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.