Variational calculation of the hyperfine Stark effect in atomic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mmultiscripts> <mml:mi>Rb</mml:mi> <mml:mprescripts/> <mml:none/> <mml:mn>87</mml:mn> </mml:mmultiscripts> <mml:

An electronically variational approach to the calculation of atomic hyperfine structure transition energies under the influence of static external electric fields is presented. The method avoids the calculation of intermediate atomic states entirely and requires only the wave functions of the electronic states involved in the respective hyperfine levels. These wave functions are obtained through relativistic general-excitation-rank configuration-interaction theory. A variant of the method also enables calculations on atoms with the most complicated of shell structures. Applications to $^{87}\mathrm{Rb}, ^{133}\mathrm{Cs}$, and a specific clock transition in $^{169}\mathrm{Tm}$ are presented. The final results ${k}_{\text{Rb}}=\ensuremath{-}1.234\ifmmode\pm\else\textpm\fi{}0.023\phantom{\rule{0.28em}{0ex}}[{10}^{\ensuremath{-}10}\phantom{\rule{0.28em}{0ex}}\mathrm{Hz}/{(\mathrm{V}/\mathrm{m})}^{2}]$ and ${k}_{\text{Cs}}=\ensuremath{-}2.347\ifmmode\pm\else\textpm\fi{}0.084\phantom{\rule{0.28em}{0ex}}[{10}^{\ensuremath{-}10}\phantom{\rule{0.28em}{0ex}}\mathrm{Hz}/{(\mathrm{V}/\mathrm{m})}^{2}]$ obtained under inclusion of up to quintuple excitations in the atomic wave-function expansion are compatible with previous calculations and, in the case of Cs, confirm that one of the earlier experimental measurements is not reliable. For $^{169}\mathrm{Tm}$ that is used in the development of atomic clocks the differential static scalar electric dipole polarizability between ground levels $J=\frac{7}{2}$ and $\frac{5}{2}$ is calculated to be $\mathrm{\ensuremath{\Delta}}{\ensuremath{\alpha}}_{0}^{s}=\ensuremath{-}0.134\ifmmode\pm\else\textpm\fi{}0.122$ a.u. This result from a pure ab initio calculation supports the result of $\mathrm{\ensuremath{\Delta}}{\ensuremath{\alpha}}_{0}^{s}=\ensuremath{-}0.{063}_{\ensuremath{-}0.005}^{+0.01}$ a.u. obtained recently [A. Golovizin, E. Fedorova, D. Tregubov, D. Sukachev, K. Khabarova, V. Sorokin, and N. Kolachevsky, Nat. Commun. 10, 1724 (2019)] where a combination of measurement and theoretical modeling has been used.