Frequency shift of hyperfine transitions due to blackbody radiation

We have performed calculations of the size of the frequency shift induced by a static electric field on the clock transition frequencies of the hyperfine splitting in ${\mathrm{Yb}}^{+}$, Rb, Cs, ${\mathrm{Ba}}^{+}$, and ${\mathrm{Hg}}^{+}$. The calculations are used to find the frequency shifts due to blackbody radiation which are needed for accurate frequency measurements and improvements of the limits on variation of the fine-structure constant $\alpha$. Our result for Cs $[\delta \nu ∕E^2=-2.26\left(2\right)\times {10}^{-10}\mathrm{Hz}∕{(\mathrm{V}∕\mathrm{m})}^2]$ is in good agreement with early measurements and ab initio calculations. We present arguments against recent claims that the actual value might be smaller. The difference $(\sim 10\%)$ is due to the contribution of the continuum spectrum in the sum over intermediate states.