# Electron Transport in Carbon Nanotubes with Adsorbed Chromium Impurities
## 2 Hamiltonian of an Electron-Phonon System for a Disordered Crystal  
The many-body Hamiltonian of a disordered crystal consists of a single-particle Hamiltonian of the electrons in the external potential of the (disordered) ionic cores, the potential energy of the electron-electron interaction, the quadratic Hamiltonian for the phonons, the contribution from the electron-phonon interaction, and the anharmonic phonon potential terms. We represent the Hamiltonian in the basis of free neutral atoms. In the Wannier representation, the system Hamiltonian is [31]  
\[H=H_{0}+H_{\text{int}} \tag{1}\]  
where the zeroth-order Hamiltonian  
\[H_{0}=H_{e}^{(0)}+H_{ph}^{(0)} \tag{2}\]  
consists of the single-particle Hamiltonian of the electrons in the field of the ionic cores  
\[H_{e}^{(0)}=\sum_{ \begin{subarray}{c}ni\gamma\\ n^{\prime}i^{\prime}\gamma^{\prime}\end{subarray} }h_{ni\gamma,n^{\prime}i^{ \prime}\gamma^{\prime}}^{(0)}a_{ni\gamma}^{+}a_{n^{\prime}i^{\prime}\gamma^{ \prime}} \tag{3}\]  
and the harmonic phonon Hamiltonian for the motion of the ion cores  
\[H_{ph}^{(0)}=\sum_{ni\alpha}\frac{P_{ni\alpha}^{2}}{2M_{i}}+\frac{1}{2}\sum_ { \begin{subarray}{c}nia\\ n^{\prime}i^{\prime}\alpha^{\prime}\end{subarray} }\Phi_{nia\alpha,n^{\prime}i ^{\prime}\alpha^{\prime}}^{(0)}u_{nia}u_{n^{\prime}i^{\prime}\alpha^{\prime}}. \tag{4}\]  
Here, the ion cores are located on a periodic lattice (i.e., the unperturbed system is periodically ordered and has no disorder). The symbol \(n\) denotes the unit cell, \(i\) denotes the \(i\)th basis vector in the \(n\)th unit cell, and \(\gamma\) denotes all of the other quantum numbers for the orbital, including spin. Disorder will enter for the species of ion at a particular lattice site, which need not be periodic via a perturbed Hamiltonian term (see below). The symbol \(h^{(0)}\) denotes the \"hopping integral\" that connects the respective orbitals. For the phonon Hamiltonian, \(n\) and \(i\) are the same as before, namely the unit cell and basis site within the unit cell, while \(\alpha\) is a spatial direction (\(x,y\), or \(z\)). \(P\) is the ionic momentum, \(M\) is the ionic mass, \(u\) is the deviation of the ion from the equilibrium position of the lattice site, and \(\Phi^{(0)}\) is the corresponding spring-constant matrix. The interaction Hamiltonian in Equation (1) is the perturbation of the system due to all of the effects we will be including. It is composed of six pieces:  
\[H_{\text{int}}=\delta\Phi+H_{ei}+H_{eph}+H_{ee}+H_{phi}+H_{phph}. \tag{5}\]  
\(\delta\Phi\) is the modification of the ion-core--ion-core Coulomb interaction due to the disordered ions added to the system; it is the difference between the original ion-ion repulsion Hamiltonian and the new one. The single-particle electronic Hamiltonian is modified by the change in the ion core and the extra term:\[H_{el}=\sum_{ \begin{subarray}{c}ni\gamma\\ n^{\prime}i^{\prime}\gamma^{\prime}\end{subarray} }w_{ni\gamma,n^{\prime}i^{ \prime}\gamma^{\prime}}a^{+}_{ni\gamma}a_{n^{\prime}i^{\prime}\gamma^{\prime}}, \tag{6}\]  
which is the difference between the new hopping Hamiltonian and the original periodic one. The electron-phonon interaction is given by  
\[H_{eph}=\sum_{ \begin{subarray}{c}ni\gamma\\ n^{\prime}i^{\prime}\gamma^{\prime}\end{subarray} }v^{\prime}_{ni\gamma,n^{ \prime}i^{\prime}\gamma^{\prime}}a^{+}_{ni\gamma}a_{n^{\prime}i^{\prime}\gamma^ {\prime}}. \tag{7}\]  
This is described in more detail below. The screened Coulomb interaction between electrons is given by the different multiband interaction terms, including density-density interactions and exchange interactions  
\[H_{ee}=\frac{1}{2}\sum_{ \begin{subarray}{c}ni\gamma\\ n^{\prime}i^{\prime}\gamma^{\prime}\end{subarray} }v^{(2)nir\gamma,n^{\prime} i^{\prime}\gamma^{\prime}}_{n^{\prime}i^{\prime}\gamma^{\prime}}\gamma^{ \prime\prime}a^{+}_{ni\gamma}a^{+}_{n^{\prime}i^{\prime}\gamma^{\prime}}a_{n^ {\prime}i^{\prime}\gamma^{\prime\prime}}a_{n^{\prime\prime}i^{\prime\prime} }\gamma^{\prime\prime}.