Expansion of the density-functional energy components $E_c$ and $T_c$ in terms of moments of the electron density

Starting from known identities in density-functional theory, it is shown that for finite electronic systems the density functionals $E_c^{\lambda }\left[\rho \right]$ and $T_c^{\lambda }\left[\rho \right]$ can be expanded in terms of powers of the classical moments $〈x^q{y}^r{z}^s\rho 〉$ of the electron density, with the expansion coefficients of the two series related in a simple way. For neutral atoms and coupling constant $\lambda =1,$ two-term universal linear expansions are demonstrated to be reasonably accurate: $E_c\left[\rho \right]=-0.1659〈\rho 〉\ln Z+0.000\mathrm{}40〈r\rho 〉Z$ and $T_c\left[\rho \right]=0.1659\mathrm{}〈\rho 〉\ln Z-0.00080〈r\rho 〉Z,$ where Z is the atomic number. The factors $\ln Z$ and Z in these formulas are rationalized.