Hellmann-Feynman, virial, and scaling requisites for the exact universal density functionals. Shape of the correlation potential and diamagnetic susceptibility for atoms

By the Hellmann-Feynman theorem, the density n(r) of many electrons in the presence of external potential v(r) obeys the relationships F $d^3$r n(r)∇v(r)=0 and F $d^3$r n(r)r×∇v(r)=0. By the virial theorem, the interacting kinetic and electron-electron repulsion expectation values obey 2T[n]+$V_{\mathrm{ee}}$[n]=-F $d^3$r n(r)r⋅∇[δT/δn(r)+δ$V_{\mathrm{ee}}$/δn(r)]. The exchange energy functional $E_x$[n] and potential $v_x$([n];r)≡δ$E_x$/δn(r) must satisfy $E_x$[n]+F $d^3$r n(r)r⋅∇$v_x$([n];r)=0, while the correlation energy and potential must satisfy $E_c$[n]+F $d^3$r n(r)r⋅∇$v_c$([n];r)<0. Somewhat counterintuitively, it is not true that T[$n_{\gamma }$]=${\gamma }^2$T[n] and $V_{\mathrm{ee}}$[$n_{\gamma }$]=γ$V_{\mathrm{ee}}$[n], where $n_{\gamma }$(r)≡${\gamma }^3$n(γr) is a scaled density with scale factor γ≠1. In fact, it is impossible to partition the exact Hohenberg-Kohn functional into a piece that scales as ${\gamma }^2$ and a piece that scales as γ, even if complete freedom with the partitioning is allowed. Instead there are universal scaling inequalities.