Kohn-Sham (KS) density functional theory (DFT) is an extremely popular, in-principle exact method, which can describe any many-electron system by introducing an auxiliary system of noninteracting electrons with the same density. When the number of electrons, N, changes continuously, taking on both integer and fractional values, the density has to be piecewise-linear, with respect to N. In this article, I explore how the piecewise-linearity property of the exact interacting density is reflected in the KS system. In particular, I suggest to express KS quantities using the two-point Taylor expansion in N and find how the expansion coefficients are restricted by the piecewise-linearity requirement. Focus is given to the total electron density, the KS subdensities, and the highest occupied (HOMO) orbital density. In addition to exact analytical results, common approximations for the HOMO, namely, the frozen and the linear regimes, are analyzed. A numerical investigation using various exchange-correlation approximations is performed to test the analytical findings. The outcomes of this work will help to remove density-driven errors in DFT calculations for open systems and ensembles.

中文翻译：

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密度的分段线性要求如何影响 Kohn-Sham 系统中的物理量。


Kohn-Sham （KS） 密度泛函理论 （DFT） 是一种非常流行的原则上精确方法，它可以通过引入具有相同密度的非相互作用电子的辅助系统来描述任何多电子系统。当电子数 N 连续变化，同时具有整数和分数值时，密度相对于 N 必须是分段线性的。在本文中，我将探讨精确相互作用密度的分段线性特性如何反映在 KS 系统中。特别是，我建议使用 N 中的两点泰勒展开来表示 KS 量，并找出膨胀系数如何受到分段线性要求的限制。重点放在总电子密度、KS 亚密度和最高占据 （HOMO） 轨道密度上。除了精确的分析结果外，还分析了 HOMO 的常见近似值，即冻结状态和线性状态。使用各种交换相关近似进行数值调查以检验分析结果。这项工作的结果将有助于消除开放系统和集成 DFT 计算中的密度驱动误差。