  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.
