Phase transitions and (p–T–X) behaviour of centrosymmetric perovskites: modelling with transformed crystallographic data

The structures of centrosymmetric perovskites (ABX 3) are modelled by transforming crystallographic data into a Cartesian space defining BX 6 octahedral tilting and distortion. Structural evolution and phase transitions under variable (p–T–X)-conditions are analysed and predicted.


Introduction: the modelling of octahedral tilting in perovskites
Synthetic perovskite-related compounds continue to attract the attention of many scientists and technologists, irrespective of whether they are working, for example, on the development of lead-free piezoelectric ceramics (Shrout & Zhang, 2007) or on organolead halide ABX 3 nanocrystals for photochemical cells (Jena et al., 2019). In general, the current availability of high-quality structural data evokes the need for all the nuances of structural change under varying conditions of pressure, temperature and composition] (p-T-X) to be precisely modelled. This expectation is particularly poignant in the case of perovskite-related compounds, ABX 3 , since, according to Mitchell (2002a), all elements, apart from the noble gases, can be found in a variant of these. In the pioneering work of Megaw (1973), their structures were regarded as comprising three essential features: (a) tilting of the anion octahedra; (b) displacements of the cations; (c) octahedral distortions. Definitive work by Glazer (1972) led to a system of classification based on the sense of tilt of the three tetrad axes of corner-linked regular octahedra about three perpendicular, or nearly perpendicular crystal axes.
A regular octahedron is characterized by three pairs of opposite vertices that are linked by stalks of equal length passing through its mid-point at right angles to one another. Two lines of regular octahedra are shown in Figs. 1(a) and 1(b), as they arise in space group Pbnm.
Zigzag chains of adjacent octahedral stalks of uniform length s are formed, these lying in planes PQRS and TUVW. When viewed along the y PC axis-of-tilting, the red and blue octahedra are seen to be rotated in opposite senses [Fig. 1(c)], leading to Glazer notation b À . By comparison, the red and green octahedra are rotated in the same sense [Fig. 1(d)]. This leads to the notation c þ , since the tilt angle is different. As tilting around the x PC and y PC axes is equivalent, the threedimensional tilt system in Pbnm is denoted by a À a À c þ (Glazer, 1972).
In this now well established a # b # c # nomenclature, superscripts # can be +, À or 0, denoting the angles of tilt of neighbouring octahedra along the three pseudocubic axes as in-phase (+), anti-phase (À) or zero. Importantly, Glazer (1972) also showed that the 23 tilt systems for regular octahedra correlated with 15 alternative space groups. As well as aiding the correct interpretation of perovskite diffraction patterns (Glazer, 1975), these correlations stimulated grouptheoretical analysis of perovskites, their being confirmed significantly later by Howard & Stokes (1998. The latter Tilting of regular octahedra about the x PC and y PC axes, as in space group Pbnm. Pseudocubic axes x PC , y PC and z PC are directed parallel to orthorhombic vectors 1 1 10 Â Ã , 110 ½ and [001], respectively. (a) Tilting around the y PC axis (in clinographic projection) with y the angles of inclination of stalks of length s to the axis; (b) Tilting around the z PC axis (head-on view showing mirror planes); (c) y PC -axis tilting [as in (a)] viewed along the y PC axis towards the origin; (d) z PC -axis tilting [as in (b)] viewed along the z PC axis towards the origin. work led to minor modifications to the space groups originally assigned by Glazer and established a top-down, hierarchical system of group-subgroup relationships starting from the cubic Pm3m aristotype for perovskite structures with tilted, regular octahedra (Fig. 2).
Two independent contributions concerning octahedral tilting in perovskites were made by Woodward (1997a,b) and Thomas (1996). Woodward (1997a) concluded that perfectly regular octahedra could not be linked together in some tilt systems. The contentious space groups were subsequently narrowed down to Cmcm (tilt system a 0 b þ c À ) and P4 2 /nmc (a þ a þ c À ) by Howard & Stokes (1998), with only the latter requiring irregular octahedra. The work of Thomas (1996), by comparison, was primarily concerned with the dependence of AX 12 :BX 6 polyhedral volume ratio, V A /V B , on octahedral tilting in orthorhombic and tetragonal perovskites. Inclination angles y and z in Figs. 1(a) and 1(b), and by extension x , were used to relate octahedral stalk lengths s and tilting to the lengths of pseudocubic cell axes 1, 2 and 3 and therefore cell volume [equation (1)].
This form results from the coupling of inclination angles 1 and 2 when pseudocubic axes 1 and 2 are oriented at approximately 45 to the crystal axes. In untilted structures (with 1 = 2 = 3 = 0), V A /V B is exactly equal to five (Thomas, 1989). Tamazyan & van Smaalen (2007) subsequently argued that inclination angles 1 , 2 and 3 defined by Thomas (1996) were unnecessarily influenced by octahedral distortion. They therefore proposed an alternative method of calculating tilt angles, although this was at the expense of losing the simple link to polyhedral volume ratio expressed by equation (1). Later work by Wang & Angel (2011) restored the link between octahedral tilting and ratio V A /V B , employing a group-theoretical, rather than a geometrical approach to separate octahedral tilting and distortion. These authors expressed the ratio V A /V B as a function of the amplitudes of the normal modes in a cubic perovskite, rather than by direct calculation from anion coordinates in experimentally determined crystal structures.
The conflict between geometrical, i.e. crystal-chemical and group-theoretical methods in describing perovskite structures is somewhat contrived, since both have legitimate fields of application and ultimately have similar aims. This was made clear in seminal work by Knight (2009), who utilized grouptheoretical methods to develop a general parameterization of centrosymmetric perovskites based on symmetry-adapted basis vectors of the Pm 3 3m phase. He also pointed to difficulties with a method earlier proposed by this author for a general crystal-chemical parameterization of centrosymmetric perovskites (Thomas, 1998). Two specific problems were identified, (a) that the method was 'geometrically complex '; and (b) that it relied 'totally on an empirical analysis of known crystal structures'. He also made the important point, (c), that the A site has a coordination number less than 12 due to a geometrically complex coordination polyhedron (Knight, 2009). In the current work, point (a) is addressed by focusing on the distortion of centrosymmetric octahedra as independent geometrical forms. Point (b) is addressed by deriving analytical expressions dependent on space group for the three stalk vectors defining octahedral geometry. Point (c) is addressed by developing a simple parameterization based on AX 8 sub-polyhedra that correlates with the tilt classification system of Glazer (1972). Details of these methodological improvements are given in the following section.
2. Elements of the revised crystal-chemical method 2.1. Pseudocubic representations of octahedra In order that an object can be described as an octahedron, it requires three recognisable pairs of opposite vertices that may be linked to each another by vectors. If the ends of the stalks of a regular octahedron [Fig. 3(a)] are randomly displaced by limited amounts [ Fig. 3(b)], the resulting form is still recognisable as an octahedron [Fig. 3(c)]. The six independent vertices lead to six sets of x; y; z ½ triplets in Cartesian space, i.e. 18 parameters. If only the form of the octahedron is relevant, and not its absolute position or orientation, these 18 parameters are reduced to 12 by subtracting six parameters: three to fix one vertex in space at [0,0,0] and three to define the orientation of the octahedron. In Fig. 3(c), one vertex has been fixed at [0,0,0] and the octahedron so rotated that the opposite vertex is fixed at [0 ,0, z 1 ]. A further rotation of the octahedron about the z axis has been carried out to fix a vertex at [0, y 2 , z 2 ]. The total of six coordinate-components equal to zero here signifies a reduction in independent parameters from 18 to 12, i.e. (z 1 ; y 2 z 2 ; x n y n z n ; n = 3,5). If, however, the translations at opposite ends of the three stalks are equal and opposite [Fig. 3(d)], a centrosymmetric octahedron results [ Fig. 3(e)], in which the three stalks bisect one other at the centre of symmetry. Only one end of a stalk needs to be fixed in space in order to define the position of the opposite end, so that the number of independent parameters is halved to six [Fig. 3(e)]: ðz 1 ; y 2 z 2 ; x 3 y 3 z 3 Þ. This situation applies to all perovskites with B ions located at centres of symmetry.
The six independent parameters of a centrosymmetric octahedron may be assigned to three stalk lengths a 1 , a 2 ; a 3 and three angles of intersection of the stalks, 12 , 23 and 31 . In order to visualize the extent of its distortion, a parallelepiped enclosing the octahedron may be constructed by displacing the three stalk vectors to a common origin (Fig. 4).
As the three independent edge lengths of the parallelipiped are approximately equal, it may also be termed a pseudocube. Such a pseudocube is defined by the same six parameters as the octahedron, i.e. a 1 , a 2 ; a 3 ; 12 ; 23 and 31 . It follows that regular octahedra would lead to pseudocubic representations of octahedra (PCRO) of cubic form.
In the case of the non-centrosymmetric octahedron of Fig. 3(c), the six octahedral vertices do not touch their parallelogram faces at the meeting points of the four quadrants [Fig. 4(d)]. The 12 independent parameters may be accommodated by noting the 2D polar coordinates r; ½ of the three emergent octahedral vertices, giving rise to additional parameters r 1 , r 2 ; r 3 ; 1 ; 2 and 3 . Thus the PCRO construction is of general validity for visualizing distorted octahedra. Its parameterization may prove to be useful for characterizing non-centrosymmetric and polar perovskites in future work. The transformation between octahedron and PCRO is also reversible. An advantage of the PCRO visualization method is that the aggregate distortion parameters representing normal and shear distortion developed for pseudocubic representations of tetrahedra (PCRT), and , (Reifenberg & Thomas, 2018;Fricke & Thomas, 2021) may be taken over without modification [equations (2), (3)]. The number of independent parameters of a generalized, isolated centrosymmetric octahedron is equal to six. Views (a) to (e) discussed in x2.1.
In this connection, the following values are obtained for the centrosymmetric and non-centrosymmetric octahedra of Figs [0.0370, 7.10 ]. It follows that the centrosymmetric octahedron displays a higher degree of angular distortion here. Furthermore, if the angular distortion parameters are expressed in radians instead of degrees, the relative degree of shear versus normal distortion can be quantified. Since 11.90 correspond to 0.21 and 7.10 to 0.12 radian, it follows that the shear distortion is greater than the normal distortion in both cases.

Parameterization of PCRO in terms of three vectors
The octahedra in perovskites are not isolated but form a three-dimensional corner-sharing network. Their crystal structural parameters, i.e. space group, unit-cell parameters and atomic coordinates, deliver full information on octahedral distortion, tilting and connectivity. Importantly, space group symmetry ensures connectivity. Once the distortion and tilting of a single octahedron in a structure with only one symmetry-independent octahedron has been defined, the distortions and tilting of all the other octahedra in the unit cell follow. To convey full information on the tilting of this single octahedron in a structure, its PCRO may be described by the three Cartesian stalk vectors, a 1 , a 2 and a 3 along its pseudocube edges. Generalized, analytical expressions for these vectors in different space groups may be derived inductively from known crystal structures. This process is described in x2.2.1 for space group Pbnm with B ions located at 4b special positions. The three vectors defining the axes of the pseudocubic unit cell are likewise defined in this Cartesian space, so that inclination angles of the stalks to the three pseudocubic axes, 1 , 2 and 3 may straightforwardly be calculated [see Fig. 1, equation (1) and x2.3]. In addition, the method also allows calculation of tilt angles a , b and c of the octahedra around the three pseudocubic axes, even when the octahedra are distorted.
2.2.1. Space group Pbnm with B ions at 4b positions. The octahedral cage coordinating the B ion at 0 1 2 0 in the unit cell of orthorhombic space group Pbnm may be taken, this being the cab setting of space group 62 with standard symbol Pnma. 0 1 2 0 is one of the 4b special positions, with X1 ions in 4c special positions and X2 ions in 8d general positions (see Table S1 in the supporting information). 1 Derivation of the analytical form of stalk, or equivalently PCRO vectors a 1 , a 2 and a 3 starts by taking an example structure, such as CaTiO 3 at 296 K [Yashima & Ali (2009); ICSD code 162908]: a = 5.3709 Å , b = 5.4280 Å , c = 7.6268 Å ; x(O1) = 0.0708, y(O1) = 0.4830, x(O2) = 0.7113, y(O2) = 0.2891, z(O2) = 0.0375; z(O1) has the fixed value of 1 4 . The essential step is to convert from numerical fractional coordinates of the titanium ion and its six oxygen neighbours (Table 1, columns 2-4) to analytical fractional coordinates ( Table 1, columns 8-10). One way to generate the numerical fractional coordinates would be to use a simple computer program to generate the Cartesian coordinates of the titanium and oxygen ions. (Table 1, columns 5-7). The numerical fractional coordinates (Table 1, columns 2-4) would then follow by multiplying these by the inverse orthogonalization matrix, in this trivial case Letters a to f of the oxygen ions correspond to the atom labels in Fig. 1(a). They are based on the principle that the octahedral stalk from a to b is oriented closest to the x PC axis in the positive direction, with the stalks from c to d and from e to f oriented closest to the positive y PC and z PC directions, respectively. The analytical expressions in columns 8 to 10 of Table 1 are obtained by inspection: the values in columns 2 to 4 are compared with the starting values of x(O1), y(O1), x(O2), y(O2) and z(O2). It is important to note that the results depend on the convention used for expressing these para-   Table 2 by taking the differences in the analytical fractional coordinates for atom pairs (a,b), (c,d) and (e,f) given in Table 1. Cartesian vectors are formed by multiplying these differences by the orthogonalization matrix The Cartesian axes are parallel to the axes of the orthorhombic unit cell in this case.
By adopting this analytical representation, vectors a 1 , a 2 and a 3 are no longer tied to the example structure of CaTiO 3 : they apply to all perovskites in space group Pbnm, provided that the B ions occupy 4b sites and that the correct convention in choosing the values parameters x(O1), y(O1), x(O2), y(O2) and z(O2) has been followed. All eight crystallographic parameters are involved in defining the geometry of the PCRO in space group Pbnm: a, b, c, x(O1), y(O1), x(O2), y(O2), z(O2). For regular octahedra, six of these eight degrees of freedom (d.o.f.) would be used up in forming octahedra of a particular volume, with five defining the regular form and the sixth the volume. The remaining two d.o.f. would be used to define the unit-cell parameters and octahedral tilting. If the octahedra were only approximately regular, as is generally the case, more d.o.f. would be available for optimizing the octahedral tilting and unit-cell volume, in response to different A and B ion radii or to changing temperature or hydrostatic pressure. Thus the idea of structural compromise in forming connected octahedral networks in perovskites can be modelled in response to varying (p-T-X) conditions.

2.2.2.
Other space groups. The other space groups analysed in this work are those that commonly arise in experimentally determined crystal structures. They span the following space groups: Pbmn (B ions in 4a sites), Cmcm, Ibmm, P4/mbm, P4 2 /nmc, I4/mcm, R 3 3c, Pm 3 3m. Tables similar to Table 2 are generated for them in xS2 of the supporting information.

Implementation of the crystallographic to structural transformation in the Microsoft Excel Solver environment
Structural analysis requires a one-way transformation from crystallographic to structural parameters. For example, crystallographic parameters a, b, c, x i , y i , z i are transformed to structural parameters such as PCRO parameters and tilt angles. By comparison, structural prediction requires a reversible transformation between the two parameter sets. The modeller will seek to establish systematic variations in the structural parameters with (p-T-X). If successful, interpolations and extrapolations to other (p-T-X) values can be made before reverse-transforming to crystallographic parameters. This technique has been demonstrated for olivines (Thomas, Table 1 Calculation of analytical fractional coordinates of the six oxygen anions coordinating the titanium ion in CaTiO 3 located at 0, 1 2 ,0.

Figure 5
Screenshot of the Pbnm worksheet, which contains the reference data for CaTiO 3 at 296 K (Yashima & Ali, 2009). No refinement has taken place, since the entries in the 'Refined data' and 'Reference data' boxes are identical.

Table 2
The three vectors defining the PCRO with mid-point 0, 1 2 ,0 in space group Pbnm.
Cartesian components Nearest pseudocubic axes 2017), coesite (Reifenberg & Thomas, 2018) and quartz (Fricke & Thomas, 2021). The requirements of ease and flexibility of use together with a reversible transformation suggest that the Microsoft Excel Solver environment is appropriate for applying the method. In this connection, an Excel datafile already programmed is provided in the supporting information. This consists of eight worksheets for the different space groups. A screenshot of the worksheet for Pbnm with B ions in 4b positions is given in Fig. 5.
The user enters the crystallographic data in the 'Refined data' box in blocks K3:K5 (light-green background), N4:N8 and N10:N11 (mustard background), whereupon all the structural parameters are automatically calculated by Excel. The computational core of the spreadsheet is in blocks R9:T11 (deep-blue background) and U9:W11 (dark-green background). The formulas in these cells correspond to the entries in Table 2 and the pseudocubic axes in Cartesian coordinates, respectively. Values of all the structural parameters relating to octahedral distortion and tilting (light-blue background) can be seen as ways of describing the numerical values in these two blocks in a structurally meaningful way.
The purpose of the 'Reference data' box is threefold. First, it acts as a repository for the source, reference data (cells with orange background). Secondly, the dependent structural parameters can be calculated for this reference data by clicking on the button in cell C15. 2 Thirdly, both reference data and dependent structural parameters can be used as constraints in structural refinements, which apply to the cells in the 'Refined data' box.
The different types of structural parameters, all with lightblue background, are summarized as follows. a) PCRO parameters a 1 , a 2 , a 3 , 23 , 31 , 12 are calculated as the lengths and intersectional angles of the vectors a 1 , a 2 , a 3 in block R9:T11. For example, 23 ¼ arccos a 2 Á a 3 =a 2 a 3 ð Þ . Aggregate distortion parameters and [equations (2) and (3)] are calculated in block P13:P16, the latter quoted in both degrees and radians. b) Pseudocubic angle PC is calculated as the angle between pseudocubic axes a PC and b PC in cell P11: c) Inclination angles 1 , 2 , 3 of the octahedral stalks are quoted in block X9:X11. These describe the relationship of the two sets of vectors in blocks R9:T11 and U9:W11 to one another. With respect to vector a 1 in block R9:T9 and its nearest pseudocubic axis vector a PC in block U9:W9, the following applies: The subscripts X; Y; Z here denote the Cartesian X; Y and Z components, respectively. It follows that d) The method of calculation of angles of tilt a and c in the a À a À c þ tilt system of space group Pbnm is shown in Fig. 6. These angles constitute an alternative to the inclination angles for relating the two sets of vectors in blocks R9:T11 and U9:W11 to each other. Since tilt angles are to be calculated for distorted octahedra, the method evaluates the projections of oxygen atoms O a, O b, O c and O d in the planes perpendicular to pseudocubic axes a PC , b PC and c PC . These four atoms also define PCRO vectors a 1 and a 2 : vector a 3 does not influence the calculated tilt angles.
A tilt of type a À around a PC is the angle between vector (O c 0 -O d 0 ) and its projection in the xy plane. Similarly, an a À tilt around b PC is the angle between vector (O a 0 -O b 0 ) and its projection in the xy plane. Calculated values are shown in cells Y9 and Y10 in Fig. 5, with the mean value in cell Y11 (8.53 , 8.51 and 8.52 , respectively). The observed difference of 0.02 in calculated values results from an interplay of the deviation of PC from 90 and the octahedral distortions.
The in-phase c þ tilting causes the projections of the a 1 and a 2 octahedral vectors in the xy plane to be rotated away from pseudocubic axes x PC and y PC (Fig. 6). The two resulting tilt angles are given in cells Y12 and Y13 (8.82 and 8.87 ), with the mean value in Y14 (8.84 ).
The algorithms for calculating these tilt angles are described in xS3 of the supporting information, where reference is also made to their implementation in the Excel file in the supporting information.
e) Unit-cell volume V UC , polyhedral volumes V B , V A and volume ratio V A / V B are calculated in blocks P3:P4 and P6:P7.
a 1x a 1y a 1z a 2x a 2y a 2z a 3x a 3y a 3z : The latter calculation method is facilitated by the centrosymmetry of the octahedron. Volume ratio V A /V B follows from these parameters as ð Þ. f) Parameters g A and g B are likewise extracted from the core computational data (blocks R9:T11 and U9:W11) and defined in x2.6. g) Parameters L and (block N18:19 with brown background) relate to the A ion positions and are independent of the anionic network. They are defined in x2.7.

Example Solver refinements
Initial insight into use of the Solver is gained here by addressing an issue relevant to the historical development of perovskite structural chemistry: the ability of different space groups to accommodate regular octahedra. A comparison is also made between tilt angles calculated as in point d) of x2.3 and the values yielded by commonly used approximations.
The five degrees of freedom required to define the regular octahedral form (see x2.2.1) result in the constraints a) a 2 = a 1 ; b) a 3 = a 2 ; c) 23 = 90 ; d) 31 = 23 ; and e) 12 = 31 . Since there are eight d.o.f. in total, two further constraints may be applied in the refinement. For example, it may be stipulated that the unit cell and the octahedral volumes remain unchanged: f) V UC = V UC,reference and g) V B = V B,reference . The coding of these constraints in the Solver in shown in Fig. 7(a), whereby constraints f) and g) also refer to cells $H$3 and $H$4 in Fig. 5.
The end-point of this tightly defined refinement is shown in Fig. 7(b): a cubic PCRO of side length 3.9014 Å is generated, corresponding to a regular octahedron with perpendicular stalks. The values of the other structural parameters described in x2.3 appear in the cells with a light-blue background.
The crystal structures obtained by applying the same refinement conditions to reference structures in all the space groups are given in Table 3. The Solver constraints for the respective space groups are pre-programmed in the Excel file in the supporting information.
For space group P4 2 /nmc, the obtaining of regular octahedra in tilt system a + a + c À is at variance with the conclusion of Howard & Stokes (1998). A resolution of this discrepancy is thought likely to depend on the observed interdependence of the coordinates of atoms O2 and O3 in the refined structure (Table 3), which implies a refinement into higher symmetry or pseudo-symmetry than P4 2 /nmc.
The values of tilt angles below the dividing line in Table 3    Tilted octahedra and basal parallelogram planes, e.g. PQRS, of AX 12 polyhedra (in yellow) viewed along the orthorhombic z axis in space group Pbnm. The + and -signs in black denote z heights of the octahedral vertices relative to the z = 0 plane of the diagram. These are due to a À tilting. Green dashed lines show the directions of pseudocubic axes x PC and y PC , the reference directions for the a À tilting. Red dotted lines represent the perpendicular planes to the green dashed lines, which are projected as lines in the xy plane. Red circles lie above the red dotted lines at the same z height as the octahedral vertices O a to O d. They are denoted by O a 0 to O d 0 and result from the four projections (O a ! O a 0 ), . c þ tilting causes the octahedra to be rotated about the orthorhombic z axis (or equivalently, the z PC axis). The relevant tilt angles, c,1 and c,2 , are shaded orange. (Vectors p and q are defined in x2.6.) angles by Megaw (1973), a more developed treatment according to space group was given by Kennedy, Prodjosantoso et al. (1999) as follows. The four parameterizations a) to d) here correspond to headers a) to d) in the row 'Method' in Table 3. a) Pbnm (B ions in 4a): O2: Mountstevens et al. (2003) augmented this set with a result for space group Imma, which upon resetting in Ibmm, is as follows: d) Ibmm: O2: ð 1 4 ; 1 4 ; wÞ. a ¼ arctan In general, the agreement between values generated in this work and approximations a) to d) above is satisfactory for the in-phase tilts, c . Antiphase angles a for space groups Pbnm and Ibmm show poorer agreement. In order to investigate whether this is a systematic discrepancy, the structural data of Kennedy, Prodjosantoso et al. (1999) for CaTiO 3 (Pbnm) at 1273 K were fed into the Excel program, yielding the following two sets of tilt values and inclination angles 3 : a = 6.53 ; a,refined = 6.59 ; 3 = 8.92 ; 3,refined = 9.24 . These are to be compared with the tilt values given by the Kennedy approximation: a = 9.19 ; a,refined = 9.24 . These results and those in the Pbnm (B: 4a) and Ibmm columns of Table 3 lead to the following two conclusions: (1) The a angles calculated by the method of x2.3 are systematically lower than those generated by the Kennedy approximation; (2) The Kennedy approx-imation generates exact values of inclination angle for refined structures with regular octahedra, i.e. 3,refined , and not tilt angle a as defined in this work.
In the case of Cmcm, good agreement for a between x2.3 and Kennedy, Prodjosantoso et al. (1999) is obtained for regular octahedra in the refined structure, but not for the distorted octahedra in the unrefined structure.
A rationalization for the smaller values of a generated by the method of x2.3 is that the inclination angle 3 of a regular octahedron will always be larger than an angle within a plane of projection perpendicular to a pseudocubic axis (Fig. 6). The former angle is generated by the approximations of Kennedy, Prodjosantoso et al. (1999) and the latter angle by the method of x2.3.

PCRO parameters, inclination angles and tilt systems with numbers of degrees-of-freedom by space group
Whether regular octahedra can be formed in a particular space group depends on whether the octahedra have sufficient degrees of freedom (d.o.f.). The number of d.o.f. defining the octahedral anionic network is N(UC) + N(X), of which N(tilt) are used to define independent tilt angles. In P4 2 /nmc with irregular octahedra, the tilt system is a þ b þ c À , i.e. N(tilt) = 3. For regular octahedra, this is reduced to a þ a þ c À with N(tilt) = 2. The parameter denoting the remaining d.o.f. available to construct the octahedra, N(PCRO), is equal to N(UC) + N(X) À N(tilt) (  Table 3 Summary of Solver refinements by space group corresponding to structures with regular octahedra.
The octahedral and unit-cell volumes are identical to those in the reference structures.

Space group
Pbnm ( and R 3 3c, this does not preclude the formation of regular octahedra, since the space group symmetry itself provides partial regularity: in Ibmm: a 1 = a 2 and 23 = 31 ; in I4/mcm and P4/mbm: a 1 = a 2 with 23 = 31 = 12 = 90 ; in R 3 3c: a 1 = a 2 = a 3 with 23 = 31 = 12 . In space group P4 2 /nmc, the N(PCRO) value of less than 6 signifies that the two tilt angles in the refined structure will be interdependent.
The final two rows in Table 4 relate to the d.o.f. assigned to A and B cations. The structures associated with the A cations are discussed in x2.7, after the primary structures associated with the parameter N(tilt) have been considered in the following sub-section.

Additional anionic network parameters related to octahedral tilting
Since octahedral tilting per se is a secondary structural feature relative to the BX 6 octahedra, the primary structures resulting from this tilting necessarily concern the coordination of the A ions. It has long been customary to assume AX 8 polyhedra in GdFeO 3 -type perovskites such as CaTiO 3 in space group Pbnm (Liu & Liebermann, 1993). Mitchell & Liferovich (2004), adopting the perspective of coordination chemistry, reported an increase in coordination number from 8 to 9 with x in the solid solution series Ca 1-x Na x Ti 1-x Ta x O 3 . Mitchell (2002b) states generally that coordination numbers of 8, 9, 10 and 12 are possible, and calculates as an example the volume of uncoordinated space in SrZrO 3 (I4/mcm) as 8.7%. This is the difference between V A , the volume assigned to AX 12 polyhedra from volume filling with BX 6 octahedra, and the volume of a more appropriate AX 8 polyhedron, V AX 8 ð Þ. From a functional viewpoint, the AX 8 polyhedra in tilted perovskites provide scaffolding to support the eight octahedral faces of the full AX 12 polyhedra. In Pbnm, the inner AX 8 polyhedron is formed by the top and bottom parallelogram AX 12 faces and four vertical struts (Fig. 8).
It is now appropriate to interpret Fig. 6 in terms of the effect of octahedral tilting on A-ion coordination. The c + tilting causes parallelograms PQRS to be formed instead of rectangles, with a concomitant reduction in area A(PQRS). The a À tilting, by comparison, leads to equal and opposite displacements of vertices P, Q, R and S along z (denoted by + and À). The volume contributions made by plane PQRS and the other five faces of the AX 8 polyhedron to V AX 8 ð Þ are not affected by this tilting, since changes brought about by the equal and opposite displacements relative to the centre-of-symmetry at [0,0,0] cancel one another out. 3 The mirror plane through A, B, C and D in all space groups with c + tilting means that the horizontal positions, in this case the x and y coordinates of the bottom and top parallelograms in Fig. 8, are identical. It follows that the formula V(AX 8 ) = A(PQRS)(c/2) could be used, since the centres of both parallelograms are separated by c/2 in all space groups with in-phase c-axis tilting apart from P4/mbm, where the separation is c. In space groups P4 2 /nmc and I4/mcm, to which anti-phase, c À tilting applies, the same formula is valid, since the mean cross-section ? z is A PQRS ð Þ .  3-axis in equation (1) z -z z z --Inclination angles 1 6 ¼ 0 Since the c AE component of the tilting is solely responsible for the reduction in A(PQRS) and V(AX 8 ), the converse applies that A(PQRS) quantifies the extent of this tilting. Without c AE tilting, A(PQRS) would be equal to one quarter of the unit cell cross-sectional area, i.e. (ab)/4 in crystal systems with perpendicular x and y axes. A dimensionless coefficient A to quantify the amount of c AE tilting may therefore be defined as follows.
Although the b À tilting about the x PC and y PC axes in Pbnm, more generally the a AE b AE tilting component when applied to all space groups, does not affect V AX 8 ð Þ, it does affect V B . For a given perovskite compound with unit-cell cross-section ab, this tilting allows larger V B values than would otherwise have been the case. The greater the degree of tilting, the larger the V B value. A dimensionless coefficient B to quantify this is given by relating V B to the volume of an upright, untilted octahedron of equal basal area in xy projection, V B;ref . The in-plane components of vectors p and q in Fig. 6 define the waist of this octahedron of perpendicular height c/2. These depend in turn on PCRO vectors a 1 and a 2 .
In the presence of a AE b AE tilting, B > 1. V B is calculated as one-sixth of the volume of the PCRO and V B,ref as twice the volume of a right pyramid of height c=4 with parallelogram base of area p x p y q x q y ; i.e.
1 3 Á c 2 p x p y q x q y : Analytical expressions for A and B are quoted in Table 5. The form of B is similar for all space groups, and A can generally be traced back to simple expressions involving fractional coordinates. Particularly significant are the fixed, limiting values of A ¼ 1 and B ¼ 1 in space groups Ibmm and I4/mcm, respectively. These govern the sequence of phase transitions in series with increasing V A volume (see x3.1). The corresponding derivations are in xS4 of the supporting information, along with diagrams of the c AE and a AE b AE tilt patterns determining A and B , respectively. These parameters are also calculated in the Excel datafile in the supporting information.
Since c AE tilting reduces V AX 8 ð Þand therefore V A , whereas a AE b AE tilting increases V B , use of V A /V B as an indicator of the overall degree of tilting is vindicated. By using parameters A and B to quantify the tilting, which are simply calculated, the difficulties of defining and calculating tilt angles unequivocally can be circumvented. As may be inferred from the complexity of Fig. 6 and the comparison of calculated tilt angles in the lower part of Table 3, this could be seen as an advantage.

A cation positions
In earlier work on the parameterization of centrosymmetric perovskites (Thomas, 1998), difficulties were experienced in fixing the positions of the A ions by reference to their coordinating X ions in the AX 12 polyhedra. However, subsequent work by Magyari-Kö pe et al. (2001,2002) demonstrated that their positions could be treated as a function of the V A /V B ratio.
The essential objective is to quantify the small displacements from geometrically regular positions [0, 0] and [ 1 2 , 1 2 ] in the relevant plane of projection (Fig. 9). The parameterization applied here is limited to a transformation to polar coordi-  Table 5 Analytical expressions for structural parameters A and B to quantify c AE and a AE b AE tilting.

Analysis of structures at variable temperature and chemical composition
The structural parameters defined in x2 are used here to characterize the sequences of phase transitions observed with increasing temperature and varying chemical composition in centrosymmetric perovskites. Li et al. (2004) identified the following commonly occurring sequence with increasing temperature or mean A ion radius: Pbnm ! Ibmm ! I4/mcm ! Pm 3 3m. This applies to Sr 1Àx Ba x HfO 3 and Sr 1-x Ba x ZrO 3 perovskites , as well as to the systems Sr 1-x Ba x SnO 3 and Ca 1-x Sr x SnO 3 (Mountstevens et al., 2003). Identification of the intermediate Ibmm phase was significant, as it had previously been regarded as rare . The temperature range of stabilization of this phase is highly variable, with values greater than 570 K in BaPbO 3 (Fu et al., 2007), of 140 K in SrRuO 3 (Kennedy et al., 2002), 90 K in SrHfO 3 (Li et al., 2004) and less than 20 K in SrRhO 3 . It is characterized by a fixed value of parameter A equal to one (Table 5).
A second sequence was identified by Ahtee & Darlington (1980) for NaTaO 3 : Pbmn ! Cmcm ! P4=mbm ! Pm 3 3m, with further structural work by Kennedy, Prodjosantoso et al. (1999) reporting transition temperature ranges of 738-753 K, 823-863 K and 883-913 K. More recently, Mitchell et al. (2014) reported the same series of phase symmetries for the mineral lueshite, NaNbO 3 . Both sequences are characterized by an increasing V A /V B ratio with T or x, which may be attributed either to a larger volume expansion coefficient of V A relative to V B or to a compositionally induced expansion of the A-site volume relative to the B site. The term 'A-ion perturbation' may be applied in both cases, since a change in the A-site geometry is the dominant driving force causing the whole structure to respond.
A solid solution series with a rising V A /V B ratio brought about by reducing V B has been established by Yang (2008)  A cation positions in space groups (a) Pbnm and (b) Cmcm in 2D projection. Green circles: height + 1 4 ; brown circles: height À 1 4 . A network with dashed lines is shown in (a). Table 6 Structural parameters for two solid solutions spanning space group Ibmm at room temperature. LaCr 1-x Ni x O 3 at room temperature. Upon increasing x from 0 to 0.7, a Pbnm ! R 3 3c phase transition is observed. In this case, a B-ion perturbation is being applied.

x-and T-series with A-ion perturbations
3.1.1. Sr x Ba 1-x SnO 3 and Sr x Ba 1-x HfO 3 . Table 6 contains structural parameters calculated from the refinements of Mountstevens et al. (2003) for the solid solution series Sr x Ba 1-x SnO 3 and of Li et al. (2004) for the Sr x Ba 1-x HfO 3 series.
On reading Table 6 from top to bottom, values of V A confirm that the increasing mean A-ion radius is the principal driving force for the structural changes in both systems. V A rises by 7.2% between x = 0 and x = 1 in the tin-containing and by 8.4% in the hafnium-containing series. Furthermore, the larger ionic radius of Hf 4+ compared to Sn 4+ in sixfold coordination [0.71 cf. 0.69 Å ; Shannon (1976)] causes the V A value at a given Sr:Ba ratio to be systematically larger in the system with hafnium as B ion. By comparison, the volume of the B site increases by 1.4% between x = 0 and x = 1 in the tincontaining series compared system compared to 2.3% in the hafnium series. The following ranges of the V   Reading these two diagrams from left to right, the Pbnm ! Ibmm transition is triggered by A reaching the limiting value of 1 whilst B is still > 1. The range of stability of the Ibmm phase is determined by the x value at which B becomes equal to 1, whereupon a phase transition to I4/mcm takes place. The diagrams suggest that the approach of A to 1 takes place continuously at the Pbnm-Ibmm boundary. By comparison, the Ibmm-I4/mcm boundary is characterized by a discontin-uous step in B down to 1. Also characteristic of the latter boundary is a fall in A to a value less than 1. This may be understood by considering that A and B can only both be equal to one in the aristotype Pm 3 3m phase. However, at the start of the range of stability of the I4/mcm phase, V A / V B has not reached the limiting value of 5. The transition from I4/mcm to Pm 3 3m takes place as soon as V A has increased sufficiently for this V A / V B limit to be reached.
3.1.2. BaPbO 3 . The above rationalization can be transferred without modification to series where temperature is responsible for an increasing V A volume. For example, the Ibmm phase is stabilized over a wide temperature range in BaPbO 3 perovskites, these being of technological relevance due to superconductivity in the Ba-Pb-Bi-O system. Following earlier structural refinements by Moussa et al. (2001) and Ivanov et al. (2001) in the monoclinic space group C2/m on cooling and at room temperature, respectively, Fu et al. (2005) interpreted this monoclinic distortion as probably due to twinning. They also provided four structural refinements in Ibmm at room temperature and at 4.2 K. Upon heating, Fu et al. (2007) reported phase changes to tetragonal I4/mcm and thereafter to Pm3m at approximate temperatures of 573 and 673 K respectively. Fig. 10(c) is generated from experimental points calculated from the structural refinements of Fu et al. (2005Fu et al. ( , 2007. The absence of Pbnm symmetry in BaPbO 3 can be ascribed to the V A /V B ratio remaining above 4.8320 (8) over the temperature range from 4.2 K to 773 K, this being higher than the threshold values of 4.81 and 4.824 noted in x3.1.1 for Ibmm symmetry. This interpretation is supported by the stabilization of the strontium compound SrPbO 3 in Pbnm (Fu & Ijdo, 1995). The calculated V A / V B value here is significantly lower, at 4.418 (2).
Despite the absence of a Pbnm phase in BaPbO 3 , Fig. 10(c) shows that the Ibmm ! I4/mcm transition is also governed by discontinuous jumps in B to 1 and A to less than 1, as in Figs. 10(a) and 10(b). The approach to the aristotype phase follows the same principles as noted earlier.
3.1.3. CaTiO 3 . The mineral perovskite, CaTiO 3 , has not been observed in space group Ibmm, although it exists in Pbnm at room temperature (Sasaki et al., 1987) and at temperatures up to 1373 K (Liu & Liebermann, 1993). Redfern (1996) paved out the sequence of phase transitions at yet higher temperatures through I4/mcm to and Pm 3 3m aristotype, reporting transition temperatures of 1373-1423 K and at $1523 K. Although Kennedy, Howard & Chakoumakos (1999) subsequently raised the possibility of an orthorhombic Cmcm structure being stabilized at around 1380 K, this was refuted by Ali & Yashima (2005), who later generated the nineteen structural refinements upon which the experimental points of Fig. 10(d) are based (Yashima & Ali, 2009). It is logical that no intermediate phase of symmetry Ibmm is formed with increasing temperature, since A remains significantly below one over the whole Pbnm range. This means that the degree of AO 8 expansion induced thermally is smaller than the compositionally induced AO 8 -expansion analysed in x3.1.1. Stabilization of the Pbnm phase is curtailed by B falling to 1 at the Pbnm-I4/mcm boundary. After the   Table 10. temperature of this phase transition has been reached, the overall expansion of the unit cell allows the volumetric requirements of the TiO 6 octahedron to be satisfied without a À tilting in space group I4=mcm. The remaining tilting allows further comparatively greater AO 8 expansion with increasing temperature up to the temperature of the phase transition to Pm 3 3m. In all four series upon which Fig. 10 is based, the changes in tilt systems taking place at the phase transitions, these being critical events, are largely determined by the systematic changes in tilt angles taking place over wide compositional or temperature ranges. Parameters A and B are ideally suited for encapsulating these changes.
3.1.4. NaTaO 3 and NaNbO 3 . These compounds provide an experimental basis for assembling the factors governing stabilization of the Cmcm phase, and more widely, the progression from Pbnm to Pm 3 3m via Cmcm and P4/mbm instead of Ibmm and I4/mcm. I4/mcm and P4/mbm structures are very similar, the only difference being anti-phase or inphase c axis tilting, respectively. Furthermore, structural parameters V A /V B , A and B cannot discriminate between these two structures. It is therefore appropriate to regard Cmcm and Imma, in the first instance, as alternative precursor phases to a generic phase of symmetry I4/mcm or P4/mbm.
An analysis of the phases of Cmcm symmetry in terms of V A /V B , A and B within these two compounds is given in Table 7.
The definitive characteristic of the Cmcm phase is the proximity of B values to one in Table 7. This parameter has much larger values in the Ibmm phase, with B values greater than 1.03 observed [see Table 6 and Figs. 10(a), 10(b) and 10(c)]. This difference is due to there being only one tilt of type a AE b AE in Cmcm, compared to two in Ibmm (see xS4 of the supporting information). Values of B close to one confirm Cmcm as a precursor to an I4/mcm or P4/mbm phase, in which B is exactly equal to one. The existence of Cmcm symmetry in CaTiO 3 at $1380 K, as proposed by Kennedy, Howard & Chakoumakos (1999), is indeed possible, since B is close to one at this temperature. Also to be noted in Table 7 is the splitting of A values made possible by there being two symmetry-independent A sites in Cmcm. Geometry simply requires a mean value h A i = ( A1 + A2 )/2 of less than one. This splitting is considerably more marked in NaNbO 3 . The ability for the lower A value to be significantly less than one could go some way towards rationalizing the phase coexistence of Pbnm and Cmcm phases at room temperature observed in NaTaO 3 by Knight & Kennedy (2015). A further contributory factor towards stabilization of Cmcm in 1 mol% K-doped samples of NaTaO 3 at room temperature (Arulnesan et al., 2016) could be preferential occupation of the sites with higher A value by the larger potassium ions, i.e. a partial ordering. It is speculated that the proximity of B to one in Cmcm is conducive to a to a higher temperature phase transition to P4/mbm, whereas the sudden fall in B within Ibmm or Pbnm phases observed in Figs. 10(a)-10(d) favours a transition to I4/mcm.

X-series with B-ion perturbations
Calculated structural parameters for the LaCr 1-x Ni x O 3 solid solution series (Yang, 2008) are given in Table 8.
Unlike the structures perturbed by A ions, the variation in A and B parameters is small and also unsystematic for structures in space group Pbnm. There is no tendency of either parameter to approach one. The expected cross-correlations between parameter pairs A $ c and B $ a are observed within the Pbnm phase field: the greater the magnitude of the deviations of A and B from one, the larger the h c i and h a i angles. The a tilt angle falls to 4.64 and the mean tilt angle (h a i + h c i)/2 to 5.50 in Pbnm just before the phase transition, with a rising to 5.97 in the R 3 3c phase at x = 0.7. A gradual reduction in a to 5.72 is observed up to x = 1, this still being a stable rhombohedral phase far from a transition to the cubic aristotype. The gradual fall in V B with x is consistent with the smaller ionic radius of Ni 3+ compared to Cr 3+ (Shannon, 1976) and is responsible for the systematic increase in V A /V B with x, which peaks at 4.841 in Pbnm before the phase transition to R 3 3c and falls to 4.809 thereafter. The Pbnm ! R 3 3c transition is examined further in x4.4.

Modelling the structural variation of Pbmn perovskites with increasing pressure
The structures adopted by perovskites under increasing hydrostatic pressure rest on the different compressibilities of the AX 12 and BX 6 polyhedra (Angel et al., 2005), such that the volume ratio V A /V B may either increase or decrease. In  Table 8 Structural data relevant to the Pbnm ! R 3 3c transition at x $ 0.7 in LaCr 1-x Ni x O 3 .  (1) general, experimental difficulties limit the pressure range over which full structure refinements can be obtained, a common practice being to report the variation of unit-cell parameters over a wider pressure range by means of the third-order Birch-Murnaghan equation of state [equation (7)].
Here, coefficient K T is the bulk modulus and K 0 0 is its derivative with respect to pressure. V 0 represents the reference unit-cell volume. The appropriate notation to use for the variation of cell parameters a, b, c with pressure would be K a0 , K 0 a0 , a 0 etc., whereby cubes a 3 ; b 3 ; c 3 will have been used for the Birch-Murnaghan fitting. It is also common to quote linear compressibilities a , b , c , whereby a ¼ ð@=@PÞða=a 0 Þ T , etc., these generally holding over a limited pressure range. By setting K 0 0 values equal to 4, the term in curly brackets is equal to one, giving rise to the second-order Birch-Murnaghan equation.
The primary motivation for studying perovskites under pressure has been to simulate MgSiO 3 perovskite in the lower mantle, for which a phase transition to a post-perovskite phase at pressures > 125 GPa and a temperature of 2700 K has been proposed (Murakami et al., 2004). This system is simulated by assuming regular octahedra in the following sub-section.

Simulation of MgSiO 3 under pressures of up to 125 GPa
It may be assumed that MgSiO 3 remains in space group Pbnm at all pressures up to $125 GPa. Accordingly, the 12 sets of a, b, c unit-cell parameters reported by Vanpeteghem et al. (2006) at pressures up to 10 GPa were used to derive the following Birch-Murnaghan constants to second order for the   unit-cell volume and a and c unit-cell parameters: K T = 252.90 GPa; V 0 = 162.52 Å 3 ; K T;a = 232.11 GPa; V 0;a = 109.08 Å 3 ; K T;c = 240.78 GPa; V 0;c = 328.33 Å 3 .
The assumption was made that these constants could extended to 125 GPa, thereby yielding theoretical a, b and c unit-cell parameters over this whole range. This process was validated by comparing the values obtained ('Extrapolated B-M') with values quoted by Murakami et al. (2004) and Fiquet et al. (2000). Acceptable agreement is found in Table 9.
200 equally spaced pressures were taken within the range up to 125 GPa and a, b, c values calculated from the above Birch-Murnaghan constants. Values of x(O1), y(O1), x(O2), y(O2) and z(O2) were allowed to vary within the core Excel Solver functionality for space group Pbnm (Fig. 7), so as to generate regular octahedra. The resulting variation of parameters A and B with pressure is shown in Fig. 11(a).
Parameter A decreases from 0.9576 at 0 GPa to 0.9529 at 125 GPa, which indicates an increase in c þ tilting. By comparison, B increases from 1.0649 at 0 GPa to 1.0939 at 125 GPa, indicating an increase in a À tilting. The effect of pressure is to drive the structure ever further into the Pbnm phase field. It is likely that the imminent phase transition to post-perovskite at 125 GPa is triggered by the strong OÁ Á ÁO repulsions that will be associated with this degree of octahedral tilting. Ross et al. (2004) elucidated the response of the perovskite compound YAlO 3 at pressures up to 8 GPa by providing eight structural refinements of a synthetic single crystal. A decrease in octahedral tilting with pressure was found, which was ascribed to the AlO 6 octahedral compressibility being greater than that of the YO 12 site. It follows that an approach towards cubic symmetry will occur, similar to the T-series with A ion perturbation in x3.1. The Birch-Murnaghan constants to third order quoted by Ross et al. (2004) were used to generate unitcell parameters at different pressures, as in x4.1. Two alternative simulations were carried out for pressures up to 8 GPa, the first assuming regular octahedra as in x4.1, and the second exploiting the octahedral distortions reported by Ross et al. (2004) from an analysis of their structural refinements. For this purpose, the linear, decreasing trends in the three different Al-O bond lengths with increasing pressure observed by them were translated into linear relationships for PCRO parameters a 1 to a 3 . In addition, linear relationships for PCRO angle parameters 23 , 31 and 12 were derived [equations (8)

Simulation of YAlO 3 at pressures up to 52 GPa
Solver-based refinements in Pbnm were carried out at equally spaced pressure values, the spacing determined by covering the pressure range from 0 to 60 GPa with 200 values. Parameters x(O1), y(O1), x(O2), y(O2) and z(O2) were allowed to vary in refinements with end-point determined by the minimum deviation from conditions (8). The deviation was observed to range between 0.001 and 0.050%. The results of both simulations are shown in Fig. 11(b) and in that part of Fig. 11(c) with the blue-shaded background. The slope of both B curves is negative, compared to the positive slope of B for MgSiO 3 . The slope of A for the simulation with regular octahedra is positive, compared to the negative slope for MgSiO 3 . The opposite behaviour to MgSiO 3 is shown by YAlO 3 , which shows a gradual progression away from Pbnm towards higher symmetry. The curve for A for distorted octahedra has a shallow minimum at $3.4 GPa, which is more clearly seen in Fig. 11(c). Thereafter A rises, as for undistorted octahedra.  Table 11 Structural parameters for (La 1-x Nd x )GaO 3 in space groups Pbnm and R 3 3c.
x Space group p (GPa)  (7) 4.46 (7) 8.3 (1) 6.39 (7) 48.136 (7) 10.146 (7)  In a computational experiment, the pressure was allowed to rise towards 60 GPa, with unit-cell constants calculated from the Birch-Murnaghan constants of Ross et al. (2004) for the pressure range up to 8 GPa. Fig. 11(c) shows that the curvature of both pairs of curves changes sign as the octahedra change from regular to distorted. Since the constraints in equations (8) only apply up to 8 GPa, it became increasingly difficult to reach low deviations at the end-points of the refinements. The maximum deviation amounted to 1.08%, which was obtained at the highest pressure.
The ascending curve for A with distorted octahedra in Fig. 11(c) reaches zero at a pressure of $41.5 GPa. This would be consistent with a phase transition to Ibmm [cf. Figs. 10(a) and 10(b)]. By comparison, the B curve is the first to reach zero in the simulation with regular octahedra. The corresponding pressure is $51.6 GPa. This would be consistent with a phase transition to I4/mcm or P4/mbm [cf. Fig. 10(d)]. The asymptotic approach of the B curve to 1 for regular octahedra gives rise to a broad pressure range in which Cmcm could be stabilized as well as Pbnm prior to this phase transition. This phenomenon has been observed in NaTaO 3 by Knight & Kennedy (2015) over a broad temperature range, as discussed in x3.1.4. Assuming pattern similarity, a phase change to P4/mbm would be anticipated at 51.6 GPa. More generally, the simulations of YAlO 3 , both with and without octahedral distortion, indicate how this distortion is expected to have a direct effect on the sequence of phase transitions and the pressures at which they occur.
In summary, the analysis of (p-T-X)-induced phase transitions in terms of A and B parameters is a stimulus to further, targeted experimental work on the systems that have been analysed and simulated in x3.1, x4.1 and x4.2.

Crystal structures generated in the high-pressure simulations
A by-product of the simulations in x3.1 is the generation of full sets of oxygen ion coordinates. Table 10 contains these data corresponding to the conditions denoted by dashed lines in Figs. 11(a) and 11(c). The space group is Pbmn with the B ions in 4b positions. The full set of associated structural parameters is also quoted below the line in the table.
In all cases, the effect of increasing pressure is to reduce V A and V B . However, the ratio V A /V B is reduced in MgSiO 3 and increased in YAlO 3 . This signifies movement further into the Pbnm phase field in the former case [ Fig. 11(a)] and movement away towards higher symmetry in the latter [ Fig. 11(c)]. This is borne out by the changes in inclination angles x , x , x , tilt angles a , c and tilt-related parameters A , B .
4.4. Analysis of (La 1-x Nd x )GaO 3 structures at pressures of up to 12 GPa Apart from the above simulations, the structural parameters used in this work are applied to the structural refinements of Angel et al. (2007) for this 3:3 perovskite solid solution. In carrying out A-ion substitutional perturbations, it was found that a phase transition from Pbnm to R 3 3c takes place under pressure for x values up to 0.20, but not for x = 0.62 or x = 1. The question arises as to whether a particular pattern of structural evolution within the Pbnm phase is associated with a phase transition to R 3 3c. Data for the Pbmn and R 3 3c phases at their lowest and highest investigated pressures for a given x value are given in Table 11.
The direct influence of the A-ion perturbations is seen in the values of V A , which decrease downwards in the table with increasing x for the p = 0.0001 GPa values. Since V B remains approximately constant, a parallel trend of decreasing V A /V B ratio with increasing x is generally observed. This is consistent with greater stabilization within the Pbnm phase field. Pressure induces the opposite trend to raising x, since the V A /V B ratios for maximum pressures within the Pbnm phase field are uniformly higher than at atmospheric pressure. A parallel increase in A and decrease in B is observed. At lower x values up to 0.20, which are associated with reduced Pbnm stabilization, the increased pressure induces a phase transition to R 3 3c. Angel et al. (2007) report the following approximate pressures for this phase transition: x = 0: 2.2 GPa; x = 0.06: 5.5 GPa; x = 0.12: 7.8 GPa; x = 0.20: 12 GPa. The expected cross-correlations between parameter pairs A $ c and B $ a are observed within the Pbnm phase field: the greater the magnitude of the deviations of A and B from one, the larger the h c i and h a i angles. It is proposed that the observed fall in h c i to $4.2-4.4 with increasing pressure is the principal driving force for the phase transitions to R 3 3c. After the transitions for x = 0 and x = 0.12, i.e. within the R 3 3c phase field, the a tilt angle is larger and approximately equal to the mean tilt angle (h a i + h c i)/2 in the Pbnm field beforehand. For x = 0.62 and x = 1.00, the critical range of c between 4.2 and 4.4 is not reached at the pressures investigated, so that these compounds remain stabilized in space group Pbnm. 4.5. Structural parameters for YAl 0.25 Cr 0.75 O 3 with locked octahedral tilting Ardit et al. (2017) subsequently took up the theme of locked octahedral tilting in orthorhombic 3:3 perovskites (A ion +3; B ion +3) by reference to the solid solution YAl 0.25 Cr 0.75 O 3 in space group Pbnm. Although in general agreement with Zhao et al. (2004) and Angel et al. (2005) in assuming compressibility ratios (AO 12 )/(BO 6 ) < 1 for a 3:3 perovskite and > 1 for a 2:4 perovskite, their compound showed a compressibility ratio approximately equal to one (Table 12).
In spite of consistently falling V A and V B values with increasing pressure, the V A / V B ratio remains approximately constant, as do parameters A , B , h c i and h a i. Discrepancies are observed between the values of the tilt angles calculated according to x2.3 and the values quoted by Ardit et al. (2017) without declaring the method of calculation. This is not unexpected, as discussed in x2.4. Since values of h c i and h a i calculated according to x2.3 show the expected cross-correlations between parameter pairs A $ c and B $ a , indirect support is given for the correctness of these calculations.

Discussion
The ability of the method to analyse experimental structural data and the sequences of phase transitions between space groups has been demonstrated in x3 and x4. By comparison with the group-theoretical approach, it has not been necessary to relate these explicitly to the cubic aristotype, since deviations of V A /V B from the limiting value of 5 in the aristotype provide a direct indication of how far away a given structure is from the aristotype. Parameters V A , V B and V A /V B continue to yield valuable insight, for example in Table 11. However, the direct calculation of Glazer tilt angles c and [ a , b ] and associated derived parameters A and B has allowed a deeper analysis of the structural factors leading to phase transitions than a consideration of these volumes alone. The significance of the generalized algorithms introduced here for the Glazer tilt angles may be assessed by reference to Wang & Angel (2011): ' . . . the decomposition of a perovskite structure including tilted and distorted octahedra by geometric analysis does not result in an unambiguous definition of the Glazer (1972) tilts and the problem is more acute in perovskites with lower space-group symmetries'. These authors noted further that 'unambiguous expressions for both the Glazer tilts and their relationship to the V A / V B ratio are still to be determined explicitly for each space group, and in a general form'. These observations led Wang & Angel (2011) to resort to grouptheoretical methods in order to relate the amplitudes of symmetry-adapted modes to V A /V B ratio. The current work, by comparison, has remained strictly based on unit-cell parameters and atomic coordinates. Although it has not provided generalized analytical expressions for these tilt angles, it has led to two generalized algorithms (a) for all centrosymmetric space groups apart from R 3 3c and (b) for space group R 3 3c (and more generally, triclinic space groups). An analytical link between c , [ a , b ] and V A / V B has not research papers 58 Noel W. Thomas Phase transitions and (p-T-X) behaviour in perovskites been provided, since inclination angles 1 to 3 are more suitable for this purpose [equation (1)]. However, the linkedcell implementation within Excel allows the empirical derivation of numerical relationships between Glazer tilt angles and V A /V B .
The methodological innovations of the current work and their benefits are summarized in Table 13.
The question of the relative stability of alternative perovskite phases has been addressed by several workers employing quantum-mechanical methods (Zagorac et al., 2014) as well as the semi-empirical bond-valence method. Woodward (1997b) initiated this discussion by evaluating the ionic and covalent bonding in perovskites and analysing the conditions of stabilization of favoured tilt systems a À a À c þ , a À a À a À and a 0 a 0 a 0 . In later work (Lufaso & Woodward, 2001), an algorithm was developed to minimize the so-called global instability index (GII) in alternative tilt systems, this being the r.m.s. deviation between calculated bond valences and ideal cationic valences. The bond-valence method was also the favoured approach of Zhao et al. (2004) in rationalizing the relative compressibilities of the AO 12 and BO 6 polyhedra in perovskites. It led to a clear differentiation in behaviour between 2:4 and 3:3 perovskites, which has remained a feature of experimentally led investigations of perovskites under pressure, for example by Ardit et al. (2017).
The potential of direct transformation of crystallographic data into structural parameters for the analysis of sequences of phase transitions has been demonstrated in this work. The set of structural parameters has been extended and a baseline provided for future investigations of non-centrosymmetric perovskites. The intention is to promote more detailed interaction between experimental crystallography and modelling in developing new materials by atomic and molecular design.

Related literature
The following references are cited in the supporting information: Hahn (1995), Williams (1971).