### 15.1 Theoretical background

At the effective single-particle level, the Hamiltonian of the coupled system of electrons and
vibrations is given by [160]

| (15.1) |

where the first term Ĥ^{e} describes the electronic system and the second term Ĥ^{v} the vibrational
degrees of freedom. The last term in the Hamiltonian

| (15.2) |

describes the first order electron-vibration (EV) interaction. The EV coupling constants are given
as

| (15.3) |

where χ = (k,u) is a shorthand notation that refers both to the displacement of atom k from the
equilibrium value of the position _{k} along the Cartesian component R_{k,u} with u = x,y,z as well
as the index pair itself. Furthermore, _{χ}^{α} = _{χ}^{α}∕ are the mass-normalized normal modes,
obtained from the eigenvectors _{χ}^{α} of the dynamical matrix as calculated from the aoforce
module [160].