Pauli potential in terms of kinetic energy density and electron density in the leading Coulombic term of the nonrelativistic $1/Z$ expansion of spherical atomic ions

The Pauli potential $V_P$ in density functional theory is known to be the difference between the functional derivative of the single-particle kinetic energy $T_s[n]$ with respect to the electron density $n$ and its von Weizsäcker counterpart. For the leading Coulombic term in the $1/Z$ expansion for spherical atomic ions, $V_P[n]$ is written in terms of the kinetic energy density plus $n(r)$ and its low-order derivatives. For comparison, the example of an arbitrary number of closed shells with purely harmonic confinement is also treated.