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A stereochemical study of polyhedral eight-vertex structures is presented, based on continuous shape measures (CShM). Reference polyhedra, shape maps, and minimal-distortion interconversion paths are presented for eight-vertex polyhedral and polygonal structures within the CShM framework. The application of these stereochemical tools is analyzed for several families of experimental structures: 1) coordination polyhedra of molecular transition-metal coordination compounds, classified by electron configuration and ligands; 2) edge-bonded polyhedra, including cubane structures, realgar, and metal clusters; 3) octanuclear transition-metal supramolecular architectures; and 4) coordination polyhedra in extended structures in inorganic solids. Structural classification is shown to be greatly facilitated by these tools, and the detection of less common structures, such as the gyrobifastigium, is straightforward.

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.

More than 750 d 0 transition metal oxide octahedra have been examined in order to better understand the out-of-center distortion occurring with these cations. A continuous symmetry measures approach was used to quantify the magnitude and direction of the distortion. Using this approach we were able to divide the d 0 transition metals into three categories: strong (Mo 6+ and V 5+ ), moderate (W 6+ , Ti 4+ , Nb 5+ , and Ta 5+ ), and weak (Zr 4+ and Hf 4+ ) distorters. We also examined and discussed the directional preference of the distortion for each cation.

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Ab initio periodic calculations based on local Gaussian basis sets have been performed to investigate the site preference and cation selectivity of H, Li, Na, and K in Al-substituted high-silica dehydrated chabazite. Two cation sittings were considered:  the 6-membered ring (SII) and the 8-membered ring (SIII‘) pores. Structures have been fully relaxed at the Hartree−Fock level of theory, and the relative stabilities were verified with density functional single point calculations employing the B3-LYP hybrid functional. Computed ion site preferences for Li and K are in agreement with experimental evidence:  Li is more stable at SII whereas K prefers the SIII‘ site. For Na, our calculations show that the more stable location is the SII site, at variance with the experimentally observed site preference which depends on the Si/Al ratio. Computed zeolite framework selectivity toward hydrated cations follows the sequence K > Na > Li in line with observed trends. Computed electric field gradients have also been used to evaluate the quadrupolar parameters as a further characterization of the cation positions.