Density-functional exchange correlation through coordinate scaling in adiabatic connection and correlation hole

The exact exchange-correlation functional ${\mathit{E}}_{\mathrm{xc}}$[n] must be approximated in density-functional theory for the computation of electronic properties. By the coupling-constant integration (adiabatic-connection) formula we know that ${\mathit{E}}_{\mathrm{xc}}$[n]=${\mathcal{F}}_0^1$(${\mathit{V}}_{\mathit{e}\mathit{e}}^{\mathrm{\alpha }}$[n]-U[n])dα, where ${\mathit{V}}_{\mathit{e}\mathit{e}}^{\mathrm{\alpha }}$[n] is the electron-electron repulsion energy of ${\mathrm{\Psi }}_{\mathit{n}}^{\mathrm{m}\mathrm{i}\mathrm{n},\mathrm{\alpha }}$, which is that wave function that yields the density n and minimizes 〈T^+αV${\mathrm{^}}_{\mathit{e}\mathit{e}}$〉. Here α is the coupling constant. Consequently, knowledge of the behavior of ${\mathit{V}}_{\mathit{e}\mathit{e}}^{\mathrm{\alpha }}$[n] as a function of α ensures knowledge of ${\mathit{E}}_{\mathrm{xc}}$[n]. With this in mind and for the purpose of approximating ${\mathit{E}}_{\mathrm{xc}}$, it was previously established that (∂${\mathit{V}}_{\mathit{e}\mathit{e}}^{\mathrm{\alpha }}$/∂α)≤0. The present paper reveals that ${\mathit{V}}_{\mathit{e}\mathit{e}}^{\mathrm{\alpha }}$[n]=α${\mathit{V}}_{\mathit{e}\mathit{e}}^1$[${\mathit{n}}_{1/\mathrm{\alpha }}$], where ${\mathit{n}}_{\mathrm{\beta }}$(x,y,z)=${\mathrm{\beta }}^3$n(βx,βy,βz), and where β is a coordinate scale factor.