Octahedron vertices

At the outset of this chapter, we noted that the beauty of symmetry and pattern is ultimately the beauty of simplicity. The elegance of the chemistry of these supra-molecular capsules, too, lies in the profound chemical consequences of simple changes wrought by the defined microenvironments within these assemblies. The earliest examples of altered chemical activity within supramolecular coordination compounds come from Fujita and coworkers, in which they employed their palla-dium-vertexed octahedra (Fig. 16) in the Diels-Alder cycloaddition of isoprene with naphthoquinone [26], accelerating this bimolecular addition 113-fold. 

Structures of heteropolytungstate and isopolytungstate compounds have been determined by x-ray diffraction. The anion stmctures are represented by polyhedra that share corners and edges with one another. Each W is at the center of an octahedron, and an O atom is located in each vertex of the octahedron. The central atom is similarly located at the center of an XO tetrahedron or XO octahedron. Each such polyhedron containing the central atom is generally surrounded by octahedra, which share corners, edges, or both with it and with one another. Thus, the correct total number of... 

In the families of heptanuclear clusters, two geometries are found the capped octahedron that is typical for 98-valence electrons, and the vertex-sharing open tetrahedral (butterfly) stmctures typical for 106-valence electrons. An example of the former is Osy(CO)22 (51) an example of the latter is [H2AuOsg(CO)2Q] (52). In the AuOs cluster anion, the gold atom is at the vertex-sharing position. 

Suppose you have six balls with three different colors, three red, two blue, one yellow. Balls of the same color cannot be distinguished. In how many ways can you assign the six balls to the six vertices of an octahedron which moves freely in spacel If the octahedron is fixed in space in such a way that the vertices are designated as upper, lower, front, back, left, and right vertex, then the number is determined by basic permutation principles as... 

To answer the question one has to examine carefully the permutations which correspond to the 24 rotations of the octahedron. We partition these permutations into cycles and assign to each cycle of a certain order k the symbol f. assign to a cycle of order 1 (vertex which is invariant under rotation), f to a cycle of order two (transposition), /g to a cycle of order three, etc. A permutation which is decomposed into the product of cycles with no common elements is represented by the product of the symbols /. associated with the corresponding cycles. Thus the rotations of the octahedron are described by the following products ... 

A further decrease in the X Me ratio, to 4, leads to linkage of the octahedral units by sharing more than one ligand so as to achieve coordination saturation. Sharing of two vertexes (two comers of the each octahedron) leads to the formation of compounds with layered-type structures. 

The structure of LiTa02F2, as reported by Vlasse et al. [218], is similar to a ReC>3 type structure and consists of triple layers of octahedrons linked together through their vertexes. The layers are perpendicular to the c axis, and each layer is shifted, relative to the layer below, by half a cell in the direction (110). Lithium atoms are situated in the centers of the tetragonal pyramids (coordination number = 5). The other lithium atoms are statistically distributed along with tantalum atoms (coordination number = 6) at a ratio of 1 3. The sequence of the metal atoms in alternating layers is (Ta-Li) - Ta - (Ta-Li). Positions of oxygen and fluorine atoms were not determined. The main interatomic distances are (in A) Ta-(0, F) - 1.845-2.114 Li-(0, F) - 2.087-2.048 (O, F)-(0,F) - 2.717-2.844. 

The crystal structure of KTa02F2 is made up of Ta(0, F)6 octahedrons that are linked via six hexagonal rings. The rings are connected via their vertexes forming infinite empty channels along the c axis [139]. 

K3Ta40gF7 [225] too has a layered structure and contains clusters of the Ta3X,5 type. Each cluster is composed of three TaX6 type octahedrons that are linked to one another via their vertexes. 

The compounds characterized by X Me = 3.5 have a common formula of M2Me205F2 and crystallize either in a pyrochlore [192] or a veberite [229] type structure. According to X-ray powder diffraction patterns, the structure of Na2Nb205F2 can be regarded as a super-structure of pyrochlore, which is made up of octahedrons connected in layers and arranged in the (111) direction. The layers are linked via octahedrons so that each octahedron in one layer shares three vertexes with an octahedron in the adjacent layer. 

The crystal structure of tantalum and niobium dioxyfluorides, TaC F and Nb02F, consists of oxyfluoride octahedrons linked via their vertexes to form a three-dimensional lattice with a Re03 type structure, as demonstrated by Andersson and Astrom [233] and by Frevel and Rinn [234]. Fig. 39 shows the structure of NbC F. 

The lowest coordination number of tantalum or niobium permitted by crystal chemistry formalism is 6, which corresponds to an octahedral configuration. X Me ratios that equal 3, 2 or 1 can, therefore, be obtained by corresponding substitutions in the cationic sub-lattice. A condition for such substitution is no doubt steric similarity between the second cation and the tantalum or niobium ion so as to enable its replacement in the octahedral polyhedron. In such cases, the structure of the compound consists of oxyfluoride octahedrons that are linked by their vertexes, sides or faces, according to the compound type, MeX3, MeX2 or MeX respectively. Table 37 lists compounds that have a coordination-type structure [259-261]. 

CoNbOF5 [129] can also be considered an MeX3 type compound due to the steric similarity of cobalt and niobium ions. This compound crystallizes in tetragonal syngony with cell parameters a = 7.81 and c = 9.02 A (Z = 4 p = 3.19 g/cm3), and can be considered to have a distorted cubic Re03 structure. Both cobalt and niobium occur in the center of oxyfluoride octahedrons that are linked via their vertexes. 

According to the above classification, the structures of LiNb(Ta)F6 and Li2Nb(Ta)OF5 should be composed of lithium cations and isolated octahedral complex ions, Nb(Ta)F6 or Nb(Ta)OF52, respectively. It is known, however, that the structure of these compounds consists only of octahedrons linked via their vertexes in the first case, and via their sides in the second case. The same behavior is observed in compounds containing bi- and trivalent metals. 

If there are not enough electrons for all of the polyhedron edges, 3c2e bonds on the triangular polyhedron faces can be the next best solution to compensate for the lack of electrons. This solution is only possible for deltahedra that have no more than four edges (and faces) meeting at any vertex. These include especially the tetrahedron, trigonal bipyramid and octahedron. 

KT1 does not have the NaTl structure because the K+ ions are too large to fit into the interstices of the diamond-like Tl- framework. It is a cluster compound K6T16 with distorted octahedral Tig- ions. A Tig- ion could be formulated as an electron precise octahedral cluster, with 24 skeleton electrons and four 2c2e bonds per octahedron vertex. The thallium atoms then would have no lone electron pairs, the outside of the octahedron would have nearly no valence electron density, and there would be no reason for the distortion of the octahedron. Taken as a closo cluster with one lone electron pair per T1 atom, it should have two more electrons. If we assume bonding as in the B6Hg- ion (Fig. 13.11), but occupy the t2g orbitals with only four instead of six electrons, we can understand the observed compression of the octahedra as a Jahn-Teller distortion. Clusters of this kind, that have less electrons than expected according to the Wade rules, are known with gallium, indium and thallium. They are called hypoelectronic clusters their skeleton electron numbers often are 2n or 2n — 4. 

MX4 layer of vertex-sharing octahedra, and the packing of such layers in the K2NiF4 type. The packing in SnF4 is obtained by leaving out the K+ ions and shifting the layers towards each other in such a way that every octahedron apex of one layer comes to be between four apexes of the next layer... 

Some forms of rings and chains of vertex-sharing tetrahedra in silicates. How the chain conformations adapt to the size of the cation octahedra is shown for two chains (the octahedron chain is a section of a layer)... 

Take pairs of face-sharing coordination octahedra and join them by common vertices to form a chain, with every octahedron taking part in one common vertex not belonging to the shared face. What is the composition of the resulting chain ... 

Figure 4.6 Relationships of idealized sd -1 -hybridized ML molecular shapes to simple polyhedra. Each panel shows the hybrid-orbital axes in dumbbell dz2 -like form embedded within the polyhedron, together with the associated allowed (no-hms-vertex) dispositions of ligands on the polyhedral vertices (with the unmarked metal atom occupying the polyhedral centroid in each case) (a) sd1 square, (b) sd2 octahedron, (c) sd3 cube, and (d) sd5 icosahedron.

An unusual type of deltahedral cluster found in organoindium chemistry but not in deltahedral borane chemistry is the eight-vertex In8[Si(CMe3)3]6 (see Chapter 2.3.4.2) its structure is based on a bicapped octahedron with idealized D34 sym-... 

Figure 4.34. The cluster [Mo6C1i4]2 is shown. Numerous compounds are known which contain this group as a negative ionic cluster. Six Cl atoms were added to the six vertexes of the white octahedron shown in Fig. 4.32.

Fig. 11.3 Surface complex of octahedrally coordinated Cd on a) goethite and b) lepidocrocite. The Cd surface complex can be bonded to either the edge (E) or vertex of one octahedron. For each type of complex (e. g. C(ooi)), the super-...

The approach adopted is to view the molecule in three dimensions, imagining each atom or group to be placed at a vertex of hn appropriate polyhedron. In organic chemistry this is usually the tetrahedron with carbon at the centre. Table 3.3 (p. 18) shows the polyhedra normally encountered in organic and inorganic chemistry. It also includes for each polyhedron the polyhedral symbols to denote shape and coordination number. It is to be noted that these polyhedra are often presented in a highly formalised fashion. An octahedron is often represented with the apices rather than the octahedral faces depicted, thus ... 

The existence of new periodicities setting up along directions different from [001] implies a change of the copper coordination in one row, parallel to b. Such a change can be ensured, if we except three-fold coordination, either by a copper vacancy which can be occasionally encountered, but not in a systematic and periodic way, or by the interconnection of the rows. This interconnection can be ensured by an octahedron, a pyramid or a tetrahedron, all in agreement with the usual coordination of Cu(n) (or Cu(HI)). A model is proposed for the supercell 2a x a%/To in Figure 21a CuOB pyramids are lined up along the a-axis, with alternated positions of the vertex... 

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