Electric field of a hydrogen ground-state electron cloud ($Q_e(r)=-e[1-e^{-2r/a_0}(1+2r/a_0+2r^2/a_0^2)]$)

For spherical charge density ${\rho }_e=-e{e}^{-2r/a_0}/(\pi a_0^3)$, integrate $4\pi s^2{\rho }_e(s)$ from zero to $r$ to obtain the displayed enclosed charge. Gauss's law then gives $E_e=Q_e(r)/(4\pi {ϵ}_0{r}^2)$, directed inward. Near the origin $E_e\simeq -er/(3\pi {ϵ}_0{a}_0^3)$; at large radius it approaches the field of charge $-e$.

Adding a point nucleus of charge $+e$ leaves the unperturbed total fieldIt is exponentially small rather than identically zero, because the electron cloud has no compact boundary. This frozen-density field alone does not describe the dominant long-range response to another charge: charge-induced dipole attraction requires allowing the atom to polarize.