H3O+ tetrahedron induction in large negative linear compressibility

Despite the rarity, large negative linear compressibility (NLC) was observed in metal-organic framework material Zn(HO3PC4H8PO3H)∙2H2O (ZAG-4) in experiment. We find a unique NLC mechanism in ZAG-4 based on first-principle calculations. The key component to realize its large NLC is the deformation of H3O+ tetrahedron. With pressure increase, the oxygen apex approaches and then is inserted into the tetrahedron base (hydrogen triangle). The tetrahedron base subsequently expands, which results in the b axis expansion. After that, the oxygen apex penetrates the tetrahedron base and the b axis contracts. The negative and positive linear compressibility is well reproduced by the hexagonal model and ZAG-4 is the first MOFs evolving from non re-entrant to re-entrant hexagon framework with pressure increase. This gives a new approach to explore and design NLC materials.

diffraction and observed NLC 36,37 . The b axis of ZAG-4 increases almost 2% in the range of 1.65-2.81 GPa, indicating a strong NLC (K l ≈ −16 TPa −1 ). Due to the inherently small atomic scattering factor of hydrogen, the exact positions of H 2 O in ZAG-4 can not be easily detected by X-ray diffraction technique 38 . Aurelie U Ortiz et al. calculated the ZAG-4 structure under pressure, found a proton transfer and attributed the NLC (from 1.65 to 2.81 GPa) to this structural transition 39 . However, PLC was observed after NLC in experiment. This explanation did not answer why NLC and PLC occurs subsequently after proton transfer. Therefore, an unambiguous mechanism for the NLC in ZAG-4 is still an unresolved matter.
Density functional calculation, an integral part of MOFs research, is complementary to experimental techniques and offers invaluable information in characterization and understanding of systems 27,[40][41][42] . In order to elucidate the NLC mechanism of ZAG-4, we performed the density functional calculation using both PBE and Wu-Cohen (WC) functional 43 as implemented in the Quantum Espresso package 44 to determine their atomic structures under pressure. The PBE functional, usually overestimating lattice parameters, has been widely used in density functional calculations and the WC functional is known to be accurate in predicting solid volumes [45][46][47] . The wave function was expanded in a plane-wave basis set with an energy cutoff of 70 Ry and the first Brillouin zone was sampled on a 3 × 3 × 4 mesh. The ultra-soft pseudopotential was used to represent the electron-ion interaction.
The conventional unit cell of ZAG-4 is depicted in Fig. 1 21,35,36 . It has a base-centered monoclinic lattice with the b axis perpendicular to the a-c plane. Herein, the three building blocks of ZAG-4 are the inorganic Zn-O-P-O chains and two bridging ligands (C 4 H 8 and H 2 O). The 1D Zn-O-P-O chains orient along the c axis and are linked with each other along the b axis by H 2 O molecules. So as to give a detailed description of the linkage between H 2 O molecules and Zn-O-P-O chains, we enlarge the bridging zone in Fig. 1(c). The ZnO 4 and PO 3 C tetrahedrons are linked along the c axis by sharing oxygen atoms and form the Zn-O-P-O chains. The water molecules are located between two Zn-O-P-O chains. As a result, Zn-O-P-O chains bridged by H 2 O form an inorganic 2D structure parallel to the b-c plane. Furthermore, the inorganic planes are linked with each other along the a axis by C 4 H 8 , which is directly bonded with the P cations. In brief, the 3D framework is established with the inorganic Zn-O-P-O chains extending along the c axis linked with each other by C 4 H 8 chains and H 2 O molecules along the a and b axis, respectively.
Firstly, we use the experimental crystal structure at zero pressure as our initial point and fully relax the lattice parameters and atomic positions with PBE functional. The calculated lattice parameters (a = 19.00 Å, b = 8.41 Å, c = 8.18 Å, and volume = 1214.27 Å 3 ) agree well with experimental results (a = 18.51 Å, b = 8.29 Å, c = 8.27 Å, and volume = 1160.55 Å 3 ). As the calculated volume is slight larger than the experimental value, our calculated bulk modulus of (11.6 GPa) is slightly lower than the experimental result (11.7 GPa). Although it is about one thirty-fifth of the bulk modulus of sp 3 carbon allotrope (around 400 GPa) 48,49 , this value is higher than that of porous MOFs MIL-53 and NH 2 -MIL-53 30,50 , but is lower than that of dense MOFs 51-54 . In order to explore its NLC mechanism, we applied hydrostatic pressure to ZAG-4 and investigate its structure variation. The calculated lattice parameters, accompanied with available experimental data 36 , are shown in Fig. 2. We increase pressure and take the previously optimized structure as the initial point of higher pressure condition. In this way, we increase pressure to 8 GPa and optimize the structure step by step.
As can be seen in Fig. 2, the experimental lattice parameters are well reproduced by our calculation based on the PBE functional. It is well-known that PBE calculation typically overestimates the lattice parameters by 1-2%. As far as the overestimation is concerned, our calculated results agree excellently with the experimental data. As pointed by Gagnon et al., the alkyl chains serve like a spring cushion and hence contract much under pressure 36 . With pressure increasing from zero to 2 GPa, the calculated volume compressibility is around 71 TPa −1 , and that of the experimental value from zero to 1.65 GPa is around 69 TPa −1 . The calculated b and c axis have a jump from 2.25 to 2.5 GPa, which is accompanied with the proton transfer. Below 2.25 GPa, the H 1 is close to the PCO 3 octahedron and far away from the H 2 O (Fig. 1(c)). When pressure increases to 2.5 GPa, the proton turns to be close to the H 2 O and forms the H 3 O + tetrahedron. Aurélie U Ortiz et al. also found this proton transfer and the H 3 O + tetrahedron formation 36,39 . They used the wine-rack motif to explain the NLC of ZAG-4 after the proton transition. However, as show in Fig. 2b, NLC does not occur immediately after proton transfer. Instead, the b axis smoothly expands from 5 to 6.25 GPa (blue shaded area in Fig. 2(b)), which is far away from the proton transfer pressure (2.5 GPa). This indicates that proton transfer is not enough to lead to the NLC of ZAG-4. There must be something new.
We repeat these calculations with the WC functional, because the lattice constants of solids as determined by it are between LDA and PBE results and on average closer to experiment [45][46][47] . In the WC results of ZAG-4, the H 3 O + tetrahedron is formed at zero pressure. Consequently, the jump from 2.25 to 2.5 GPa in Fig. 2 does not exist in WC functional results (Supplementary Information). Although the H 3 O + is formed at zero pressure, the NLC does not take place immediately from zero pressure. Instead, the NLC of the b axis occurs in WC functional results from 3.5 to 4.75 GPa. This means the H 3 O + tetrahedron and the NLC of b axis are reproduced in WC functional results as well.
We now pay our attention to the changes under pressure of the b axis. The experimentally observed NLC is from 1.65 to 2.81 GPa with the b axis increasing 1.8% (red shaded area in Fig. 2(b)) 36 . The experimental data are too few (only two) to give a detailed description of the lattice parameters evolution. So as to draw a complete picture, we calculate the lattice parameters from 5 to 7 GPa with a small pressure step of 0.25 GPa. As shown in Fig. 3(a), the b axis increases from 5 GPa and reaches its maximum at 6.25 GPa. Accordingly, the average NLC from 5 to 6.25 GPa is −11 TPa −1 , as strong as that of KMn[Ag(CN) 2 ] 3 (−12 TPa −1 , from 0 to 2.2 GPa) 13 . After that, the b axis decreases with pressure increase, and hence the linear compressibility turns to be positive. It is the deformation of the H 3 O + tetrahedron that leads to the NLC of the b axis. As shown in Fig. 1(d,e), the apex oxygen of the H 3 O + tetrahedron is above the H triangle at 5 GPa. The distance (d O-H triangle ) between the apex oxygen and the H triangle is defined to be positive at this condition (left inset of Fig. 3(a)). Comparing the left and right inset of Fig. 3(a) (or Fig. 1(e,f)), we find that d O-H triangle turns from positive to negative with pressure increase. At the same time, evolution of d O-H triangle is completely accompanied with the evolution of the b axis from expansion to contraction. We also calculate the structure of dehydrated ZAG-4. By deleting the H 2 O molecules in ZAG-4 at zero pressure, we get the initial structure of dehydrated ZAG-4. Following the process of Fig. 2, we increase pressure and optimize the structure step by step. The calculated results (Supplementary Information) show that the b axis of dehydrated ZAG-4 decreases smoothly with pressure increase. This means that the H 3 O + tetrahedron deformation is essential to the NLC of ZAG-4.
It is easy to understand the NLC mechanism of ZAG-4 with the following picture in mind. Initially, d O-H triangle of the H 3 O + tetrahedron is positive at low pressure. The apex oxygen moves towards the H triangle with pressure increase. It leads to the decrease of d O-H triangle (Fig. 3(a)), the area expansion of hydrogen triangle (Fig. 3(b)) and the expansion of the b axis ( Fig. 3(a)). Therefore, the NLC of the b axis is directly results from the H 3 O + tetrahedron deformation. With further pressure increase, the apex oxygen penetrates the H triangle and d O-H triangle turns to be negative (right inset of Fig. 3(a)). From then on, the apex oxygen moves away from the H triangle. Consequently, both the area of the H triangle and the b axis decrease with pressure increase. The calculated results using Wu-Cohen functional also show these characteristic features of H 3 O + tetrahedron deformation. In brief, the b axis is expanded by the apex oxygen approaching the H triangle, and is contracted by the apex oxygen moving away from the H triangle.
There are four microscopic mechanisms frequently used: (i) ferroelastic phase transition, (ii) polyhedral tilt, (iii) helical system, and (iv) wine-rack, honeycomb or related topology. 6 The NLC and PLC mechanism of ZAG-4 can be explained by the hexagonal model 55,56 . The non re-entrant and re-entrant hexagons are illustrated in the left and right inset of Fig. 4, respectively. The linear compressibility K l of this mode can be given analytically as 56 (1) l h s in which, as illustrated in the left inset of Fig. 4, i and h are the lengths of the inclined and horizontal ribs, respectively; θ is the angle between the inclined rib and the vertical direction; ϕ h and ϕ s are the hinging and stretching force constants per unit thickness of the hexagonal plane, respectively. The first term of equation (1) assumes that ribs are rigid along their lengths and the deflection totally originates from the change of angle θ, while the second one represents that hexagon deforms solely through stretching of ribs and θ does not change at all. With pressure increase, θ deceases from positive to negative, which is similar to the H 3 O + tetrahedron deformation depicted in the inset of Fig. 3(a). Figure 4 shows the calculated linear compressibility with i/h = 3 and ϕ s /ϕ h = 10. Similarly with the b axis evolution in Fig. 3(a), the left side of Fig. 4 is NLC (34° > θ > 3°) and the right side is PLC (θ < 3°). There are two H 3 O + tetrahedrons with positive d O-H triangle in Fig. 1(d). This condition is similar to the non re-entrant hexagon with NLC. When d O-H triangle becomes negative, it turns to be similar to the re-entrant hexagon with positive NL. Therefore, the NLC mechanism of ZAG-4 can be simplified to the hexagonal model.
In conclusion, based on the first-principle calculations, we investigated the crystal structure of ZAG-4 under pressure. Our calculated evolutions of the lattice parameters with pressure excellently agree with the experimental results. The NLC of ZAG-4 occurs in the pressure span of 1.5 GPa (1.16 GPa) with the b axis increasing 1.2% (1.8%) in calculation (experiment). By inspecting the evolution of atomic position with pressure, we found that the H 3 O + tetrahedron deformation is the key point of understanding the large NLC. Initially, with pressure increase, the apex oxygen moves toward the tetrahedron base and expands the H triangle and the b axis. After penetrating the H triangle, the apex oxygen moves away and then the b axis contracts with pressure increase. The NLC characteristic of ZAG-4 is well reproduced by the hexagonal model, which gives a vivid explanation for the linear compressibility switch from negative to positive values. Usually, crystal has either non re-entrant or re-entrant hexagon framework. ZAG-4 is the first MOFs evolving from non re-entrant to re-entrant hexagon framework with pressure increase. Our finding prescribes a general way to obtain the rare NLC property in complex structures. As H 2 O molecules abundantly exist in inorganic and organic materials, it will prompt more investigations on negative compressibility in complex materials 57 .