= Finite proton-shell correction to hydrogen ground energy
Model a <proton> as a uniformly charged thin <spherical shell> of radius $b$. Write $C=e^2/(4\pi\epsilon_0)$. The <shell theorem> gives an electron <potential energy> $-C/b$ inside the shell and $-C/r$ outside. Relative to a point <Coulomb potential>, the perturbation is consequently $C(1/r-1/b)$ for $r<b$ and zero otherwise.
For the <hydrogen atom> <ground state>, <time-independent perturbation theory> gives the <first-order energy correction>
$$
\Delta E=\frac{4C}{a_0^3}\int_0^b e^{-2r/a_0}\left(r-\frac{r^2}{b}\right)\,dr
=\frac{2Cb^2}{3a_0^3}+O\left(\frac{Cb^3}{a_0^4}\right).
$$
Since $E_{1s}=-C/(2a_0)$, the signed fractional change is $\Delta E/E_{1s}=-4b^2/(3a_0^2)+O((b/a_0)^3)$. The correction $\Delta E$ is positive: distributing the positive charge weakens the attraction near the origin and reduces the magnitude of the binding energy. Here $a_0$ is the <Bohr radius> and $b\ll a_0$ is the small-radius approximation.
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