Density dependent embedding potentials for piecewise exact densities

Frozen Density Embedding Theory (FDET) [Wesolowski {\it Phys. Rev. A} {\bf 77}, 012504 (2008)] provides the interpretation of the eigenvalue equations for an embedded $N'$-electron wavefunction, in which the embedding operator is multiplicative, as the Euler-Lagrange equation corresponding to the constrained minimisation of the Hohenberg-Kohn energy functional. The constraint is given by a non-negative function integrating to an integer $N-N'>0$ with $N$ being the total number of electrons in the whole system ($\min_{\rho\rightarrow\forall_{\mathbf{r}}\big(\rho({\mathbf r})\ge \rho_2({\mathbf r}\big)} E^{HK}_v[\rho]=E^{HK}_v[\rho_1^{FDET}+\rho_2]\ge E^{HK}[\rho_v^{o}]=E^o_v$). The exact FDET eigenvalue equations are analysed for $\rho_2$ such that it is equal to the exact ground-state density $\rho_v^{o}({\mathbf r})$ in some measurable volume. It is shown that, the stationary ($\rho_1^{FDET}$) obtained from the FDET eigenvalue equations - if it exists - differs from $\rho_1^o=\rho_v^{o}-\rho_2$ leading to the sharp inequality $E^{HK}[\rho_1^{FDET}+\rho_2]> E^o_v$ for such densities $\rho_2$. The result is discussed in the context of subsystem DFT, pseudopotential theory, and exact density-dependent embedding potentials.