 c

Using a spherical Bohr-radius volume for the atom, the calculations below give a cutoff electromagnetic vacuum energy density of about $6.2\times 10^{15}\mathrm{J}{\mathrm{m}}^{-3}$, compared with a hydrogen binding-energy density of about $3.5\times 10^{12}\mathrm{J}{\mathrm{m}}^{-3}$. The cutoff vacuum estimate is about $1.8\times 10^3$ times larger. Alternative reasonable definitions of atomic volume change this ratio by a factor of a few, not the main order-of-magnitude conclusion.

The cutoff result is a bare zero-point mode sum. It is not a classical radiation bath incident on the atom, nor a calculation of an observable absolute vacuum energy without a subtraction or renormalization prescription.