research papers
Van Vleck analysis of angularly distorted octahedra using VanVleckCalculator
^{a}Cavendish Laboratory, University of Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom
^{*}Correspondence email: lavn2@cam.ac.uk
Van Vleck modes describe all possible displacements of octahedrally coordinated ligands about a Q_{2}, Q_{3}) along the bond axes of the octahedron, disregarding the angular distortion and losing information on the octahedral shear modes (Q_{4}, Q_{5} and Q_{6}) in the process. In this paper, the validity of assuming bond lengths to be orthogonal in order to calculate the Van Vleck modes is discussed, and a method is described for calculating Van Vleck modes without disregarding the angular distortion. A Python package for doing this, VanVleckCalculator, is introduced and some examples of its use are given. Finally, it is shown that octahedral shear and angular distortion are often, but not always, correlated, and a parameter η is proposed as the shear fraction. It is demonstrated that η can be used to predict whether the values will be correlated when varying a tuning parameter such as temperature or pressure.
They are a useful analytical tool for analysing the distortion of octahedra, particularly for firstorder Jahn–Teller distortions, but determination of the Van Vleck modes of an octahedron is complicated by the presence of angular distortion of the octahedron. This problem is most commonly resolved by calculating the bond distortion modes (Keywords: crystal structure; Jahn–Teller distortion; octahedral distortion; octahedral tilting; crystallographic software.
1. Introduction
The Van Vleck distortion modes (Van Vleck, 1939) describe all possible displacements of octahedrally coordinated ligands about a They are particularly useful in the context of the Jahn–Teller (JT) effect (Jahn & Teller, 1937), which in general occurs when a highsymmetry coordination is destabilized with respect to a deviation to lower symmetry as a consequence of electronic degeneracy. The JT effect distorts the via the JT distortion. While the JT distortion is not unique to octahedra in bulk crystalline materials, it was in octahedra that it was first observed experimentally (Bleaney & Bowers, 1952), and it was in materials with JTdistorted octahedra that colossal magnetoresistance (Millis et al., 1996) and hightemperature superconductivity (Fil et al., 1992; Keller et al., 2008) were discovered.
A transition metal cation in an octahedral configuration will have its d orbitals split into three t_{2g} orbitals at lower energy and two e_{g} orbitals at higher energy. [Here we use the notation that lowercase symmetry descriptors (such as e_{g} or t_{2g}) refer to orbitals with this symmetry and uppercase descriptors (such as E_{g} or T_{2g}) refer to the symmetry more generally.] It will have a number n of electrons in these d orbitals (hereafter described as d^{n}). For certain values of n and, where applicable, certain low or highspin characters,^{1} there will exist multiple orbitals that could be occupied by an electron or an electron hole with equal energy. This degeneracy is destabilizing, resulting in the most stable configuration of atomic sites being one in which the ligands distort from their highsymmetry positions in order to rearrange the orbitals into a nondegenerate system with minimized energy. This is shown for a lowspin d^{7} transition metal cation (such as Ni^{3+} or Co^{2+}) in Fig. 1, though such distortions may occur for any value of n in d^{n} where there is a degenerate occupancy. The stabilization energy due to the JT effect is larger for e_{g} degeneracy than t_{2g} degeneracy, and so the effect will remain prominent up to higher temperatures, and is hence more widely studied, in JTactive materials with e_{g} degeneracy (CastilloMartínez et al., 2011).
In the literature, various techniques for parameterizing the JT distortion are used. An oftenused example (Kimber, 2012; Lawler et al., 2021; NagleCocco et al., 2022; GenreithSchriever et al., 2023) is the bondlength distortion index, defined by Baur (1974) as
where n is the number of ligands (6 for an octahedron), l_{i} is the distance between the core ion and the ith coordinated ion, and l_{av} is the average of all the distances between the core ion and coordinated ions.
A similar parameter (Shirako et al., 2012; Sarkar et al., 2018; NagleCocco et al., 2022) is the effective ECoN, which for an octahedron deviates from 6 only when there is bondlength distortion. ECoN is defined by Hoppe (1979) as
where is a modified average distance defined as
Here, l_{min} is the smallest core–ligand bond length in the octahedron.
Finally, a third parameter used to quantify the JT distortion (Schofield et al., 1997; Kyono et al., 2015; Mikheykin et al., 2015) is the quadratic elongation 〈λ〉, defined by Robinson et al. (1971) as
where l_{0} is the centretovertex distance of a regular polyhedron of the same volume.
More recently, an alternative approach to modelling polyhedral distortion has been described (Cumby & Attfield, 2017), involving fitting an ellipsoid to the positions of the ligands around a calculating the three principal axes of the ellipsoid R_{1}, R_{2} and R_{3}, where R_{1} ≤ R_{2} ≤ R_{3}, and using the variance of these three radii as a metric for the distortion. This has been applied to the firstorder JT distortion by Pughe et al. (2023).
These parameterizations each have merits. However, they are not sensitive to the symmetry of the octahedral distortion. The Van Vleck modes are conceptually different from each of these for quantifying the JT distortion because they can be used to quantify distortion with the precise symmetry of the transition metal e_{g} orbitals. This is important because JT distortions typically follow a particular symmetry. When the distortion is due to degeneracy in the e_{g} orbitals it will be of E_{g} symmetry; when it is due to degeneracy in the t_{2g}degenerate orbitals it may be either E_{g} or T_{2g} symmetry (Child & Roach, 1965; Bacci et al., 1975; Holland et al., 2002; Halcrow, 2009; Teyssier et al., 2016; Schmitt et al., 2020; Streltsov et al., 2022), although there is relatively little unambiguous experimental evidence for a JTinduced shear compared with more typical E_{g} distortion.
In this paper, we present a Python (Van Rossum & Drake, 2009) package, VanVleckCalculator, for calculating the Van Vleck distortion modes. We show that the approach to calculating the modes which is commonly used in the literature is a reasonable approximation for octahedra with negligible angular distortion, but results in the loss of information in other cases. We propose a new metric, the shear fraction η, for understanding the correlation between octahedral shear and angular distortion. Finally, we reanalyse some previously published data in terms of the Van Vleck modes to show that these can be an effective way of understanding octahedral behaviour.
2. Theory
Within an octahedron, we can split the six ligand ions into three pairs, where the two ions within the pair are opposite one another. In the absence of angular distortion (i.e. assuming all ligand–core–ligand angles are an integer multiple of 90°), there would exist a basis where each of the three axes exist directly along the x, y and z axes, and where the origin in space is defined as the centre of the octahedron.
Each pair within an octahedron can therefore be assigned to an axis and labelled as the a, b or c pair, respectively. Within a pair, ions can be labelled as − or + depending on whether they occur at a negative or positive displacement from the origin, along the axis, respectively. This notation is demonstrated in Fig. 2, where each pair of ions is represented by a different colour.
For each of the six ligands, we define a set of coordinates , and , where α is a, b or c denoting the pair in which the ligand is, and β is − or + denoting the ion within the pair.
The ideal positions of the six ligands are (R, 0, 0), (−R, 0, 0), (0, R, 0), (0, −R, 0), (0, 0, R) and (0, 0, −R), where R is defined as the distance between the centre of the octahedron and the ligand in an ideal octahedron (in practice, this is taken as the average of the core–ligand bond distances). This results in 18 independent variables. Using these, we further define a set of Van Vleck coordinates (capitalized to distinguish from true coordinates) which give the displacement of the ion within an axis away from its ideal position. For instance, for the ion with α = a and β = −, = , = and = . See Fig. 2 for clarification of the ion notation.
Using these coordinates, the first six Van Vleck modes (Q_{j}, j = 1–6) are defined as follows (Van Vleck, 1939):
We only discuss these first six Van Vleck modes, which are shown in Fig. 2. Q_{1} to Q_{3} describe bondlength distortions, whereas Q_{4} to Q_{6} describe octahedral shear distortions. Q_{1} is a simple expansion/contraction mode which does not affect symmetry and will not be discussed further.
Q_{2} and Q_{3} are a planar rhombic distortion and a tetragonal distortion, respectively; they are considered degenerate due to the Hamiltonian, which is discussed for instance by Kanamori (1960). These two modes form a basis for distortions describing different octahedral configurations with the symmetry of the transition metal e_{g} orbitals (Goodenough, 1998; Khomskii & Streltsov, 2021). These modes are of most relevance for firstorder JT distortions occurring due to degenerate e_{g} orbitals. A phase space of possible octahedral configurations can be constructed using these two parameters (Kanamori, 1960), as shown in Fig. 3. Here the magnitude of the distortion ρ_{0} can be calculated as
and the angle ϕ of this distortion, being of purely Q_{3} character, can be calculated by
(Note that this angle does not represent a physical angle within the octahedron.)
All possible combinations of the Q_{2} and Q_{3} modes correspond to a particular angle ϕ and hence a particular configuration as shown in Fig. 3. The structural effect of a rotation of ϕ within a range of 120° can be quite significant, as shown in Fig. 3; such changes can manifest as a JTelongated (compressed) octahedron with four short (long) and two long (short) bonds [such as NiO_{6} in NaNiO_{2} (NagleCocco et al., 2022)] or two short, two medium and two long bonds [such as LaMnO_{3} (RodriguezCarvajal et al., 1998)].
A characteristic of JT distortion is that, in the absence of external distortive forces, the symmetry of the structure matches the symmetry of the orbitals involved. Typically, any dorbital JT distortion will have some planar rhombic (Q_{2}) or tetragonal (Q_{3}) character. However, sometimes when the degeneracy occurs in the t_{2g} orbital, there may instead be a trigonal component to the symmetry of the distortion, which manifests as an angular distortion instead (Child & Roach, 1965; Bacci et al., 1975; Holland et al., 2002; Halcrow, 2009; Teyssier et al., 2016; Schmitt et al., 2020; Streltsov et al., 2022). For the more commonly studied case of a degeneracy in the e_{g} orbitals, the effect of a rotation of ϕ similarly changes the symmetry of the d orbitals. Fig. 1 shows the splitting of the d orbitals in an octahedrally coordinated d^{7} transition metal due to an elongationtype firstorder JT distortion, where the tetragonal elongation occurs along the z axis. Note that the unpaired e_{g} electron occupies the d_{z2} orbital. In the opposite case of a compressiontype firstorder JT distortion along the z axis, the lowerenergy, and hence singly occupied, orbital would be the ; this would correspond to a rotation in ϕ of 180°. More generally, as a function of ϕ, there exist a set of special angles separated by a 60° rotation corresponding to a particular e_{g} orbital being singly occupied by a d electron. These are tabulated in Table 1. An octahedron for which ϕ does not correspond to one of these special angles exhibits orbital mixing (RodriguezCarvajal et al., 1998; Zhou & Goodenough, 2008b).

The Q_{4} to Q_{6} modes describe shear of the octahedra, i.e. the effect whereby paired ligands on opposite sides of a central ion are displaced in opposite directions and have trigonal T_{2g} character. The shear modes may be used to quantify the JT distortion in octahedra where the degeneracy occurs within t_{2g} orbitals (Child & Roach, 1965; Teyssier et al., 2016). The magnitude of the calculated shear is typically correlated with angular distortion, which is commonly quantified using the metric called the bondangle variance (BAV) (Robinson et al., 1971), defined here as
m is the number of bond angles (i.e. 12 for an octahedron), ζ_{i} is the ith bond angle and ζ_{0} is the ideal bond angle for a regular polyhedron (i.e. 90° for an octahedron). However, for direct comparison with the shear modes, it is more appropriate to use the standard deviation σ_{ζ}.
For an octahedron with nonzero T_{2g}(Q_{4}, Q_{5}, Q_{6}) modes, increasing their magnitude will increase the angular distortion, but an octahedron may have angular distortion without exhibiting octahedral shear. To analyse the extent to which angular distortion in an octahedron is due to shear, we propose a shear fraction parameter η, demonstrated in Fig. 4 and defined below.
First, we must define a set of shear and `antishear' angular indices, which are modifications of equations (8) to (10) in terms of angles rather than displacements. The indices are represented with Δ and a subscript corresponding to the plane in which rotation occurs: the ab plane corresponds to the Q_{4} mode, the ac plane to the Q_{5} mode and the bc plane to the Q_{6} mode. The absence or presence of a prime symbol ′ designates whether the index represents shear or antishear, respectively. Finally, the δ angle is the rotation of the ligand from its ideal Van Vleck coordinate in a clockwise direction, within the plane in which the corresponding Van Vleck shear (Q_{4} to Q_{6}) would occur. These are defined thus (see the supporting information, Fig. S7):
We then quantify the shear and antishear distortions using the following equations:
From here, we define the shear fraction η as
This η parameter will be important in interpreting the relation between the angular distortion σ_{ζ} and the Van Vleck shear modes Q_{4} to Q_{6}.
3. Implementation
In this section, the algorithm used to calculate the Van Vleck distortion modes is discussed. It is written using Python 3 (Van Rossum & Drake, 2009) as a package called VanVleckCalculator, with the full code available on GitHub (NagleCocco, 2023) and also presented with annotations in the supporting information. Data handling and some calculations make use of NumPy (Harris et al., 2020) and crystal structures are handled using PyMatGen (Ong et al., 2013).
A flow chart showing the octahedral rotation algorithm can be found in Fig. S1.
Besides calculating the Van Vleck modes and the angular shear modes described in this paper, VanVleckCalculator can also calculate various other parameters as described in the supporting information.
3.1. Selecting an origin
Selection of the origin is a key step in calculating Van Vleck modes. The most common approach, for an MX_{6} octahedron, is to take the M ion as the origin. This is a reasonable approach, given that M ions are typically positioned at, or very close to, the centre of an octahedron. This is particularly appropriate for unit cells derived from (Rietveld, 1969) of Bragg diffraction data, where the M ion is likely to occur on a highsymmetry Wyckoff site. A third, similar, option would be to choose the average position of the six ligands as the origin in space. An example of when this may be a desirable choice would be for systems exhibiting a pseudoJT effect (also called the secondorder JT effect), where the central cation is offset from the centre of the octahedron.
In some instances, a et al., 2007) and (MD) (Bocharov et al., 2020) simulations. Such a typically retains the periodicity which is an axiom of a typical crystallographic but will exhibit local variations. For instance, a obtained by analysis of Bragg diffraction data is typically regarded as an `average' structure, insensitive to local phenomena such as thermally driven atomic motion or disordered atomic displacements such as a noncooperative JT distortion. In a crystallographic thermal motion of atoms is typically represented by variable atomic displacement parameters (ADPs) (Peterse & Palm, 1966). In contrast, a should reflect local phenomena, for instance exhibiting local JT distortions in a system with a noncooperative JT distortion, and representing thermal effects not with ADPs but rather by distributing equivalent atoms in adjacent repeating units in slightly different positions. In this regard, a can be considered a `snapshot' of a at a point in time. It may therefore not be appropriate to set the core ion as the centre of the octahedron in a as the positioning of both core and ligand ions is in part due to thermal effects and so the `centre' of the octahedron will be displaced as a result of random motion. The alternative option would simply be to use the crystallographic site of the central ion and fix this as independent of the precise motion of the central ion locally.
may be simulated using a Examples include `bigbox' pair distribution function (PDF) analysis (TuckerIn VanVleckCalculator, the user has the option to take as the centre of the octahedron either the central ion, the average position of the six ligands or a specified set of coordinates.
3.2. Calculating Van Vleck modes along bond directions
The calculation of the Van Vleck modes, as described in the Theory section, requires that the basis in space be the octahedral axes (i.e. the three orthogonal axes entering the octahedron via one vertex, passing through the central ion and exiting via the opposite vertex). For a given this may require that an octahedron be rotated about each of the three axes making up the basis until the octahedral axes perfectly align with the basis. This becomes more complicated when the octahedron exhibits angular distortion (i.e. exhibits ligand–core–ligand angles that are not integer multiples of 90°). In this case, it is impossible to define octahedral axes according to the strict criteria previously defined.
In the literature, this problem is generally evaded by simply calculating the Van Vleck modes on the basis of bond directions rather than Cartesian coordinates; see, for example, previous work on the perovskite LaMnO_{3} (Goodenough et al., 1961; RodriguezCarvajal et al., 1998; Capone et al., 2000; Chatterji et al., 2003; Zhou & Goodenough, 2008b; Zhou et al., 2011; Snamina & Oleś, 2016; Fedorova et al., 2018; Lindner et al., 2022), other perovskites (Alonso et al., 2000; Wang et al., 2002a; Tachibana et al., 2007; Zhou & Goodenough, 2008a; CastilloMartínez et al., 2011; Franchini et al., 2011; Chiang et al., 2011; Dong et al., 2012; Fedorova et al., 2015; Ji et al., 2019; Xu et al., 2020; Ren et al., 2021) or nonperovskite materials (Moron et al., 1993; Cussen et al., 2001; Wang et al., 2002b) (we note that some reports use a different variation which still uses Kanamori's approximation; papers cited here include those which use the approximation, even if the precise definitions differ). In this case, Q_{2} and Q_{3} are defined according to the following equations which were first expressed by Kanamori (1960), where l, m and s are the short, medium and long bond lengths, respectively [the equations presented here differ from Kanamori's as they have been multiplied by a factor of so that they are mathematically equivalent to equations (6) and (7)]:
This relies on the implicit assumption that the bond lengths are orthogonal. This is clearly a reasonable approximation in many cases, particularly when the angular distortion is very small. For instance, in LaMnO_{3}, the cornersharing octahedral connectivity enables mismatched polyhedra to tessellate via octahedral tilting [Fig. 5(e) in Section 4.1] rather than intraoctahedral angular distortion. However, for systems with greater angular distortion, for instance those with edge or facesharing interactions, it is not so clear that this approximation is valid.
3.3. Calculating Van Vleck modes within Cartesian coordinates
In VanVleckCalculator we have written an algorithm for rotating an octahedron about three Cartesian axes with a defined origin within the octahedron, such that the ligands are as close as possible to the axes (within the constraint that there is angular distortion). This allows for calculation of Van Vleck modes in a way that does not artificially constrain the octahedral shear modes (Q_{4}, Q_{5} and Q_{6}) to be zero.
First, three orthogonal axes are taken as the x, y and z axes.^{2} By default, these are the [1, 0, 0], [0, 1, 0] and [0, 0, 1] axes, respectively, but alternative sets of orthogonal vectors can be given by the user. For instance, for regular octahedra rotated 45° about the x axis, the user would be recommended to give as axes [1, 0, 0], and . This vector is given as a Python list with shape (3, 3). For consistency, the cross product of the first two axes should always be parallel with the third given vector; if antiparallel, the algorithm will automatically multiply all elements in the third vector by −1. The three pairs of the octahedron (as defined in the Theory section) are each assigned to one of these three axes. This assignment is performed according to the magnitude of the projection of a vector between the two atoms in a pair along a given axis, with the z axis assigned first, then the y axis from amongst the two pairs not assigned to the z axis, and finally the x axis is automatically assigned to the remaining pair. Within each pair, the ligands are then ordered such that the ligand with the negative distance is the first along the assigned vector and the ligand with the positive distance occurs second.
Second, the octahedron is rotated repeatedly about the x, y and z directions of the basis until the orthogonal axes supplied in the previous step match the basis precisely. This is performed in a `while' loop structure, with the rotation angles about the three axes summed in quadrature and compared with a defined tolerance (by default, 3 × 10^{−4} rad in VanVleckCalculator); if the total rotation exceeds the tolerance, the step is repeated.^{3} This step is usually unnecessary and can be skipped by leaving the default set of orthogonal axes, which are [1, 0, 0], [0, 1, 0] and [0, 0, 1] (meaning no rotation will occur).
Third, an automatic rotation algorithm will further minimize the effect of angular distortion. For each of the three axes, the four ligands not intended to align with that axis are selected. The angle to rotate these four ligands about the origin such that each is aligned with its intended axis within the plane perpendicular to the axis of rotation is calculated. The octahedron is then rotated about this axis by the average of these four angles. This occurs iteratively until, for a given iteration, the sum (in quadrature) of the three rotation angles is less than the alreadymentioned defined tolerance.
At this point, the octahedron is optimally aligned with the basis (given the limitation that there may be angular distortion) and the Van Vleck modes can be calculated.
3.4. Ignoring or including angular distortion: a comparison
To evaluate the utility of calculating the Van Vleck modes without disregarding the angular distortion, we perform a comparison between the two approaches. We have calculated the Van Vleck distortion modes and associated parameters for octahedra in NaNiO_{2} and LaMnO_{3} with both a method that ignores angular distortion and calculates modes along bond directions [consistent with the Q_{2} and Q_{3} equations defined by Kanamori (1960)] and a method that uses Cartesian coordinates in order to take angular distortion into account. Table 2 shows this for these two materials. Firstly, for the Van Vleck modes calculated without ignoring angular distortion, we can see that the octahedral shear modes (Q_{4}, Q_{5}, Q_{6}) are larger for the material with higher angular distortion (as quantified using bondangle variance). While the effect of ignoring angular distortion is significant for the Q_{4}, Q_{5} and Q_{6} modes, it makes negligible difference for the calculation of Q_{2} and Q_{3} modes and the associated ρ_{0} and ϕ parameters. It is therefore likely to be a reasonable approximation to take, particularly for calculation of ϕ as is common in the literature, even for octahedra which exhibit higher angular distortion. However, there is a definite loss of information in assuming that the shear modes Q_{4} to Q_{6} are zero. The impact of this is assessed in the case studies.

4. Case studies
4.1. Temperature dependence of octahedral shear in LaAlO_{3}
Perovskite and perovskitelike crystal structures are amongst the most important and most widely studied crystalline material classes in materials science today. Perovskite crystal structures have ABX_{3} chemical formulae, with A and B being ions at the centres of dodecagons and octahedra, respectively, and the X anions constituting the vertices of these polyhedra. The BX_{6} octahedra interact via cornersharing interactions. There are also perovskitelike crystal structures such as the double perovskites, A_{2}BB′X_{6} (King & Woodward, 2010; Koskelo et al., 2023), for which many of the same principles apply.
The ideal perovskite system would be cubic, with ):
, but many related structures with lower symmetry are known. This typically occurs in three situations (Woodward, 1997(i) when there is a mismatch between the ionic radii of the octahedrally coordinated B cation and the dodecagonally coordinated A cation, resulting in tilting of the octahedra [see Fig. 5(e)];
(ii) when there is displacement of the central cation from the centre of the octahedron, typically due to the pseudoJT effect;
(iii) when the ligands of the octahedron are distorted by electronic phenomena such as the firstorder JT effect.
In this case study, we focus on the first case, where a size mismatch results in octahedral tilting. Octahedra are often modelled as rigid bodies, but in practice they are not rigid in all systems and the octahedral tilting will often induce strain resulting in angular distortion. This is typically far smaller than that seen in edgesharing materials such as NaNiO_{2}, but it is large enough that it cannot be disregarded when attempting a full understanding of the structure of the material. As was noted by Darlington (1996), this angular distortion commonly manifests as shear.
LaAlO_{3} is a perovskitelike ABX_{3} material which is cubic (space group ) above around ∼830 K but exhibits a rhombohedral distortion below this temperature (with ) due to octahedral tilting (Hayward et al., 2005) [Figs. 5(e) and 5(f)]. There is no bondlength distortion in either temperature regime; a calculation of the bondlength distortion index would yield a value of zero at all temperatures. In the lowtemperature regime, the magnitude of the distortion continuously decreases with increasing temperature, reaching zero at the transition temperature. Most commonly in the literature, the tilting angle between the octahedral axis and the c axis (0° in the cubic phase) is used to quantify this distortion; for LaAlO_{3} this is shown in Fig. 5(a). The strain induced by this distortion results in intraoctahedral angular distortion. Hayward et al. (2005) modelled this in terms of strain tensors, finding a linear temperature dependence below the transition temperature. This differs from the temperature dependence of the tilting angle (which resembles an exponential decline), implying the two are related but distinct phenomena. Cumby & Attfield (2017) instead modelled the octahedral distortion for this same data set using the radii of a minimumbounding ellipsoid and also found an approximately linear temperature dependence of the long and short radii as they approach convergence [see Fig. 5(b)].
Here, we calculate the Van Vleck shear modes. Due to the symmetry of the octahedral tilting, there is only one independent shear mode and Q_{5} = −Q_{4} = −Q_{6}. We compare this with the bondangle standard deviation given in equation (13) [Fig. 5(c)]. We see that, despite being distinct parameters, they have identical temperature dependences. We attribute this to the shear fraction η being precisely 1 for all temperatures where there is angular distortion, meaning that shear is completely correlated with angular distortion.
4.2. Bigbox analysis of PDF data on LaMnO_{3}
The JT distortion in LaMnO_{3}, a perovskitelike ABX_{3} material which has the shown in Fig. 6(a), occurs as a consequence of degeneracy in the e_{g} orbitals on the highspin d^{4} Mn^{3+} ion. At ambient temperatures it is a prime example of a cooperative JT distortion, exhibiting longrange orbital order where the elongation of the JT axis alternates between the a and b directions for neighbouring MnO_{6} octahedra, never occurring along the c direction (Khomskii & Streltsov, 2021) [Fig. 6(b)]. With heating to ∼750 K, the JT distortion can no longer be observed in the average structure obtained from Bragg diffraction (RodriguezCarvajal et al., 1998). However, the JT distortion persists locally, as has been shown by PDF (Qiu et al., 2005) and (García et al., 2005; Souza et al., 2005) measurements. This transition is one of the most widely studied orbital order–disorder transitions for firstorder JT distortion. The hightemperature orbital regime has been described theoretically in terms of a threestate Potts model (Ahmed & Gehring, 2006, 2009), a view supported by bigbox analysis of combined neutron and Xray PDF data (Thygesen et al., 2017), as performed using RMCProfile (Tucker et al., 2007).
In this case study, we take a 10 × 10 × 8 _{3}, obtained using RMCProfile against total scattering data obtained at room temperature and previously published in the aforementioned work (Thygesen et al., 2017). The results are shown in Fig. 6. We repeat the analysis of this from the perspective of the E_{g}(Q_{2}, Q_{3}) Van Vleck distortion modes, using two different approaches: (i) applying the algorithm for automatically determining a set of orthogonal axes to each octahedron individually, and (ii) following the Van Vleck equations (23) and (24) proposed by Kanamori (1960) where angular distortion is disregarded. In each of these cases the crystallographic site of the is taken as the origin and so thermally driven variations in the Mn position will not affect the result.
of LaMnOAs can be seen in Figs. 6(c) and 6(d), there are two clusters of octahedra within the polar plot, occurring at ϕ ≃ ±107°. This corresponds to occupation of the d_{y2} orbitals (+) and of the d_{x2} orbitals (−). In both cases, the superposition of perpendicular Q_{3} compression and elongation modes results in an octahedron with mixed Q_{2}–Q_{3} character. This finding is consistent with previous work which placed MnO_{6} octahedra from LaMnO_{3} onto the framework of an E_{g}(Q_{2}, Q_{3}) polar plot (Zhou & Goodenough, 2008a; Zhou et al., 2011).
Fig. 6(e) shows the MnO_{6} octahedron in the average structure of LaMnO_{3} at room temperature, with the three different bond lengths plotted in Fig. 6(f) along with a histogram of all the bond lengths in the This shows how the combination of the Q_{2} and Q_{3} distortion modes manifests in the octahedral distortion.
The Q_{2} contribution to the distortion, as seen from the three different Mn—O bond lengths in LaMnO_{3}, is also present in JTdistorted ACuF_{3} (A = Na, K, Rb) (Lufaso & Woodward, 2004; Marshall et al., 2013; Khomskii & Streltsov, 2021) and even in some JTundistorted perovskites (Zhou & Goodenough, 2008a), indicating it is related to the structure. It is not intrinsic to JTdistorted manganates, as it is absent in highspin d^{4} Mn^{3+} with edgesharing octahedral interactions and collinear orbital ordering such as αNaMnO_{2} and LiMnO_{2} [checked using ICSD (Inorganic Database, FIZKarlsruhe, Germany; https://icsd.fizkarlsruhe.de/index.xhtml) references 15769 (Jansen & Hoppe, 1973) and 82993 (Armstrong & Bruce, 1996) respectively]. The Q_{2} component to the octahedral distortion is therefore probably intrinsic to the (Zhou & Goodenough, 2006, 2008a), which occurs as a result of octahedral tilting reducing the symmetry from cubic to Pnma. In LaMnO_{3}, the combination of the Q_{2} component to the distortion and the orbital ordering [Fig. 6(b)] are a possible distortion of the Pnma In this way, the orbital ordering may be coupled to the octahedral tilting, a link previously made by Lufaso & Woodward (2004).
Finally, we also calculate the Q_{4} to Q_{6} octahedral shear modes for all octahedra in the presented as a histogram in Fig. S2. We present the average and standard deviation, calculated assuming orthogonal axes and with the automated octahedral rotation: Q_{4} = −0.02 ± 0.13 Å, Q_{5} = 0.02 ± 0.10 Å and Q_{6} = −0.00 ± 0.11 Å. In each case, the magnitude of the distortion is zero within the standard deviation, and the value from the average structure presented in Table 2 also falls within the range of error. This low level of shear generally supports the validity of calculating the E_{g}(Q_{2}, Q_{3}) Van Vleck modes along bond directions rather than a Cartesian coordinate system for a system like LaMnO_{3}. It is interesting that the standard deviation is higher for Q_{4}, which quantifies the shear within the plane in which there is orbital ordering.
4.3. Effect of pressure on the JT distortion in NaNiO_{2}
In recent years, there have been several studies looking at the effect of applied pressure on the JT distortion in crystalline materials (Åsbrink et al., 1999; Loa et al., 2001; Choi et al., 2006; Zhou et al., 2008, 2011; Aguado et al., 2012; Mota et al., 2014; Caslin et al., 2016; Zhao et al., 2016; Collings et al., 2018; Bhadram et al., 2021; Lawler et al., 2021; Scatena et al., 2021; Ovsyannikov et al., 2021; NagleCocco et al., 2022). Most of these have shown that, as a general rule, pressure reduces the magnitude of the JT distortion as a consequence of the elongated bond being more compressible than the shorter bonds.
Zhou et al. (2011) used Van Vleck modes to quantify the effect of pressure on the JT distortion in the cornersharing perovskitelike compounds LaMnO_{3} and KCuF_{3}. While the application of pressure reduces the magnitude of the distortion, as quantified using ρ_{0} [equation (11)], they argue that it does not change the orbital mixing ϕ [equation (12)]. KCuF_{3} has similar orbital ordering to LaMnO_{3}, except the degeneracy is due to the d^{9} hole rather than an electron. The variablepressure crystal structures for KCuF_{3} are available from the ICSD (catalogue codes 182849–182857) and are utilized here.
We previously studied the effect of pressure on the JT distortion in NaNiO_{2} (NagleCocco et al., 2022) by performing of neutron diffraction data from the PEARL instrument (Bull et al., 2016) at the ISIS Neutron and Muon Source (Oxfordshire, UK). However, we did not utilize the Van Vleck distortion modes, instead quantifying the JT distortion using the bondlength distortion index (Baur, 1974) and the effective (Hoppe, 1979). In that study, we found no deviation from the ambientpressure C2/m (Dick et al., 1997; Sofin & Jansen, 2005), shown in Fig. 7(a), for all pressure points at room temperature up to ∼4.5 GPa. This permits only four short (long) and two long (short) bonds or six equal bond lengths, depending on the angle β, and so throughout the measured pressure range there is no Q_{2} character to the JT distortion, consistent with the principle that hydrostatic pressure does not change orbital mixing (Zhou et al., 2011).
Here, we perform a fresh analysis of the variablepressure octahedral behaviour as a function of pressure at room temperature in NaNiO_{2} in terms of the E_{g}(Q_{2}, Q_{3}) Van Vleck distortion modes. For a reference we sought a material that does not exhibit a firstorder JT distortion but does exhibit bondlength distortion; for this purpose, we selected Fe_{2}O_{3}, the pressure dependence of which was previously studied by Finger & Hazen (1980) and which exhibits bondlength distortion due to its face and edgesharing octahedral connectivity. Fe_{2}O_{3} contains highspin d^{5} Fe^{3+} cations within octahedra which interact via both face and edgesharing interactions. Note that Fe_{2}O_{3} probably exhibits some very subtle pseudoJT distortion (related to, but distinct from, the first JT effect discussed here) on account of the Fe^{3+} ions (Cumby & Attfield, 2017; Bersuker & Polinger, 2020), but this does not impact the discussion in any meaningful way.
In Fig. 7(c) we compare (for NaNiO_{2}) ρ_{0} with three other parameters (bondlength distortion index, quadratic elongation and effective coordination number) which are often used to parameterize the magnitude of the JT distortion. The trend for each is near identical, although the magnitudes differ greatly, indicating that each is a reasonable parameter for quantifying the magnitude of the JT distortion. This can be compared with Fig. 7(d), which shows the same parameters for the JTundistorted FeO_{6} octahedra in Fe_{2}O_{3}; it can be seen that ρ_{0} remains approximately at zero throughout the measured pressure range, despite a high level of bondlength distortion as represented by the bondlength distortion index, effective and quadratic elongation (a similar plot for KCuF_{3} can be seen in Fig. S3). This means that, while these parameters are valid for quantifying the magnitude of the JT distortion, they are also sensitive to other kinds of distortion. ρ_{0} is calculated using Q_{2} and Q_{3} which have E_{g} symmetry, so ρ_{0} will only be nonzero for a distortion with E_{g} symmetry. Thus, it is arguably the ideal choice for parameterizing the magnitude of this type of JT distortion. However, while ρ_{0} is more reliable than the other parameters shown in Figs. 7(c) and 7(d) for demonstrating the presence of JT distortion, it is not always strictly zero for a JTinactive octahedron as it will have a nonzero value if the octahedron is distorted with an e_{g} symmetry. For example, the NaO_{6} octahedron in C2/m NaNiO_{2} has the same symmetry as the NiO_{6} octahedron and so exhibits a value of ρ_{0} between 0.065 and 0.05 within the studied pressure range (Fig. S4), and JTinactive FeO_{6} octahedra in RFeO_{3} perovskites have nonzero ρ_{0} due to the E_{g} symmetry of the distorted octahedra, as shown by Zhou & Goodenough (2008a).
Fig. 8 shows a polar plot for the behaviour of NaNiO_{2} and KCuF_{3} in the range 0–5 GPa (the measured range for NaNiO_{2}). It can be seen that within this pressure range, the magnitude of the JT distortion decreases far more for KCuF_{3} than for NaNiO_{2}; this reflects the fact that KCuF_{3} is more compressible, with a bulk modulus of 57 (1) GPa (Zhou et al., 2011) in contrast to 121 (2) GPa for NaNiO_{2} (NagleCocco et al., 2022), as obtained by a fit to the thirdorder Birch–Murnaghan equation of state (Birch, 1947). Within this pressure range we see that ϕ does not change with pressure for either material and that this property is true regardless of whether ϕ is or is not a special angle (as in Table 1), consistent with the interpretation of Zhou et al. (2011).
Finally, in the previous study (NagleCocco et al., 2022) we showed, using specific O—Ni—O bond angles, that pressure reduces the angular distortion for NaNiO_{2}. Here, we show that pressure also reduces the related shear distortion in NaNiO_{2}. This is demonstrated in Fig. 9 where we plot the octahedral shear Q_{4}, Q_{5} and Q_{6} modes for NaNiO_{2} and Fe_{2}O_{3} against the bondangle standard deviation σ_{ζ}, defined in equation (13). Unlike the AlO_{6} octahedra in LaAlO_{3} (Fig. 5), for NiO_{6} octahedra in NaNiO_{2} there is no perfect correlation between the shear modes and angular distortion despite η ≃ 1, because there is more than one independent shear mode, but we can see that shear distortion and angular distortion are still highly correlated. However, for Fe_{2}O_{3} the shear fraction η ≪ 1 and there is no correlation between the shear distortion modes and angular distortion. This difference in behaviour probably arises because the main driver of the change is a continuous decrease in the JT distortion in NaNiO_{2}, while in Fe_{2}O_{3} the positions of the oxygen anions are determined by the reduced arising from trying to satisfy multiple face and edgesharing interactions. This result could only be achieved by calculating the Van Vleck modes in a Cartesian coordinate system as outlined in this paper, as opposed to calculating the distortion modes along bond directions, indicating the relevance of calculating the Van Vleck modes in this way and of the shear fraction η we propose in this work.
5. Conclusion
We have presented VanVleckCalculator, a code package written in Python 3 for the calculation of octahedral Van Vleck distortion modes. These modes are particularly important for understanding the behaviour of the JT distortion, and we have shown that the parameter ρ_{0} (which is based on the Van Vleck Q_{2} and Q_{3} modes) is a more reliable way of quantifying the JT distortion than other oftused parameters such as the bondlength distortion index.
We have shown the importance of using a Cartesian set of coordinates for this calculation, instead of calculating the modes along bond directions as is often done in the literature. This is because calculating the Van Vleck distortion modes along bond directions relies on the assumption that there is no angular distortion or octahedral shear, which is often a false assumption and artificially constrains the Q_{4}, Q_{5} and Q_{6} modes to be zero. We have shown that there is value in calculating these latter modes, for instance in understanding the effect of octahedral tilting on octahedra in perovskitelike materials. These shear modes will also be useful for parameterizing the JT effect when the degeneracy occurs in the t_{2g} orbitals and results in a trigonal distortion, because their symmetry matches the distortion.
We have also shown that octahedral shear correlates with angular distortion for materials under the influence of tuning parameters such as pressure or temperature where there is a continuously varying distortion, such as octahedral tilting (as in LaAlO_{3}) or firstorder JT distortion (as in NaNiO_{2}). However, there is no correlation when the distortion is caused by competing interactions due to face or edgesharing octahedra (as in Fe_{2}O_{3}). We propose a new parameter, the shear fraction η [defined in equation (22)], which can be used to predict whether there will be correlation between octahedral shear modes and angular distortion.
6. Related literature
For further literature related to the supporting information, see Halasyamani (2004), Koçer et al. (2019) and Swanson & Peterson (1980).
Supporting information
Additional information. DOI: https://doi.org/10.1107/S1600576723009925/oc5030sup1.pdf
Footnotes
^{1}In the lowspin case, t_{2g} orbitals fill fully before e_{g} orbitals gain electrons; in the highspin case, once the t_{2g} orbitals are singly occupied, the next two electrons will populate the e_{g} orbitals.
^{2}For a set of three orthogonal vectors chosen as the axes, the choice to assign each to x, y or z will not affect the value of ρ_{0}, but will affect the value of = arctan(Q_{2}/Q_{3}) by an integer multiple of 120°, plus a reflection about the nearest special angle (see Table 1) if there is Q_{2}–Q_{3} mixing.
^{3}This is because rotation operations do not commute and so a single rotation about each axis is unlikely to result in the defined axes being superimposed over the basis vectors.
Acknowledgements
The authors thank Andrew L. Goodwin (University of Oxford) for useful discussions and for sharing the LaMnO_{3} from the data of Thygesen et al. (2017). The authors thank James Cumby (University of Edinburgh) for sharing the crystal structures for LaAlO_{3} originally published by Hayward et al. (2005) and subsequently analysed by Cumby & Attfield (2017). The authors acknowledge comments on earlier drafts of this manuscript from James M. A. Steele, Fiamma Berardi and Venkateswarlu Daramalla, all at the University of Cambridge. LNC acknowledges Annalena R. GenreithSchriever (University of Cambridge) and Ben Tragheim (University of Warwick) for useful discussions. Graphs and radial plots were prepared using MatPlotLib (Hunter, 2007). figures were drawn using VESTAIII (Momma & Izumi, 2011).
Funding information
LNC acknowledges a scholarship EP/R513180/1 to pursue doctoral research from the UK Engineering and Physical Sciences Research Council (EPSRC).
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