Electron-phonon interaction in an $N$-atomic 1-D periodic chain

The study of electron-phonon interaction as a prominent inelastic effect is of great importance. In the present work, we have studied the inelastic effects due to the first order electron-phonon interactions on electronic properties of mono-atomic periodic chain, using the exact diagonalization technique. Hence, only acoustic modes are considered. To avoid the approximate results arising from Keldysh formalism and on-site electron-phonon interaction assumption, we have used the Green's function technique together with Fr\"{o}hlich Hamiltonian for the interaction part. Finally, as an example, we apply our method to the case of $N=6$ which could be considered as a Benzene-like molecule.