Variational Calculation of the Hyperfine Stark Effect in Atomic ⁸⁷Rb, ¹³³Cs, and ¹⁶⁹Tm

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 wavefunctions of the electronic states involved in the respective hyperfine levels. These wavefunctions are obtained through relativistic general-excitation-rank configuration interaction theory. A variant of the method also enables for calculations on atoms with the most complicated of shell structures. Applications to $^{87}$Rb, $^{133}$Cs and a specific clock transition in $^{169}$Tm are presented. The final results $k_{\text{Rb}} = -1.234 \pm 0.0223$ [$10^{-10}$ Hz/((V/m)$^2$)] and $k_{\text{Cs}} = -2.347 \pm 0.084$ [$10^{-10}$ Hz/((V/m)$^2$)] obtained under inclusion of up to quintuple excitations in the atomic wavefunction expansion are compatible with previous calculations and, in the case of Cs, confirm that one of the earlier experimental measurements is not reliable. For $^{169}$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 $J=\frac{5}{2}$ is calculated to be $\Delta\alpha^s_0 = -0.134 \pm 0.11$ a.u. This result from a pure {\it{ab initio}} calculation confirms the result of $\Delta\alpha^s_0 = -0.063^{+0.01}_{-0.005}$ a.u. obtained in {\it{Nat. Comm.}} {\bf{10}} (2019) 1724 where a combination of measurement and theoretical modeling has been used.