Polytypism of Compounds with the General Formula Cs{Al 2 [ T P 6 O 20 ]} ( T = B, Al): OD (Order ‐ Disorder) Description, Topological Features, and DFT ‐ Calculations

: The crystal structures of compounds with the general formula Cs{ [6] Al 2 [ [4] T P 6 O 20 ]} (where T = Al, B) display order − disorder (OD) character and can be described using the same OD groupoid family. Their structures are built up by two kinds of nonpolar layers, with the layer symmetries Pc ( n )2 ( L 2 n +1 ‐ type) and Pc ( a ) m ( L 2 n ‐ type) (category IV). Layers of both types ( L 2 n and L 2 n +1 ) alternate along the b direction and have common translation vectors a and c ( a ~ 10.0 Å, c ~ 12.0 Å). All ordered polytypes as well as disordered structures can be obtained using the following partial sym ‐ metry operators that may be active in the L 2 n type layer: the 2 1 screw axis parallel to c [– – 2 1 ] or inversion centers and the 2 1 screw axis parallel to a [2 1 – –]. Different sequences of operators active in the L 2 n type layer ([– – 2 1 ] screw axes or inversion centers and [2 1 – –] screw axes) define the formation of multilayered structures with the increased b parameter, which are considered as non ‐ MDO polytypes. The microporous heteropolyhedral MT ‐ frameworks are suitable for the migration of small cations such as Li + , Na + Ag + . Compounds with the general formula Rb{ [6] M 3+ [ [4] T 3+ P 6 O 20 ]} ( M = Al, Ga; T = Al, Ga) are based on heteropolyhedral MT ‐ frameworks with the same stoichiometry as in Cs{ [6] Al 2 [ [4] T P 6 O 20 ]} (where T = Al, B). It was found that all the frameworks have common nat ‐ ural tilings, which indicate the close relationships of the two families of compounds. The conclu ‐ sions are supported by the DFT calculation data.

In this paper we provide a complete OD-theoretical analysis of the compounds with the general formula Cs{ [6] Al2[TP6O20]} (where T = B [25], Al [26]) and derive symmetry and atom coordinates for the hypothetical MDO2 polytype. The energies of the observed and hypothetical structures of the family are calculated using the density functional theory (DFT). Possible ion-migration paths inside the microporous frameworks of the family are estimated for different alkaline ions using the topological analysis.

Methods
The symmetrical relations between the compounds have been analyzed using the OD theoretical approach [27][28][29][30] for the OD families containing more than one (M > 1) kind of layers [31]. The OD layers have been chosen in accordance with the equivalent region (ER) requirements [32]. As a reference structure for the further analysis, the MDO1 polytype observed in Cs{Al2[AlP6O20]} [26] was used. This compound was reported in the nonstandard setting of the space group C2cb [a = 10.0048(7) Å, b = 13.3008(10) Å, c = 12.1698(7) Å], which was transformed into the standard setting Aea2 using the [00-1 / 010 / 100] matrix (the resulting unit cell parameters are: a = 12.1698(7) Å b = 13.3008(10) Å, c = 10.0048(7) Å). The unit-cell parameters and space groups of the crystal structures of Cs{Al2[BP6O20]} polytypes have been transformed accordingly in order to preserve the orientation and stacking direction of the OD-layers.
Topological analysis of the frameworks was performed by means of natural tilings (the smallest polyhedral cationic clusters that form a framework) of the 3D cation nets [33]. The complexity parameters of the frameworks in different polytypes were calculated as Shannon information amounts per atom (IG) and per reduced unit cell (IG,total) [34,35]. To analyze the migration paths of alkaline cations in the structures, the Voronoi method [36], which has proven itself in the study of cationic conductors of various types [37,38], was used. Topological and complexity parameters for the whole structures as well as ion migration paths have been calculated using the ToposPro software [39].
DFT calculations on the existing MDO-, non-MDO-4O, as well as hypothetical MDO2 type polytypes (T = Al, B) were performed using the PBE exchange-correlation functional [40] of the GGA-type utilizing the projector augmented wave method (PAW) as implemented in the Vienna ab initio simulation package (VASP) [41,42]. The energy cut-off was set at 500 eV with a 10 × 8 × 8 (MDO1, MDO2), and 6 × 4 × 4 (non-MDO-4O) Monkhorst−Pack [43] k-point mesh used for Brillouin zone sampling. The convergence towards the k-point mesh was checked. Full optimization of the unit cell parameters and atomic coordinates was performed for all the structures except the MDO1 polytype of Cs{Al2[BP6O20]}, for which the original cell parameters were retained and atomic coordinates optimized (as the compound was found to have the lowest energy, cell parameter optimization was deemed unnecessary). For the optimization, the structures were converted to the space group P1.

OD (Order−Disorder) Relationships
The crystal structures of Cs{ [6] Al2[ [4] TP6O20]} (where T = B [25], Al [26]) belong to the same OD family of category IV [31] with two types of nonpolar OD layers and can be described by an OD groupoid [27]. The layers are as following: 1. Nonpolar L2n+1 type with the layer symmetry pcn2 [or Pc(n)2 in terms of the OD notation, where braces indicate the direction of missing periodicity [44]] was reported previously [20] and is represented by the tetrahedral [ [4]   Layers of both types (L2n and L2n+1) alternate along the b direction and have common translation vectors a and c (a ~ 10.0 Å, c ~ 12.0 Å), with b0, the distance between the two nearest equivalent layers, corresponding to one half of the b parameter of the compound studied by Lesage et al. [26]. Because the symmetry of the L2n type layers is higher than that of the L2n+1 type layers, polytypic relations are possible. All ordered polytypes as well as disordered structures can be obtained using the following symmetry operators that may be active in the L2n type layer: the 21 screw axis parallel to c [--21] or inversion centers and the 21 screw axis parallel to a [21 --] (Figure 2) [20]. The symmetry relation common to all polytypes of this family are described by the OD groupoid family symbol: where r = 0; the first line contains the layer-group symbols of the two constituting layers, while the second line indicates positional relations between the adjacent layers [46]. In accordance with the NFZ relation [27,28], there is only one kind of the (L2n, L2n+1, L2n+2) triples and two kinds of the (L2n-1, L2n, L2n+1) triples. Consequently, the smallest possible number of different triples in a structure is two and only two MDO polytypes are possible: The first MDO structure (MDO1 polytype) ( Figure 3, left) can be obtained when the [--21] operator is active in L2n type layer. Through the action of this operator the asymmetric unit at x, y, z (I) is converted into the asymmetric unit at -x, ½-y, ½+z (II); the latter unit is converted by the [--2] operator in the L2n+1 layer into the asymmetric unit at x, ½+y, ½+z (III). I and III are related by the translation vector t = b0 + c/2, which is the generating operation, giving rise by the continuation to an A-centered structure with the basis vectors a, b = 2b0, c and the space group Aea2. The MDO1 polytype corresponds to the structure of Cs{Al2[AlP6O20]} with the following unit cell parameters: a = 12.1698(7) Å b = 13.3008(10) Å, c = 10.0048(7) Å [26].
The second MDO structure (MDO2 polytype) (Figure 3, right) can be obtained when the inversion centers and [21 --] operators are both active in the L2n type layer. Through the action of the operator [21 --] the asymmetric unit at x, y, z (I) is converted into the asymmetric unit ½+x, -y, ½-z (II); the latter unit is converted by the [-n -] operator in the L2n+1 layer into the asymmetric unit x, ½+y, -z (III); (I) and (III) are related by a b glide normal to c, with translational component b0, which is the generating operation: its continuation also generates an orthorhombic structure with the basis vectors a, b = 2b0, c (the same for the MDO1 polytype) and the space group Pcnb (or Pbcn in the standard setting). The MDO2 polytype has not yet been observed for the compound with the general formula Cs{ [6] Al2[ [4] TP6O20]}. The calculated atomic coordinates for the MDO2 polytype are given in Table S1 (Supplement Materials). Different sequences of operators active in the L2n type layer ([--21] screw axes or inversion centers and [21 --] screw axes) define the formation of structures with the increased b parameter, which are considered as non-MDO polytypes (because of the presence of more than one kind of (L2n-1, L2n, L2n+1) triples) [27]. The compound Cs{Al2[BP6O20]} [25] contains four L2n and L2n+1 types layers, where each L4n type layer has active [21 --] screw axes, while in the L4n+2 type the inversion centers and [--21] screw axes are active ( Figure 4). The AlO6 octahedra in the L2n+2 and L2n+4 type layers are tilted slightly differently, which can be explained by the "desymmetrization" effect of OD structures [27,47,48], when the ideal symmetry suffers slight (in some cases severe) distortions and the symmetry of OD layers in the polytype is lower than the idealized one. The orthorhombic structure of Cs{Al2[BP6O20]}-4O is characterized by the basis vectors a, b = 4b0, c (where a = 11.815(2) Å, b = 26.630(4) Å, c = 10.042(2) Å [25]) and the space group Pcab (nonstandard setting of the space group Pbca).

Topological Features
Compounds with the general formula Cs{ [6] Al2[ [4] TP6O20]} (where T = B [25], Al [26]) are characterized by the heteropolyhedral MT-frameworks [20,[49][50][51] of MO6-octahedra and TO4-tetrahedra related to classic zeolites and zeolite-type materials where all oxygen ligands are bridged between two cations only [52]. In accordance with the theory of mixed anionic radicals [53][54][55], the general crystal chemical formula of the framework (taking into account the degree of sharing of oxygen ligands) can be written as [20]: where where m and ni, VM and i T V are the valences of the M and Ti cations, respectively. If M = M 3+ , T1 = T 3+ , T2 = P 5+ , m = z, n1 = y, n2 = z, the Formula (1) can be rewritten as: Taking into account the observed ratio between the x, y, and z coefficients, the stoichiometry of the heteropolyhedral MT-framework is: Topological features of the MDO1 and non-MDO 4O polytypes have been described previously [20]. The cationic 3D net corresponding to the heteropolyhedral MT-framework of MDO2 polytype consists of four natural tiles ( Figure 5) [6] Al2[ [4] TP6O20]} (where T = B, Al).  [33]. The topologically equivalent tiles are colored in the same color.

Ion Migration Path
Migration maps of Na + cation were constructed for the MDO1, MDO2, and non-MDO 4O polytypes (Table 2). Despite the presence of large pores filled by large Cs + ions, the size of the effective windows between them is not enough for the migration of large alkaline cations. However, all the types of the microporous heteropolyhedral MT-framework are suitable for the migration of smaller ions such as Li + , Na + Ag + . The types of migration maps depend on the topological type of the MT-framework (Figure 6), in particular, for Na + ions, the maps are represented by 2D layers parallel to (100) for the MDO1 and non-MDO 4O polytypes, while for the MDO2 polytype it is represented by the system of parallel 1D channels directed along [010] (Figure 6). In the case of Li + ions, the migration 3D maps are similar for all the types of the frameworks. Table 2. The natural tiles in the MT-frameworks of the polytypes of compounds with the general formula Cs{ [6] Al2[ [4] TP6O20]} (where T = B, Al).

Polytype
Natural tiles

DFT Calculations
In order to gain more insight into the stability of various polytypes, energy-wise, we have performed DFT calculations on the existing as well as hypothetical compounds with the general formula Cs{Al2[TP6O20]} (T = Al, B) with the structures belonging to MDO1, MDO2, and non-MDO 4O type polytypes, for T=Al; B. The comparative data and optimized unit cell parameters are given in Table 3 (for MDO1, T = Al, original unit cell metrics were retained).
As seen from the comparison between the original and optimized cells of Cs{Al2[BP6O20]} of the non-MDO 4O type, they are in a very good agreement, with the difference in volume of ca. 13 Å, i.e., ca. 0.4% (see Table 3). The optimized coordinates in all structures showed only minimal shifts from their original positions, mostly associated with a very small rotation of tetrahedra. It is important to note that, despite unconstrained optimization, all the structures, observed as well as hypothetical, retained their original cell symmetries.
As seen from Table 3, for the T = Al series, the structure with the lowest energy was the MDO1-type polytype. However, the non-MDO 4O-type structure was only ca. 0.06 eV higher in energy, which corresponds to ca. 6.2 kJ/mol. This difference is not large, yet is arguably outside the margin of error for the computational method used, which is commonly estimated as 1-2 kJ/mol. The important thing here is that both experimentally observed types of structures (albeit not both of them for T = Al), showed comparable energies. Moreover, our calculations indicate that, under the right conditions, it might be possible to obtain the non-MDO 4O polytype for aluminum. Regarding the MDO2-type structure, the optimization gave us a stable minimum structure with the energy of ca. 0.5 eV (ca. 49 kJ/mol) higher than MDO1. This means that, potentially, such a structure might exist, however, the energy difference to the lowest energy structure is significant, and thus it might be difficult to stabilize such a polytype.
For the T = B series, once again the lowest energy corresponds to the experimentally observed structure, this time it is the non-MDO 4O polytype (see Table 3). In this case, however, its energy is only ca. 0.03 eV (ca. 3 kJ/mol) lower than that of the hypothetical MDO2-type structure. The difference is on the border of the perceived accuracy of the computational method, thus the MDO2 polytype appears to be a good candidate for the experimental discovery. The MDO1-type structure in this case looks like the least favorable, energy-wise, with the difference between its energy and minimal structure being ca. 0.09 eV ( ca. 8.6 kJ/mol). This is clearly outside the margin of error; however, the difference is small enough to be compensated by various effects in real crystals. It must also be noted regarding all our calculations, that by their very nature they simulate ideal periodic crystals in their ground state at 0 K. In addition, in our computations we cannot account for potential kinetic hindrance of certain paths of compound formation.

Discussion
The heteropolyhedral MT-frameworks with similar stoichiometry (3) have been found in compounds with the general formula Rb{ [6]   Despite of the absence of the tetrahedral layers, the MT-framework can also be considered as the result of alternation along b of two types of nonpolar OD layers parallel to (010): 1. The first one corresponds to a layer with the symmetry P2(2)21 consisting of tetrahedral chains. The tetrahedral layer in Cs{Al2[TP6O20]} and tetrahedral pseudolayer in Rb{M2[TP6O20] are formed by the same FBU and demonstrate the symmetrical relationship ( Figure 8) indicating the possible OD-character as was previously shown for compounds with tetrameric [57] and pentameric [20]

Conclusions
The polytypism of compounds with the general formula Cs{Al2[TP6O20]} (T = Al, B) has been described using the OD theory approach. The crystal structure of the hypothetical MDO2 polytype has been proposed and optimized using DFT calculations. It was shown that the heteropolyhedral MT-frameworks of all the polytypes contain similar natural tilings. The compounds with the general formula Rb{ [6] M 3+ 2[ [4] T 3+ P6O20]} (M = Al, Ga; T = Al, Ga) have the heteropolyedral MT-frameworks with the same stoichiometry. It was found that all the frameworks had common natural tilings, which indicates the relationship of both families of compounds. Our computational data agree well with those which are experimentally available and, we believe, provide a reasonable basis for an internally consistent picture which supports crystallographic considerations concerning the formation of the polytypes of compounds with the general formula Cs{Al2[TP6O20]} (T = Al, B). Thus, it is seems possible to synthesize the MDO2 polytype as well as the "missing" members, such as MDO1 polytype of Cs{Al2[BP6O20]} and non-MDO 4O polytype of Cs{Al2[AlP6O20]} using hydrothermal techniques.