Spatially dependent generalization of Kato’s theorem for atomic closed shells in a bare Coulomb field

For a bare Coulomb potential energy -${\mathrm{Ze}}^2$/r, it is shown that the total electron density ρ(r) for an arbitrary number of closed shells is given by ρ(r)=(2Z/$a_0$) ${\mathcal{F}}_r^{\infty }$ ${\rho }_s$(r)dr where ${\rho }_s$ is the s-state contribution to ρ(r). This yields Kato’s theorem [∂ρ(r)/∂r${]}_{r=0\mathrm{}}$=(-2Z/$a_0$)ρ (r=0) as a limiting case. That ∂ρ/∂r is always negative follows, for all distances r. For the nth closed shell, with density ${\rho }_n$(r), it is further shown that $R_{n0}$(r)=[(-$a_0$/2Z)∂${\rho }_n$/∂r] ${}_{}{}^{1/2}{}_{}{}^{}$, with $R_{n0}$ the s-state radial wave function. This result can be used to construct an explicit differential equation for ${\rho }_n$(r).