Density dependent embedding potentials for piecewise exact densities

In Frozen Density Embedding Theory (FDET) [T. A. Wesolowski, Phys. Rev. A 77, 012504 (2008)], the total N-electron density is represented as a sum of two components ρ1 and ρ2, where ρ1 is obtained from a Schrödinger-like N'-electron eigenvalue equation with N' < N and ρ2 is an arbitrary non-negative real function integrating with respect to N - N'. It is shown that the exact total ground-state electron density ρvo cannot be obtained from FDET even if ρ2 is piecewise equal to ρvo (i.e., equal to ρvo on some measurable volume element). The result is discussed in the context of subsystem approach in density functional theory, pseudopotential theory, and embedding potentials derived from inverted Kohn-Sham equation.