  Grimme's DFT-D dispersion contribution to interatomic forces constants,
required for the computation of the phonon band structures in
density-functional perturbation theory, has been derived analytically. The
implementation has then been validated with respect to frozen phonons, and
applied on materials where weak cohesive forces play a major role i.e. argon,
graphite, benzene. We show that these dispersive contributions have to be
considered to properly reproduce the experimental vibrational properties of
these materials, although the lattice parameter change, coming from the
ground-state relaxation with the proper functional, induces the most important
change with respect to a treatment without dispersion corrections. In the
current implementation, the contribution of these dispersion corrections to the
dynamical matrices (with a number of elements that is proportional to the
square of the number of atoms) has only a cubic scaling with the number of
atoms. In practice the overload with respect to density-functional calculations
is small, making this methodology promising to study vibrational properties of
large dispersive systems.
