Density-functional formulas for atomic electronic energy components in terms of moments of the electron density

Reasoning from known identities arising from scaling considerations, formulas in terms of only the moments of the electron density, $M_k=〈r^k\rho 〉,$ are developed for all energy components in the density-functional theory of the ground states of atoms. In particular, employing Hartree-Fock data for the first 54 atoms and many of their ions, it is demonstrated that formulas of the form $T_S={\sum }_{k=1\mathrm{}}^4{A}_k{(M}_{-2/k}{)}^k$ and $E_x={\sum }_{k=1\mathrm{}}^4{C}_k{(M}_{-1/k}{)}^k$ fit the neutral atoms He through Xe with standard deviations of 0.26% and 0.70%, respectively.