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Physical Review A

Equation satisfied by the energy-density functional for electron-electron mutual Coulomb repulsion

Daniel P. Joubert

Published Oct 14, 2011• 4 citations
Publisher

Abstract

It is shown that the electron-electron mutual Coulomb repulsion energy-density functional ${V}_{ee}^{\ensuremath{\gamma}}[\ensuremath{\rho}]$ satisfies the equation${V}_{ee}^{\ensuremath{\gamma}}[{\ensuremath{\rho}}_{N}^{1}]\ensuremath{-}{V}_{ee}^{\ensuremath{\gamma}}[{\ensuremath{\rho}}_{N\ensuremath{-}1}^{\ensuremath{\gamma}}]=\ensuremath{\int}{d}^{3}r\frac{\ensuremath{\delta}{V}_{ee}^{\ensuremath{\gamma}}[{\ensuremath{\rho}}_{N}^{1}]}{\ensuremath{\delta}{\ensuremath{\rho}}_{N}^{1}(\mathbf{r})}[{\ensuremath{\rho}}_{N}^{1}(\mathbf{r})\ensuremath{-}{\ensuremath{\rho}}_{N\ensuremath{-}1}^{\ensuremath{\gamma}}(\mathbf{r})],$where ${\ensuremath{\rho}}_{N}^{1}(\mathbf{r})$ and ${\ensuremath{\rho}}_{N\ensuremath{-}1}^{\ensuremath{\gamma}}(\mathbf{r})$ are $N$-electron and $(N\ensuremath{-}1)$-electron densities determined from the same adiabatic scaled external potential of the $N$-electron system at coupling strength $\ensuremath{\gamma}.$

DOI
10.1103/physreva.84.042508

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It is shown that the electron-electron mutual Coulomb repulsion energy-density functional Vee\ensuremathγ[\ensuremathρ]{V}_{ee}^{\ensuremath{\gamma}}[\ensuremath{\rho}]Vee\ensuremathγ​[\ensuremathρ] satisfies the equationVee\ensuremathγ[\ensuremathρN1]\ensuremath−Vee\ensuremathγ[\ensuremathρN\ensuremath−1\ensuremathγ]=\ensuremath∫d3r\ensuremathδVee\ensuremathγ[\ensuremathρN1]\ensuremathδ\ensuremathρN1(r)[\ensuremathρN1(r)\ensuremath−\ensuremathρN\ensuremath−1\ensuremathγ(r)],{V}_{ee}^{\ensuremath{\gamma}}[{\ensuremath{\rho}}_{N}^{1}]\ensuremath{-}{V}_{ee}^{\ensuremath{\gamma}}[{\ensuremath{\rho}}_{N\ensuremath{-}1}^{\ensuremath{\gamma}}]=\ensuremath{\int}{d}^{3}r\frac{\ensuremath{\delta}{V}_{ee}^{\ensuremath{\gamma}}[{\ensuremath{\rho}}_{N}^{1}]}{\ensuremath{\delta}{\ensuremath{\rho}}_{N}^{1}(\mathbf{r})}[{\ensuremath{\rho}}_{N}^{1}(\mathbf{r})\ensuremath{-}{\ensuremath{\rho}}_{N\ensuremath{-}1}^{\ensuremath{\gamma}}(\mathbf{r})],Vee\ensuremathγ​[\ensuremathρN1​]\ensuremath−Vee\ensuremathγ​[\ensuremathρN\ensuremath−1\ensuremathγ​]=\ensuremath∫d3r\ensuremathδ\ensuremathρN1​(r)\ensuremathδVee\ensuremathγ​[\ensuremathρN1​]​[\ensuremathρN1​(r)\ensuremath−\ensuremathρN\ensuremath−1\ensuremathγ​(r)],where \ensuremathρN1(r){\ensuremath{\rho}}_{N}^{1}(\mathbf{r})\ensuremathρN1​(r) and \ensuremathρN\ensuremath−1\ensuremathγ(r){\ensuremath{\rho}}_{N\ensuremath{-}1}^{\ensuremath{\gamma}}(\mathbf{r})\ensuremathρN\ensuremath−1\ensuremathγ​(r) are NNN-electron and (N\ensuremath−1)(N\ensuremath{-}1)(N\ensuremath−1)-electron densities determined from the same adiabatic scaled external potential of the NNN-electron system at coupling strength \ensuremathγ.\ensuremath{\gamma}.\ensuremathγ.
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