= Electric field of a hydrogen ground-state electron cloud
{title2=$Q_e(r)=-e[1-e^{-2r/a_0}(1+2r/a_0+2r^2/a_0^2)]$}
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 $E_e=Q_e(r)/(4\pi\epsilon_0r^2)$, directed inward. Near the origin $E_e\simeq-er/(3\pi\epsilon_0a_0^3)$; at large radius it approaches the field of charge $-e$.
Adding a point nucleus of charge $+e$ leaves the unperturbed total field
$$
E_{\mathrm{atom}}(r)=\frac{e e^{-2r/a_0}}{4\pi\epsilon_0}\left(\frac1{r^2}+\frac2{a_0r}+\frac2{a_0^2}\right).
$$
It 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.
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