The Critical Angle Criterion: A Law To Predict Anomalous Compressive Behaviors in Framework Materials

In this study, we challenge the previously commonly used empirical rule of thumb to explain the abnormal compressive behaviors observed in Zn M (CN) 4 (M = Ni, Pd) compounds. Instead, we propose the critical angle criterion that allows for direct prediction of compressive behaviors in various framework structures based on four structural models. According to this criterion, structures with a framework angle smaller or larger than the critical angle will exhibit negative linear compressibility or negative area compressibility (NLC/NAC). A systematic analysis reveals that the criterion accords with all previous reports. The application of the criterion on another 22 materials indicates that 20 of them will exhibit NLC along the c -axis, while the other two materials will exhibit NAC in the ab -plane, which is confirmed by the further density functional theory (DFT) calculations. The critical angle criterion will provide valuable insights and serve as a foundation for exploring materials exhibiting desired abnormal compressive behaviors.