Controlling multiple orderings in metal thiocyanate molecular perovskites Ax{Ni[Bi(SCN)6]}

We report four new A-site vacancy ordered thiocyanate double double perovskites, , A = K+, NH4+, CH3(NH3)+ (MeNH3+) and C(NH2)3+ (Gua+), including the first examples of thiocyanate perovskites containing organic A-site cations. We show, using a combination of X-ray and neutron diffraction, that the structure of these frameworks depends on the A-site cation, and that these frameworks possess complex vacancy-ordering patterns and cooperative octahedral tilts distinctly different from atomic perovskites. Density functional theory calculations uncover the energetic origin of these complex orders and allow us to propose a simple rule to predict favoured A-site cation orderings for a given tilt sequence. We use these insights, in combination with symmetry mode analyses, to show that these complex orders suggest a new route to non-centrosymmetric perovskites, and mean this family of materials could contain excellent candidates for piezo- and ferroelectric applications.


Introduction
Molecular perovskites, perovskites of composition AMX 3 where at least one of A, M or X is molecular, have additional degrees of freedom which can produce orderings impossible in atomic perovskites. 1 These new orderings provide novel routes for materials to respond to external stimuli. One area of particular interest is using the molecular components to create electrical polarisation, without the need for the second-order Jahn-Teller distortions or stereoactive lone pairs that drive piezo-and ferroelectricity in atomic perovskites, e.g. BaTiO 3 or Pb(Zr, Ti) O 3 . 2 Molecular perovskites now possess polarisations and transition temperatures which approach those of inorganic perovskites. 3 However, their polarity is typically produced by the orientational order of polar A-site cations. 4 The functionality oen bestowed by the MX 3 framework, e.g. ferromagnetism or ferroelasticity, therefore usually couples weakly to the polarisation, limiting the scope for multiferroicity.
Generating polarisation via collective distortions of MX 3 framework is difficult as the conventional cooperative tilts of the MX 6 octahedra are intrinsically non-polar. However, by combining octahedral tilts with other symmetry-breaking orders, such as A-site or M-site occupational order, we can generate polar structures: the so-called hybrid improper ferroelectrics. 5 Furthermore, recent work has shown that the unusual framework distortions possible in molecular perovskites, such as unconventional tilts and columnar shis, while non-polar, offer new routes to polarity. 1 Creating materials capable of sustaining new ordering types and sustaining multiple simultaneous orders is therefore a powerful method for generating novel function. 6 Gaining control over these orders, both individually and separately, remains one of the challenges of solid-state chemistry. One key guiding parameter is the tolerance factor where r A is the radius of the A cation, r M is the radius of the M cation and r X is the radius of the X anion, which quanties our intuition that the A-site cation has to t well into the MX 3 cage. It indicates whether AMX 3 is likely to be a perovskite, rather than (s > 1) 1D or 2D structure types (e.g. hexagonal 'perovskite') or (s < 0.8) other, dense non-perovskite structure types (e.g. ilmenite). Although originally developed for atomic perovskites, the tolerance factor approach can rationalise the structures of a wide-range of molecular perovskites, including formates and alkylammonium metal halides, 8,9 and its fundamental geometric insight has been generalised to other systems. 10,11 s is also linked to the size of the octahedral tilts, as smaller s tends to require large tilts to retain a dense structure. However, creating new function requires controlling the relative sense of the tilts, i.e. whether each layer of octahedral tilts rotates in same sense as the next (a + in the Glazer notation 12 ), or opposite sense (a À ), not just their magnitude. This remains challenging to predict for new perovskites. 13 s can be readily tuned by creating solid solutions of cations (or cations and vacancies) on the A or M site, as the entropy of mixing stabilises these phases at the high synthesis temperatures used. Conversely, this means that cation-ordering is uncommon, particularly on the A-site. 14 A-site order is most oen stabilised by large size differences between A-site cations, especially the extremal size difference between a vacancy and a cation, and therefore typically produces layered order which minimises the local strain, e.g. ,2 3 . Msite order is stabilised by large charge differences, which favours rocksalt order (the 'double perovskite' structure) for electrostatic reasons. 14 Simultaneous control of these A-site and M-site occupational orders to make so called 'double double' perovskites requires therefore specic chemical compositions, but can produce new function e.g. polarity in NaLaMnWO 6 . 18 Rarer A-site occupational orders are typically stabilised by coupling the A-site order to octahedral tilt distortions of the MX 3 framework. Notably, CaFeTi 2 O 6 has the unusual a + a + c À tilt sequence which facilitates columnar A-site order 19 and the a + a + a + tilt sequence found in CaCu 3 Ti 4 O 12 stabilises 3 : 1 Cu 3 Au-type A-site order. 20 These challenges mean the synthesis of double double perovskites oen requires specialist conditions such as high pressure. 14,21 Molecular perovskites are fertile ground for the exploration of multiple simultaneous orders because of their chemical diversity, low temperature syntheses, and the toolbox of crystal engineering (e.g. H-bonding). 22 We focus in this paper on the family of perovskite-like materials derived from thiocyanate, A x {M [M 0 (SCN) 6 ]}, of particular interest for their catalytic and optical function. 7,23-26 These NCS-perovskites have complete M-site order, due to the difference between N-and S-termini of the ligand, and have large tilts due to the frontier molecular orbitals of the NCS À ligand. 7 The robustness of these distortions means NCSperovskites are an ideal platform for exploring complex orderings.
Like the related cyanide Prussian blue analogues, NCSperovskites are stable in the 'empty perovskite' ReO 3 structure. 27 Indeed, there are only two reported NCS-perovskites containing Asite cations: Cs{Cd(NCS) 3  can be incorporated into NCS-perovskites. We demonstrate that the identity of A-site cation plays a critical role in the structure of thiocyanate perovskites, and that new and unusual combinations of A-site order, M-site order and octahedral tilt patterns can be readily achieved in these materials. In particular, we show using a combination of X-ray and neutron diffraction and density functional theory (DFT) calculations that the A-site cation order and octahedral tilts are strongly coupled. Inspired by these structures, we use symmetry analysis and DFT calculations to suggest the combination of complex orders found in thiocyanate perovskites could be used to produce cooperative properties such as piezoelectricity. We were able to grow large single crystals of phase 1 and 2 by slow evaporation of butanone solutions of the desired stoichiometry. Solution of the structure using single crystal X-ray diffraction (SCXD) showed that the phases are isostructural, as is oen found for K + and NH 4 + compounds [ Fig. 2 3 , and its structure derives from this Pm 3m structure (using the setting with the A-site at the origin) through four symmetry-lowering distortions. The M-site cations have rocksalt order and this order transforms as the R 2 À irreducible representation The combination of octahedral tilting and rocksalt M-site order leaves all pseudocubic Ni 4 Bi 4 (SCN) 12 cages still equivalent by symmetry, meaning that the A-site cation ordering may not be viewed being drive by these three distortions alone. The A-site cation ordering therefore lowers the space-group symmetry further, from P2 1 /n to P 1, and in addition produces a large shear strain compared to the M[Bi(SCN) 6 ] structures (a z 97 vs. a ¼ 90 ).
SCXD renement allowed us to tentatively locate the positions of the H atoms and demonstrate that the orientation of the NH 4 + cation in 1 is ordered. Single crystal neutron diffraction (SCND) measurements on a large single crystal (16 mm 3 ) at 20 K, carried out using instrument D19 at the ILL, allowed accurate determination of the H atom positions and its anisotropic atomic displacement parameters, which were consistent with those observed via SCXD. Variable temperature unit cell measurements between 20 K and 260 K and an additional full collection at 260 K found no evidence of any structural phase transitions in this range. Renement of the 260 K dataset conrmed the presence of NH 4 + orientational order throughout this temperature range. The ordering of the NH 4 + cation does not lower the symmetry of 1 beyond the symmetry of compound 2.
We further investigated the energetic driving force for the observed A-site order using DFT calculations of K{Ni[Bi(SCN) 6 ]}. We carried out geometry optimisations of supercells containing the seven simplest A-site cation orders: rocksalt, layered (with layer normals along the a, b and c directions) and columnar (with columns running along the a, b and c directions), generated from supercells of the Fe[Bi(SCN) 6 ] structure [ESI Section 4 ‡]. The lowest energy structure was the observed columnar [001] order [ Table 1], which also had signicantly more anisotropic strain than all other orderings [ESI Table 4 ‡]. The stability of each cation order thus depends on the how easily the parent framework can deform to accommodate a given order.

(MeNH 3 ){Ni[Bi(SCN) 6 ]}, 3
We obtained single crystals of MeNH 3 {Ni[Bi(SCN) 6 ]}, 3, using a route analogous to that used for 1 and 2. SCXD studies of MeNH 3 {Ni[Bi(SCN) 6 ]} revealed that it also crystallises as a double double vacancy-ordered perovskite which, like 1 and 2, has a structure derived from an ReO 3 -type parent by introducing cations into half of the pseudocubic cages [ Fig. 3]. However, 3 has a more complex structure than 1 and 2 and its unit cell is a (2 Â 6 Â 4) monoclinic P2/n supercell of the Pm 3m aristotype (i.e. is 12 times larger than the structures 1 and 2), due to an unusual ordering of the MeNH 3 + cations and complex octahedral tilting.
The complexity of the order, together with the high metric pseudosymmetry c We were not able to locate the hydrogen atoms on the MeNH 3 + cation and our assignment of the polarity of MeNH 3 + cation, i.e. which atom was carbon and which nitrogen, was thus tentative. We therefore carried out a series of SCND studies on large (z1 mm 3 ) single crystals using instrument D19 at the ILL. These measurements did not allow us to denitively answer these questions because we were unable to obtain an untwinned crystal of sufficient size, but did conrm both the space group symmetry and broad structural features observed via SCXD.  We therefore carried out DFT geometry optimisations to understand the energy scales of the disorder in this system. We created an ordered model of the structure with P1 symmetry derived from our diffraction model and geometry optimised it to conrm its stability. Next, we systematically swapped the carbon and nitrogen atoms of each the eight symmetry independent MeNH 3 + cations, one cation at a time, and geometry We used ISODISTORT 35 to carry out symmetry mode analysis of 3. We rst investigated the Ni[Bi(SCN) 6 ] À framework and found that the distortion of the structure from the hypothetical parent Pm 3m structure (from the rocksalt M-site ordered Fm 3m structure) could be described well by six symmetry-adapted distortion modes in addition to rocksalt M-site order, one of which describes the global contraction of the structure  Table 2].
The two T 2 modes are notable as they are not zone corner Brillouin modes, and correspond to complex, but conventional, octahedral tilts. All conventional tilting modes will produce  a doubling of the unit cell in the tilt plane (as the rotations of adjacent octahedra within the plane have opposite senses), but adjacent layers need not tilt with the same sense. The two highest symmetry octahedral tilting modes are: all layers being in phase, a + in the Glazer tilt notation 12 and [C] in the notation of Peel et al. 36 which transforms as a M 2 + distortion mode, and each layer alternating in its sense rotation, a À , [CA] and R 5 À . In 3, the tilts normal to the b and c axes repeat aer six and four layers (respectively) and are therefore complex. In total, the tilt sequence for this perovskite is rocksalt M-site order (R 2 À ) was sufficient to produce the observed P2/n (2 Â 6 Â 4) structure. These modes are therefore likely the primary order parameters, with the R 5 À (G 4 + ) mode being a secondary order parameter. The A-site order can only be described by secondary order parameters arising from all three tilts, with any pairwise combination being insufficient, which suggests that it is the nal distribution of anions ordering produced by the complete octahedral tilt pattern which is responsible for the observed ordering. Slow evaporation of a butanone solution containing Gua(SCN), Ni 2+ and Bi(SCN) 6 3À in a 6 : 1 : 1 ratio yielded large single crystals of compound 4. We were again able to determine its structure using SCXD, which revealed that it was also a vacancy ordered double perovskite, in space group Pn 3, a (2 Â 2 Â 2) supercell of the Pm 3m aristotype [ Fig. 4]. However, 4 contains only half the number Gua + cations anticipated and renement of the occupancies showed that this structure contained a signicant fraction of M-site vacancies À 1 6 Á , rened formula Gua 0.5 Ni[Bi 0.900(4) (S 0.860(6)) N 0.841(12) N) 6 ], corresponding to , 9 -Gua 3 Ni 6 [Bi(SCN) 6 ] 5 . This reduced A-site occupancy and the presence of vacancies also accords with the lower volume of compound 4 compared to 3, despite 4 containing a signicantly larger A-site cation [ Table 3].
The observed space group of Pn 3 is that expected for the a + a + a + tilt sequence, 37 and indeed analysis using ISODISTORT conrmed this tilt sequence is adopted by compound 4. This tilt sequence is well known for other perovskites with 1 : 3 A-site cation ratio. 14 In addition each Gua + cation is disordered over four positions. Our single crystal diffraction measurements are consistent with both static and dynamic disorder, but the absence of any A-site order at 120 K, well-below typical ordering temperatures for Gua + containing molecular perovskites, 38,39 suggests that this disorder is static.
Synchrotron single crystal X-ray diffraction measurements showed the presence of weak structured diffuse scattering, consisting of rods lying along h100i * type directions [ESI Fig. 2 ‡]. The intensity of the diffuse scattering decayed with increasing scattering vector, Q, implying that the diffuse scattering is produced primarily by correlated substitutional disorder, most likely vacancy ordering, rather than displacive disorder. The asymmetric distribution of intensity around each Bragg peak additionally suggests that the structure relaxes around these vacancies. 40 Future analysis will focus on gaining quantitative understanding of vacancy order.

Gua(SCN) and hydrogen bonding
Thiocyanate is a hydrogen bond (H-bond) acceptor, but there are comparatively few studies of its hydrogen bonding propensity. 41 To benchmark the hydrogen bonding between the A-site cations and the NCS À in these materials we therefore examined a hydrogen bond rich material, Gua(SCN). 42 We redetermined the structure using a crystal present in commercially supplied GuaSCN. Gua(SCN) crystallises in space group P 1, with Z 0 ¼ 2. Its structure arises largely from the need to optimise its H-bonding, as it comprises H-bonded layers in the bc plane slip-stacked along the a direction [ Fig. 5]. These layers consist of a honeycomb lattice of NCS À ions, which lie approximately normal to the layer, and a bilayer of Gua + cations positioned at the centre of the honeycomb voids and which form a triangular lattice. Half of the NCS À ions point up and half down, and this up-down pattern is stripe-ordered along the b* direction.
Each Gua + cation forms charge-assisted bifurcated hydrogen bonds to three NCS À ions: one to an N-terminus and two to an Sterminus. Likewise, the NCS À forms hydrogen bonds to three Gua + cations, one through its N-terminus and two through its Sterminus. These hydrogen bonds also cause the NCS À ions to   41 These distances, together with a search of short contacts present in the Cambridge Structural Database, guided our investigation into the presence of H-bonding in 1, 3 and 4. We searched for all close contacts from the donor nitrogen to NCS (d NH/N < 3.2 A, d NH/S < 3.6 A), as donor hydrogen atoms were only accurately located in 1. We found that strong hydrogen bonds are present for each compound, and are likely to be structure-directing.
There  Fig. 4(b)]. As each cage contains four distinct orientations of the Gua + cation, and is surrounded by twelve NCS À ligands, this means one quarter of all NCS À will be H-bond acceptors.  38 . 25 Perovskites containing organic A-sites cations are of particular interest as these organic cations can possess intrinsic electric dipoles (e.g. MeNH 3 + ) and quadrupoles (e.g. Gua + ). The orientational order of organic A-site cations can thus generate electrical polarisation, either directly through ferrodipolar order, as in the formate perovskites, 4 or indirectly through coupling of ferroquadrupolar order to other order parameters, for example (Gua) {Cu(HCO 2 ) 3 }. 1, 38 We did not nd polar orientational order in these new perovskites, and complete orientational order was only present in 1, as 3 shows partial disorder and 4 complete disorder. Our variable temperature diffraction studies found no evidence of any phase transitions below 260 K, implying that the observed A-site disorder is static, which for compound 4 is likely related to the presence of M-site vacancies. Our DFT calculations suggest that orientational order in 3 is moderately favourable as DE CN,av ¼ 16 kJ mol À1 (0.17 eV z 6 kT at room temperature). Careful structural examination revealed that hydrogen bonding is an important factor in the structures of these materials, as in other molecular perovskites, 43 and indeed, DE CN,av is comparable to the H-bonding energies found in formate perovskites. 22 This suggests that temperature-induced phase transitions might be uncovered with careful comprehensive variable temperature structural and calorimetric studies, as in (NH 4 ) 2 {Ni [Cd(SCN) 6 ]}, which undergoes an order-disorder transition associated with the NH 4 + cation at around 120 K. 23 Optimisation of the orientational order of A-site cations towards ferroic order might be possible through crystal-engineering, by tuning the hydrogen-bonding or introducing halogen-bonding moieties, 44 and by deepening our understanding of the role of framework entropy in NCS-perovskites. 45

Coupling between A-site occupational order and octahedral tilting
In atomic perovskites, the tolerance factor has been successfully used not only to suggest whether a composition will be a perovskite, but also to provide a rst indication of the magnitude of octahedral tilts, e.g. CaTiO 3 adopts the distorted a À a À c + tilt system at room temperature, whereas SrTiO 3 is cubic with no tilts. Although the tolerance factor approach explains which molecular AMX 3 frameworks are likely to crystallise with a perovskite structure, 8,9 it does not account for either the magnitudes or kinds of framework distortions observed. 46,47 For example, in the series of formate perovskites A{Mn(HCO 2 ) 3 } where A + ¼ Rb + , 48 CH 3 NH 3 , 49 (CH 3 ) 2 NH 2 (ref. 50) and (CH 2 ) 3 NH 51 (arranged in increasing size of A + /increasing s), there is no systematic trend in the size or pattern of the octahedral tilting, respectively: a À a À c À , a À a À c + , a À a À c À and a À a À c + . The scarcity of NCS-perovskites has thus far prevented investigation of the relationship between cation size and tilts. We nd, contrary to simple geometric arguments, that the average size of the NiN 6 octahedral tilt (measured by the :N-Ni-Bi angle) and the BiS 6 tilt (:S-Bi-Ni) change very little for these four compounds from the parent M[Bi(SCN) 6 ] frameworks. This conforms to the general nding that the metalthiocyanate bond-angles do not vary in NCS-perovskites and that guest-framework interactions exert only second-order effects. 7,23,25,26 Compound 4 crystallises with both A-and Msite vacancies, suggesting that there is a maximum average size of A-site cation that can be incorporated within the {Ni [Bi(SCN) 6 ]} À framework and providing further evidence of the ease of formation of [Bi(SCN) 6 ] 3À vacancies in these materials. The tolerance factor therefore may provide a useful upper bound on cation size for NCS-perovskites (the lower bound not being meaningful due to the variety of ReO 3 structure NCSframeworks), but we have not found it to be predictive of the tilts or A-site ordering-just as for other molecular perovskites.

A-site vacancy order
A-site vacancy ordered molecular perovskites are rare, as the additional structural degrees of freedom oen mean other structure types are favoured for high vacancy concentrations: for example AM II M III (HCO 2 ) 6 compounds adopt niccolite-type structures. 52 The Prussian blue analogue cyanides can accommodate the complete range of A-site compositions, which has been exploited for their potential as battery electrode materials, 53 but the A-sites are typically disordered. Some degree of rocksalt A-site cation order has been observed in a number of frameworks of approximate composition AM II M III (CN) 6 (ref. 54) but this is typically incomplete, perhaps due to the high symmetry of these phases. 55 47 It is therefore noteworthy that A-site vacancy order appears to be the rule in NCS-perovskites, rather than the exception. 1-4 all possess complete A-site order and these orderings are unusual for perovskites molecular or otherwise: in 1 and 2 the cations have columnar order; in 3 the MeNH 3 + order into 3 Â 2 Â 1 blocks and in 4 the cations are present in one quarter of the cages with Cu 3 Au order. The block-order of cations in MeNH 3 {Ni[Bi(SCN) 6 ]} is to the best of our knowledge unknown in any other perovskite. It can be related to the nanochequerboard/nanochessboard phases observed in compositionally complex analogues of the rare-earth vacancy perovskites, such as ,1 3 þ4x Nd1 3 Àx Li 3x fTiO 3 g; x z 0:1 (ref. 57) and , 0:2 Nd 0:6 Ca 0:1 fTiO 3 g. 58 These phases have a modulation in the occupancy of the A-site on a ca. 5 nm lengthscale. In addition the combination of M-site rocksalt and A-site columnar order found in 1 and 2 has only been reported previously for the high-pressure oxides MnLnMnSbO 6 , Ln ¼ La, Pr, Nd, Sm, 21 and CaMM 0 ReO 6 , M ¼ Mn or (Mn 0.5 Cu 0.5 ) and M 0 ¼ Mn or Fe. 59 We nd that for this family of compounds the A-site order and tilts are strongly coupled: each tilt sequence has its own cation order. Columnar order in 1 and 2 accompanies the a À a À c + tilt, the unique A-site order in 3 is accompanied by the unique complex a + b ++À++À c +À+À tilt, and the Cu 3 Au order occurs with a + a + a + tilt. One possible reason for this can be seen in the distribution of NCS À anions between pseudocubic cages [ Fig. 6]. Each NCS À must lie within one of four adjacent pseudocubic cages, with which cage it lies within determined by the tilting of two metal octahedra it is connected to [ Fig. 6(a)]. Each cage is bounded by 12 thiocyanates, so on average a cage contains three thiocyanates. In 4 one quarter of the cages contain no NCS À , with 3 4 containing four thiocyanates [ Fig. 6(d)]. We nd that the pseudocubic cages containing no NCS À are the cages containing Gua + cations, whereas the pseudocubic cages containing four NCS À contain no A-site cations. This correlation likely arise from simple reasons of sterics: there is not enough space in the cages containing four thiocyanates for an A-site cation. This approach is in agreement with previous rationalisations of the structures of CaCu 3 Ti 4 O 12 -type perovskites, which also have a + a + a + tiles and Cu 3 Au A-site order, where the largest cation (e.g. Ca 2+ ) sits in the cages containing no O 2À anions. 20 The pseudocubic cages in 3 contain 0, 2, 3 and 4 NCS À anions in the ratio 1 : 3 : 2 : 6; each and every pseudocubic cage which does not contain an A-site cation contains four NCS À anions, and every cage containing fewer than four anions also contains an A-site cation [ Fig. 6(c)]. This suggests that the complex tilt pattern derives, in part, from the need to rearrange the NCS À anions to accommodate the larger MeNH 3 + cations in the pseudocubic cages. This ability of octahedral tilts to increase the available volume in some cages, at the expense of others, provides an explanation for why the average, rather than maximum, A-site cation size appears to be the key factor for perovskite stability. In contrast, all the cages in 1 and 2 contain three NCS À and so cooperative framework shear therefore is necessary to accommodate the A-site cations. We have applied this anion-in-cage counting method to each of the four simplest 3-tilt patterns (in the approximation that all tilts have equal magnitude) [ Table 4]. We nd that these tilts, aside from the previously mentioned a + a + a + tilt sequence, would not be expected to stabilise any particular A-site cation order according to this counting method, as all cages contain three NCS À , even if the cages are symmetry distinct. This suggests that complex tilts may be well be favoured in molecular perovskites with large Asite cation size disparities.

Complex tilts
The tilts along the b and c directions in 3 are complex, that is, the repeat distance for the tilt pattern is greater than two unit cells (i.e. the tilts transform as a [00k]T 2 irrep) [ Fig. 7(a)]. There are very few examples of perovskites with complex tilts, 60,61 and perhaps the best known is LiNbO 3 , which forms phases possessing tilts with periodicities of four (S-phase) and six (Rphase) unit cells along a single axis. 36 Complex tilting has also been observed in molecular perovskites, 46 with incommensurate tilting discovered in Me 2 NH 2 {Co(HCO 2 ) 3 } 62 and complex tilts along multiple axes in Me 2 NH 2 {Mn(H 2 PO 2 ) 3 }. 47 In both cases the tilts are unconventional as they are formed from shis and/or out-of-phase tilts (where adjacent octahedra tilt in the same sense). Compound 3 is the rst perovskite of any kind, to our knowledge, to show complex conventional tilts along multiple axes.

Breaking centrosymmetry with complex tilts
It is well known that simple conventional cooperative octahedral tilts cannot produce non-centrosymmetric structures. 12 This is only true, however, for Brillouin zone-corner tilts. We show here that tilt patterns containing complex tilts can generate non-centrosymmetric structures, by carrying out symmetry analysis of the structures derived from the simplest example of a complex tilt, the 1 2 T 2 tilt along the c axis, with in-phase M 2 + tilts along the a and b axes generates a non-centrosymmetric structure with space-group P 42c [ESI Fig. 7 ‡]. We generated a model of a hypothetical Fe[Bi(SCN) 6 ] polymorph possessing these distortions using ISODISTORT and then examined the distribution of NCS À anions. The pseudocubic cages contain 0, 2 or 4 NCS À in the ratio 1 : 2 : 6, suggesting that a structure in which 3 8 of the cages were occupied by A-site cations would stabilise this distortion [ Fig. 7(b)]. We therefore lled the cages containing 0 or 2 NCS À anions with Cs + cations, producing a model with composition Cs 3 {Fe[Bi(SCN) 6 ] 4 } [ Fig. 7(c and d)]. Very encouragingly our model, which was constructed only taking into account symmetry analysis and counting cage occupancies, was found to be stable with DFT geometry optimisation. We found that moving a Cs + cation into any of the other cages incurred a signicant energetic penalty, suggesting that this cation ordering is energetically preferred. This structure is piezoelectric, with the piezoelectricity arising from the complex tilts along the c-axis. The investigation of other stoichiometries and A-site cations will likely be a fruitful route to generating new polar and ferroelectric perovskites.

Conclusion
In this study we have investigated how the identity of the A + cation determines the coupled A-site occupational order and octahedral tilt distortions of , x A 2Àx Ni½BiðSCNÞ 6 xþ2 3 , A ¼ K + , NH 4 + , MeNH 3 + and Gua + , vacancy-ordered perovskites. We Table 4 No. of NCS À present in pseudocubic octahedral cages for the four simplest three tilt sequences Tilt n cage Â n NCS À/cage a + a + a + 3 Â 4 : 1 Â 0 a + a + a À 1 Â 3 a + a + a À 1 Â 3 : 1 Â 3 a À a À a À 1 Â 3 have shown that organic cations can be hosted as A-site cations in NCS-perovskites, and that the synthesis of large single crystals of these materials can be achieved via facile solution methods. NCS-perovskites have robust M-site order and nearly xed magnitude conventional octahedral tilts: in this work we have shown that A-site occupational order can also be readily achieved in NCS-perovskites. We demonstrate that, by controlling these three orders at once, we can produce new and unprecedentedly complex perovskite structures, notably the non-Brillouin zone corner tilts and block A-site order found in (MeNH 3 ){Ni[Bi(SCN) 6 ]}. We have devised a simple counting method for predicting the coupling between octahedral tilts and A-site occupational order, supported by DFT calculations. Finally, we have shown how complex conventional tilts can produce new routes to noncentrosymmetric materials. These results suggest that exploration of NCS-perovskites and complex tilts more generally may uncover further functional behaviour, including ferroelectricity, and anomalous mechanical properties such as negative thermal expansion or negative linear compressibility.