A continuous symmetry study of the structures of transition metal six-vertex polyhedra is presented, considering both molecular models and experimental structural data. The concept of symmetry map is introduced, consisting of a scatterplot of the symmetry measures relative to two alternative ideal polyhedra. In the case of hexacoordinated complexes, we take as reference shapes the octahedron and the equilateral trigonal prism and study different distortions from these two extremes, including the Bailar twist that interconverts one into another. Such a symmetry map allows us to establish trends in the structural chemistry of the coordination sphere of hexacoordinated transition metal atoms, including the effects of several factors, such as the electron configuration or the presence of bidentate, terdentate or encapsulating ligands. Also introduced is the concept of a symmetry constant, which identifies a distortive route that preserves the minimum distance to two reference symmetries. A wide variety of model distortions are analyzed, and the models are tested against experimental structural data of a wide variety of six-coordinated complexes.

The application of continuous symmetry measures to the description of the structures of penta-coordinate complexes is discussed. The characteristic values of the trigonal bipyramid symmetry measure, S(TBP), for molecules with square pyramidal (SP) or intermediate geometries corresponding to a Berry pseudorotation coordinate are established. The experimental data from a reference set comprising 71 structures of more than 40 homoleptic transition metal complexes (including organometallic molecules and inorganic solids) and four EPh₅ derivatives of Group 15 elements are consistent with the Berry pathway. The structures of several families of transition metal complexes are analyzed in the light of the variations of S(TBP) corresponding to other distortions of the ideal TBP, including bond stretch or non-Berry angular distortion modes. The families of compounds analyzed are represented by the general formulae [OMX₄]⁻, [NMX₄]⁻, [Cu(bipy)₂X]⁺, [Cu(phen)₂X]⁺, [M(dppe)₂X], [M(tripod)X], [M(terpy)X₂], [M(NO)L₄] with the {MNO}⁸ electron count, vanadium-(IV) and -(V) compounds including several oxides and vanadyl complexes. The S_N2 substitution pathway in Sn compounds that had been previously shown to be well represented by the structural data of a series of compounds with, e.g., a XSnC₃O core are also shown to be elegantly described by the simultaneous change in their TBP and T_d continuous symmetry measures.