Exact condition on the non-interacting kinetic energy for real matter

From the analysis of the asymptotic expansion [1,2] for the total energies of neutral atoms, we suggest a modified gradient expansion approximation to the kinetic energy which satisfies the exact asymptotic condition as the number of electrons N $\rightarrow$ $\infty$. The resulting functional determines the small gradient limit of any generalized gradient approximation, and conflicts with the standard gradient expansion. We apply this new functional to the atoms up to Z $\sim$ 88 in comparison with the 2nd and 4th gradient expansion approximations. We also give a modern, highly accurate parametrization of the Thomas-Fermi density of neutral atoms.\newline \newline $[1]$ Thomas-Fermi model: The second correction, J. Schwinger, Phys. Rev. A {\bf 24}, 2353 (1981).\newline $[2]$ Semiclassical atom, B.-G. Englert and J. Schwinger, Phys. Rev. A {\bf 32}, 26 (1985).