Comparative study of the gradient expansion of the atomic kinetic energy functional-neutral atoms

In a previous work, it has been shown that for the noble gas atoms, the ground state kinetic energy functional Tρ, where ρ is the electronic density, can be computed to surprising accuracy from the truncated gradient expansion: Tρ=T0ρ+T2ρ+T4ρ, with T0ρ =3/10(3π2) ℱρ5/3dτ, T2ρ=1/72ℱ (∇ρ)2/ρdτ, and T4ρ given by the formula of Hodges. In this work, we report systematic calculations for improved near-Hartree–Fock atomic densities up to Z=54. We find that the truncated gradient expansion of Tρ can also lead to very good kinetic energies for open-shell and unpaired-electron atoms. The value given by TρHF generally gets better, as compared with the Hartree–Fock result, as Z increases. For the Xe atom, T0+T2+T4 is within 0.08% of the exact Hartree–Fock kinetic energy with T2/T0=0.048, T4/T2=0.16. It is interesting to note that some minima appear in the deviation curves for T0, T0+T2, and T0+T2+T4, as compared with Hartree–Fock results. For the near-degenerate ground and lower excited states of some atoms, the truncated kinetic energy expansion can reproduce the relative kinetic energies qualitatively. On the other hand, because the cusp condition at the atomic nucleus and the exponentially vanishing density are valid for every Hartree–Fock density, it may not be possible to reproduce the Hartree–Fock or exact results with the gradient expansion in a systematic way.