(Z^*-Z)\sum_{k = 1}^{N} A_{\lambda_k}(Z^*) \label{eq:10}
  \\
  &+ \sum_{k<l = 1}^{N}
    \left(B_{\lambda_k,\lambda_l}^{\lambda_k,\lambda_l}(Z^*) -
    B_{\lambda_k,\lambda_l}^{\lambda_l,\lambda_k}(Z^*)\right),
    \nonumber\end{aligned}$$ where we have defined: $$\begin{aligned}
  A_{\lambda_{k}}
  &= \int
    \frac{|\Psi_{\lambda_{k}}^{\epsilon}({\boldsymbol}{r})|^2}{|{\boldsymbol}{r}|}
    d{\boldsymbol}{r}, \label{eq:11}
  \\
  B_{\lambda_3,\lambda_4}^{\lambda_1,\lambda_2}
  &=
    \int\frac{\Psi_{\lambda_1}^{\epsilon*}({\boldsymbol}{r})
    \Psi^{\epsilon*}_{\lambda_3}({\boldsymbol}{r}')
    \Psi_{\lambda_2}^{\epsilon}({\boldsymbol}{r})
    \Psi_{\lambda_4}^{\epsilon}({\boldsymbol}{r}')}{|{\boldsymbol}{r}-{\boldsymbol}{r}'|}
    d{\boldsymbol}{r} d{\boldsymbol}{r}', \label{eq:12}\end{aligned}$$ with the implied dependence on the effective charge ommited for simplicity. The $A_{\lambda_{k}}$ describes the contribution from the single-particle operator $\hat{W}_{1}$, while $B_{\lambda_3,\lambda_4}^{\lambda_1,\lambda_2}$ from the double-particle one $\hat{W}_{2}$. The last term in Eq. (\[eq:10\]) is the difference of the Coulomb and exchange integrals. Both $A_{\lambda_{k}}$ and $B_{\lambda_3,\lambda_4}^{\lambda_1,\lambda_2}$ can be evaluated analytically for a given set of occupation numbers (See Appendix \[sec:appendix-a.-details\]).

![(Color online) The relative difference between the values of ground state energies of the first 118 neutral atoms obtained via the effective charge model ($E^{(0)}$) and by solving the Dirac-Hartree-Fock equations ($E_{\mathrm{DHF}}$)[@DESCLAUX1973311].[]{data-label="accuracy"}](error_comparison.pdf){width="80mm"}

In addition, we mention here that fermionic operators of negative energies do not contribute to the energy of the system in the zeroth- and first-order as we are considering the corrections only to the electronic states, that is the states that do not contain $\mu_{i}$ occupation numbers. However, starting from second-order perturbation theory, the negative energy states will contribute to the observable characteristics, since there exist nonvanishing matrix elements due to the 