The Distortion Theorem implies that the irregularity of bond distances in a distorted coordination polyhedron causes an increase of mean bond distance. Examination of 40 polyhedra containing the lone-pair cation Te(IV) shows that this does not imply an increase in polyhedral volume. Volumes of these polyhedra are 10.3-23.7 Å(3), compared with the 12.83 Å(3) expected for a hypothetical regular octahedron. There is little correlation between volume and measures of polyhedral distortion such as quadratic elongation, bond-angle variance or vector bond valence. However, the oxygens of our polyhedra lie very close to a sphere of best fit, centred at ~ 1 Å from the Te(IV) atom. The Te(IV)-centre distance is an index of lone-pair stereoactivity and is linearly related to the radius Rsph of the sphere; this is explained by a more localized lone pair repelling the anions more strongly, leading to a longer non-bonded distance between the lone pair and anions. Polyhedral volume still varies considerably for a given Rsph, because the oxygen ligands may be distributed over the whole sphere surface, or confined to a small portion of it. The uniformity of this distribution can be estimated from the distance between the sphere centre and the centroid of the O6 polyhedron. Te(IV)-centre and centroid-centre distances alone then account for 95% of the variation observed in volume for polyhedra which are topologically octahedral. Six of the polyhedra studied that are outliers are closer in shape to pentagonal pyramids than octahedra. These have short distances from the central Te(IV) cation to other Te(IV) and/or to large, polarizable cations, suggesting additional weak bonding interactions between these species and the central lone pair. The flexibility of lone-pair polyhedra is further enhanced by the ability of a single polyhedron to accommodate different cations with different degrees of lone-pair activity, which facilitates more diverse solid solution behaviour than would otherwise be the case.
Keywords: distortion theory; lone-pair stereoactivity; polyhedral distortion.