We investigate the behavior of three-dimensional (3D) exchange-correlation
energy functional approximations of density functional theory in anisotropic
systems with two-dimensional (2D) character. Using two simple models, quasi-2D
electron gas and two-electron quantum dot, we show a {\it fundamental
limitation} of the local density approximation (LDA), and its semi-local
extensions, generalized gradient approximation (GGA) and meta-GGA (MGGA), the
most widely used forms of which are worse than the LDA in the strong 2D limit.
The origin of these shortcomings is in the inability of the local (LDA) and
semi-local (GGA/MGGA) approximations to describe systems with 2D character in
which the nature of the exchange-correlation hole is very nonlocal. Nonlocal
functionals provide an alternative approach, and explicitly the average density
approximation (ADA) is shown to be remarkably accurate for the quasi-2D
electron gas system. Our study is not only relevant for understanding of the
functionals but also practical applications to semiconductor quantum structures
and materials such as graphite and metal surfaces. We also comment on the
implication of our findings to the practical device simulations based on the
(semi-)local density functional method.Comment: 21 pages including 9 figures, to be published in Phys. Rev.
