Density dependent embedding potentials for piecewise exact densities

TL;DR

This paper analyzes the exact eigenvalue equations in Frozen Density Embedding Theory (FDET) when the embedded density matches the true ground-state density in some region, revealing inequalities and implications for subsystem DFT and embedding potentials.

Contribution

It provides a detailed analysis of the eigenvalue equations in FDET for densities equal to the exact ground-state density in certain regions, highlighting differences from the true density and resulting inequalities.

Findings

Stationary densities from FDET eigenvalue equations differ from the true density minus the embedded density.

An inequality $E^{HK}[ ho_1^{FDET}+ ho_2]> E^o_v$ is established for certain densities.

Results have implications for subsystem DFT, pseudopotential theory, and density-dependent embedding potentials.

Abstract

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.