Past exam of the atomic and laser physics course of the University of Oxford / 2021 / 4 / e / ii

The finite nuclear size correction perturbation is unchanged for the same lead atomic nucleus. Its leading effect on one occupied $1s$ orbital is still proportional to that orbital's contact density,Other electrons screen the nuclear attraction and alter the self-consistent $1s$ wavefunction, usually reducing its contact density and hence its positive shift relative to the bare hydrogen-like ion. The reduction is modest for a deep $1s$ orbital: it penetrates well inside most other shells and still samples nearly the full $Z=82$ attraction. Replacing $Z$ by the neutral atom's asymptotic charge one would be inappropriate for this core orbital.

A neutral lead atom has two $1s$ electrons. Their contributions add in a frozen-orbital estimate of total atomic energy, roughly twice the per-electron shift, with relaxation and correlation corrections. An experimentally observed transition frequency again depends on the difference between the finite-size shifts of both participating many-electron quantum states. A quantitative neutral-lead result requires a screened relativistic many-electron calculation, not the one-electron formula with a single universally valid effective charge.