How the Piecewise-Linearity Requirement for the Density Affects Quantities in the Kohn-Sham System

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.