Ellipsoidal analysis of coordination polyhedra

The idea of the coordination polyhedron is essential to understanding chemical structure. Simple polyhedra in crystalline compounds are often deformed due to structural complexity or electronic instabilities so distortion analysis methods are useful. Here we demonstrate that analysis of the minimum bounding ellipsoid of a coordination polyhedron provides a general method for studying distortion, yielding parameters that are sensitive to various orders in metal oxide examples. Ellipsoidal analysis leads to discovery of a general switching of polyhedral distortions at symmetry-disallowed transitions in perovskites that may evidence underlying coordination bistability, and reveals a weak off-centre ‘d5 effect' for Fe3+ ions that could be exploited in multiferroics. Separating electronic distortions from intrinsic deformations within the low temperature superstructure of magnetite provides new insights into the charge and trimeron orders. Ellipsoidal analysis can be useful for exploring local structure in many materials such as coordination complexes and frameworks, organometallics and organic molecules.

It is unclear whether the assigned low temperature symmetries of I4/mcm for CeAlO 3 and I2/m for PrAlO 3 are describing the same phase or not, but both materials show a symmetrydisallowed orthorhombic Imma → 3c transition. Thermal variations of the CeAlO 3 ellipsoid radii, their average <R> and standard deviation (R) are shown in Supplementary Fig. 1.
Transitions from orthorhombic Pbcn or Pnma symmetry to Cmcm are more complex as the latter structure has two independent A cation sites, but changes in either s(S) A or s(S) B are still observed.

Supplementary Note 2. Iron Oxides
Ellipsoidal analysis provides a useful approximation for comparing the distortions and shapes of polyhedra independent of coordination number and geometry. These aspects are illustrated with reference to iron oxide polyhedra from 390 structures in the ICSD (499 polyhedra in total) where 98 Fe 2+ and 401 Fe 3+ sites have coordination numbers between 4 and 6. Plots and histograms are in Supplementary Fig. 2 and summary versions of the plots are in Fig. 4. Supplementary Figs. 2a and 2b show plots of ellipsoid standard deviation (R) against mean radius <R> and shape parameter S . The (R) vs. <R> distribution in Supplementary Fig. 2a shows a large cluster of points at <R> ≈ d and (R) ≈ 0, where d is an average Fe-O distance, corresponding to the large number of reported regular tetrahedra and octahedra. Ideal values based on 6-coordinate ionic radii are d = 2.18 and 2.05 Å for high spin Fe 2+ and Fe 3+ respectively, and the displacement between the histograms shown reflects this size difference. The magnitudes D of the displacement vector for Fe relative to the ellipsoid centre of FeO n polyhedra are plotted against <R> in Supplementary Fig. 2b. The off-centre d 5 effect is apparent but is seen more clearly for the analysis of 6-coordinate polyhedra in Fig. 5c, as described in the main paper.
T p f σ(R) vs. ellipsoid shape S in Supplementary Figs. 2c and 2d is another useful way to observe the spectrum of iron oxide polyhedra. S varies from near -1 for square planar complexes in the oblate distortion limit through many distorted and regular tetrahedra and octahedra at S ≈ -0.2 to 0.3 (the distribution for FeO 6 polyhedra alone is in Fig. 5e). The Vshaped appearance of the distribution results from the most symmetric ellipsoid shapes p g w σ( ); (R) = -(√2/3)R 2 S while prolate ellipsoids have (R) = (√2/3)R 2 S/(1-S). These limits are shown on the plot taking the median ellipsoid radius to be R 2 = 1.81 Å. The line for geometrically scalene ellipsoids, which have S = 0 but are non-spherical with R 3 /R 2 = R 2 /R 1 is also shown. This coordination with two long, two medium, and two short Fe-O bonds is rather rare compared to oblate and prolate coordinations.
The σ(R) vs. ellipsoid shape S plot for magnetite in Fig. 4f shows remarkable structure giving insights into the electronic distortion. The same plot is reproduced as Supplementary Fig. 3 to show how distortions within trimerons are correlated.