16.1 Theoretical background

At the effective single-particle level, the Hamiltonian of the coupled system of electrons and vibrations is given by [317]

\[\begin{equation} \hat{H}=\hat{H}^{\text{e}}+\hat{H}^{\text{v}}+\hat{H}^{\text{ev}}, \end{equation}\](16.1) where the first term \(\hat{H}^{\text{e}}\) describes the electronic system and the second term \(\hat{H}^{\text{v}}\) the vibrational degrees of freedom. The last term in the Hamiltonian \[\begin{equation} \hat{H}^{\text{ev}}=\sum_{\mu\nu}\sum_{\alpha}\hat{d}_{\mu}^{\dagger}\lambda_{\mu\nu}^{\alpha}\hat{d}_{\nu}(\hat{b}_{\alpha}^{\dagger}+\hat{b}_{\alpha}) \end{equation}\](16.2) describes the first order electron-vibration (EV) interaction. The EV coupling constants are given as \[\begin{equation} \lambda_{\mu\nu}^{\alpha}=\left(\dfrac{\hbar}{2\omega_{\alpha}}\right)^{1/2}\sum_{\chi}\bigl\langle\mu\bigl|\dfrac{\mbox{d}\hat{H}^{\text{e}}_1}{\mbox{d}\chi}\bigr|\nu\bigr\rangle \mathcal{A}_{\chi}^{\alpha}, \end{equation}\](16.3) where \(\chi=(k,u)\) is a shorthand notation that refers both to the displacement of atom \(k\) from the equilibrium value of the position \(\vec{R}_k\) along the Cartesian component \(R_{k,u}\) with \(u=x,y,z\) as well as the index pair itself. Furthermore, \(\mathcal{A}_{\chi}^{\alpha}=\mathcal{C}_{\chi}^{\alpha}/\sqrt{M_k}\) are the mass-normalized normal modes, obtained from the eigenvectors \(\mathcal{C}_{\chi}^{\alpha}\) of the dynamical matrix as calculated from the aoforce module [317].