密度泛函理论中坐标标度与粒子数标度的相关性
Relevance of coordinate and particle-number scaling in density functional theory
摘要：
我们在密度泛函理论的背景下讨论了$β$依赖的电子密度标度族形式$n_λ(\R)=λ^{3β+1}\; n(λ^β\R)$.特别地，我们考虑了以下特殊情况：（$β=1/3$和$λ\gg 1$），这是至关重要的中性原子的半经典理论;均匀电子气标度（$β=-1/3$和$λ\gg 1$），这在金属团簇的半经典理论中是重要的;均匀密度标度（$β=0$），它可以与密度泛函理论中的自相互作用问题联系起来，当$λ\leq 1$ ;分数标度（$β=1$和$λ\leq 1$），这对原子和分子碎裂很重要;以及强相关标度（$β=-1$和$λ\gg 1$），这对描述强相关极限很重要。我们的工作结果提供了证据的重要性，这个家庭的标度在电子系统的半经典和量子理论，并表明，这些标度性质必须考虑作为重要的约束条件，在建设新的近似密度泛函。我们还表明，使用统一的电子气标度，金属团簇的曲率能与动力学和交换相关能的二阶梯度膨胀。
英文摘要：
We discuss a $β$-dependent family of electronic density scalings of the form $n_λ(\R)=λ^{3β+1}\; n(λ^β\R)$ in the context of density functional theory. In particular, we consider the following special cases: the Thomas-Fermi scaling ($β=1/3$ and $λ\gg 1$), which is crucial for the semiclassical theory of neutral atoms; the uniform-electron-gas scaling ($β=-1/3$ and $λ\gg 1$), that is important in the semiclassical theory of metallic clusters; the homogeneous density scaling ($β=0$) which can be related to the self-interaction problem in density functional theory when $λ\leq 1$; the fractional scaling ($β=1$ and $λ\leq 1$), that is important for atom and molecule fragmentation; and the strong-correlation scaling ($β=-1$ and $λ\gg 1$) that is important to describe the strong correlation limit. The results of our work provide evidence for the importance of this family of scalings in semiclassical and quantum theory of electronic systems, and indicate that these scaling properties must be considered as important constraints in the construction of new approximate density functionals. We also show, using the uniform-electron-gas scaling, that the curvature energy of metallic clusters is related to the second-order gradient expansion of kinetic and exchange-correlation energies.
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