the sum of single-electron $\hat{W}_1$ and double-electron $\hat{W}_2$ operator components. Moreover, we do not take into account the Breit part in the potential of the electron-electron interaction and radiative quantum electrodynamics corrections, thus limiting ourselves to the description of the multi-electron atom within DHF approximation. However, we mention here that for large $Z$ these corrections become relevant and would have to be included for all models. We stress here that Eq. (\[eq:1\]) is the exact transformation of the Hamiltonian of the multi-electron atom, since we just added and subtracted the term proportional to $Z^{*}$. The eigenvalues of $\hat{H}$ define the values of the energy $E$ of a multi-electron atom.

The secondary-quantized wave function $\hat{\Psi}({\boldsymbol}r)$ is expanded in the complete Dirac hydrogen-like basis, which is dependent on $Z^{*}$ $$\hat{\Psi}({\boldsymbol}r) = \sum_\nu\left( \hat{a}_\nu \Psi_\nu^\epsilon(Z^{*},
    {\boldsymbol}r) + \hat{b}^\dag_\nu  \Psi_\nu^{-\epsilon}(Z^{*}, {\boldsymbol}r)\right), \label{eq:5}$$ with $\Psi^\epsilon_{\nu}(Z^{*}, {\boldsymbol}r)$ and $\Psi^{-\epsilon}_{\nu}(Z^{*}, {\boldsymbol}r)$ being the positive and negative energy eigenstates of the single-particle Dirac Hamiltonian respectively, identified by the set of collective quantum numbers $\nu=n_r l j m_{j}$ and $\nu = {\boldsymbol}p l j m_{j}$ for the discrete and continuous spectra correspondingly (actual expressions for the functions are given in Appendix \[sec:appendix-a.-details\]). The fermionic operators of positive and negative 