Polyhedra parent

Series Parent Formula Skeletal Electrons Cluster Geometry (deltahedron = closed triangulated polyhedron)... 

Nest-like, nonclosed polyhedral structure molecular formula B H + 4 ( + 2) skeletal electron-pairs n vertices of the parent (n + l)-atom c/oro-polyhedron occupied. 

Consideration of the description of leapfrogging in terms of edge crossings suggests formulas for the permutational representations of the leapfrog polyhedron Lf in terms of structural representations of the parent, P. From... 

Fig. 5.27. Idealised 10-coordination polyhedra (1) bicapped square antiprism (D ) (ii) bicapped dodecahedron (D2) (iii) C2V polyhedron based on a dodecahedron. The dotted lines A B in (iii) indicate the location of the original trapezoidal plane in the parent dodecahedron. (From Al-Karaghouli and Wood, by courtesy of the...

Cluster Total electron count (TEC) No. of skeletal electron pairs (S) Vertices of parent polyhedron Structural conclusion... 

Let us now ask how we could predict the correct total electron count, as just defined, for a stable cluster of known structure (i.e., closo, nido, or arachno). To do this for metal carbonyl clusters, it is postulated that in addition to the electrons necessary for skeletal bonding each metal atom will also have 12 nonskeletal electrons. The basis for this assumption is that in the pyramidal M(CO)3 unit each M—CO bond will comprise two formally carbon tr electrons that are donated to the metal atom and two formally metal it electrons that backbond, at least partially, to the CO ligand. Thus, in predicting the total electron count for a closo polyhedral cluster of n vertices, the result would be 12n + 2 n + 1). Similarly, for nido and arachno clusters that are derived from an n-vertex polyhedron (their parent polyhedron) by removal of one or two vertices, respectively, there will be 12 and 24 fewer total electrons, respectively. 

The predictions for TEC can therefore be stated in the following equations (where n is the number of vertices in the parent polyhedron for the nido and arachno cases—not the actual number of metal atoms in the cluster itself) ... 

For Rh6(CO)16 the TEC is (6 X 9) + (2 X 16) = 86. The number of skeletal pairs (S) will be obtained by subtracting 6 X 12 from 86 and dividing by 2 the result is 7. Since the number of vertices of the parent polyhedron is S - 1, we conclude that this will be a six-vertex polyhedron (e.g., an octahedron). Since there are six metal atoms, we conclude that a closo structure, probably an octahedron, is appropriate. This is correct. In general 86-electron clusters consisting of six metal atoms are octahedral. 

The next two examples in Table 16-2, Os5C(CO)15 and [Fe4C(CO)12]2 show how nido and arachno structures are predicted. The nido structure already shown in Fig. 16-12(a) for the iron analogue, Fe5C(CO)i5, is derived by removal of one vertex from the parent polyhedron, which is an octahedron. The [Fe4C(CO)i2]2 cluster should be isostructural with the Fe4C(CO)13 cluster, whose structure has been shown in Fig. 1642(d). It is derived from a parent octahedron by removal of two vertices. 

The utility of this approach lies in its ability to predict the framework connectivity of the child compound. Connectivity is defined as the average number of distinct M-X-M linkages around the metal centers or, alternatively, the average number of bonds that must be broken to liberate a discrete polyhedron. For frameworks with only one kind of metal and one kind of anion the connectivity of the parent is given by ... 

As Figure 15-2 shows, each of these fragments has a single electron in a hybrid orbital at the vacant site of the parent polyhedron. These orbitals are sufficiently similar to meet Hoffmann s isolobal definition. Using Hoffmann s symbol to designate... 

Organic Inorganic Organo- metallic Example Vertices Missing from Parent Polyhedron Electrons Short of Filled Shell... 

Note that all the examples shown above are one ligand short of the parent complex. Fe(CO)5 is isolobal with CH3, for example, because both have filled electron shells and both are one vertex short of the parent polyhedron. By contrast, Fe(CO)5 and CH4 are not isolobal. Both have filled electron shells (18 and 8 electrons, respectively), but CH4 has all vertices of the tetrahedron occupied, whereas Fe(CO)5 has an empty vertex in the octahedron. 

Coordination Number of Transition Metal for Parent Polyhedron... 

In addition, it is sometimes useful to relate the total valence electron count in boranes to the structural type. In closo boranes, the total number of valence electron pairs is equal to the sum of the number of vertices in the polyhedron (each vertex has a boron-hydrogen bonding pair) and the number of framework bond pairs. For example, in there are 26 valence electrons, or 13 pairs (= 2n + 1, as mentioned previously). Six of these pairs are involved in bonding to the hydrogens (one per boron), and seven pairs are involved in framework bonding. The polyhedron of the closo structure is the parent polyhedron for the other structural types. Table 15-8 summarizes electron counts and classifications for several examples of boranes. 

Vertices in Parent Polyhedron Classification Boron Atoms in Cluster Valence Electrons Framework Electron Pairs Examples Formally Derived From... 

Vertices Framework Atoms in in Parent Electron Cluster Polyhedron Pairs... 

A more unified perspective can be developed by starting with the polyhedra derived from the regular orbits in the two parent groups, cubic (Oh) and icosahedral (Ih). The various Archimedean and Platonic solids then follow by a process of collapsing vertices of these regular-orbit polyhedra. Each polyhedron appears as the realization of an orbit of Oh/Ih or a subgroup. 

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