Two-dimensional limit of exchange-correlation energy functional approximations

We investigate the behavior of three-dimensional (3D) exchange-correlation energy functional approximations of density-functional theory in anisotropi c systems with two-dimensional (2D) character. Using two simple models, the quasi-2D electron gas and two-electron quantum dot, we show a fundamental limitation of the local density approximation (LDA) and its semilocal exten sions, generalized gradient approximation (GGA) and meta-GGA (MGGA), the mo st 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) a nd semilocal (GGA/MGGA) approximations to describe systems with 2D characte r in which the nature of the exchange-correlation hole is very nonlocal. No nlocal functionals provide an alternative approach, and explicitly the aver age density approximation 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 s tructures and materials such as graphite and metal surfaces. We also commen t on the implication of our findings to the practical device simulations ba sed on the (semi)local density-functional method.