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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 ,
(15.1)

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)
(15.2)

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, χ
(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 ⃗ 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].

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