  In metal-organic-framework (MOF) perovskites, both magnetic and ferroelectric
orderings can be readily realized by compounding spin and charge degrees of
freedom. The hydrogen bonds that bridge the magnetic framework and organic
molecules have long been thought of as a key in generating multiferroic
properties. However, the underlying physical mechanisms remain unclear. Here,
we combine neutron diffraction, quasielastic and inelastic neutron scattering,
and THz spectroscopy techniques to thoroughly investigate the dynamical
properties of the multiferroic MOF candidate [CH$_3$NH$_3$][Co(HCOO)$_3$]
through its multiple phase transitions. The wide range of energy resolutions
reachable by these techniques enables us to scrutinize the coupling between the
molecules and the framework throughout the phase transitions and interrogate a
possible magnetoelectric coupling. Our results also reveal a structural change
around 220 K which may be associated with the activation of a nodding donkey
mode of the methylammonium molecule due to the ordering of the CH$_3$ groups.
Upon the occurrence of the modulated phase transition around 130 K, the
methylammonium molecules undergo a freezing of its reorientational motions
which is concomitant with a change of the lattice parameters and anomalies of
collective lattice vibrations. No significant change has been however observed
in the lattice dynamics around the magnetic ordering, which therefore indicates
the absence of a substantial magneto-electric coupling in zero-field.
