parts of the PhD thesis work of Kamil Dzikowski to be submitted to the Heidelberg University, Germany.

Details of the calculation {#sec:appendix-a.-details}
==========================

Here, we present all formulas needed for the fully analytical evaluation of all integrals encountered in the calculation of the first-order correction to the energy of the system, as well as the computation of atomic scattering factors and electronic densities.

The key idea, allowing for the analytic calculation is the observation that both the basis wave functions and all appearing potentials can be split into their radial and angular components by simple expansion in spherical harmonics. Thus we get the hydrogen-like Dirac wave functions as [@flugge_practical_1971] $$\begin{aligned}
  \Psi_{n_{r}ljm}
  &({\boldsymbol}{r},Z^*) \nonumber
  \\
  &=
  \frac{1}{r} \left(
  \begin{array}{c}
    G_{n_{r},\kappa}(r,Z^*)\Omega_{\kappa,m}(\theta,\phi)
    \\
    F_{n_{r},\kappa}(r,Z^*)\Omega_{-\kappa,m}(\theta,\phi)
  \end{array}
  \right), \label{eq:28}\end{aligned}$$ with the angular part given by spherical harmonics $$\begin{aligned}
  \Omega_{\kappa,m}
  &(\theta,\phi) \nonumber
  \\
  &= \left(
  \begin{array}{c}
    \sqrt{\frac{1}{2} - \frac{m}{2\kappa+1}}
    Y_\kappa^{m-1/2}(\theta,\phi)
    \\
    -\frac{\kappa}{|\kappa|}\sqrt{\frac{1}{2} + \frac{m}{2\kappa+1}}
    Y_\kappa^{m+1/2}(\theta,\phi)
  \end{array}
  \right), \label{eq:29}\end{aligned}$$ and the radial part by the Whittaker functions $M_{\kappa,\mu}(z)$ of the first kind [@whittaker_expression_1903]

$$\begin{aligned}
    \begin{pmatrix}
      G_{n_r,\kappa}(Z^{*}, r)
      \\
      F_{n_r,\kappa}(Z^{*}, r)
    \end{pmatrix}
     = N\left\{\frac{n_r}{2\gamma}\left(
      \begin{array}{c}
        Z^*\alpha
        \\
        i (\gamma-\kappa)
      \end{array}\right) M_{\gamma+n_r,\gamma+\frac{1}{2}}(2 \chi  Z^*r)
    -\rho(1+2\gamma)\left(
      \begin{array}{c}
        \frac{\gamma-\kappa}{Z^* \alpha}
        \\
        i\end{array}
    \right) M_{\gamma+n_r,\gamma-\frac{1}{2}}(2 \chi
    Z^*r)\right\}, \label{eq:30}
  \end{aligned}$$

where $$\begin{aligned}
  \rho
  