# ZIF-75 under Pressure: Negative Linear Compressibility and Pressure-Induced Instability
## 3 Results
### Elasticity Tensor and Mechanical Stability  
The computed elasticity tensor for ZIF-75 is given in Equation (1). Crystals of the tetragonal space group \(I41/a\) belong to the tetragonal (II) class (4/m). In this case, the elastic tensor has only 11 nonvanishing elastic constants, of which only six are independent [126]\((C_{11}=C_{22};C_{44}=C_{55};C_{13}=C_{23};C_{16}=-C_{26})\).  
\[C=\left(\begin{array}{cccccc}5.22&3.45&10.17&0.00&0.00&0.04\\ 3.45&5.22&10.17&0.00&0.00&-0.04\\ 10.17&10.17&13.35&0.00&0.00&0.00\\ 0.00&0.00&0.00&0.36&0.00&0.00\\ 0.00&0.00&0.00&0.00&0.36&0.00\\ 0.00&0.00&0.00&0.00&-0.29\end{array}\right) \tag{1}\]  
For a given material, the general Born condition for mechanical stability is that the corresponding elastic matrix should be positive definite [126], i.e., all the eigenvalues of the elastic matrix should be positive. For crystals belonging to the tetragonal (II) class (4/m), there is a simple set of equivalent conditions for mechanical stability [126]:  
\[C_{11}>|C_{12}| \tag{2}\]  
\[2\cdot C_{13}^{2}>(C_{11}+C_{12}) \tag{3}\]  
\[C_{44}>0 \tag{4}\]  
\[2\cdot C_{16}>C_{66}(C_{11}-C_{12}) \tag{5}\]  
The last condition is not satisfied; therefore, the crystal structure of ZIF-75 is not mechanically stable. The reason for the mechanical instability of ZIF-75 will be unveiled in the next section, where the deformation of its crystal structure under pressure is investigated.  
Behavior of ZIF-75 under the Effect of Pressure. Pressure-Induced Phase Transition and Negative Linear Compressibility  
The crystal structure of ZIF-75 was fully optimized under the effect of 14 different isotropic pressures within the range going from \(P=-1.0\) to 1.0 GPa. The computed lattice parameters and unit cell volumes at these pressures are tabulated in Table S2 of the Supplementary Materials and plotted in Figure 4. As can be clearly seen, the effect of pressure induces a phase transition at a pressure near \(P=0.1\) GPa. The presence of this pressure-induced phase transition at a small pressure close to zero GPa is the main reason for the mechanical instability in ZIF-75, as found in Section 3.2.  
Figure 3: Comparison of the X-ray diffraction patterns of ZIF-75 derived from the experimental [2] and theoretical tetragonal crystal structures.  
From the computed elasticity matrix, the dependence of the compressibility of ZIF-75 as a function of the orientation of the external pressure was obtained and plotted on the E1 AM computer program [127]. As can be seen in Figure 5, the compressibility is negative for uniaxial pressures applied in any direction contained in the (1 1 0) plane. As it will be shown, indeed, the compressibility along \(c\) direction is also negative when ZIF-75 is submitted to isotropic pressures.  
As can be observed in Figure 4, the behavior of the \(a\) and \(b\) lattice parameters, while presenting a rapidly decreasing behavior near the phase transition, is normal in the sense that they decrease invariably along the full pressure range. However, the \(c\) lattice parameter presents an anomalous behavior and sharply increases before the phase transition in the pressure range going from \(P=-0.06\) to 0.1 GPa. Therefore, ZIF-75 presents a remarkable negative linear compressibility effect in this pressure range since the linear compressibility along the \(c\) direction, \(k_{c}=-1/c\cdot(\partial c/\partial P)_{p}\), is strongly negative.  
Figure 4: Computed unit cell volumes and lattice parameters of ZIF-75 as a function of the applied isotropic pressure: (**A**) unit cell volume; (**B**) \(a\) lattice parameter; (**C**) \(b\) lattice parameter; (**D**) \(c\) lattice parameter.  
Figure 5: Compressibility of ZIF-75 as a function of the applied strain. The negative and positive values are displayed in red and green, respectively.