therefore, cannot be compared with the band structure and the DOS from experiments. We apply the many-body perturbation theory within Hedin’s GW approximation for the electronic self-energy operator.[@hedin] It can be demonstrated that one has to solve the QP equation:[@hl; @onidarubioreining; @cappellinissc92; @Stankovski] $$\begin{aligned}
 \label{eq:1}
\nonumber
 \left\{ -\frac{\hslash^{2}}{2m_{o}}\nabla_{\mathbf{r}}^{2}+\nu_{H}(\mathbf{r})+\nu_{ext}(\mathbf{r}) \right\}\Phi_k^{QP}(\mathbf{r})\\
+\int d^{3}\mathbf{r'}\varSigma(\mathbf{r},\mathbf{r'};\varepsilon_k^{QP})\Phi_k^{QP}(\mathbf{r'})=\varepsilon_k^{QP}\Phi_k^{QP}(\mathbf{r}).\end{aligned}$$ Equation (\[eq:1\]) has a similar structure as the KS equation of DFT. The fundamental difference is that the *XC* potential of DFT is replaced by the non-Hermitian, nonlocal energy dependent self-energy operator $\varSigma$ resulting in QP eigenvalues $\epsilon_k^{QP}$ and QP wave functions $\Phi_k^{QP}$. Due to the fact that the Hamiltonian is non-Hermitian, QP eigenvalues are usually complex numbers. The real part describes resonance energies (excitation energies) and the imaginary part defines the lifetime of excited states. Equation (\[eq:1\]) should be solved self-consistently; however, the most common GW approach avoids this procedure. Since empirically one often finds that QP wave functions are similar to the KS DFT-LDA ones,[@hl] it is natural to take them as a starting point for a GW calculation and, hence, to calculate QP corrections in the sense of a perturbation treated within first-order perturbation theory.[@hl; @onidarubioreining; @cappellinissc92] The following equation, $$\label{eq:2}
\varepsilon_k^{QP}=\varepsilon_k+\langle  \varphi_{k} \lvert \varSigma(\epsilon_k^{QP})-\nu_{XC} \rvert 