We investigate the behavior of three-dimensional 3D exchange energy functional of density-functional theory in anisotropic systems with two-dimensional 2D character and 1D character. The local density approximation (LDA), the generalized gradient approximation (GGA), and the meta-GGA behave as functions of quantum well width. We use the infinite-barrier model (IBM) for the quantum well. In the first section, we describe the problem of three-dimensional exchange functional, in the second section we introduce the quasi-2D IBM system, in the third section we introduce the quasi-1D IBM system. Using that an exact-exchange functional provides the correct approach to the true two-dimensional limit, we want to show that the 2D limit can be considered as a constraint on approximate functionals. For the 1D limit case we also propose a new functional obtained with methods completely similar to those of 2D limit.

我们研究了密度泛函理论的三维 3D 交换能量泛函在具有二维二维特征和一维特征的各向异性系统中的行为。局部密度近似 (LDA)、广义梯度近似 (GGA) 和元 GGA 表现为量子阱宽度的函数。我们对量子阱使用无限势垒模型 (IBM)。第一部分描述三维交换泛函问题，第二部分介绍准二维IBM系统，第三部分介绍准一维IBM系统。使用精确交换泛函为真正的二维极限提供了正确的方法，我们想证明二维极限可以被视为对近似泛函的约束。