Abstract: Kirzhnits method is extended in the gradient expansion of the Dirac density matrix of a many electron system to the two dimension case. The zeroth-order and second-order expansion for the Dirac density matrix in two dimensions is derived. From it, the expansion for the density and kinetic energy are further derived. The results show that, in contrast to the case of the three dimensions, to the second order of the gradient expansion, there is only the Laplacian type of term Δ²k_F², while the gradient term of (Δk_F²)² is absent in the two dimensional case. The result should be useful in the application of density functional theory to two dimensional systems.

Key words: Thomas-Fermi theory, Hohenberg-Kohn theory, Kohn-Sham method, kinetic energy density

摘要： 把传统的Kirzhnits方法扩展到了二维，获得了Dirac密度矩阵的零阶和二阶展开的表达式,进而得到密度和动能的直到二阶的梯度展开表达式。发现和三维情况不同的是,直到二阶的梯度展开的表达式,二维情况只有拉普拉斯类型的项Δ²k_F²,而梯度平方形式的项(Δk_F²)²是不存在的。

关键词: Thomas-Fermi理论, Hohenberg-Kohn理论, Kohn-Sham方法, 动能密度