The factor G$_{a}$ is defined by the quantum numbers of the valence state, G$_{a}$ = n $^{3}$(2 l + 1) j ( j + 1) 2( αZ ) $^{3}$m$_{j}$ , (73) where n is the principal quantum number, j and m$_{j}$ are the angular momentum and its projection, and l defines the parity of the state. We note that T$_{0}$ is the effective one-particle operator, which is employed in calculations of various contributions to the hyperfine splitting. The full Hamiltonian of the hyperfine interaction is the well-known Fermi-Breit o perator. To the first order in α and 1 /Z the screened QED correction x$_{SQED}$ to the hyperfine splitting is given by x$_{SQED}$ = x SE SQED + x VP $_{SQED}$, (74) x SE / VP SQED = Δ E SE / VP SQED /E$_{F}$ . (75) In Table IV the screened self-energy and vacuum-polarization corrections to the hyperfine splitting are presented for several values of Z in the range Z = 20 – 83 . The calculations are performed with the Fermi model for the finite nuclear charge distributi on. The finite nuclear magnetization distribution is introduced via an additional factor F ( r ) in the operator T$_{0}$ [47]. The individual contributions to x SE SQED and x VP SQED for lithiumlike bismuth $^{209}$Bi 80+ are presented in Table II in the Feynman and Coulomb gauges. Perfect agreement is found between the total results in the differ- ent gauges. This is also true for the other values of Z . We mention, however that the contribution of the screened vacuum-polarization diagram F , which is gauge invariant itself, is calculated in the Feynman gauge only. For the screened Wichmann-Kroll magnetic-loop part we have employed the hydrogenic 2 s value from Ref. [57], assuming that it enters with the same screening ratio as the Uehling terms. In Table V we present the total value of the hyperfine splittin g in bismuth in terms of the specific difference of the ground state hyperfine splitting in the H-l ike ion ( 1 s ) and in the Li-like ion ( 2 s ): Δ $^{′}$E = Δ E (2 s ) − ξ Δ E (1 s $^{)}$. It was proposed in Ref. [40] to consider this difference in order to overcome the problem of a large uncertainty of the Bohr-Weisskopf (BW) effect, originating from the nuclear magnetization distribution. The parameter ξ is chosen to cancel the BW correction, and the accuracy of the specific difference Δ $^{′}$E appears to be much higher than the accuracy of the splittings Δ E (1 s ) and Δ E (2 s ) themselves. The value of ξ = 0 . 16886 has been found for bismuth, taking into account the BW effect on all of the contributions, presented in Table V. For the related discussion we refer to our Letter [48]. We only mention here that the rms radius was taken to 18