# Effects of Electron Correlation inside Disordered Crystals
## 4 Density of Electronic and Phononic States
\begin{array}{l}\Sigma_{emi_{1}\gamma_{1},n_{2}i_{2}\gamma_{2}}^{\lambda m_ {1}\mu i}(\varepsilon)=\upsilon_{n_{1}i_{1}\gamma_{1},n_{2}i_{2}\gamma_{2}}^{ \lambda n_{1}i_{1}}+\\ +\sum\limits_{n_{3}i_{3}\gamma_{3}}\widetilde{\upsilon}_{n_{3}i_{3}\gamma_{3},n_{2}i_{2}\gamma_{2}}^{(2)m_{1}i_{1}\gamma_{1},n_{4}i_{4}\gamma_{4}}\left( \chi_{n_{3}i_{3}\gamma_{3},n_{4}i_{4}\gamma_{4}}^{\lambda m_{3},n_{4}i_{4} \gamma_{4}}^{\lambda m_{3},n_{4}i_{4}\gamma_{4}}\right)\\ n_{4}i_{4}\gamma_{4}\end{array}  \tag{61}\]  
where  
\[ \begin{array}{l}Z_{n_{3}i_{3}\gamma_{3},n_{4}i_{4}\gamma_{4}}^{\lambda m_{ 1}\mu i}=-\frac{1}{\pi}\int\limits_{-\infty}^{\infty}f(\varepsilon,\varepsilon _{F})\;\operatorname{Im}\Bigl{\langle}G_{n_{3}i_{3}\gamma_{3},n_{4}i_{4} \gamma_{4}}^{\alpha n}(\varepsilon)\Bigr{\rangle}\Bigr{|}_{(ni)\in\lambda m_{ \lambda\lambda}}\;d\varepsilon\end{array}  \tag{62}\]  
The value of \(\widetilde{Z}_{n_{3}i_{3}\gamma_{3},n_{4}i_{4}\gamma_{4}}^{\lambda m_{1}i}\) in Equation (61) is obtained from Equation (62) via replacing of the fullGreen's function via the effective medium Green's function. The matrix diagonal elements \(Z_{n_{3}i_{3}\gamma_{3},n_{4}i_{4}\gamma_{4}}^{\lambda m_{3},n_{4}i_{4}\gamma_ {4}}\) in Equation (62) are similar to the occupation numbers for the electron states \(Z_{n_{i}\delta\sigma}^{\lambda m_{1}i}\) (see Refs. [28; 29]). Using Equations (35) and (60),we obtain the expression for the intrinsic energy part \(\Sigma_{phini\alpha,n_{i}i_{1}}^{\lambda m_{1}i_{1}}(\varepsilon)\), which describes the scattering of phonons:  
\[\Sigma_{phini\alpha,n_{i}i_{1}}^{\lambda n_{1}i_{1}}(\varepsilon)=\frac{ \varepsilon^{2}}{R^{2}}(M_{i_{1}}-M_{\lambda})\delta_{n\mu i}\delta_{i\mu} \delta_{\alpha\alpha\prime} \tag{63}\]  
It should be noted that, in the infinite crystal limit, on the right-hand side of Equations (61) and (63), the terms inversely proportional to the number of lattice sites are neglected. We require the fulfilment of the condition  
\[\left\langle t^{0i_{1}}\right\rangle=0 \tag{64}\]  
from which follows the system of coupled equations for the operator in Formulas (53) and (54)  
\[\begin{array}{l}\sigma_{e}^{0i_{1}}(\varepsilon)=\left\langle[1-(\Sigma_{e}^{0 i_{1}}(\varepsilon)-\sigma_{e}^{0i_{1}}(\varepsilon))\widetilde{G}^{aa^{+}}( \varepsilon)]^{-1}\right\rangle^{-1}\times\\ \times\left\langle[1-(\Sigma_{e}^{0i_{1}}(\varepsilon)-\sigma_{e}^{0i_{1}}( \varepsilon))\widetilde{G}^{aa^{+}}(\varepsilon)]^{-1}\Sigma_{e}^{0i_{1}}( \varepsilon)\right\rangle^{  
\[ \begin{array}{l}\sigma^{0i_{1}}_{ph}(\epsilon)=\left\langle\left[1-(\Sigma^{0i_{1}}_ {ph}(\epsilon)-\sigma^{0i_{1}}_{ph}(\epsilon))\widetilde{G}^{uu}(\epsilon) \right]^{-1}\right\rangle^{-1}\times\\ \times\left\langle\left[1-(\Sigma^{0i_{1}}_{ph}(\epsilon)-\sigma^{0i_{1}}_{ph }(\epsilon))\widetilde{G}^{uu}(\epsilon)\right]^{-1}\Sigma^{0i_{1}}_{ph}( \epsilon)\right\rangle\end{array}  \tag{66}\]  
The matrix elements for Green's function for the electron subsystem of the effective medium can be effectively computed by means of Fourier transformation:  
\[\widetilde{G}^{aa^{+}}_{ni^{+}\gamma,ni^{+}\gamma^{\prime}}(\epsilon)=\frac{1 }{N}\sum_{\mathbf{k}}\widetilde{G}^{aa^{+}}_{i^{+}j_{\gamma}\gamma}(\mathbf{ k},\epsilon)e^{i\mathbf{k}(\mathbf{r}_{ni}-\mathbf{r}_{ni})}, \tag{67}\]  
\[\widetilde{G}^{aa^{+}}(\mathbf{k},\epsilon)=\left(\epsilon-\widetilde{H}( \mathbf{k},\epsilon)\right)^{-1}, \tag{68}\]  
where  
\[\widetilde{H}(\mathbf{k},\epsilon)=h^{(0)}(\mathbf{k})+\widetilde{\Sigma}_{ cuph}(\mathbf{k},\epsilon)+\widetilde{\Sigma}_{ee}(\mathbf{k},\epsilon)+\sigma_{e}( \mathbf{k},\epsilon) \tag{69}\]  
\(N\) is the number of primitive unit cells.