# Effects of Electron Correlation inside Disordered Crystals
## 2 Hamiltonian of an Electron-Phonon System of a Disordered Crystal
\begin{array}{c}H_{\alpha e}=\frac{1}{2}\sum\limits_{n\alpha}v_{n_{3},n_{4}}^{(2 )n_{1}n_{2}}a_{n_{1}}^{+}a_{n_{2}}^{+}a_{n_{3}}a_{n_{4}},\\ n_{3},n_{4}\\ n=(ni\gamma).\end{array}  \tag{8}\]  
The adjustment of the interplay of the phonons with theores generated by the disordering of the atoms is described by the following expression:  
\[ \begin{array}{c}H_{\beta nc}=\frac{1}{2}\sum\limits_{n\alpha}\Delta M_{n \alpha,n^{\prime}\sigma^{\prime}\alpha^{\prime}}^{-1}P_{n\alpha}P_{n^{\prime} \sigma^{\prime}\alpha^{\prime}}+\\ \begin{array}{c}n^{\prime}i^{\prime}\alpha^{\prime}\\ n\alpha i^{\prime}\alpha^{\prime}\\ n^{\prime}i^{\prime}\alpha^{\prime}\end{array},\end{array}  \tag{9}\]  
where  
\[\Delta M_{ni\alpha,n^{\prime}i^{\prime}\alpha^{\prime}}^{-1}=\left(\frac{1}{ M_{ni^{\prime}}}-\frac{1}{M_{i}}\right)\delta_{nn^{\prime}}\delta_{ii^{\prime}} \delta_{\alpha\alpha^{\prime}}, \tag{10}\]  
\(\Delta\Phi_{ni\alpha,n^{\prime}i^{\prime}\alpha^{\prime}}=\Phi_{ni\alpha,n^{ \prime}i^{\prime}\alpha^{\prime}}-\Phi_{ni\alpha,n^{\prime}i^{\prime}\alpha^ {\prime}}^{(0)}\) and \(M_{ni}\), \(M_{i}\) are the atomic masses at the site \((ni)\) for disordered alloys as well as for the ordered alloys. The cubic anharmonic potential expressions are also included for the phonons. We assume that phonons stand small and they can be handled perturbatively through  
\[ \begin{array}{c}H_{phph}=\frac{1}{3!}\sum\limits_{ni\alpha}\Phi_{ni\alpha,n^ {\prime}i^{\prime}\alpha^{\prime},n^{\prime\prime}i^{\prime\prime}\alpha^{ \prime\prime}}^{-1}u_{ni\alpha}\times\\ \begin{array}{c}n^{\prime}i^{\prime}\alpha^{\prime}\\ n^{\prime\prime}i^{\prime\prime}\alpha^{\prime\prime}\\ n^{\prime\prime}i^{\prime\prime}\alpha^{\prime\prime}\end{array}&\\ \times u_{n^{\prime}i^{\prime}\alpha^{\prime}}u_{n^{\prime\prime}i^{\prime\prime }}^{-1}\alpha^{\prime\prime}.\end{array}  \tag{11}\]  
The operators \(a_{ni\gamma}^{+}\), \(a_{ni\gamma}\) produce and dismantle electrons in the state described by Vane's function. \(\Phi_{ni\gamma}(\xi)=\left\langle\xi|ni\gamma\right\rangle\), where \(\xi=(\mathbf{r},\sigma^{\prime})\) is the spatial as well as the \(z\)-component of the wave function of the spin coordinates. To build up the Wannier functions, we employ analytical interpretations for the wave functions of an electron in the field of atomic nuclei of kind \(\lambda\), restricted in the lattice sites \((ni)\) for an ideally arranged crystal:  
\[ \begin{array}{c}\psi_{ni\delta}(\mathbf{r}-\mathbf{r}_{ni})=R_{\xi l}(| \mathbf{r}-\mathbf{r}_{ni}|)Y_{lm}(\mathbf{r}-\mathbf{r}_{ni}),\\ Y_{lm}(\mathbf{r}-\mathbf{r}_{ni})=Y_{lm}(\theta,\varphi),\end{array}  \tag{12}\]  
where \(\theta,\varphi\) are the spherical angular coordinates for the vector \(\mathbf{r}-\mathbf{r}_{ni}\). Here, \(\delta=\bar{\varepsilon}lm\) is a superindex that includes the quantum numbers for the basis energy eigenvalue \(\widetilde{\epsilon}\), the normal angular momentum quantum numbers \(l\) and \(m\), \(\mathbf{r}\) is the electron location vector, and \(\mathbf{r}_{ni}\) is the location vector for the atom at place \((ni)\) in balance. \[ \begin{array}{c}\mathbf{r}_{ni}=\mathbf{r}_{n}+\mathbf{\rho}_{i},\\ \mathbf{r}_{n}=\sum\limits_{\
u}l_{\
u}\mathbf{a}_{\
u},\end{array}  \tag{13}\]  
\(\mathbf{r}_{n}\) is the location vector of the junction n of the crystal lattice, \(\mathbf{\rho}_{i}\) is the vector of the respective location of the junction of the sublattice \(i\) at the unit cell \(n\). Vectors \(\mathbf{\rho}_{i}\) are labelled as fundamental relocation vectors of the crystal lattice, the character of which is decided by the dimension of the crystal. The coordinates \(l_{\
u}\) of the radius vector \(\mathbf{r}_{n}\) of the crystal lattice junction are integers. The number \(\
u\) takes on values \(\
u=1,2,3\) for three-dimensional crystals, \(\
u=1,2\) for two-dimensional crystals, and \(\
u=1\) for one-dimensional crystals.