singlet excitations in matrix form as:[@onidarubioreining; @cappellini2] $$\begin{aligned}
 \label{eq:3}
\nonumber
{\hat H}(v c\mathbf{k},v' c'\mathbf{k'})=[\varepsilon_c^{QP}(\mathbf{k})-\varepsilon_v^{QP}(\mathbf{k})]\delta_{v v'}\delta_{c c'}\delta_{\mathbf{k} \mathbf{k'}} \\
\nonumber
-\int d^{3}\mathbf{r} \int d^{3}\mathbf{r'} \varphi_{c\mathbf{k}}^{\ast}(\mathbf{r})\varphi_{c'\mathbf{k'}}(\mathbf{r})W(\mathbf{r},\mathbf{r'})\varphi_{v\mathbf{k}}(\mathbf{r'})\varphi_{v'\mathbf{k'}}^{\ast}(\mathbf{r'})\\
+2\int  d^{3}\mathbf{r}\int d^{3}\mathbf{r'} \varphi_{c\mathbf{k}}^{\ast}(\mathbf{r})\varphi_{v\mathbf{k}}(\mathbf{r}) \bar{v}(\mathbf{r},\mathbf{r'})\varphi_{c'\mathbf{k'}}(\mathbf{r'})\varphi_{v'\mathbf{k'}}^{\ast}(\mathbf{r'}),\end{aligned}$$ where matrix elements between KS or gKS wave functions of CB states and VB states occur. Contributions to Eq.(\[eq:3\]) which destroy particle number conservation have been omitted. The first term describes the noninteracting quasi-electron quasi-hole pairs. The second term accounts for the screened electron-hole Coulomb attraction with the statically screened Coulomb potential $W(\mathbf{r},\mathbf{r'})$. The third contribution, governed by the nonsingular part of the bare Coulomb interaction $\bar{v}(\mathbf{r},\mathbf{r'})$, represents the electron-hole exchange or crystal local-field effects.[@onidarubioreining; @roedl] The QP eigenvalues entering the first term and $W(\mathbf{r},\mathbf{r'})$ entering the second term have been directly extracted from the scQP-GW results. After diagonalization of the exciton matrix, or, more precisely, solving the homogeneous BSE or stationary two-particle Schrödinger equation with the eigenvalues $E_{\varLambda}$ and the eigenfunctions $A_{\varLambda}(v c\mathbf{k})$ of the pair states $\Lambda$, frequency dependent macroscopic dielectric function including excitonic effects can be written as: $$\begin{aligned}
 \label{eq:4}
\nonumber
\varepsilon_{\alpha\alpha}(\omega)&=&\delta_{\alpha\alpha}+\frac{16\pi e^{2}\hslash^{2}}{V}\sum_{\varLambda}\left\lvert \sum_{cv\mathbf{k}}\frac{\langle c\mathbf{k} \lvert v_{\alpha} \rvert v\mathbf{k}\rangle}{\varepsilon_c(\mathbf{k})-\varepsilon_v(\mathbf{k})}A_{\varLambda}(v c\mathbf{k}) \right\rvert^{2}\\
&\times& \left[ \frac{1}{E_{\varLambda}-\hslash(\omega+i\gamma)}+\frac{1}{E_{\varLambda}+\hslash(\omega+i\gamma)}  \right] \end{aligned}$$ where $v_{\alpha}$ is the corresponding Cartesian component of the single-particle velocity operator 