# Optical Response of an Interacting Polaron Gas in Strongly Polar Crystals
## 3 Semianalytic Approximations
In the present calculation, we use the model of independent oscillators [25; 26] which correspond to the LO- and TO-phonon modes:  
\[\varepsilon_{L}\left(\mathbf{q},\Omega\right)=\varepsilon_{\infty}\prod_{j=1 }^{n}\left(\frac{\Omega^{2}-\omega_{L,j}^{2}\left(\mathbf{q}\right)}{\Omega^ {2}-\omega_{T,j}^{2}\left(\mathbf{q}\right)}\right). \tag{23}\]  
The extended RPA thus takes into account the dynamic screening of the Coulomb electron-electron interaction by the lattice polarization. The resulting retarded density-density Green's function is:  
\[G^{R}\left(\mathbf{q},\Omega\right)=\frac{\hbar VP^{\left(1\right)}\left( \mathbf{q},\Omega\right)}{1-\frac{4\pi e^{2}}{\varepsilon_{L}\left(\mathbf{q},\Omega\right)q^{2}}P^{\left(1\right)}\left(\mathbf{q},\Omega\right)}, \tag{24}\]  
where \(P^{\left(1\right)}\left(\mathbf{q},\Omega\right)\) is the Lindhard polarization function,  
\[P^{\left(1\right)}\left(\mathbf{q},\Omega\right)=\frac{1}{V}\sum_{\mathbf{k}, \sigma,\lambda}\frac{f_{F}\left(\varepsilon_{\lambda}\left(\mathbf{k}+ \mathbf{q}\right)-\mu_{\lambda}\right)-f_{F}\left(\varepsilon_{\lambda}\left( \mathbf{k}\right)-\mu_{\lambda}\right)}{\hbar\Omega+\varepsilon_{\lambda} \left(\mathbf{k}+\mathbf{q}\right)-\varepsilon_{\lambda}\left(\mathbf{k} \right)+i0^{+}} \tag{25}\]  
with the Fermi distribution function \(f_{F}\left(\varepsilon\right)\):  
\[f_{F}\left(\varepsilon\right)=\frac{1}{e^{\beta\left(\varepsilon-\mu\right)}+ 1}. \tag{26}\]  
The function \(\mathcal{G}\left(\mathbf{q},\Omega\right)\) is obtained from \(G^{R}\left(\mathbf{q},\Omega\right)\) using the analytic identity,  
\[\left(1-e^{-\beta\hbar\Omega}\right)\text{Im}\,\mathcal{G}\left(\mathbf{q}, \Omega\right)=\text{Im}\,G^{R}\left(\mathbf{q},\Omega\right), \tag{27}\]  
and then the Kramers-Kronig dispersion relation for \(\text{Re}\,\mathcal{G}\left(\mathbf{q},\Omega\right)\). For a comparison with experiment, the Green's functions are calculated here accounting for damping within the Mermin-Lindhard approach [27; 28; 29; 30] (where the damping is introduced in such a way to conserve the local electron number). This leads to the modification of the retarded density-density Green's function as follows:  
\[V_{\mathbf{q},j}=\frac{\hbar\omega_{L,j}}{q}\left(\frac{4\pi\alpha_{j}}{V} \right)^{1/2}\left(\frac{\hbar}{2m_{b}\omega_{L,j}}\right)^{1/4}, \tag{18}\]0  
where \(\gamma\) is the phenomenological damping factor. The values of \(\gamma\) found in the literature are of the order of the Fermi energy of electrons [31; 32]. Here, \(\gamma\) is a fitting parameter of the same order of magnitude (the _only_ fitting parameter which is in fact used here). The calculation is performed with the values \(\gamma=1.2\varepsilon_{F,1}\) for \(T=7\,\mathrm{K}\) and \(\gamma=2\varepsilon_{F,1}\) for \(T=300\,\mathrm{K}\). It should be noted that the results appear to be only slightly sensitive to chosen values of \(\gamma\). The calculation of the Green's functions for non-parabolic bands requires knowledge of overlap integrals [33] for the Coulomb and electron-phonon interactions, which is not yet reliably known and needs a microscopic calculation. In order to simplify the computation keeping main features of the non-parabolic band dispersion, we perform two approximations. First, we apply the density-of-states approach already successfully used in Ref. [34]. The approximation consists of the replacement of the true band energy \(\varepsilon_{\lambda}\left(\mathbf{k}\right)\) by the model isotropic band energy \(\varepsilon_{\lambda}\left(k\right)\) which provides _the same density of states_ as that for the true band energy \(\varepsilon_{\lambda}\left(\mathbf{k}\right)\).