Collapse of the Electron Gas to Two Dimensions in Density Functional Theory

Local and semilocal density functional approximations for the exchange-correlation energy fail badly in the zero-thickness limit of a quasi-two-dimensional electron gas, where the density variation is rapid almost everywhere. Here we show that a fully nonlocal fifth-rung functional, the inhomogeneous Singwi-Tosi-Land-Sjölander (STLS) approach, which employs both occupied and unoccupied Kohn-Sham orbitals, recovers the true two-dimensional STLS limit and appears to be remarkably accurate for any thickness of the slab (and thus for the dimensional crossover). We also show that this good behavior is only partly due to the use of the full exact exchange energy.

Figure 1

Exchange-correlation energy per particle of an IBM quasi-2D electron gas of fixed 2D electron density ($r_s^{2\mathrm{D}}=2/\sqrt{3}$), as a function of the quantum-well thickness $L_{\max }/\lambda$ ($L_{\max }=4.44$). The 2D limit is from Eqs. (8, 9). Various density functional approximations have been used (LSDA, PBE-GGA, and HGGA), as well as the fifth-rung RPA and ISTLS. The exact exchange energy per particle ${\epsilon }_x$ is also plotted, for comparison. While the local and semilocal density functionals diverge in the 2D limit, RPA and ISTLS calculations nicely recover the corresponding xc energy of a 2D electron gas. The HGGA recovers the 2D exact exchange limit; i.e., as in the case of the LSDA and the PBE-GGA, the HGGA correlation energy per particle goes to zero in the 2D limit.

Figure 2

As in Fig. 1, but now for $r_s^{2\mathrm{D}}=4$ ($L_{\max }=15.39$).