# Effects of Electron Correlation inside Disordered Crystals
## 2 Hamiltonian of an Electron-Phonon System of a Disordered Crystal  
The disordered system's Hamiltonian (disordered semiconductor or alloy)contains the Hamiltonian of electrons at the outer nucleus field, the Hamiltonian of electron-electron interplay, and the Hamilton of the nucleus, as well as the Hamiltonian of electron-nucleus interchange. The movement of an ion subsystem lessens to nucleus oscillations nearby thebalance position under the influence of the nucleus interplay force, as well as their indirect interplay via electrons. Using the Wannier representation, we can describe the system'sHamiltonian as follows [25]:  
\[H=H_{0}+H_{\text{int}} \tag{1}\]Consequently, the zero-order Hamiltonian is defined as:  
\[H_{0}=H_{e}^{(0)}+H_{ph}^{(0)} \tag{2}\]  
Itcomprisesthe electron's Hamiltonian inside the field of atom cores of the perfectl-yarranged crystal:  
\[H_{e}^{(0)}=\sum_{ \begin{array}{c}ni\gamma\\ n\
mid\gamma\\ n^{\prime}i^{\prime}\gamma^{\prime}\end{array} }h_{n\
mid\gamma,n^{\prime}i^{ \prime}\gamma^{\prime}}^{(0)}a_{n^{\prime}i^{\prime}\gamma^{\prime}}^{+} \tag{3}\]  
Additionally, it comprises the phonon harmonic Hamiltonian of the movement for theores of atoms in the ideally arranged crystal:  
\[H_{ph}^{(0)}=\sum_{ \begin{array}{c}ni\alpha\\ n^{\prime}i^{\prime}\alpha^{\prime}\end{array} }\frac{1}{2}\sum_{ \begin{array} []{c}ni\alpha\\ n^{\prime}i^{\prime}\alpha^{\prime}\end{array} }\Phi_{n\alpha,n^{\prime}i^{ \prime}\alpha^{\prime}}^{(0)}u_{ni\alpha}u_{n^{\prime}i^{\prime}\alpha^{ \prime}} \tag{4}\]  
In this place, cores of the atoms are placed in the periodic lattice (namely, the unperturbed structure is periodically arranged, and it is without disorder). The character \(n\) indicates the unit cell, \(i\) indicates the \(i\)th basis vector in the \(n\)th unit cell, and \(\gamma\) indicates another quantum number for the orbital, inclusive of spin. Disorder will be introduced-for the sort of core at the specific lattice area, whichdoes not needto be periodic through theperturbed Hamiltonian term (view below). The character \(h^{(0)}\) symbolises the \"hopping integral\" that links the relevant orbitals. For the phonon Hamiltonian, \(n\) and \(i\) are the same variables as previously, that is, the unit cell and basis site inside the unit cell. At the same time, \(\alpha\) is the spatial direction (\(x\), \(y\), or \(z\)). Namely, \(P_{ni\alpha}\) describes the core momentum, \(M_{i}\) describes themass of the core, \(u_{ni\alpha}\) describes the variation of the core from the balance position of the lattice site, and \(\Phi_{ni\alpha,n^{\prime}i^{\prime}\alpha^{\prime}}^{(0)}\) describes the consonant springconstant matrix. The interplay Hamiltonian in Equation (1) is the disturbance of the system caused by all, includingby our effects. It contains six parts:  
\[H_{\text{int}}=\delta\Phi+H_{ce}+H_{eph}+H_{ee}+H_{phph} \tag{5}\]  
\(\delta\Phi\) is the adjustment of the core-core Coulomb interplay caused by the addition to the system's disordered atoms; it is the distinction betweenthe initial core-core repugnance Hamiltonian and the new Hamiltonian. The electronic Hamiltonian is altered by the term  
\[H_{ce}=\sum_{ \begin{array}{c}ni\gamma\\ n^{\prime}i^{\prime}\gamma^{\prime}\end{array} }w_{ni\gamma,n^{\prime}i^{ \prime}\gamma^{\prime}}a_{ni\gamma}^{+}a_{n^{\prime}i^{\prime}\gamma^{\prime}} \tag{6}\]  
This term is the difference betweenthe new hopping Hamiltonian as well as the original periodic Hamiltonian. The electron-phonon interaction is described by the following equation:  
\[H_{eph}=\sum_{ \begin{array}{c}ni\gamma\\ n^{\prime}i^{\prime}\gamma^{\prime}\end{array} }v^{\prime}u_{ni\gamma,n^{\prime }i^{\prime}\gamma^{\prime}}a_{n\gamma}^{+}a_{n^{\prime}i^{\prime}\gamma^{ \prime}} \tag{7}\]A more detailed description is underneath. The Coulomb interaction Hamiltonian among electrons is described by the following expression:  
\[