Polyhedral structures with an odd number of vertices: nine-coordinate metal compounds

The stereochemistry of nine-coordinate transition-metal and rare-earth compounds has been studied by means of continuous shape measures (CShM) and related tools. Several reference nine-vertex polyhedra have been defined and their minimal distortion interconversion paths established. A theoretical shape map is presented in which the structures can be placed according to their distances in CShM space to the capped square antiprism and the tricapped trigonal prism, which are the most common polyhedra in nine-coordinate compounds. The structures of almost 2000 metal coordination spheres in molecular and extended solid-state compounds have been analyzed. Clear stereochemical trends can be established for subsets of these compounds grouped according to the nature of their ligands, which include families of compounds spread along the interconversion paths between the capped square antiprism and the capped cube, or between the tricapped trigonal prism and the tridiminished icosahedron.     

The rich stereochemistry of eight-vertex polyhedra: a continuous shape measures study

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.     

Polyhedral structures with an odd number of vertices: nine-atom clusters and supramolecular architectures

The geometries of metal clusters and supramolecular architectures that contain nine metal atoms are analyzed within the framework of continuous shape measures (CShM). The most common polyhedra in nine coordinate complexes, the capped square antiprism and the tricapped trigonal prism, are also found among these families of compounds, even if much more scarcely. In addition, a variety of new shapes, not found among coordination polyhedra, can be identified and their proximity to the ideal geometries quantified. These include a linear chain, two types of trigonal columns, the planar regular enneagon, two-dimensional hexagonal and square grids, fragments of a close-packed structure, the triangular cupola, the tridiminished icosahedron or different fragments of the icosahedron. Among the nine-atom boranes and related clusters of the groups 13 and 14 elements, those having between 18 and 20 framework electrons present the structure of the tricapped trigonal prism, the expected closo structure. However, clusters with 21 and 22 framework electrons present a variety of structures with geometries covering nearly all the path that takes one from the capped square antiprism (nido form) to the tricapped trigonal prism (closo form).     

Shape and symmetry of heptacoordinate transition-metal complexes: structural trends

The stereochemistries of heptacoordinate transition-metal complexes are analyzed by using continuous symmetry and shape measures of their coordination spheres. The distribution of heptacoordination through the transition-metal series is presented based on structural database searches including organometallic and Werner-type molecular complexes, metalloproteins, and extended solids. The most common polyhedron seems to be the pentagonal bipyramid, while different preferences are found for specific families of compounds, as in the complexes with three or four carbonyl or phosphine ligands, which prefer the capped octahedron or the capped trigonal prism rather than the pentagonal bipyramid. The symmetry maps for heptacoordination are presented and shown to be helpful for detecting stereochemical trends. The maximal symmetry interconversion pathways between the three most common polyhedra are defined in terms of symmetry constants and a large number of experimental structures are seen to fall along those paths.     

Topological aspects of chemically significant polyhedra

This paper considers both static and dynamic properties of chemically significant polyhedra. Static properties of polyhedra consider relationships between the numbers and degrees/sizes of polyhedral vertices, edges, and faces; polyhedral symmetries; and numbers of topologically distinct polyhedra of various types. Dynamic properties of polyhedra involve studies of polyhedral isomerizations from both macroscopic and microscopic points of view. Macroscopic aspects of polyhedral isomerization can be described by graphs called topological representations in which the vertices correspond to different permutational isomers and the edges to single degenerate polyhedral isomerization steps. Such topological representations are presented for isomerizations of polyhedra having five, six, and eight vertices. Microscopic aspects of polyhedral isomerizations arise from consideration of the details of polyhedral topology, such as the topological aspects of diamond-square-diamond processes. In this connection, Gale diagrams are useful for describing isomerizations of five- and six-vertex polyhedra, including the Berry pseudorotation of a trigonal bipyramid through a square pyramid intermediate and the Bailar or Ray and Dutt twists of an octahedron through a trigonal prism intermediate.     

Chemical applications of topology and group Theory. XXIV: Chiralization of chemically significant polyhedra [1]

Permutation group-theoretical methods are used to study the chiralization of achiral polyhedral skeletons with v vertices by successive ligand replacement. Starting from the fully symmetrical ligand partition (), such chiralization processes may be characterized either by the minimum number of ligand replacement steps m, or the minimum number of different kinds of ligandsi, required to destroy all improper rotations. These parameters are trivially related to the lowest degree chiral ligand partition(s) as determined by the subduction of the skeleton point group G into the corresponding symmetric groupS by the procedure of Ruch and Schönhofer. Two different chiralization pathways with different values ofm andi are found for the octahedron, cube, hexagonal bipyramid, and icosahedron. Many less symmetrical chemically significant polyhedra have the degree 2 ligand partition (v - 2, 2) as the lowest degree chiral ligand partition and thus have only one chiralization pathway. Such polyhedra include the bicapped tetrahedron, trigonal prism, capped octahedron, bisdisphenoid, square antiprism, 4, 4, 4-tricapped trigonal prism, 4-capped square antiprism, 4,4-bicapped square antiprism, and the cuboctahedron.     

Density functional theory study of nine-atom germanium clusters: effect of electron count on cluster geometry

Density functional theory (DFT) at the hybrid B3LYP level has been applied to the germanium clusters Ge(9)(z) clusters (z = -6, -4, -3, -2, 0, +2, and +4) starting from three different initial configurations. Double-zeta quality LANL2DZ basis functions extended by adding one set of polarization (d) and one set of diffuse (p) functions were used. The global minimum for Ge(9)(2)(-) is the tricapped trigonal prism expected by Wade's rules for a 2n + 2 skeletal electron structure. An elongated tricapped trigonal prism is the global minimum for Ge(9)(4)(-) similar to the experimentally found structure for the isoelectronic Bi(9)(5+). However, the capped square antiprism predicted by Wade's rules for a 2n + 4 skeletal electron structure is only 0.21 kcal/mol above this global minimum indicating that these two nine-vertex polyhedra have very similar energies in this system. Tricapped trigonal prismatic structures are found for both singlet and triplet Ge(9)(6)(-), with the latter being lower in energy by 3.66 kcal/mol and far less distorted. The global minimum for the hypoelectronic Ge(9) is a bicapped pentagonal bipyramid. However, a second structure for Ge(9) only 4.54 kcal/mol above this global minimum is the C(2)(v)() flattened tricapped trigonal prism structure found experimentally for the isoelectronic Tl(9)(9)(-). For the even more hypoelectronic Ge(9)(2+), the lowest energy structure consists of an octahedron fused to two trigonal bipyramids. For Ge(9)(4+), the global minimum is an oblate (squashed) pentagonal bipyramid with two pendant Ge vertices.     

Chemical applications of topology and group theory. 17. An information theoretical approach to polyhedral symmetry [1]

Information theoretic parameters are described which measure the asymmetry of polyhedra based on partitions of their vertices, faces, and edges into orbits under action of their symmetry point groups. Such asymmetry parameters are all zero only for the five regular polyhedra and are all unity for polyhedra having no symmetry at all, i.e. belonging to the C ₁ symmetry point group. In all other cases such asymmetry parameters have values between zero and unity. Values for such asymmetry parameters are given for all topologically distinct polyhedra having five, six, and seven vertices; all topologically distinct eight-vertex polyhedra having at least six symmetry elements; and selected polyhedra having from nine to twelve vertices. Effects of polyhedral distortions on these asymmetry parameters are examined for the tetrahedron, trigonal bipyramid, square pyramid, and octahedron. Such information theoretic asymmetry parameters can be used to order site partitions which are incomparable by the chirality algebra methods of Ruch and co-workers.     

Polyhedral structures with three-, four-, and five fold symmetry in metal-centered ten-vertex germanium clusters

Studies using density functional theory (DFT) at the hybrid B3LYP level indicate that the relative energies of structures with three-fold, four-fold, and five-fold symmetry for centered 10-vertex bare germanium clusters of the general type M@Ge(10) (z) depend on the central metal atom M and the skeletal electron count. For M@Ge(10) clusters with 20 skeletal electrons the DFT results agree with experimental data on the isoelectronic centered 10-vertex bare metal clusters. Thus the lowest energy structure for Ni@Ge(10), isoelectronic with the known Ni@In(10) (10-), is a C(3v) polyhedron derived from the tetracapped trigonal prism. However, Zn@Ge(10) (2+) is isoelectronic with the known cluster Zn@In(10) (8-), which has the lowest energy structure, a D(4d) bicapped square antiprism. For the clusters Ni@Ge(10) (2-), Cu@Ge(10) (-), and Zn@Ge(10) that have 22 skeletal electrons the lowest energy structures are the D(4d) bicapped square antiprism predicted by the Wade-Mingos rules. For the clusters Ni@Ge(10) (4-), Cu@Ge(10) (3-), and Zn@Ge(10) (2-) that have 24 skeletal electrons the lowest energy structures are C(3v) polyhedra with 10 triangular faces and 3 quadrilateral faces derived from a tetracapped trigonal prism by extreme lengthening of the edges of the capped triangular face of the underlying trigonal prism. For the clusters Cu@Ge(10) (5-) and Zn@Ge(10) (4-) that have 26 skeletal electrons the lowest energy structures are the D(5d) pentagonal antiprisms predicted by the Wade-Mingos rules and the C(3v) tetracapped trigonal prism as a somewhat higher energy structure. However, for the isoelectronic Ni@Ge(10) (6-) the relative energies of these two structure types are reversed so that the C(3v) tetracapped trigonal prism becomes the global minimum. The effects of electron count on the geometries of the D(5d) pentagonal prism and D(4d) bicapped square antiprism centered metal cluster structures are consistent with the bonding/antibonding characteristics of the corresponding HOMO and LUMO frontier molecular orbitals.     

The Coordination Polyhedra OsN n in Crystal Structures

Characteristics of the Voronoi-Dirichlet polyhedra were used to perform the crystal-chemical analysis of 53 compounds containing osmium atoms surrounded by nitrogen atoms. Depending on the metal oxidation state, which varies from Os(II) to Os(VII), the coordination polyhedra formed by osmium atoms can be octahedra or trigonal prisms (OsN₆), square pyramids (OsN₄Os), tetrahedra (OsN₄), or triangles (OsN₃). The parameters of the Voronoi-Dirichlet polyhedra allow one to estimate quantitatively the main stereochemical features of Os atoms, depending on their oxidation state, and in controversial cases, they can be used to determine the oxidation state of osmium in the crystal structures.     

Structural isomers and reactivity for Rh6 and Rh6+

The structure, energetics, and interconversion of isomers of Rh(6) and Rh(6)(+) are studied by using density functional theory with Gaussian basis sets, using guess structures derived from basin-hopping simulations, and obtained by using the Sutton-Chen potential. A large range of spin multiplicities is considered for each isomer. Our calculations suggest two low-lying structures as possible structural isomers: a square bipyramid and a trigonal prism. The reactivity of these two candidate structural isomers with respect to adsorption of nitric oxide is studied via location of reaction transition states and calculation of reaction barriers. Similarities and differences with surface reaction studies are highlighted. These data provide powerful evidence that structural isomerism, and not different spin states, is responsible for the observed biexponential reaction kinetics.     

Tetrastrontium dimanganese copper nonaoxide,Sr4Mn2CuO9

Single crystals of Sr₄Mn_2.09Cu_0.91O₉ have been grown by flux synthesis and the structure, closely related to the hexagonal perovskite 2H, was solved from single‐crystal X‐ray data in space group P321. The structure of Sr₄Mn₂CuO₉ is composed of chains of face‐sharing polyhedra with a sequence of two octahedra and one trigonal prism. The octahedra are filled by Mn atoms and the Cu atoms are randomly distributed at the centres of the square faces of the trigonal prism. A stacking fault is observed within one of the two chains, which can be attributed to a shifting of the chain along the c axis.     

A Phosphine Functionalized β-Diketimine Ligand for the Synthesis of Manifold Metal Complexes

A bis(diphenyl)-phosphine functionalized β-diketimine (PNac-H) was synthesized as a flexible ligand for transition metal complexes. The newly designed ligand features symmetrically placed phosphine moieties around a β-diketimine unit, forming a PNNP-type pocket. Due to the hard and soft donor atoms (N vs. P) the ligand can stabilize various coordination polyhedra. A complete series ranging from coordination numbers 2 to 6 was realized. Linear, trigonal planar, square planar, tetrahedral, square pyramidal, and octahedral coordination arrangements containing the PNac-ligand around the metal center were observed by using suitable metal sources. Hereby, PNac-H or its anion PNac⁻ acts as mono-, bi- and tetradendate ligand. Such a broad flexibility is unusual for a rigid tetradentate system. The structural motifs were realized by treatment of PNac-H with a series of late transition metal precursors, for example, silver, gold, nickel, copper, platinum, and rhodium. The new complexes have been fully characterized by single crystal X-ray diffraction, NMR, IR, UV/Vis spectroscopy, mass spectrometry as well as elemental analysis. Additionally, selected complexes were investigated regarding their photophysical properties. Thus, PNac-H proved to be an ideal ligand platform for the selective coordination and stabilization of various metal ions in diverse polyhedra and oxidation states.     

Crystal structure of compounds [Ln(H2O)8]3+·4CCl3COO?·4H2O·2(4-bpy)·(4-bpyH)+(Ln(III) = Gd, Tb, Er, AND Tm)

Novel compounds of Gd(III), Tb(III), Er(III), and Tm(III) with 4,4??-bipyridine (4-bpy) and trichloroacetates are prepared. The title compounds are isomorphic and isostructural in the solid state. All atoms in the studied compound lie in general positions of the Cc space group. The coordination polyhedra around central atoms can be described as bicapped trigonal prisms slightly disordered towards a square antiprism. All 4-bpy molecules are located in the outer coordination sphere together with four trichloroacetate anions and four water molecules. One of the nitrogen atoms of one 4-bpy is protonated due to stoichiometry.     

Crystallographic and structural characterisation of heterometallic platinum compounds: Part I. Heterobinuclear Pt compounds

This review covers almost 290 heterobinuclear Pt derivatives. When the heterometals (M) are non transition and the binuclear are found both with and without a metal to metal bond. Where M is a transition metal or actinide, only those with a metal-metal bond have been included here. There are thirteen non-transition metals (Sn, Hg, Ge, Sb, Tl, Zn, Pb, Cd, Na, K, Ga, Ca and In). The shortest Pt-M bond distance is 235.2(1) (Pt-Ge). There are eighteen transition metals (Fe, W, Rh, Re, Pd, Ag, Ir, Mo, Mn, Re, Co, Cu, Cr, Au, Ni, Ti, Ta and V). The shortest Pt-M bond distance is 249.5(2) pm (Pt-Cr). There is one example of an actinide, Pt-Th at 298.4(1) pm. The Pt atom has oxidation numbers 0, +2 and +4. The Pt coordination geometries include square planar (most common), trigonal bipyramidal, pseudo octahedral (Pt(IV)) and a few prevalently capped trigonal prismatic seven coordinate species. There are at least two types of isomerism distortion and polymerisation. Factors affecting bond lengths and angles are discussed and some ambiguities in coordination polyhedra are outlined.      

Crystal Chemistry of Ternary and Quaternary Transition Metal Nitrides

Nitrides are fascinating materials. Although the air we breathe is 78% nitrogen, essentially 100% of the land we walk upon are made of oxides. It would take an encyclopaedia to review oxide crystal chemistry, on the other hand we can present here a comprehensive review of ternary and quaternary solid state nitrides. In the ternary and quaternary solid state nitrides (IA,IIA)-M-N, unusual low valence M⁺ were found for copper, nickel, cobalt, iron and manganese, and various MN_n transition metal-nitrogen coordination polyhedra with CN=2(linear), 3(trigonal-planar), 4(tetrahedral and square planar), 5(square pyramidal), and 6(octahedral and trigonal prismatic) and different fashions of connection make nitrides an important family of inorganic solid state materials. A suitable classification is made on the basis of the correlation between coordination numbers (CN), the connection fashions and oxidation states of the anionic principal structures.     

Geometry of the Fluorides, Oxofluorides, Hydrides, and Methanides of Vanadium(V), Chromium(VI), and Molybdenum(VI): Understanding the Geometry of Non-VSEPR Molecules in Terms of Core Distortion

This paper describes a study of the topology of the electron density and its Laplacian for the molecules VF(5), VMe(5), VH(5), CrF(6), CrMe(6), CrOF(4), MoOF(4), CrO(2)F(2,) CrO(2)F(4)(2)(-) and CrOF(5)(-) all of which, except VF(5,) CrF(6), and CrOF(5)(-) have a non-VSEPR geometry. It is shown that in each case the interaction of the ligands with the metal atom core causes it to distort to a nonspherical shape. In particular, the Laplacian of the electron density reveals the formation of local concentrations of electron density in the outer shell of the core, which have a definite geometrical arrangement such as four in a tetrahedral arrangement or five in a square pyramidal or trigonal bipyramidal and six in an octahedral arrangement. Ligands that are predominately covalently bonded are found opposite regions of charge depletion between these core charge concentrations. In VH(5), VMe(5), CrOF(4), and MoOF(4), these core charge concentrations have a square pyramidal arrangement, and the regions of charge depletions have the corresponding inverse square pyramidal arrangement so that these molecules have a square pyramidal geometry rather than a trigonal prism geometry. In CrMe(6), there are five core charge concentrations with a trigonal bipyramidal arrangement so that the regions of charge depletion have a trigonal prismatic arrangement and the molecule has the corresponding trigonal prism geometry rather than an octahedral geometry. In contrast, molecules in which the only ligand is the more ionically bound fluorine are less affected by core distortion and have VSEPR-predicted structures. The unexpected bond angles in CrO(2)F(2) and the preference of CrO(2)F(4)(2)(-) for a cis structure are also discussed in terms of the pattern of core charge concentrations.     

Shape and spin state in four-coordinate transition-metal complexes: the case of the d(6) configuration

Generalized polyhedral interconversion coordinates are defined within the framework of Avnir's continuous shape measures. The application of such interconversion coordinates to the study of the potential energy surfaces that define the stereochemical choice in four-coordinate transition metal complexes with different spin states is presented, and the correlation between potential energy curves and distribution of experimental structures along the tetrahedron to square interconversion path is shown for the case of the d(6) transition-metal complexes.