Convolution representation of the relation between total electron density and that of s states in closed-shell atoms

Motivated by the Coulomb-field result that the derivative of the density ρ(r) for an arbitrary number of closed shells is directly proportional to the s-state density ${\rho }_s$(r), we have explored for closed-shell atoms a convolution relation between ${\rho }_s$(r) and ∂ρ/∂r. This relation is most readily expressed in K space, and we thereby establish certain relations between the scattering factors f(K) and $f_s$(K) corresponding to total density and s density, respectively. The method is illustrated by using near-Hartree-Fock accuracy data of Clementi for closed-shell atoms Ne and Ar. For the Hartree-Fock theory, it is shown that at large r, ${\rho }_s$(r)∝$r^{\mathrm{-}4}$ρ(r). Use is made in the convolution representation of the electron-nuclear potential energy of the closed-shell atom and the second derivative ${\partial }^2$ρ/∂$r^2$ evaluated at the nucleus.