Radiative corrections to the long-wavelength optical-mode spectrum of the electron-phonon model: Absence of mode-splitting effects and hardening of the mode

Renormalization of the phonon spectrum of the Einstein form due to a weak coupling with electrons is calculated. No splitting of the phonon mode is found. A $q$-dependent hardening of the long-wave optical phonons contrasts with the softening of acoustic phonons in a weakly coupled electron-phonon system and with the softening of all phonons in the strong-coupling polaronic regime.