Non--adiabatic effects lead to the breakdown of the semi-classical phonon concept

Phonon properties of realistic materials are routinely calculated within the Density Functional Perturbation Theory,(DFPT). This is a semi–classical approach where the atoms are assumed to oscillate along classical trajectories immersed in the electronic Kohn–Sham density, treated quantistically. In this work I demonstrate that, in metals, non–adiabatic effects induce a gap between the DFTP phonon frequencies and the fully quantistic solution of the phonon Dyson equation. A gap that increases with the phonon energy width reflecting the breakdown of the semi–classical DFPT description. The final message is that non–adiabatic phonon effects can be included only by using a fully quantistic approach.