Local Distortions in a Prototypical Zeolite Framework Containing Double Four‐Ring Cages: The Role of Framework Composition and Organic Guests**

Abstract Cube‐like double four‐ring (d4r) cages are among the most frequent building units of zeolites and zeotypes. In materials synthesised in fluoride‐containing media, the fluoride anions are preferentially incorporated in these cages. In order to study the impact of framework composition and organic structure‐directing agents (OSDAs) on the possible occurrence of local distortions of fluoride‐containing d4r cages, density functional theory (DFT) calculations and DFT‐based molecular dynamics simulations were performed for AST‐type zeotypes, considering four different compositions (SiO2, GeO2, AlPO4, GaPO4) and two different OSDA cations (tetramethylammonium [TMA] and quinuclidinium [QNU]). All systems except SiO2‐AST show significant deformations, with a pyritohedron‐like distortion of the d4r cages occurring in GeO2‐ and GaPO4‐AST, and a displacement of the fluoride anions towards one of the corners of the cage in AlPO4‐ and GaPO4‐AST. While the distortions occur at random in TMA‐containing zeotypes, they exhibit a preferential orientation in systems that incorporate QNU cations. In addition to providing detailed understanding of the local structure of a complex host‐guest system on the picosecond timescale, this work indicates the possibility to stabilise ordered distortions through a judicious choice of the OSDA, which might enable a tuning of the material's properties.


Introduction
Double four-ring cages (d4r units, face symbol 4 6 , t-cub tile in the nomenclature of natural tilings [1] ) are a prototypical building unit of zeolites and zeotypes. A recent statistical analysis of the zeolite frameworks included in the IZA (International Zeolite Association) Database of Zeolite Structures [2] showed that the tcub tile is the second most frequent tile, occurring in 36 out of 239 zeolite frameworks. [3] As d4r units consist of four-membered rings, the TÀ OÀ T angles (where T = tetrahedrally coordinated atoms such as Si, Ge, Al, P, …) along the edges are relatively small, typically below 140 degrees. Because the equilibrium Si-OÀ Si angle is closer to 150 degrees, [4,5] these building units are strained in all-silica zeolites. [6] The strain can be reduced through an incorporation of heteroatoms, especially germanium, at some corners of the cage, [4,7] or through encapsulation of fluoride anions inside the cages. [8,9] In fact, many all-silica zeolites containing d4r units have, so far, not been obtained in the absence of fluoride. While the formation of a particular framework type is primarily governed by the organic structuredirecting agents (OSDAs), which are encapsulated in larger cavities, this indicates that the fluoride anions play an important structure-directing role in the formation of d4r cages. [10] Examples of d4r-containing all-silica zeolites include octadecasil (AST framework type in the IZA nomenclature [2] ), [11] ITQ-7 (ISV), [12] ITQ-12 (ITW), [13] ITQ-13 (ITH), [14] ITQ-29 (LTA), [15] HPM-1 (STW), [16] and IM-17 (UOV). [17] Some of these neutral-framework materials have been proposed, for example, for applications in adsorption-based gas separations (ITQ-12: propene/propane separation; [18] ITQ-29: carbon dioxide/methane separation [19] ) and in hydrogen-selective membranes, [20] for the storage of mechanical energy through water intrusion/extrusion, [21,22] and as low-k dielectrics. [23] Synthesis in the presence of fluoride is also widely used in the field of germanosilicates, where various extra-large pore zeolites containing d4r units have been reported ("extra-large pore zeolites" have pore apertures formed by rings of at least 14 T atoms). [24,25] The materials IM-12 (UTL), [26] ITQ-33 (ITT), [27] ITQ-37 (-ISV), [28] ITQ-44 (IRR), [29] ITQ-54 (-IFU), [30] and CIT-13 (*CTH) [31] fall in this category. Systems containing Ge-rich d4r units have received particular attention due to the possibility to selectively remove Ge from the framework. [32,33] Selective Ge removal forms the basis of the socalled ADOR (assembly-disassembly-organisation-reassembly) mechanism, a versatile strategy to prepare new frameworks that are inaccessible through direct synthesis. [25,34] Finally, several phosphate-based zeotypes with d4r cages have been synthesised in fluoride-containing media, among them the extra-large pore gallophosphate (GaPO) cloverite (À CLO) [35] and LTA-type aluminophosphates (AlPOs). [36,37] While practical applications of cloverite are severely limited due to its low germanosilicates. [67] It was found that the fluoride anions reside at the cage centre in the pure end members SiO 2 -AST and GeO 2 -AST, but that they are displaced towards one Ge atom, or towards GeÀ OÀ Ge edges, in mixed (Si,Ge)O 2 systems, often forming directional GeÀ F bonds with a length of 2.2 to 2.4 Å. The formation of these bonds leads to a reduction in the freedom of motion that is clearly visible in the root mean square displacements (RMSDs) of fluoride and in the FÀ Ge radial distribution functions (RDFs). Building on this previous study, the present work uses an analogous methodological approach to investigate several new aspects: (1) As pointed out above, off-centre displacements have been reported for some, but not all, AlPO 4 and GaPO 4 zeotypes containing d4r cages, and diffraction methods can provide only limited insights into the preferred local environments.
In order to study the dependence of the equilibrium position of fluoride anions, and of their dynamic behaviour, on the composition of the framework, AST-type zeotypes having four different compositions (SiO 2 , GeO 2 , AlPO 4 , GaPO 4 ) are included. (2) The previous computational study dealt exclusively with models containing the highly symmetric tetramethylammonium (TMA) cation as OSDA. However, AST-type systems like octadecasil [11] and AlPO 4 -16 [50,68] can also be synthesised using the less symmetric quinuclidinium (QNU) cation. Given its very different molecular structure, notably the presence of a terminal À NH group, different framework-OSDA interactions can be anticipated, especially in terms of hydrogen bonding. To this end, TMA-and QNU-containing models are compared to investigate if and how the organic cation affects the equilibrium position and dynamics of the fluoride anions. (3) Previous work on TMA-containing octadecasil postulated the existence of weak CÀ H···O "hydrogen bonds" between methyl groups and framework oxygen atoms. [62,69] The computations performed in this study can give further insights into the potential existence of these bonds, and their significance at finite temperatures. (4) In order to investigate the effect of temperature, AIMD simulations were performed for three temperatures (150 K, 298 K, 573 K), whereas previous work exclusively studied the behaviour at 298 K (room temperature). The central motivation of the present work is the development of a more detailed atomic-level picture of the local structure of these systems, as the insights obtainable with commonly used experimental methods are limited. While this is primarily of fundamental interest, the findings could eventually contribute to an improved understanding of the formation of zeolites and zeotypes during fluoride-mediated synthesis, which might aid the development of new or improved synthesis strategies. Furthermore, the presence of local distortions could also affect potential applications of these materials, especially those related to the dielectric properties.
It has to be noted that AST-type zeolites and zeotypes are of little relevance for applications because their pores are inaccessible to most guest molecules. However, they constitute an ideal model system in the present context for the following reasons: First, AST-type materials have been synthesised in SiO 2 , GeO 2 , and AlPO 4 composition, so only GaPO 4 -AST is a purely hypothetical system. Second, prior experimental studies have shown that each of the larger octadecahedral (ast) cages is occupied by a single OSDA molecule, rather than several molecules. This facilitates the construction of models for the calculations. Given the similarity of the local environment, it can be expected that the findings obtained for AST in the present work are, to a large degree, transferable to other zeolites/ zeotypes with d4r cages (probably excepting those where the cages are strongly distorted).

Structure Models
Throughout this work, labels including both framework composition and OSDA are used to distinguish different systems (e. g., SiO 2 -AST_TMA = all-silica AST incorporating TMA cations; as fluoride anions are present in all models, they are not included in the label). The starting models of the framework structure of SiO 2 -, GeO 2 -, and AlPO 4 -AST were taken from experimental crystal structure data. [11,42,46,50] Since GaPO 4 -AST has, to the authors' knowledge, not yet been synthesised, this model was constructed on the basis of AlPO 4 -AST. In all starting structures, the fluoride anions were located at the centre of the d4r cages. The cationic OSDAs in the larger ast cages are disordered in the experimental structures. As this disorder had to be removed prior to the DFT calculations, some assumptions regarding the orientation of OSDAs in adjacent cages with respect to each other were necessary. The TMA cation has no dipole moment, so it seems plausible to assume that the orientation of a cation in one cage does not have a strong influence on cations in other cages. Therefore, it was assumed that all TMA molecules have the same orientation, preserving the body-centering of the structure and leading to I � 4 symmetry ( Figure 1). [67] The situation is different for the QNU cation, which has a dipole moment. An essentially random orientation of the QNU molecules would require the use of a large supercell, and an identical orientation of all molecules (which would preserve the body-centering) would lead to an overall polarisation, which appears unlikely. In order to avoid both issues, an arrangement was used in which all QNU molecules in one plane perpendicular to the c axis (around z = 0) point in one direction, whereas those in the other plane (around z = 0.5) point in the opposite direction ( Figure 1).

Computational Details
DFT structure optimisations and DFT-based AIMD simulations were performed using the Quickstep part [70] of the CP2K code, [71] installed on the HLRN-III/HLRN-IV facilities of the North-German Supercomputing Alliance. All calculations employed the PBE exchange-correlation functional [72] in conjunction with Grimme's D3 dispersion correction. [73] This PBE-D3 functional delivered fairly accurate results in a recent benchmarking study of zeolites and zeotypes, [74] and it was successfully used in previous AIMD investigations of fluoride-containing zeolites. [67,75] A plane-wave energy cutoff of 600 Ry was used, and the first Brillouin zone was sampled at the Γ point, only, in keeping with previous work. [67] The calculations employed Goedeker-Teter-Hutter pseudopotentials devised by Krack. [76] Gaussian triplezeta (TZVP/TZVP-SR) basis sets were used in the structure optimisations, whereas the AIMD simulations used double-zeta (DZVP-SR) basis sets. [77] These basis sets are available in the Basis_Molopt_Ucl (TZVP-SR basis sets for Al, Ga, Ge) and Basis_Molopt (all others) files distributed with the CP2K code. All structure optimisations were carried out using the tetragonal unit cell of AST-type zeolites (a � 9 Å, c � 14 Å). The atomic positions and unit cell parameters were optimised, fixing the unit cell to a tetragonal metric (a = b, α = β = γ = 90°), and using the following convergence criteria: maximal geometry change (step size) = 2 × 10 À 5 bohr, maximal residual force (gradient) = 10 À 6 Ha bohr À 1 , and maximal pressure deviation = 0.01 GPa. The AIMD simulations, which started from the optimised structures, employed a 2 × 2 × 1 supercell. These simulations were performed in the canonical (NVT) ensemble for temperatures of 150 K, 298 K, and 573 K, using a Nosé-Hoover thermostat [78,79] with a timestep of 0.5 fs and a time constant of 50 fs. Three independent AIMD simulations covering 10 ps (20,000 steps) were run for each system at each temperature. The first 2.5 ps (5,000 steps) from each trajectory were discarded (equilibration phase), and the remaining 7.5 ps (15,000 steps) were used in the analysis (production phase). The analysis made use of the VMD code, version 1.9.3, [80] to calculate radial distribution functions (RDFs) for selected combinations of elements, root mean square displacements (RMSDs), and average structures (= average atomic positions over the whole 7.5 ps). All RDFs and RMSDs discussed throughout this work correspond to averages over three independent trajectories. The numerical results are compiled in an EXCEL file that has been deposited in the Figshare repository under https:// doi.org/10.6084/m9.figshare.12981557.v1. Archives containing the trajectories in PDB format (production phase only), DFToptimised structures and AIMD average structures in CIF format, and sample input files have been deposited in the same repository. All structure visualisations in this article were prepared using Vesta. [81]

Results and Discussion
The first part of this section presents the results of the DFT optimisations, with most emphasis on the local structure of the d4r units. The second part addresses the AIMD simulation results. After giving a brief overview of the RMSDs of framework atoms and fluoride anions, the local environments of fluoride anions are investigated in more detail by analysing RDFs and AIMD average structures. As the behaviour differs considerably as a function of framework composition, these results are presented separately for each zeotype. The role of OSDA dynamics and framework-OSDA hydrogen bonds is discussed in the last part.

DFT Structure Optimisations
Prior to discussing the optimisation results, a brief description of the AST framework is warranted. In the cubic aristotype (space group Fm � 3m), the T sites at the corners of the d4r cages, labelled T1, are equivalent. The d4r cages are connected to each other via linkages to another type of T atom, labelled T2 ( Figure 1). While the edges of the d4r cages are formed by T1À OÀ T1 linkages, T1À OÀ T1 and T1À OÀ T2 linkages together form the edges of larger ast (octaedecasil) cages (face symbol 4 6 · 6 12 ). The T1À OÀ T2 angle in the cubic aristotype is fixed to 180°for symmetry reasons. [68] Due to the instability of such straight linkages, [82] a reduction in symmetry occurs in real ASTtype materials: Whereas materials with T = Si and Ge possess I4=m symmetry, [11,42,83] those with two different species at the T sites, like AlPO 4 -16, crystallise in space group I � 4. [50,84] When performing DFT calculations, a further reduction in symmetry is often necessary because the disorder of the OSDA molecules needs to be removed, as discussed above.
The unit cell parameters of the DFT-optimised AST systems are summarised in Table 1. When comparing pairs of systems having the same framework composition, but containing different OSDAs, it is worth noting that the incorporation of QNU leads to an expansion of the a axis and a concurrent contraction of the c axis when compared to the corresponding TMAcontaining system. Clearly, this is related to the elongated shape of the QNU molecules, which are oriented in a way that the longest molecular dimension lies in the ab plane. Among systems containing the same OSDA, but having different framework composition, replacing elements from the 3 rd period (Si, Al) by elements from the 4 th period leads to an expansion of the unit cell, especially along the c axis, in line with the increase of the atomic radius of the T atoms. Experimental cell parameters are available for three of the eight systems, and the corresponding values are listed in Table 1. Compared to the experimental values, the DFT calculations deliver a shorter a axis and a longer c axis. Similar tendencies were observed and discussed in previous DFT studies of calcined SiO 2 -AST. [85,86] The effect is fairly prominent for AlPO 4 -AST_QNU, where the relative deviation in a exceeds À 2 %, and moderately pronounced for SiO 2 -AST_TMA. In contrast, agreement with experiment is excellent for SiO 2 -AST_QNU.
As the CP2K optimisations do not make use of symmetry information, slight deviations from ideal symmetry occur in the DFT-optimised structures (up to a few 1/100 Å). In order to restore the symmetry prior to the analysis, a symmetry search was carried out using Materials Studio. [87] The results, tabulated in Table 1, show that all TMA-containing systems have I � 4 symmetry, the highest possible symmetry permitted by the chosen arrangement of the OSDA molecules in the ast cages. [67] All QNU-containing systems have symmetry P2 1 after the symmetry search. The reduction in symmetry is related to a) the lower symmetry of the OSDA, which removes the fourfold rotoinversion axis, and b) the relative orientation of the QNU molecules in planes perpendicular to c, visualised in Figure 1, which removes the body-centering. The d4r cages of the DFT-optimised AST_TMA systems are shown in Figure 2, which also reports the ranges of TÀ O and TÀ F distances and TÀ O-T angles along the cage edges (i. e., T = T1). The TÀ O distances fall into the respective ranges determined in a statistical analysis of experimental crystal structures, [88] they are not further discussed here. In SiO 2 -and GeO 2 -AST_TMA, the fluoride anions are located at the cage centre, as evidenced by a very narrow distribution of the TÀ F distances. The distances agree well with those determined experimentally for AST-type SiO 2 and GeO 2 systems (d(SiÀ F) exp = 2.63 Å, d(GeÀ F) exp = 2.75 Å). [42,46] The GeÀ OÀ Ge angles are significantly smaller than the SiÀ OÀ Si angles, in line with the smaller equilibrium angle. [4,5,45,89,90] In AlPO 4 -and GaPO 4 -AST_ TMA, the distributions of Al/GaÀ F and PÀ F distances are also very narrow, with the Al/GaÀ F distances being systematically shorter than the PÀ F distances. The same trend has been observed in crystallographic studies of d4r-containing aluminoand gallophosphates, [35,36,50,51] and quantitative agreement with experimental TÀ F distances is satisfactory. An inspection of OÀ TÀ O angles (for O atoms occupying edges of the d4r cage) shows larger deviations from the ideal tetrahedral angle for AlO 4 /GaO 4 tetrahedra compared to PO 4 tetrahedra, with individual OÀ AlÀ O/OÀ GaÀ O angles reaching 118°/121°, respectively (OÀ P-O angles do not exceed 114°). Apparently, the AlO 4 /GaO 4 tetrahedra are less rigid than the PO 4 tetrahedra, and attractive electrostatic interactions with the fluoride anions cause a certain displacement of the metal cations into the cage. The higher degree of flexibility can be explained with the more ionic (= less directional) nature of the AlÀ O/GaÀ O bonds in comparison to the dominantly covalent PÀ O bonds. [91] A visualisation of the TMA cations and the surrounding ast cages is provided in the Supporting Information, Figure S1. This figure shows that two or three hydrogen atoms of each methyl group form relatively close (< 2.7 Å) contacts to framework oxygen atoms, with H···O distances of 2.57/2.66 Å in SiO 2 -AST_ TMA, 2.44/2.52 Å in GeO 2 -AST_TMA, 2.68/2.64 Å in AlPO 4 -AST_ TMA, and 2.63/2.51/2.60 Å in GaPO 4 -AST_TMA. These distances appear to be too long to serve as unambiguous evidence for the presence of CÀ H···O hydrogen bonds, which have been postulated in a previous NMR study of SiO 2 -AST_TMA. [69] On the other hand, they are up to~10 % smaller than the sum of van der Waals radii of hydrogen and oxygen of 2.70 Å, [92] indicating a non-negligible electrostatic interaction. Figure 3 shows one d4r cage of the DFT-optimised AST_ QNU structures as well as the adjacent QNU cation that is hydrogen-bonded to a framework oxygen atom via the À NH group (each d4r cage is surrounded by six QNU cations, but only one of them is hydrogen-bonded to an O atom belonging to the cage). In all four cases, the TÀ F distances show a slightly broader distribution than in the AST_TMA systems. When looking at the individual distances, it becomes apparent that the fluoride anions are slightly displaced from the cage centre towards the hydrogen-bonded QNU cation; however, this displacement amounts only to a few 1/100 Å. The TÀ OÀ T angles along the edges are rather similar to those in AST_TMA systems, with the exception of GeO 2 -AST_QNU, where they are systematically smaller. This reduction in angles coincides with a pronounced distortion of the d4r cage. While the GeÀ Ge distances are hardly affected, the OÀ O distances measured across the faces show a bimodal distribution, with "short" OÀ O distances of about 3.4 Å and "long" OÀ O distances of about 4.7 Å ( Figure S3). In contrast, all OÀ O distances in GeO 2 -AST_ TMA fall close to 4.2 Å. Similarly distorted d4r cages have been previously observed in a crystallographic study of the germanate ASU-7, where they were described as rectangular prisms. [93] A closer look at these units reveals that the idealised symmetry is higher than that of a general rectangular prism (point group mmm), because three faces meeting at each corner are related by a threefold rotation axis (point group m � 3). This feature can be illustrated by drawing auxiliary OÀ O connections across the faces. As shown in Figure S3, the resulting shape resembles a pyritohedron, i. e., an irregular pentagonal dodecahedron consisting of identical, but irregular pentagons. In the following, these distorted d4r cages will be referred to as "pyritohedronlike" in order to distinguish them from undistorted "cube-like" cages. It is worth noting that published crystal structures of the GeO 2 zeotypes ASU-7 and FOS-5 contain d4r units whose geometry deviates only slightly from point group symmetry m � 3. [47,93] Prior to a further analysis of the framework distortion in GeO 2 -AST_QNU, it is useful to take a look at the NÀ H···O hydrogen bonds. The H···O distances are included in Figure 3. This distance is longest in SiO 2 -AST_QNU, with 1.98 Å, and shortest in GeO 2 -AST_QNU, with 1.70 Å, indicating a pronounced difference in the hydrogen bond strength. The relatively short, strong hydrogen bond in GeO 2 -AST_QNU can be explained with the less obtuse GeÀ OÀ Ge angles, which render the oxygen atom more exposed, resulting in an increased hydrogen bond acceptor ability. Besides, electron density maps show a significantly higher valence electron density along the H···O connection in comparison to the other QNU-containing zeotypes, pointing to a more covalent character of the hydrogen bond ( Figure S4).
The symmetry search was repeated for models in which the extra-framework content (fluoride anions, OSDA cations) was removed (last column of Table 1). Removal of the extra-framework species results in space group I4=m for SiO 2 -/GeO 2 -AST models with the exception of GeO 2 -AST_QNU, and space group I � 4 for AlPO 4 -and GaPO 4 -AST. The reduction in symmetry of GeO 2 -AST_QNU to I2=m is related to the presence of the pyritohedron-like d4r cages, which have lost the fourfold rotation symmetry. A comparison of the two GeO 2 -AST frame-works ( Figure S5) reveals that the GeO 4 tetrahedra in GeO 2 -AST_ QNU are rotated with respect to their orientation in the TMAcontaining system. The rotation of tetrahedra occurs at both the T1 and the T2 sites, and results in the distortion of the d4r cages observed above. It is interesting to note that the T2À OÀ T1 angles increase as a consequence of these concerted rotations, from~127°in GeO 2 -AST_TMA to~131°in GeO 2 -AST_ QNU (Figure S1/S2).
Taken together, there are two possible explanations for the framework distortion observed in GeO 2 -AST_QNU, and its absence in GeO 2 -AST_TMA: 1) The distorted framework (I2=m) is intrinsically more stable, possibly due to the narrower distribution of GeÀ OÀ Ge angles. The DFT optimisation of GeO 2 -AST_TMA found only a local minimum, but not a global minimum, due to the high symmetry of the starting structure.
2) The stabilisation of the I2=m structure over the I4=m structure is related to framework-OSDA interactions, specifically, the formation of hydrogen bonds between QNU molecules and framework oxygen atoms. This point will be revisited when discussing the MD results for the two GeO 2 -AST systems.

RMSDs of Framework Atoms and Fluoride Anions
Prior to discussing radial distribution functions and AIMD average structures for the individual systems, it is useful to take a look at the RMSDs computed for framework atoms and fluoride anions (large standard deviations and significant scatter severely restrict the possibilities of a meaningful analysis of the OSDA atoms' RMSDs). They are tabulated in the Supporting Information (Table S1)

SiO 2 -AST
The FÀ Si RDFs of SiO 2 -AST systems are shown in Figure 4 and the average structures computed from individual trajectories are visualised in Figures S6a and S7a. The FÀ Si RDFs of SiO 2 -AST_TMA show a symmetric maximum centred at~2.65 Å, which corresponds to oscillations of fluoride about its equilibrium location at the centre of the cage, replicating the findings from the earlier AIMD study of this system. [67] Due to increased thermal motion, the maximum becomes broader with increasing temperature. The lower part of Figure 4 overlays the trajectories of one individual fluoride anion inside a d4r cage with the time-averaged positions of the surrounding framework atoms. It is apparent that the number of short SiÀ F contacts (below 2.3 Å) increases with temperature due to the more vigorous motion of the fluoride anion, and that such short contacts occur without any preferential direction of displacement.
Compared to the TMA-containing system, the FÀ Si RDFs of SiO 2 -AST_QNU show some rather intricate differences, with more prominent "shoulders" at both smaller and larger FÀ Si distances that are best visible at 150 K. This observation can be attributed to a certain off-centre displacement of fluoride. The slight change in the equilibrium location is hardly visible in the average structures ( Figure S7a), but it becomes more apparent when looking at the trajectory of an individual fluoride anion (lower part of Figure 4): At a temperature of 150 K, the fluoride anion is preferentially displaced towards the SiÀ OÀ Si linkage whose oxygen atom participates in the hydrogen bond. With increasing temperature, this effect becomes less pronounced due to the increased overall motion.

GeO 2 -AST
The FÀ Ge RDFs as well as trajectories of individual fluoride anions are visualised in Figure 5. Altogether, the findings mirror those discussed above for SiO 2 -AST systems: The RDFs show a single maximum, centred at a distance of about 2.75 Å. In GeO 2 -AST_TMA, the fluoride anions oscillate about the cage centre, whereas they are slightly displaced towards the oxygen atom participating in the hydrogen bond in GeO 2 -AST_QNU. Again, this displacement is best visible at 150 K. Neither system delivers any indications for the presence of GeÀ F bonds that have been found to occur in mixed (Si,Ge) d4r cages. [67] More interesting observations can be made when looking at the shape of the d4r cages. As discussed above, the DFT optimisations deliver a highly symmetric (cube-like) cage in GeO 2 -AST_TMA, but a distorted pyritohedron-like d4r cage in GeO 2 -AST_QNU. An inspection of the average structures of GeO 2 -AST_TMA ( Figure S8a), for which a representative example is shown in Figure 6, reveals that all d4r cages exhibit this pyritohedron-like distortion, causing the loss of tetragonal symmetry described above. This indicates that GeO 2 -AST containing pyritohedron-like cages is intrinsically more stable than the higher-symmetry form with cube-like d4r cages, and that the distortion is not governed by the OSDA. A reoptimisation of a selected 150 K average structure and the I � 4 structure, using the same supercell, confirms this, as the structure with pyritohedron-like d4r cages is favoured by À 2.5 kJ mol À 1 per GeO 2 formula unit (À 26 kJ mol À 1 per fluoride anion).
Due to the reduced symmetry, two different orientations are possible for the pyritohedron-like d4r cages. In GeO 2 -AST_ TMA, both orientations occur randomly, and such a statistical distribution has been dubbed "orientational glass" in previous studies of AST-and ASV-type systems. [45,93] In contrast, a strict ordering is observed in GeO 2 -AST_QNU at 150 and 298 K, which can be attributed to the interaction with the ordered QNU cations. Only at the highest temperature, some d4r cages in GeO 2 -AST_QNU have a different orientation. Additionally, a few of the d4r cages in the 573 K average structures show no pyritohedral distortion at all, which probably indicates that a change in orientation occurs during the AIMD run of 7.5 ps. Although the published crystal structure of GeO 2 -AST contains cube-like d4r cages, it should be noted that the observed  preference for distorted d4r cages agrees with the report of similarly distorted cages in the structurally related germanate ASU-7. [93] This raises the question to what extent such deformations are a general feature of d4r-containing GeO 2 zeotypes (or Ge-rich zeolites). The coexistence of different orientations, together with thermal motion, would render it difficult to pin this down with diffraction methods.

AlPO 4 -AST
The FÀ Al RDFs of AlPO 4 -AST zeotypes, shown in Figure 7, exhibit very different features than those calculated for SiO 2 -and GeO 2 -AST: Rather than having one broad maximum, there are two maxima, a sharp one centred at~1.9 Å, and a considerably broader one centred at~2.8 Å. A closer look at the trajectories of individual anions (Figure 7) and at the average structures (Figures S10a and S11a) confirms an off-centre displacement of the fluoride anions towards one of the Al atoms at the corners of the d4r cage, leading to the formation of AlÀ F bonds. A reoptimisation of AIMD average structures obtained at 150 K, and comparison to the initial structures with fluoride at the cage centre delivers energy differences of À 2.2/À 3.8 kJ mol À 1 per AlPO 4 formula unit (À 11/À 19 kJ mol À 1 per fluoride anion) for AlPO 4 -AST_TMA/AlPO 4 -AST_QNU, corroborating that the formation of AlÀ F bonds is energetically favoured over a centre-ofcage position of fluoride. The optimised AlÀ F bond lengths of 1.90 Å agree with values observed in aluminophosphates where fluoride anions bridge between two Al atoms. [53,55,57] Due to the displacement of fluoride towards one Al corner, the coordination number of that Al atom increases to five, and an analysis of the distances and angles around a representative Al V atom shows only relatively minor deviations from an ideal trigonal-bipyramidal coordination environment (  similar to that of the framework oxygen atoms due to the similar bond strength, as is indeed observed for 150 K. At higher temperature, the jumps between different Al atoms lead to a drastic increase in the RMSD values. While the formation of AlÀ F bonds occurs in both AlPO 4 -AST_TMA and AlPO 4 -AST_QNU, the nature of the OSDA has a pronounced effect on the direction of fluoride displacements: In the TMA-containing system, the displacement pattern is random, without detectable preference for any of the four available Al atoms (Figure 9). The situation is different in AlPO 4 -AST_QNU, where fluoride anions are preferentially bonded to the Al atom neighbouring the oxygen atom that acts as hydrogen bond acceptor. This preference is almost perfectly realised at 150 and 298 K ( Figure S11a). It can be explained as a consequence of the local perturbations that are related to the NÀ H···O hydrogen bond: Because the oxygen atom participates in the hydrogen bond, electron density is withdrawn from the Al atom, leading to an increased positive polarisation which favours the formation of an AlÀ F bond. Additionally, the oxygen atom is somewhat displaced into the ast cage, rendering the Al atom more amenable to an expansion of its coordination environment to trigonal-bipyramidal. The preferred direction of fluoride displacement resembles the ordering of the pyritohedron-like d4r cages observed in GeO 2 -AST_QNU: In both cases, the reduced symmetry of the OSDA, together with its ability to form hydrogen bonds, causes an energetic preference for a particular orientation of the fluoride-containing d4r cages.
It is worth pointing out that neither previous crystallographic studies nor the DFT optimisations of the present work pointed to the presence of AlÀ F bonds. The inability of diffraction methods to observe this off-centre displacement can be attributed to a) the lack of ordering of the displacements in real crystals, where the OSDA molecules are likely to exhibit orientational disorder, and b) the high mobility of the fluoride anions. It is worth noting that NMR studies of AlPO 4 -16 synthesised in the presence of fluoride anions reported two signals in the 19 F MAS NMR spectra, one of which was attributed to "terminal" fluoride anions. [50,84] Tentatively, this signal could be due to fluoride anions that remain bonded to a single Al atom for relatively extended periods (on the timescale of NMR experiments). The fact that the DFT optimisations reported in 3.1 did not deliver AlÀ F bonds indicates that the structures with fluoride at the cage centre correspond to local minima on the potential energy surface, which the DFT optimisation algorithm cannot leave.

GaPO 4 -AST
The FÀ Ga RDFs obtained for GaPO 4 -AST show two maxima, indicating an off-centre displacement of the fluoride anions that is comparable to that observed in AlPO 4 -AST ( Figure 10). However, the maxima are less well separated, and significant g(r) values are observed between them even at 150 K, especially for GaPO 4 -AST_TMA. A visualisation of the average structures (Figures S12a and S13a) shows that d4r cages in which fluoride is bonded to a single Ga atom are more typical in GaPO 4 -AST_ QNU, whereas configurations where fluoride is displaced towards one face with similar distances to two Ga atoms occur more frequently in GaPO 4 -AST_TMA. This distinct difference, which is qualitatively visible in the visualisation of individual fluoride trajectories (lower part of Figure 10), also manifests as a shift of the first maximum in the FÀ Ga RDF towards lower distances in GaPO 4 -AST_QNU. To corroborate this further, average structures obtained at 150 K were re-optimised. For GaPO 4 -AST_TMA, the energy difference with respect to the I � 4 structure amounts to À 2.7 kJ mol À 1 per GaPO 4 formula unit (À 13.5 kJ mol À 1 per fluoride anion). An inspection of the optimised structure shows many d4r cages with two GaÀ F contacts of similar length (typically~2.3 Å), indicating that such "bridging" modes of fluoride correspond to local minima (Figure 11, top). An analogous calculation for GaPO 4 -AST_QNU delivers an energy difference of À 2.8 kJ mol À 1 per GaPO 4 formula unit (À 14 kJ mol À 1 per fluoride anion). In this case, configurations having a single GaÀ F bond with a length of 2.08 Å dominate (Figure 11, bottom). This dependence of the equilibrium position of fluoride on the OSDA, together with the coexistence of different binding modes within the same structure, point to a very shallow potential energy surface with different local minima. Again, the DFT-optimised GaÀ F distances are close to those observed in experimental structures of systems containing GaÀ FÀ Ga bridges. [10,54] In addition to the off-centre displacement of the fluoride anions, the d4r cages in the GaPO 4 -AST zeotypes also show a pyritohedron-like distortion, as observed above for GeO 2 -AST (bottom part of Figure 10). In the TMA-containing system, neither fluoride displacements nor the distortions of the d4r cages are ordered in any way ( Figure S12a). In GaPO 4 -AST_QNU, the same displacement pattern of the fluoride anions as in the corresponding AlPO 4 -AST system is observed, which, while being most pronounced at 150 K, persists up to 573 K (Figure S13a). In contrast, the distortions of the d4r cages have a random orientation, at variance with the findings for GeO 2 -AST_ QNU. Apparently, framework-OSDA interactions have a larger influence on the preferential formation of GaÀ F bonds than on the deformation of the d4r units in GaPO 4 -AST, and the two phenomena are not strictly coupled to each other.

OSDA Dynamics and Hydrogen Bonds
The final part of the analysis deals with the dynamic behaviour of the OSDA molecules, with special emphasis on the presence of hydrogen bonds and their evolution with temperature. Due to significant disorder of the OSDAs at 298 K and 573 K, the AIMD average structures deliver only a limited amount of information in this regard. The analysis presented here relies on the FÀ N and HÀ O RDFs, because these quantities provide a convenient means to assess the most relevant features of the OSDA dynamics. For the FÀ N RDFs, only fluoride anions and nitrogen atoms lying approximately in one plane perpendicular to the c axis were considered, leading to four, rather than six, FÀ N contacts in the distance range up to~10 Å (Figure 1).
With the exception of the FÀ N RDF of AlPO 4 -AST_TMA, which will be addressed separately below, the FÀ N and HÀ O RDFs of all other TMA-containing systems show very similar features (see Supporting Information). They are visualised for the representative example of GeO 2 -AST_TMA in Figure 12. The symmetric maximum in the FÀ N RDF, which broadens with temperature, can be attributed to oscillations of the TMA cations (and, to a lesser extent, the fluoride anions) about their equilibrium positions. The HÀ O RDF shows a broad shoulder that starts to rise at distances of about 2.1 to 2.2 Å at 150 K, and moves progressively towards lower distances with increasing temperature. In their combined DFT and NMR study of SiO 2 -AST_TMA, Dib et al. concluded that weak CÀ H···O "hydrogen  bonds" are formed between some TMA hydrogen atoms and framework oxygen atoms. [69] If such bonds were present, it would be reasonable to expect an elongation of the H···O distance with increasing temperature, as increased thermal motion should weaken and -at sufficiently high temperaturebreak the bonds. The observation of shorter, rather than longer H···O distances upon increasing temperature provides no evidence for the presence of hydrogen bonds, indicating instead that non-directional interactions dominate.
Since the features of FÀ N and HÀ O RDFs of the QNUcontaining systems having different framework composition are also qualitatively similar, the discussion will largely focus on the results obtained for GeO 2 -AST_QNU ( Figure 12). The FÀ N RDF obtained for 150 K shows four distinct maxima, which correspond to the distances from a given fluoride anion to the nitrogen atoms belonging to QNU cations in the four surrounding ast cages (Figure 1). With increasing temperature, the four maxima become increasingly blurred, leading to a single broad maximum with a poorly developed substructure at 573 K. The broadening of the FÀ N RDF is considerably more pronounced than that of the FÀ Ge RDF (Figure 5), which can be attributed to the increased thermal motion of the OSDA cations. This is corroborated when looking at the HÀ O RDFs: At 150 K, there is a distinct first maximum centred at~1.7 Å, together with a second, smaller maximum centred at~2.1 Å. The respective positions of these maxima agree with the lengths of the NÀ H···O/CÀ H···O hydrogen bonds from the terminal H atoms of the QNU molecule to framework O atoms in the DFT-optimised structure ( Figure S2). A third, broad maximum at~2.6 Å corresponds to other H···O contacts. With increasing temperature, the first maximum decreases significantly, and the second maximum disappears altogether (the first maximum also disappears in other QNU-containing systems at 573 K). This indicates that the hydrogen bonds are weakened, and eventually broken, due to more vigorous movements of the QNU cations. Once the hydrogen bonds are broken, the OSDA molecules are prone to reorient within the ast cages. Indeed, the visualisation of representative last AIMD frames obtained for SiO 2 -AST_QNU and GeO 2 -AST_QNU (for T = 573 K) shows that several QNU cations have changed their orientation with respect to their perfect ordering in the starting structures ( Figure S14). Given the impact of framework-OSDA interactions on the ordering of pyritohedral distortions in GeO 2 -and GaPO 4 -AST_QNU, and on the ordered displacements of fluoride anions in AlPO 4 -and GaPO 4 -AST_QNU, it is straightforward to link the reorientation of some QNU cations to a reduced degree of ordering at 298 K and, especially, 573 K. The pronounced orientational disorder of the OSDA in the experimental crystal structures of SiO 2 -AST_QNU and AlPO 4 -AST_QNU (AlPO 4 -16), obtained at room temperature, agrees with these AIMD results. [11,50] Finally, it is worth taking a separate look at the FÀ N RDFs of AlPO 4 -AST_TMA and AlPO 4 -AST_QNU. In the former system, the FÀ N RDF shows a distinct two-peak maximum at 150 K that develops into an unusually broad maximum at higher temperatures ( Figure S10d). This observation is clearly related to offcentre displacements of the fluoride anions towards Al atoms, which lead to shorter and longer distances to the central nitrogen atoms of the surrounding TMA molecules. In the case of AlPO 4 -AST_QNU, the FÀ N RDF obtained for 150 K shows a first main maximum centred at~4.9 Å and a secondary maximum centred at~5.3 Å. This secondary maximum stems from those local environments where fluoride anions are displaced towards Al atoms that do not neighbour the hydrogen bond acceptor O atom. Although these displacements account for only a minor fraction of all fluoride anions (2 out of  Figure S11a), their contribution is clearly visible in the FÀ N RDF. Even at the lowest temperature considered, the ordered displacement of the fluoride anions imposed by the ordering of QNU cations is not strictly realised.

Conclusions
The AIMD simulations have revealed a distinctly different behaviour of the fluoride-containing d4r units depending on the framework composition: In SiO 2 -AST, the fluoride anions oscillate about the cage centre, and the d4r cages retain their cube-like shape (apart from transient distortions). Although the fluoride anions possess a larger freedom of motion in GeO 2 -AST due to the larger dimensions of the cage, their average positions remain close to the cage centre. However, the d4r cages show a pronounced pyritohedral distortion, which goes hand-in-hand with a concerted rotation of the GeO 4 tetrahedra and a narrower distribution of the GeÀ OÀ Ge angles. In AlPO 4 -AST, the fluoride anions are displaced towards one of the Al corners of the cage, forming AlÀ F bonds having a length of 1.9 Å. Apart from the formation of an AlO 4 F trigonal bipyramid at one corner, the d4r cages are only slightly distorted. Finally, GaPO 4 -AST combines both phenomena observed in GeO 2 -AST and AlPO 4 -AST, exhibiting both a pyritohedron-like distortion and a displacement of fluoride anions towards one or two Ga corners. Although these distortions occur, in the first place, independently of the nature of the OSDA, pronounced differences are observed with regard to the ordering of the distorted configurations: If the structures contain highly symmetric TMA cations, for which non-directional interactions with the framework dominate, the distortions occur in an essentially random fashion. If less symmetric QNU cations are incorporated, these form (at least) one hydrogen bond to the framework. An ordered arrangement of the QNU cations, assumed in the starting structures, then leads to an ordered pattern of hydrogen bonds which causes, in turn, an ordering of the pyritohedron-like distortions (GeO 2 -AST) or the displacements of fluoride anions towards one corner of the cage (AlPO 4 -AST, GaPO 4 -AST). In other words, ordering of the OSDAs triggers collective deformations of the framework. The degree of ordering decreases with temperature due to increased thermal motion.
It has to be conceded that the assumption of a fully ordered arrangement of the QNU cations is a significant approximation. It cannot be expected that such a strict ordering would occur in real AST-type zeotypes, which are typically synthesised at temperatures of 450 to 480 K, where thermal motion of the OSDAs will be considerable. [46,50,83] Nevertheless, the present work provides important new insights into the local structure of these systems, and it highlights the theoretical possibility to induce collective framework deformations through an ordering of the extra-framework species. It could be envisaged to judiciously choose other OSDAs that stabilise such distortions, which could be proposed on the basis of further computations. External fields might provide another possibility to induce an ordering of the OSDAs. While this work is not aimed at the prediction of any specific property of the investigated zeotypes, it can be anticipated that some material properties could be tuned through ordered distortions, which might be relevant for potential applications (e. g., in dielectrics).
It is worth emphasising that DFT optimisations starting from symmetric structures did not reveal the majority of local distortions, which became apparent only in the AIMD simulations. Except in the case of GeO 2 -AST_QNU, the undistorted structures are local minima, and the optimisations will not leave these local minima. This highlights that it is advisable to either employ AIMD simulations, or -if these are not feasible -to generate a set of perturbed starting structures prior to the DFT optimisations in cases where local distortions are expected.
Finally, it is useful to revisit previous experimental findings in the context of the present work, and to discuss possible ways to verify the findings experimentally. With regard to GeO 2 zeotypes, some previous studies have already shown a tendency of the d4r units to distort. [44,47,49,93] This distortion is sometimes visible as a disorder of the edge oxygen atoms over different positions. On the other hand, the formation of short AlÀ F or GaÀ F bonds has, so far, not been observed in AlPO 4 or GaPO 4 zeotypes where fluoride anions are encapsulated in the d4r cages. Clearly, the lack of long-range ordering in real systems will hamper a detection of such bonding scenarios with (powder) diffraction methods, as the occupancy of the fourfolddisordered F position would only amount to 0.25. The highly dynamic behaviour at room temperature might render it difficult to detect AlÀ F/GaÀ F bonds with solid-state NMR methods, but investigations at cryogenic temperatures could provide more insights. Furthermore, it could be interesting to assess whether the presence of these bonds should give rise to distinct signals in the vibrational spectra, which could possibly serve as "fingerprints". In any event, the partly unexpected findings of the present study show that it should be worthwhile to revisit these zeotypes with state-of-the-art experimental methods.

Supporting Information
The Supporting Information to this article includes a PDF file providing additional figures (details of DFT-optimised structures, AIMD average structures, radial distribution functions).