General method for the description, visualization and comparison of metal coordination spheres: geometrical preferences, deformations and interconversion pathways.
The coordination sphere geometry of metal atoms (M) in their complexes with organic and inorganic ligands (L) is often compared with the geometry of archetypal forms for the appropriate coordination number, n in ML(n) species, by use of the k = n( n- 1)/2 L-M-L valence angles subtended at the metal centre. Here, a Euclidean dissimilarity metric, R(c)(x), is introduced as a one-dimensional comparator of these k-dimensional valence-angle spaces. The computational procedure for R(c)(x), where x is an appropriate archetypal form (e.g. an octahedron in ML(6) species), takes account of the atomic permutational symmetry inherent in ML(n) systems when no distinction is made between the individual ligand atoms. It is this permutational symmetry, of order n!, that precludes the routine application of multivariate analytical techniques, such as principal component analysis (PCA), to valence angle data for all but the lowest metal coordination numbers. It is shown that histograms of R(c)(x) values and, particularly, scatterplots of R(c)(x) values computed with respect to two or more different appropriate archetypal forms (e.g. tetrahedral and square-planar four-coordinations), provide information-rich visualizations of the observed geometrical preferences of metal coordination spheres retrieved from, e.g. the Cambridge Structural Database. These mappings reveal the highly populated clusters of similar geometries, together with the pathways that map their geometrical interconversions. Application of R(c)(x) analysis to the geometry of four- and seven-coordination spheres provides information that is at least comparable to, and in some cases is more complete than, that obtained by PCA.