Functional relations for the density-functional exchange and correlation functionals connecting functionals at three densities

Abstract

It is shown that the density-functional-theory exchange and correlation functionals satisfy $0=\ensuremath{\gamma}{E}_{hx}[{\ensuremath{\rho}}_{N}]+2{E}_{c}^{\ensuremath{\gamma}}[{\ensuremath{\rho}}_{N}]\ensuremath{-}\ensuremath{\gamma}{E}_{hx}[{\ensuremath{\rho}}_{N\ensuremath{-}1}^{\ensuremath{\gamma}}]\ensuremath{-}2{E}_{c}^{\ensuremath{\gamma}}[{\ensuremath{\rho}}_{N\ensuremath{-}1}^{\ensuremath{\gamma}}]+\phantom{\rule{0.16em}{0ex}}2\ensuremath{\int}{d}^{3}{r}^{\ensuremath{'}}[{\ensuremath{\rho}}_{N\ensuremath{-}1}^{0}(\mathbf{r})\ensuremath{-}\phantom{\rule{0.16em}{0ex}}{\ensuremath{\rho}}_{N\ensuremath{-}1}^{\ensuremath{\gamma}}(\mathbf{r})]{v}^{0}([{\ensuremath{\rho}}_{N}];\mathbf{r})+\phantom{\rule{0.16em}{0ex}}\ensuremath{\int}{d}^{3}{r}^{\ensuremath{'}}[{\ensuremath{\rho}}_{N\ensuremath{-}1}^{0}(\mathbf{r})\ensuremath{-}{\ensuremath{\rho}}_{N\ensuremath{-}1}^{\ensuremath{\gamma}}(\mathbf{r})]\mathbf{r}\ifmmode\cdot\else\textperiodcentered\fi{}\ensuremath{\nabla}{v}^{0}([{\ensuremath{\rho}}_{N}];\mathbf{r})+\ensuremath{\int}{d}^{3}{r}^{\ensuremath{'}}{\ensuremath{\rho}}_{N}(\mathbf{r})\mathbf{r}\phantom{\rule{0.16em}{0ex}}\ifmmode\cdot\else\textperiodcentered\fi{}\phantom{\rule{0.16em}{0ex}}\ensuremath{\nabla}{v}_{c}^{\ensuremath{\gamma}}([{\ensuremath{\rho}}_{N}];\mathbf{r})\phantom{\rule{0.16em}{0ex}}\ensuremath{-}\phantom{\rule{0.16em}{0ex}}\ensuremath{\int}{d}^{3}{r}^{\ensuremath{'}}{\ensuremath{\rho}}_{N\ensuremath{-}1}^{\ensuremath{\gamma}}(\mathbf{r})\mathbf{r}\ifmmode\cdot\else\textperiodcentered\fi{}\ensuremath{\nabla}{v}_{c}^{\ensuremath{\gamma}}([{\ensuremath{\rho}}_{N\ensuremath{-}1}^{\ensuremath{\gamma}}];\mathbf{r})\phantom{\rule{0.16em}{0ex}}\ensuremath{-}\phantom{\rule{0.16em}{0ex}}\ensuremath{\int}{d}^{3}{r}^{\ensuremath{'}}{f}^{\ensuremath{\gamma}}(\mathbf{r})\mathbf{r}\ifmmode\cdot\else\textperiodcentered\fi{}\ensuremath{\nabla}{v}_{hxc}^{\ensuremath{\gamma}}([{\ensuremath{\rho}}_{N}];\mathbf{r})\ensuremath{-}2\ensuremath{\int}{d}^{3}{r}^{\ensuremath{'}}{f}^{\ensuremath{\gamma}}(\mathbf{r}){v}_{hxc}^{\ensuremath{\gamma}}([{\ensuremath{\rho}}_{N}];\mathbf{r})$. In the derivation of this equation the adiabatic connection formulation is used, where the ground-state density of an $N$-electron system ${\ensuremath{\rho}}_{N}$ is kept constant independent of the electron-electron coupling strength $\ensuremath{\gamma}$. Here ${E}_{hx}[\ensuremath{\rho}]$ is the Hartree plus exchange energy, ${E}_{c}^{\ensuremath{\gamma}}[\ensuremath{\rho}]$ is the correlation energy, ${v}_{hxc}^{\ensuremath{\gamma}}[\ensuremath{\rho}]$ is the Hartree plus exchange-correlation potential, ${v}_{c}[\ensuremath{\rho}]$ is the correlation potential, and ${v}^{0}[\ensuremath{\rho}]\phantom{\rule{4pt}{0ex}}$is the Kohn-Sham potential. The charge densities ${\ensuremath{\rho}}_{N}$ and ${\ensuremath{\rho}}_{N\ensuremath{-}1}^{\ensuremath{\gamma}}$ are the $N$- and $(N\ensuremath{-}1)$-electron ground-state densities of the same Hamiltonian at electron-electron coupling strength $\ensuremath{\gamma}$. ${f}^{\ensuremath{\gamma}}(\mathbf{r})={\ensuremath{\rho}}_{N}(\mathbf{r})\ensuremath{-}{\ensuremath{\rho}}_{N\ensuremath{-}1}^{\ensuremath{\gamma}}(\mathbf{r})$ is the Fukui function. This equation can be useful in testing the internal self-consistency of approximations to the exchange and correlation functionals. As an example the identity is tested on the analytical Hooke's atom charge density for some frequently used approximate functionals.