finite two-dimensional systems

Implicit and explicit density functionals for the exchange energy in finite two-dimensional systems are developed following the approach of Becke and Roussel [Phys. Rev. A 39, 3761 (1989)]. Excellent agreement for the exchange-hole potentials and exchange energies is found when compared with the exact-exchange reference data for the two-dimensional uniform electron gas and few-electron quantum dots, respectively. Thereby, this work significantly improves the availability of approximate density functionals for dealing with electrons in quasi-two-dimensional structures, which have various applications in semiconductor nanotechnology.

Figure 1

(a) Exchange hole and (b) the exchange energy per particle for two-dimensional electron gas $(\xi =0)$ in the exact expression (dashed line) and in the present model (solid line).

Figure 2

Exchange-hole potential of a two-electron quantum dot calculated exactly (dashed line) and using the implicit density functional of Sec. 3B (solid line), standard LSDA (dotted line), and the explicit density functional of Sec. 3C (dash-dotted line). The analytic density and orbitals are used as an input for the model and LSDA.

Figure 3

Similar to Fig. 2 but for 20 electrons. The exact-exchange result was calculated using the KLI approximation. Self-consistent (correlation included) density and orbitals from the standard LSDA are used as input for our functionals and LSDA exchange.