  Metal-organic frameworks (MOFs) possess a hybrid nature, combining the
inorganic properties from the metal ions and the organic properties from the
molecular linkers. Stroppa, \textit{et al.}, showed that the perovskite-type
MOF [C(NH$_2$)$_3$]M[(HCOO)$_3$] (M = Cr, Cu) exhibits the magneto-electric
coupled multiferroicity [Angew. Chem. Int. Ed. 50, 5847 (2011) and Adv. Mater.
25, 2284 (2013)]. Moreover, their ferroelectricity arises from the hybrid
improper mechanism, which also explains the magneto-electric coupling. In this
work, we further examine the electric and magnetic properties of
[C(NH$_2$)$_3$]M[(HCOO)$_3$]. We find that the hybrid mode composed of
non-polar modes induces purely electronic polarization even without the polar
mode. The polar mode compensates for the purely electronic polarization. It
leads to a counter-intuitive argument that the inversion of the polar mode
rather enhances the polarization. We provide microscopic origin and macroscopic
analysis for this polarization property. In addition, we find that the orbital
magnetic moment is comparable to the spin contribution in the Cu-based MOF.
Finally, we establish the model for the orbital magnetic moment based on the
perturbation theory.
