Singularly isostatic and geometrically unstable rigidity of metal-organic frameworks

Metal-organic frameworks (MOFs) combine high porosity with structural fragility, raising important questions about their mechanical stability. We develop a rigidity-based framework in which spring networks parameterized by UFF4MOF are used to construct rigidity and dynamical matrices. Large-scale analysis of 5,682 MOFs from the CoRE 2019 database shows that most frameworks are formally over-constrained yet cluster sharply near the isostatic threshold, revealing accidental geometric modes and placing many MOFs near mechanical instability. In the representative case of UiO-66, we show that auxiliary long-range constraints introduced by tuning the neighbor cutoff lift these modes into soft, flat, finite-frequency bands. The results show that rigidity-matrix analysis can rapidly identify MOFs likely to remain mechanically stable. This near-criticality mirrors behavior known from topological mechanics and points to a deeper design principle in porous crystals.