we provide conclusions and outlook. Finally, the details of the calculations are summarized in Appendices \[sec:appendix-a.-details\]-\[sec:solut-relat-tf\] and values of the energies are given in Appendix \[sec:values-ground-state\] and the atomic units with $\hbar = m = e
= 1$ are employed through the paper.

Relativistic effective charge model {#sec:relat-effect-charge}
===================================

As was discussed in the introduction, we can write down the Hamiltonian of the relativistic multi-electron atom in the secondary-quantized representation, using the complete and orthonormal Dirac hydrogen-like basis set, with a single parameter — effective charge $Z^{*}$ — identical for all wave functions of the basis set. Consequently, the Hamiltonian of the system reads $$\begin{aligned}
  \hat{H}
  &= \hat{H}_0 + \hat{W} = \hat{H}_{0} + \hat{W}_{1} +
    \hat{W}_{2},\label{eq:1}
  \\
  \hat{H}_0
  &=\int \hat{\Psi}^\dag({\boldsymbol}{r}) \left(c{\boldsymbol}{\alpha} \cdot
    \hat{{\boldsymbol}{p}} + \beta c^{2} - \frac{Z^*}{|{\boldsymbol}{r}|}\right)
    \hat{\Psi}({\boldsymbol}{r})d{\boldsymbol}{r}, \label{eq:2}
  \\
  \hat{W}_1
  &=\int  \hat{\Psi}^\dag({\boldsymbol}{r}) \frac{Z^*-Z}{|{\boldsymbol}{r}|}
    \hat{\Psi}({\boldsymbol}{r})d{\boldsymbol}{r}, \label{eq:3}
  \\
  \hat{W}_2
  &= \frac{1}{2}\int \frac{\hat{\Psi}^\dag({\boldsymbol}{r})
    \hat{\Psi}^\dag({\boldsymbol}{r'}) \hat{\Psi}({\boldsymbol}{r})
    \hat{\Psi}({\boldsymbol}{r'})}{|{\boldsymbol}{r}-{\boldsymbol}{r}'|}
    d{\boldsymbol}{r}d{\boldsymbol}{r'}, \label{eq:4}\end{aligned}$$ where $c$ is the speed of light ($c = 1/\alpha = 137.035999084$), $Z$ the charge of the nucleus, $Z^{*}$ the effective charge, which will be determined later, $\hat{{\boldsymbol}p} = -{\mathrm{i}}\boldsymbol{\nabla}$ the momentum operator, ${\boldsymbol}\alpha$, $\beta$ the Dirac matrices and $\hat\Psi({\boldsymbol}r)$ the secondary-quantized wave function [@FeynmanB1998statistical]. Furthermore, we split the perturbation operator $\hat{W}$ into 