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A definition of minimum distortion paths between two polyhedra in terms of continuous shape measures (CShM) is presented. A general analytical expression deduced for such pathways makes use of one parameter, the minimum distortion constant, that can be easily obtained through the CShM methodology and is herein tabulated for pairs of polyhedra having four to eight vertexes. The work presented here also allows us to obtain representative model molecular structures along the interconversion pathways. Several commonly used polytopal rearrangement pathways are shown to be in fact minimum distortion pathways: the spread path leading from the tetrahedron to the square, the Berry pseudorotation that interconverts a square pyramid and a trigonal bipyramid, and the Bailar twist for the interconversion of the octahedron and the trigonal prism. Examples of applications to the analysis of the stereochemistries of several families of metal complexes are presented.

A continuous shape and symmetry study of tetracoordinate transition-metal complexes is presented, in an attempt to provide a systematic description of the stereochemistry of the metal coordination sphere in this important family of compounds. A tetrahedron/square-planar symmetry map has been developed, the main distortion paths of the ideal geometries are presented, and the applicability of a sawhorse shape measure is discussed. More than 13,000 structural data sets have been analyzed and the corresponding stereochemistries assigned from the values of their tetrahedral and square-planar symmetry measures. A good number of structures that are quite distant from the two ideal geometries can be adequately described as snapshots along the spread pathway for their interconversion, making use of the corresponding path deviation function. Further analysis of the structural data by metal electron configuration or by the denticity and conformation of the ligands provide general rules to describe the stereochemical preferences of tetracoordinate transition metal centers.