Finite nuclear size correction ($\Delta E_{ns}\simeq (2\pi /3)[Z{e}^2/(4\pi {ϵ}_0)]|{\psi }_{ns}(0){|}^2⟨r^2{⟩}_{\mathrm{ch}}$)

Spreading a point nuclear electric charge over a small spherical nuclear charge distribution weakens the central attraction and raises electronic energy levels. In nonrelativistic first-order time-independent perturbation theory, a slowly varying $s$-state contact density gives the displayed shift. For a uniform sphere, $⟨r^2{⟩}_{\mathrm{ch}}=3R^2/5$, and integration of the difference between spherical and point Coulomb potentials gives $\Delta E_{1s}=(4/5)Z^4\mathrm{Ry}(R/a_0{)}^2$ for a hydrogen-like ion. A nonrelativistic $p$ wavefunction vanishes at the origin, suppressing its leading shift to order $R^4$. Heavy atomic nuclei require relativistic and realistic-charge-distribution corrections. An observed transition frequency shifts by the difference of the two level corrections divided by the Planck constant.