The flexibility of the T–X–T hinges between the coordination tetrahedra in various zeolitic frameworks: an empirical structural study

In recent years, a large number of precisely determined crystal structure determinations of zeolites, that is compounds based on open frameworks of coordination tetrahedra, has accumulated. This gives us the opportunity to evaluate them in terms of the flexibility of their bond angles T–X–T which serve as hinges between the coordination tetrahedra in zeolites containing various SiO4, AlO4, PO4 and similarly sized ions. Out of about 7,000 known crystal structure descriptions of zeolites we have selected 1187 which are based on single crystal diffraction studies and have estimated standard deviations of their T–O bond length of 0.01 Å or less. The mean Si–O–Si angle in zeolitic frameworks with T-sites fully occupied by one type of T atoms is 152.9° and these angles range from 127.0° to 180°, the mean for Al–O–Si is 141.9°, ranging from 124.8° to 179.3°. Straight angles Si–O–Si do occur in some zeolites.


Introduction
The porous frameworks of zeolites with tetrahedrally coordinated T atoms, such as mostly Si 4+ , Al 3+ , P 5+ but also, among others, Zn 2+ , Be 2+ , Ge 4+ , B 3+ , As 5+ , Ga 3+ , Co 2+ , crystallize in 247 different topologies (without intergrowth types) named by different framework type codes (FTC). They are listed by Baerlocher and McCusker (2022) and maintained by them for the Structure Commission of the International Zeolite Association. The scientific and commercial interest in zeolites is based on their porosity making these compounds useful for a wide variety of applications (Li and Yu 2014). The sizes of the pores are crucial in determining their porosity which in turn depends on the topology in the variation of the unit-cell dimensions when comparing differently chemically substituted pore fillings of the frameworks or when looking at their response to changes in temperature or pressure. We take the difference between minimal and maximal values observed for an angle T-X-T of a given chemical composition as an approximate measure of the flexibility of that angle in different frameworks, see column Δ max in Table 1. Below we are also going to use the corresponding measure within one particular framework as an indicator of its flexibility under varying conditions.
Another influence on the FD is of course the value of the mean tetrahedral bond lengths T-O which vary from 1.474 Å for B-O to 1.958 Å for Co-O among the 10 kinds of ions listed above. This has been studied by us in a previous paper (Baur and Fischer 2019) and is used as a basis for the data presented here. Specifically, we explore the interplay between the variation in the T-X-T angles and some of the properties of the frameworks (FD, unit-cell dimensions, and volumes). In our previous paper (Baur and Fischer 2019), we studied the astonishing variety of shapes among the TO 4 coordination tetrahedra themselves in zeolitic frameworks. In addition, we established the fact that the mean bond lengths in the tetrahedra may depend on the individual structural topology of the framework, even after accounting for variations in the influence of the value of the displacement factors of the constituting atoms. Furthermore, it was shown that the largest distortions in O-T-O angles are mostly a function of the mean T-O bond length and not of the oxidation state of T (Baur and Fischer 2019).
In this paper, we look at the variations in the hinges T-X-T, where the X atoms serve as bridges between the coordination tetrahedra around the T atoms. We do not consider the topology of the zeolite type frameworks per se, but only its influence on the magnitude and flexibility of the angles at the T-X-T hinges. A special focus will be on the dependence of the T-X-T angle on the chemical nature of the X and T atoms, on zeolites with straight T-O-T angles, and on examples of flexible and/or collapsible tetrahedral frameworks.

Data
Inasmuch as the introductory remarks presented by Baur and Fischer (2019) apply here as well, we can be short. Suffice to say that the data we are using now are covered by us in our Landolt-Börnstein volumes (Baur and Fischer 2000Fischer and Baur 2006, 2009, 2014 and Table 1 Mean T-X-T angles in zeolites from a high-quality data set (with one exception), the range of individual (ind) T-X-T angles, and the number of observations (N) for chemically different zeolites, given separately for all FTC (including SOD-types) in which they occur and then exclusively for SOD-type frameworks, SOD.
Angle T -X-T,  Table S7 and references (Ronis et al. 1989;Schnick and Lücke 1992;Karau et al. 2007) § see Table S7 ‡ one sixth of the Zn-site is occupied by Cu Notes: All sides of the angles contain bonds with occupancies by one type of T atom only. Mixed occupancies have been omitted. Stated are the number of different FTC contributing to a given T-X-T (#) and the maximal difference, Δmax, between the extreme values of T-X-T, which is a measure of the flexibility of the framework. Shown as well are the values of the distances T-X mean (from Baur and Fischer 2019, except as indicated) in an angle T-X-T. When two chemically different T are present their average T-X distance is listed. The last column records the mean oxidation state (OS) of the T atoms participating in a given T-X-T.
in ZeoBase, our databank of zeolites (Baur and Fischer 2010) in its most recent version.
The caution about how to use data from large databases discussed by Slovokhotov (2014Slovokhotov ( , 2016 applies of course as well to the present effort. According to this author, geometrical parameters gathered from diffraction data and deposited in large crystallographic data bases may have unexpected biases. Either by having an overrepresentation of data measured by one particular method (Slovokhotov 2014) or else by certain kinds of data being very popular among researchers (Slovokhotov 2016).
Previously (Baur and Fischer 2019), we employed exclusively single crystal data in our study. Since some interesting zeolites can be prepared only as powders, we shall have to employ here in places powder diffraction data as well. We have been particularly careful to avoid refinements performed with distance or angle restraints or constraints which could have influenced the geometry of the frameworks. However, the bulk of the data used in this paper is based on 1187 high quality selected zeolite single crystal structure determinations from ZeoBase that is they have mean e.s.d.s of their T-O distances of less than 0.01 Å as documented in Table S1.
The number of available structures of zeolitic frameworks in which the ligand in the coordination tetrahedron is not an oxygen atom is limited, but wherever possible we are going to consider cases containing various X atoms, where X may be F 1− , N 3− , S 2− , Se 2− or Cl 1− . Thus, we can show the influence of the chemical identity of the bridging atom on the properties of a T-X-T framework. Similarly important is the influence of the kinds of bonds T-X on the behavior of the T-X-T angles.
Excluded from this evaluation are T-X-T angles derived from datasets recorded at high temperatures and non-ambient pressures except in the overall presentation of data in the next chapter. Whereas T-O distances show relatively small changes upon increasing temperature, T-X-T angles react very sensitively as shown in Fig. 1 for the example of haüyne measured between room temperature and 1308 K (Hassan et al. 2004). The T-O-T angles range from 150.0° to 154.4°. Similarly, the T-O-T angles decrease upon increasing pressure as shown for cancrinite  in Fig. 2. All histograms shown in the present paper have been prepared using ZeoBase (Baur and Fischer 2010). Wells (1984) discussed in a purely geometrical manner the limitations on the bond angles T-X-T for shared X atoms located in vertices between two regular coordination tetrahedra. Assuming that any distance X-X between the two tetrahedra shall not be shorter than an X-X edge length within either tetrahedron he calculated a lower limit of 102° for T-X-T. The upper limit is of course 180° for a straight T-X-T arrangement, thus we should expect actual observed values of T-X-T to range from 102° to 180°. How does this agree with the empirical data?

Dependence of the T-X-T angle on the chemical nature of the X and the T atoms
The histogram shown in Fig. 3 displays all T-X-T angles found in the high-quality data set listed in Table S1 including data obtained at high temperatures. The lowest observed value for T-X-T is 94.3° for an angle Hg-S-Hg in a SODtype framework. However, this is an extreme value. A continuum of low values begins only at about 101° with 101.4° for a Cu-S-Cu angle in a SOD framework and a 101.5° angle Ga-Se-Ga in the zeolitic RWY type Ga 48 Se 96 framework (Zheng et al. 2002;Feng et al. 2005;Fischer and Baur 2009). This is in essential agreement with Wells' theoretical estimate. The T atoms in the RWY frameworks are Ga, Ge, In, or Sn, the X atoms can be either S or Se (Fischer and Baur 2009). The highest observed value of an angle T-X-T among these selenides and sulfides of RWY type and sulfide SOD types measures 110.8° (Ga-S-Ga in an RWY framework). All sulfides and selenides of our sample are located  Table 1 all are even less flexible with Δ max values below 22°.
The same trends we can see when looking at all the available framework types together (there are up to 21 FTC contributing to one type of angle, Si-O-Al, see column 5, Table 1) are visible when we compare the T-X-T for the chemically richest zeolite type, SOD, see columns 7 to 9, Table 1. The largest deviation between the mean T-X-T for all available FTC (column 2, Table 1) and the T-X-T of the compounds of the SOD type (column 7, Table 1) is 3.5° for Ge-O-Ge. Thus, we can get a reading of the T-X-T angles from looking at the SOD type alone without having a more general sample from observations of these angles in frameworks of additional FTC.
in the elongated left foot of the histogram in Fig. 3 at T-X-T values below 111°.
The framework of the RWY (UCR-20) type can be described as composed of adamantane shaped T 4 X 10 units (Zheng et al. 2002;Feng et al. 2005;Fischer and Baur 2009). These in turn can be understood as tiny components out of the sphalerite crystal structure type in which each Zn atom is coordinated tetrahedrally by four S atoms and vice versa each S atom by four Zn atoms. In sphalerite, all angles T-X-T and X-T-X are of course tetrahedral angles of arccos(-1/3) ≈ 109.5°. The mean of the 62 angles T-S-T and T-Se-T in the RWY and SOD types contained in Fig. 3 is 103.8°, indicating a 6° reduction of that angle in the adamantane T 4 X 10 group as compared with the precisely tetrahedral angle as observed in sphalerite itself.
Two thirds of the T-X-T values are in the range between 135° and 155°. They involve mostly Si, Al and P as tetrahedral atoms and oxygen atoms as hinges (or bridges) between them. Table 1 lists some of the details of the chemically different classes of T-X-T angles displayed jointly in Fig. 3. However, mixed occupancies on the T atom sites have been omitted. The data in Table 1 refer only to angles containing complete occupancy by a given element whereas Fig. 3 also contains T atoms with different cations sharing one position. One question can be immediately answered. Do the mean  Table S1) bridging the coordination tetrahedra in 1187 zeolite type frameworks. The T atoms are mostly Si, Al and P, the X atoms are mostly O, and a few S and Se. Tetrahedral sites with statistical occupa-tions are included. For additional information see text. Labels on the ordinate refer to the frequency of occurrences as they do in the following diagrams Liebau (1961a) investigated the magnitude of the angle Si-O-Si in silicates and "…concluded that at least under normal conditions stretched Si-O-Si bonds do not exist in crystalline silicates; this result is what one would expect from the considerable covalent character of the Si-O bond." An important part of Liebau's argument was that in some cases where straight Si-O-Si bonds were described in the literature their crystal structures had been refined in space groups of too high symmetry hereby putting certain oxygen atoms into more symmetrical positions. His first example was the crystal structure of the mineral petalite, LiAlSi 4 O 10 , an aluminosilicate based on a nonzeolitic dense tetrahedral framework with an FD (for a definition of FD see Baerlocher and McCusker 2022) of 23.6 T atoms per 1,000Å 3 ) the crystal structure of which was solved by Zemann-Hedlik and Zemann (1955) in space group P2/a. One bridging oxygen atom in petalite is located on a center of symmetry and consequently displays a 180° Si-O1-Si angle. Liebau (1961b) rerefined the crystal structure in space group Pa based on the structure factors measured previously by Zemann-Hedlik and Zemann (1955) for petalite. With the center of symmetry removed from atom O1, he found the Si-O1-Si angle to be reduced to 166°. Of course, that was long before it was realized that leaving out a center of symmetry, when it really is present in a given crystal structure, can lead to strong correlations, singularities and catastrophic distortions in a refinement (Baur and Tillmanns 1970;Ermer and Dunitz 1970;Schomaker and Marsh 1979). R. E. Marsh and collaborators published between 1981 (Marsh 1981) and 2009 (Marsh 2009) numerous papers in which they corrected the space group assignments of scores of previously published crystal structures giving rise to the lab colloquialism of "marshing" a structure. Some papers attempted to explain to fellow scientists how to avoid having their crystal structures being marshed (Marsh 1981(Marsh , 1995Baur and Tillmanns 1986;Baur and Fischer 2003). Effenberger (1980) collected new diffraction data on a petalite and evaluated them very carefully (Table 2, line 1). Significant evidence for a lower symmetry in space group Pa was not detected. The thermal ellipsoid of atom O1 was found to be disk-shaped and did not suggest a splitting of the site to be necessary (Effenberger 1980). The mean distance Si1-O of all four Si1-O bonds in the Si1 tetrahedron measures 1.606 Å, the distance Si1-O1 is only a little shorter at 1.596 Å ( Table 2). The equivalent isotropic displacement parameters of atom O1 are only slightly larger than for the other oxygen atom sites in petalite. This is in keeping with Cruickshank's observation (Cruickshank 1956) of large apparent bond shortenings being a consequence of large thermal motions. Two years later, (Tagai et al. 1982) fully confirmed Effenberger's results in a neutron diffraction Interestingly, the lengths of the mean bond lengths T-X do not influence the width of the T-X-T angle at all. The three narrowest mean angles, B-O-B, Zn-Cl-Zn and Cu-S-Cu are observed for the shortest T-X (B-O) as well as for the longest T-X (Zn-Cl and Cu-S), see Table 1, the last three rows. Likewise, while there appears a general tendency for the high oxidation states (OS) of the central tetrahedral cations to be associated with the more open T-X-T angles, there is also a glaring counterexample to this in the oxidation state of P having a small value of T-X-T (Table 1).
Very likely it is not a coincidence that the most useful and interesting zeolites have frameworks with SiO 2 , (Si,Al)O 2 , or AlPO 4 compositions and they are also those with the largest mean T-X-T angles and the largest Δ max values. Having large angles means that they can have low values of their FD, that is their pores are wider than for frameworks with clearly smaller mean T-X-T. Having large Δ max values means that they can accommodate to widely different values of their T-X-T angles, that is they are more flexible than those with smaller values of Δ max . A measure of the interest in zeolites containing Si, Al or P in their framework is the very high number of their crystal structures that have been determined. These Si, Al or P containing zeolites also play a dominant role in recent zeolite reviews (Moliner et al. 2015;Speybroeck et al. 2015;Li et al. 2017;Tabacchi 2018).
Theoretical energy calculations (Dawson et al. 2014) on the dense framework of cristobalite type SiO 2 and GeO 2 also indicate that Si-O-Si angles are more open than Ge-O-Ge angles. However, these were done under the assumption that the TO 4 tetrahedra themselves are rigid, which for zeolites is not generally true (Baur and Fischer 2019). Mean Si-O-Si angles of 152.9° and Al-O-P angles of 152.5° agree approximately with the corresponding angles calculated by density functional theory (DFT) which are slightly smaller with 149.23° and 147.82°, respectively (PBSEsol-D2 values in Fischer and Angel 2017). However, we do not wish to get here at any length into the voluminous theoretical literature about T-O-T angles as we are content to explore the actually observed empirical experimental values available to us.

Do straight angles T-O-T exist?
At the right foot of the histogram shown in Fig. 3 lines 3 and 4). The central tetrahedron of the pentamer is occupied in harkerite by an Al atom and located on a 3-fold axis. Consequently, one of these Al-O-Si angles is 180°, while the three other angles would in principle be free to assume values very different from a straight angle. In fact, these three angles measure 175.8(3)°, which means that they are very close to a straight arrangement without being located on a highly symmetrical site.
The Si 5 O 16 and AlSi 4 O 16 groups can be taken as tiny separate parts of tetrahedrally coordinated T atoms in silica or in aluminosilicate framework crystal structures. Thus, they might be thought of as test cases for the existence or nonexistence of straight T-O-T angles in zeolites.
Based on an evaluation of the then recent crystal structure determinations, Baur (1980) concluded that actually "Straight Si-O-Si bridging bonds do exist in silicates and silicon dioxide polymorphs". Liebau (1985) in his book on the "Structural Chemistry of Silicates" offered a very balanced discussion of this question and concluded (see page 26 in Liebau 1985) that "…it is now clear that in a number of silicates bridging oxygen atoms do indeed reside in special positions and that straight Si-O-Si bonds do exist." study of petalite, thus verifying the straight Si1-O1-Si1 angle.
In his paper concerning the unlikeliness of straight Si-O-Si angles, Liebau (1961a) questions the space group assignments of several crystal structures containing such angles. Among them is the structure of zunyite, Al 13 Si 5 O 20 (OH,F) 18 Cl, determined by Pauling (1933) and refined by Kamb (1960). The very complicated cubic structure of zunyite contains a Si 5 O 16 group with a central Si atom forming four straight Si-O-Si angles to its four surrounding SiO 4 tetrahedra. A very precise refinement of the crystal structures of two zunyites by Baur and Ohta (1982) did not give any indication of lower space group symmetry or of static disorder of the oxygen atom involved in the straight bond (Table 2, line 2). Raman and infrared microscopy (Klopprogge and Frost 1999) identified symmetric stretching Si-O-Si modes in zunyite. Three NMR studies of zunyite by Grimmer et al. (1983), Dirken et al. (1995), and Zhou et al. (2003) agree on interpreting their data as supporting a straight or almost straight angle Si-O-Si.

Notes: Listed are: angle T-O-T, T-O distances within T-O-T, mean T-O for T-O distances within the tetrahedron containing the straight T-O-T angle, framework density FD, the equivalent isotropic displacement factor (Fischer and Tillmanns 1988) B of the bridging O within T-O-T, mean B of bridging oxygen atoms in the T-O-T angle, FTC of zeolite crystal structure containing the T-O-T angle, various remarks, references (Ref).
Only cases with full occupancy by Si or Al are considered (with the exception of high pressure BRE). Only high-quality crystal structure determinations are used, unless a footnote indicates otherwise straight angles T-O-T without suspicion of the presence of static or dynamic disorder about the highly symmetric bridging oxygen atom positions. That is in keeping with Liebau's judgment (Liebau 1985) in accepting the petalite (Effenberger 1980) and zunyite (Baur and Ohta 1982) cases as true straight angles Si-O-Si. What can we learn from the examples observed in zeolites and listed in lines 5 to 15 of Table 2? Gies (1986) solved the crystal structure of the clathrasil deca-dodecasil 3R (with zeolite code DDR) in space group R3m. The Si-O distances of 1.546 Å and 1.558 Å (see line 5, Table 2) within the straight Si-O-Si angle are rather short as compared with the mean value of 1.603 Å indicating that there might be a disorder and/or twinning problem affecting the structure determination. This was the starting point for Langer et al. (2005) to restudy the crystal structure assuming space group R3 and twinning (line 6, Table 2). The resulting model for the deca-dodecasil 3R crystal structure was much improved compared with the model employed by Gies (1986) but according to Langer et al. (2005) it is still incomplete because of the pronounced distortion of the surrounding coordination tetrahedra observed by them in the crystal structure. Therefore, they assume a dynamic disorder at the oxygen atom O2 located in the straight angle Si-O-Si. Schlenker et al. (1977) presented a precise refinement of the crystal structure of a hydrated BRE-type zeolite. The question of a possible splitting of the position of the oxygen atom located in the straight bond is not discussed in their paper. Line 7 of Table 2 shows that the two O-T distances in the straight bond are only a little shorter than the other T-O distances within their coordination tetrahedra, and that the B values are not suspiciously larger for that oxygen atom. Sacerdoti et al. (2000) studied a BRE-type structure dehydrated at 373 K. Again, the variations in the T-O distances and the B values from the room temperature values are not large, even though the volume of the unit cell has changed by 3% (compare the FD values in lines 7 and 8, Table 2). The straight angle T-O-T was not affected by the reduction in volume. Upon further heating to 603 K, in another part of the crystal structure a T-O-T bridge was broken (Sacerdoti et al. 2000), but the Si-O-Si angle of 180° remained straight and stable. A high-pressure experiment by Seryotkin (2019) (lines 9 to 11 in Table 2) shows that at a pressure of about 2 GPa a phase transition with a doubling of the c unit-cell dimension occurs, without causing a change of the space group symmetry. At 2.52 GPa the straight bond of the lowpressure phase is reduced to 144.3°. A possible splitting in the low-pressure phase of the oxygen atom position in the straight T-O-T bond into two separate half-atoms is not discussed (Seryotkin 2019).
The zeolite FER has been studied repeatedly in its natural form as the mineral ferrierite and also as a synthetic Up to this point, we talked about Si-O-Si angles wherever they occurred, but mostly not in zeolites. The case of zeolites was studied in a paper by Alberti (1986) entitled "The absence of T-O-T angles of 180° in zeolites". And when Wragg et al. (2008) performed a structural analysis of pure silica zeolite-type frameworks they concluded that "180° Si-O-Si angles are probably due to unresolved disorder or structural complexity", thus returning to Liebau's original opinion (Liebau 1961a).
Inasmuch as there is no reason to suspect that zeolitic frameworks follow different kinds of rules compared with other inorganic solids it makes sense to have a closer look at the 206 cases of T-O-T angles with values between 170° and 180° mentioned above. We must see what is hidden under the large envelope of the histogram (Fig. 3) which includes all T-X-T angles in precisely determined zeolite crystal structures. Figure 4a shows all Si-O-Si angles in our sample excluding cases with mixed occupations of the tetrahedral sites, Fig. 4b includes the mixed occupations.  Table S4) and they are mostly left out of our discussion at this point, because in these mixed occupations the distance from O to T is split into two nearly overlapping populations. For O-Si and O-Al we have a length difference of about 0.13 Å (Baur and Fischer 2019). The displacement parameters of the O atoms must reflect an additional contribution (Baur 1964) from this static disorder to the dynamic disorder present in any crystal structure. We can avoid any complications that could arise from this by mostly considering T sites without statistical occupancies. Figures 4 and 5 show that there is a continuum of Si-O-Si (and Al-O-Si) angles with values between 170° and 180° that connects without any interruption to the more frequent values below 170°. Table 2 gives us a more detailed look at a selection of these straight angles in several zeolites. It extends Table 3.3 in Liebau (1985) which lists a considerable number of straight and almost straight Si-O-Si angles but does not contain any examples from zeolitic structures. The zeolite studies quoted here in Table 2 were, with two exceptions, published after 1985.
It appears that we can accept the non-zeolitic cases listed in lines 1 to 4 of Table 2 as examples of straight or nearly Pmnn, however, above approximately 400 K it is found in space group Immm with one oxygen atom positioned in a center of symmetry. Thus, it is necessarily involved in a 180° angle. But even at 379 K, in space group Pmnn the angle for the corresponding oxygen atom (not located in siliceous compound, for detailed references see Baur and Fischer (2002). Various space groups were used in these refinements. In Table 2 (lines 12 and 13) we show the results obtained by Bull et al. (2003) for an SiO 2 composition of FER. At room temperature it crystallizes in space group  Table S3) bridging pairs of Si atoms between coordination tetrahedra in zeolite type frameworks. Tetrahedrally coordinated sites with statistical occupancies are excluded; (b) 3652 individual T-O-T angles (for details see Table S4) of bridging pairs of T atom sites populated by Si and at least one additional element occupying statistically the same T site. Thus, tetrahedral sites with statistical occupations are not excluded with an Si-O-Si angle of 178.3° at 1.35 GPa and 177.1° at 3.00 GPa. Space-group settings follow the standardized setting introduced by Baur and Fischer (2002). Trudu et al. (2019) recently argued on the basis of computer modeling a center of symmetry) is at 177.9°, that is, it is essentially straight (Table 2, line 12). Similarily, Lotti et al. (2015) determined two pressure-induced phase transitions from Pmnn to P12 1 /n1 at ~ 0.7 GPa and to P2 1 /n11 at ~ 1.24 GPa  Table S5) bridging pairs of Al and Si atoms between coordination tetrahedra in zeolite type frameworks. Tetrahedrally coordinated sites with statistical occupancies are excluded; (b) 1098 individual T-O-T angles (for details see Table S6) of bridging pairs of T atom sites populated by Si or Al and at least one additional element occupying statistically the same T site. Thus, tetrahedral sites with statistical occupations are not excluded. For additional information see text EPI (epistilbite), and MWW (MCM-22) framework types Fischer 2002, 2017;Fischer and Baur 2006).

General remarks
There are no systematic studies of the flexibility and the collapsibility of the 247 framework types listed by Baerlocher and McCusker (2022). General principles have been described, e.g., by Baur (1992) and were applied to the collapsible framework of zeolites with NAT topology (Stuckenschmidt et al. 1992(Stuckenschmidt et al. , 1996Joswig and Baur 1995). Here, we present some examples of well-described zeolite frameworks that are both flexible and collapsible.

SOD, the sodalite framework type
The first crystal structure which was later to be classified as possessing a zeolite-type framework was solved by Jaeger (1929) as discussed by Baur and Fischer (2008). It was the structure of nosean, Na 8 Si 6 Al 6 O 24 SO 4 , now known to be based on the framework type called SOD. In 1930, the year following Jaeger's paper Pauling (1930a) published a comparison of the unit-cell parameters of several SODlike frameworks where other differently sized cations and/ or anions were present in place of Na and SO 4 (as they are present in nosean). Their cell parameters varied with the sizes of these pore-filling extraframework ions. Pauling (1930a) considered this framework as strong, but not rigid, because the smaller the non-framework atoms were, the smaller the unit-cell parameters became. Consequently, he termed this a partial collapse of the framework structure (Pauling 1930a). He related this reduction of the cell lengths to a rotation of the SiO 4 and AlO 4 coordination tetrahedra around the two-fold axes on which they are located in the SOD-type framework in space group P43n. We prefer to view this more generally in terms of the values of the angles T-O-T at the bridging oxygen atoms between the coordination tetrahedra, or as we call them here the hinges between the tetrahedra. As the angles T-O-T around a coordination tetrahedron TO 4 change, they all change in the same direction, that is they corotate. In Fig. 6 we see that the range of Si-O-Al values extends over 12°, from 138° to 150°. This corresponds to the central area of all observed T-X-T values in zeolite frameworks (see Fig. 3) or to more than one third of the total range covered by all T-X-T angles as reported in Table 1. The correlation coefficient R 2 of 0.95 shows that the relationship between the of FER of SiO 2 composition "that ferrierites with Immm symmetry may be classified as metastable phases". Gies (1984) solved the crystal structure of the clathrate compound dodecasil 3C (zeolite code MTN) in space group Fd3. The very short Si-O distances (Baur and Fischer 2019;Gies 1984) of 1.526 Å and 1.546 Å (see line 14, Table 2) within the straight Si-O-Si angle point toward a massive disorder and/or twinning problem in the structure determination. Subsequently Knorr and Depmeier (1997) showed for the dodecasil 3C-tetrahydrofuran that the apparent cubic symmetry was caused by "a merohedrally twinned crystal in tetragonal space group I4 1 /a." In their refinement, the Si-O-Si angle was reduced to 148.9° and the two Si-O distances within that angle assumed the normal values (Baur and Fischer 2019) of 1.592 Å and 1.597 Å.
Four different topologies of zeolites are shown in Table 2 to contain in their crystal structures 180° angles. Two of them, BRE and FER, are as well documented as in the cases of petalite and zunyite (lines 1 and 2 of Table 2). The other two, DDR and MTN, are instances of what Liebau (1961a) had in mind. In both these cases the demonstration of twinning and its consideration in the crystal structure refinement process proved that the originally assumed straight angles were only apparent.
One could argue that finding among 247 zeolite topologies only two that allow 180° T-X-T angles is not very much. The counterargument is that we have limited our exploration only to very carefully refined cases of single crystal studies. Among those we excluded are those with statistical occupations on the T-sites. This means that the straight angles in ABW, AFY, CAS, MOR, SBN, SOF, and TER topologies were not considered. Also, we did not list here the straight angles found in quality Rietveld refinements such as those of the DOH (dodecasil 1 H), EON (ECR-1), Fig. 6 The hinges between the coordination tetrahedra in a number of SOD-types: plot of 30 individual Si-O-Al angles [°] versus framework density FD of SOD-type aluminosilicate frameworks crystallizing in space group P43n (Pearson's R 2 is 0.95). The cations in the pores are mostly Na, the anions are Cl, SO 4 and various other groups. The data are taken from the same high-quality data set described above in the "Data" chapter GIS (gismondine), one of the most flexible structures with collapsible frameworks known so far The crystal structure of the framework of the mineral gismondine, CaAl 2 Si 2 O 8 ˖4H 2 O, was determined by Fischer (1963). Baerlocher and Meier (1972) studied the isostructural synthetic phase Na-P1 and pointed out its remarkably flexible and readily distorted framework. McCusker et al. (1985) showed that the GIS-type framework can "be described as a stacking of two-dimensional arrays of double crankshaft chains". This again "allows for a flexibility in the framework which is manifested in the different symmetries observed" for the GIS types (McCusker et al. 1985). Up to the year 2017 we have identified 20 different types of space group symmetries in which GIS type frameworks have been found to crystallize (Baur and Fischer 2017). These symmetries range for the GIS type from I4 1 /amd all the way down to I1 . Only the SOD type zeolites exhibit a larger population of distinct space group types. We count altogether 27 of them, ranging in symmetry from Im3m to P¯1 (Fischer and Baur 2009). The range of Si-O-Al values in the GIS-type frameworks extends over 31°, from 126° to 157°, as compared to the case of the SOD-type where the range is 19° (from 138° to 157°, see above). This shows that the topology of the GIS framework allows more flexibility than the SOD topology does.
The distortions of GIS-type aluminosilicate frameworks upon dehydration were first studied by Vezzalini et al. (1993) and followed up by further temperature dependent investigations (Pakhomova et al. 2013;Gatta and Lotti 2011). The influence on the framework by ion exchange with differently sized cations in the pores of the framework was studied too (Bauer and Baur 1998) as was its behavior under high pressure .
We plot here in Fig. 7a and b the values of the hinges against FD [T atoms per 1,000 Å 3 ], the framework density. The scatter of the 193 data points in Fig. 7a (GIS, in space groups I4 1 /amd all the way down to I 112), is much larger than in Fig. 6 (SOD), where all data points refer to only one highly symmetrical space group, P43n. But we can standardize (Fischer and Baur 2004) the settings and the coordinates of the atoms in the various space groups of the GIS structures entered in the plot shown in Fig. 7a and look at the plot of the 17 averaged atoms in Fig. 7b. Pearson's R 2 is now 0.96 similarly high as for the SOD plot of Fig. 6, thus showing that even in the instance of a framework consisting of many crystallographically independent T-O-T hinges its property of being flexible and at the same time collapsible can be demonstrated.
angles T-O-T and FD [T atoms per 1,000 Å 3 ] is close to linear, see Fig. 6.

RHO (zeolite-rho), another example of a collapsible framework
The SOD framework appears to be a simple affair (Depmeier 2005), even though the space group symmetries reported for various members of its family range from Im3m to P1 (Depmeier 2005;Fischer and Baur 2009). When crystallizing in space group P43n, as selected in Fig. 6, there is in SOD only one symmetrically equivalent oxygen atom within each unit cell, thus all are the same and obviously the tetrahedra linked by this O atom have to corotate. The framework of RHO-type (Baerlocher and McCusker 2022;Fischer and Baur 2006;Robson et al. 1973) has a slightly more complicated crystal structure and is more open than the SOD-type. Depending on the space group type in which a particular RHO-type crystal structure crystallizes, there are at least two symmetrically different oxygen atoms in each unit cell, so in principle one of them could increase and the other could decrease in value (see the case of LTA further below) as the size of the cell changes. In fact, however, all the symmetrically different tetrahedra corotate.  and Baur et al. (1988) discussed the relationship of the T-O(2)-T angle in aluminosilicate RHO frameworks with the unit-cell constant in this cubic structure. Corbin et al. (1990) plotted the cell edges a of RHO versus T-O-T showing an excellent linear relationship (R 2 = 0.91) similar to the correlation presented here in Fig. 6 for SOD. The synthetic zeolite RHO-type samples have so far been obtained only in powder form, thus all its crystal structure determinations are based on powder diffraction data using the Rietveld method. Consequently, their precision is not as high as it would be for single-crystal work. Nevertheless, the trend for the T-O-T angles versus a appears to be the same in both instances: the smaller the mean or the individual T-O-T angles, the smaller the a cell parameters.

PAU (paulingite), a very complex collapsible framework
The framework of PAU is one of the most complicated frameworks found among zeolite type structures (Gordon et al. 1966. It has 2016 atoms in a cubic unit cell with a length of a = 35.093(2) Å and crystallizes in space group Im3m in its hydrated state (Gordon et al. 1966;Bieniok et al. 1996). Upon dehydration (Bieniok et al. 2015), the unit-cell dimension is reduced to a = 34.681(1) Å and the symmetry is lowered to I4 3m. The lowering of the unit-cell lengths is accompanied by a decrease in the value of the mean T-O-T angles, just as it is the case for the SOD and RHO framework types. Nasdala, for their comments on the manuscript.
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LTA (Linde type A), a zeolitic tetrahedral framework that is both flexible and noncollapsible
The LTA framework manages to be very flexible and at the same time avoids a collapse by possessing antirotating hinges, that is when one hinge atom T-O-T opens up its T-O-T angle another angle T-O-T closes: compare Fig. 8a and b with Fig. 8c. As a consequence, the unit-cell constants of various LTA compositions change very little (Baur 1992;Fischer and Baur 2006). Analogously, the frameworks of FAU and KFI (Baur and Fischer 2002) type zeolites change their unit-cell constants little with changes in the content of their pores.
Acknowledgements This work is dedicated to Prof. Dr. Josef Zemann on the occasion of his 100th birthday. We thank Diego Gatta, an anonymous reviewer, the guest editor, Herta Effenberger, and the editor, Lutz  Fig. 7 Plots of the hinges between the coordination tetrahedra in the GIS-type. The data are from aluminosilicate GIS frameworks crystallizing in various monoclinic, orthorhombic and tetragonal space groups and taken from the same high-quality data set described above in the "Data" chapter.