15.1 Theoretical background

At the effective single-particle level, the Hamiltonian of the coupled system of electrons and vibrations is given by [181]

ˆ    ˆe  ˆ v  ˆ ev
H = H  + H  + H  ,

where the first term Ĥe describes the electronic system and the second term Ĥv the vibrational degrees of freedom. The last term in the Hamiltonian

 ˆev  ∑  ∑   ˆ†  α ˆ ˆ†  ˆ
H   =  μν α dμλ μνdν(bα + bα)

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

 α   (  ℏ )1 ∕2∑  ⟨ |dHˆe1| ⟩ α
λμν =  2ωα-       μ|-dχ-|ν Aχ,

where χ = (k,u) is a shorthand notation that refers both to the displacement of atom k from the equilibrium value of the position ⃗
Rk along the Cartesian component Rk,u with u = x,y,z as well as the index pair itself. Furthermore, Aχα = Cχα√---
 Mk are the mass-normalized normal modes, obtained from the eigenvectors Cχα of the dynamical matrix as calculated from the aoforce module [181].