  Metal-organic frameworks (MOFs) are crystalline materials consisting of metal
centers and organic linkers forming open and porous structures. They have been
extensively studied due to various possible applications exploiting their large
amount of internal surface area. Phonon properties of MOFs are, however, still
largely unexplored, despite their relevance for thermal and electrical
conductivities, thermal expansion, and mechanical properties. Here, we use
quantum-mechanical simulations to provide an in-depth analysis of the phonon
properties of isoreticular MOFs. We consider phonon band structures, spatial
confinements of modes, projected densities of states, and group velocity
distributions. We find that more complex linkers shift the spectral weight of
the phonon density of states towards higher frequencies, while increasing the
mass of the metal atoms in the nodes has the opposite effect. Due to the high
porosity of MOFs, we observe a particularly pronounced polarization dependence
of the dispersion of acoustic phonons with rather high group velocities for
longitudinal acoustic modes. Interestingly, also for several optical phonon
modes group velocities amounting to several thousand m/s are obtained. For
heterogeneous systems like MOFs, correlating group velocities and the
displacement of modes is particularly relevant. Here we find that high group
velocities are generally associated with delocalized vibrations, while the
inverse correlation does not necessarily hold. To quantify anharmonicities, we
calculate mode Gr\"uneisen parameters, which we find to be significant only for
phonons with frequencies below ~3 THz. The presented results provide the
foundations for an in-depth understanding of the vibrational properties of MOF,
and, thus, pave the way for a future rational design of systems with
well-defined phonon properties.
