# Effects of Electron Correlation inside Disordered Crystals
## 4 Density of Electronic and Phononic States  
In Equations (34) and (35), due to the introduction of the mass operator equal to the sum of the one-site operators as well as by selecting as a zero approximation the effective medium of [PERSON]'s function, the cluster expansion for [PERSON]'s functions \(G^{aa^{+}}(\varepsilon)\) and \(G^{uu}(\varepsilon)\) was performed. The described expansion is the cluster expansion's generalisation for [PERSON]'s function \(G^{aa^{+}}(\varepsilon)\) of the single-particle Hamiltonian. [PERSON]'s functions of the effective environment are defined by the expressions:  
\[\widetilde{G}^{aa^{+}}(\varepsilon)=\left[\varepsilon-h^{(0)}-\widetilde{ \Sigma}_{eph}(\varepsilon)-\widetilde{\Sigma}_{ee}(\varepsilon)-\sigma_{ \varepsilon}(\varepsilon)\right]^{-1} \tag{53}\]  
\[\widetilde{G}^{uu}(\varepsilon)=\left[\frac{\varepsilon^{2}}{\widetilde{H}^{2 }}M^{(0)}-\Phi^{(0)}-\widetilde{\Sigma}_{phe}(\varepsilon)-\sigma_{ph}( \varepsilon)\right]^{-1} \tag{54}\]  
Expressions for the operators \(\widetilde{\Sigma}_{eph}(\varepsilon)\widetilde{\Sigma}_{phe}(\varepsilon)\), and \(\widetilde{\Sigma}_{ee}(\varepsilon)\) are obtained from the expressions for the mass operators \(\Sigma_{eph}(\varepsilon)\Sigma_{phe}(\varepsilon)\), and \(\Sigma_{ee}(\varepsilon)\) (41)-(47) by replacing the vertex parts \(\Gamma^{(0)n_{1}i_{1},\alpha_{1}}_{n_{7},\,n_{3}i_{3}j_{3}\gamma_{3}}\Gamma^{(0) n,n_{2},n_{1}}_{n_{1},n_{3}}\),\(n\equiv ni\gamma\) by their expressions for ideally ordered crystals and replacing the Green's functions \(G^{aa^{+}}(\varepsilon)\) and \(G^{uu}(\varepsilon)\) with the Green's functions of the effective medium,\(\widetilde{G}^{aa^{+}}(\varepsilon)\) and \(\widetilde{G}^{uu}(\varepsilon)\). Expressions for operators \(\sigma_{\varepsilon}(\varepsilon)\) and \(\sigma_{ph}(\varepsilon)\) in Formulas (53) and (54) will be given below. The Green's functions in Equations (34) and (35) persuade the Dyson equation, which may be written in terms of the T-matrix through  
\[G(\varepsilon)=\widetilde{G}(\varepsilon)+\widetilde{G}(\varepsilon)\ T( \varepsilon)\ \widetilde{G}(\varepsilon) \tag{55}\]  
in this expression, the \(T\)-matrix is characterised by a series, where each term expresses the scattering of clusters with different numbers of nodes schematically described as follows:  
\[T=\sum_{(n_{1}i_{1})}t^{n_{1}i_{1}}+\sum_{(n_{1}i_{1})\
eq(n_{2}i_{2})}T^{(2)\ n _{1}i_{1},n_{2}i_{2}}+\ldots \tag{56}\]Here, we have  
\[T^{(2)}\;n_{i_{1}i_{1},n_{2}i_{2}}=\left[I-t^{n_{1}i_{1}}\widetilde{G}t^{n_{2}i_{ 2}}\widetilde{G}\right]^{-1}t^{n_{1}i_{1}}\widetilde{G}t^{n_{2}i_{2}}\left[I+ \widetilde{G}t^{n_{1}i_{1}}\right] \tag{57}\]  
where \(t^{n_{1}i_{1}}\) is the on-site scattering operator, which is described as follows:  
\[t^{n_{1}i_{1}}=\left[I-(\Sigma^{n_{1}i_{1}}-\sigma^{n_{1}i_{1}})\widetilde{G} \right]^{-1}(\Sigma^{n_{1}i_{1}}-\sigma^{n_{1}i_{1}}) \tag{58}\]  
The self-energy employed in Equation (34), \(\Sigma_{e}^{n_{1}i_{1}}(\varepsilon)\), satisfies  
\[w+\Sigma_{eph}(\varepsilon)+\Sigma_{ee}(\varepsilon)-\widetilde{\Sigma}_{eph}( \varepsilon)-\widetilde{\Sigma}_{ee}(\varepsilon)=\sum_{(n_{1}i_{1})}\Sigma_{ e}^{n_{1}i_{1}}(\varepsilon) \tag{59}\]  
For the electrons. For the phonons, we have  
\[ \begin{array}{l}\frac{\varepsilon^{2}}{R^{2}}\Delta M+\Delta\Phi+\Sigma_{phe }(\varepsilon)+\Sigma_{phph}(\varepsilon)-\\ -\widetilde{\Sigma}_{phe}(\varepsilon)-\widetilde{\Sigma}_{phph}(\varepsilon)= \sum\limits_{(n_{1}i_{1})}\Sigma_{ph}^{n_{1}i_{1}}(\varepsilon)\end{array}  \tag{60}\]  
Using Equations (34) and (59),we obtain the expression for the intrinsic energy part \(\Sigma_{emi_{1}\gamma,n_{1}i_{7}\gamma}^{\lambda m_{1}i_{1}}(\varepsilon)\), which describes the scattering of electrons:  
\[