Boron icosahedra

The [BI2H 2]2- ion is a regular icosahedron of atoms, each of the twenty faces being an equilateral triangle (Fig. 16.45a). All of the hydrogen atoms are external to the boron icosahedron and are attached by terminal B—H bonds. The icosahedron itself involves a resonance hybrid of several canonical forms of the type shown in Ffg. 6.45b and c. Both two-electron, two-center B—B and two-electron, three-center B—B—B bonding are involved. 

FIGURE 1 Boron icosahedron. The arrows indicate one of the six five-fold axes. 

Regarding the electronic structure, various molecular orbital calculations on a single boron icosahedron have been performed (Longuet and Roberts, 1955 Bambakidis and Wagner, 1981 Bullett, 1982 Shirai and Nakamatsu, 1994). 

There are 12 outward-pointing radial hybrids and 13 bonding molecular orbitals for intra-cluster bonding. Therefore, the boron icosahedron within a structure is two-electron-deficient 36 - (12 +13 x 2) = -2. This can be well understood by considering (B12H12)-2, for example. It should be noted that the formation of the boron cluster is extremely "economic" in terms of electrons, since conventional pair bonding would require 60 electrons, rather than 26. 

A simple treatment of the stability requirements of a boron icosahedron in a solid was presented early. An icosahedral boron cluster was shown to require two external electrons. The icosahedron is linked to its neighbors via normal covalent bonds. This description leads to a simple understanding of the stability of the BeO and BeP stractures. 

People who make discoveries do not necessarily use the aesthetic side until afterward. Then they recognize it and say, yeah, that s beautiful. Of course, we were not the first to find the boron icosahedron either. That was found by Sevastyanov and Zhdanov in C3B12, boron carbide, in 1941. That was a missed opportunity because nobody thought then that you could make fragments of this polyhedron and arrive at a structure of decaborane. That was not an idea that was around. It s an idea we missed. 

On comparing basis functions for irreducible representations, one can correlate many of the levels of related structures. Examining the plotted waves for the boron icosahedron and for the decaborane-14 led us to the results presented in Table XV. Note that three of the levels remain very close together as A, A2. 

Figure 14.6 The boron icosahedron and one of the boranes. A, The icosa-hedral structural unit of elemental boron. B, The structure of B5H9, one of many boranes.

The most stable of the boron modifications is j -rhombohedral boron (Figure 4.8), which has a unit cell made out of boron icosahedra with each atom being connected to the top of a pentagonal pyramid. This part of the structure is similar to a buckyball (Buckminster-Fullerene or C q), but this boron ball is filled with a boron icosahedron. The unit cell is finished with a single boron atom and two B q units that are draped as three fivefold rings around a central boron atom. The total number of boron atoms per unit cell in this modification is 105. 

The 4 point group is that to which a regular icosahedron, illustrated in Figure 4.13(a), belongs. It contains 20 equilateral triangles arranged in a three-dimensional sttucture. This is the conformation of the anion, in which there is a boron atom, with a hydrogen atom... 

The a-rhombohedral form of boron has the simplest crystal stmcture with slightly deformed cubic close packing. At 1200°C a-rhombohedral boron degrades, and at 1500°C converts to P-rhombohedral boron, which is the most thermodynamically stable form. The unit cell has 104 boron atoms, a central B 2 icosahedron, and 12 pentagonal pyramids of boron atom directed outward. Twenty additional boron atoms complete a complex coordination (2). 

The a-tetragonal form of boron has a unit cell B qC2 or B qN2 it always has a carbon or nitrogen in the crystal. The cell is centered a single-boron atom is coordinated to four icosahedrons (4Bj2 + 2B). The -tetragonal form has a unit cell of 192 boron atoms but is not, as of this writing, totally defined. 

Each boron atom is bonded to five others ia the icosahedron as well as either to a carbon atom or to a boron atom ia an adjacent icosahedron. The stmcture is similar to that of rhombohedral boron (see Boron, elemental). The theoretical density for B22C2 is 2.52 g/mL. The rigid framework of... 

Boron is unique among the elements in the structural complexity of its allotropic modifications this reflects the variety of ways in which boron seeks to solve the problem of having fewer electrons than atomic orbitals available for bonding. Elements in this situation usually adopt metallic bonding, but the small size and high ionization energies of B (p. 222) result in covalent rather than metallic bonding. The structural unit which dominates the various allotropes of B is the B 2 icosahedron (Fig. 6.1), and this also occurs in several metal boride structures and in certain boron hydride derivatives. Because of the fivefold rotation symmetry at the individual B atoms, the B)2 icosahedra pack rather inefficiently and there... 

Figure 6.1 The icosahedron and some of its symmetry elements, (a) An icosahedron has 12 vertices and 20 triangular faces defined by 30 edges, (b) The preferred pentagonal pyramidal coordination polyhedron for 6-coordinate boron in icosahedral structures as it is not possible to generate an infinite three-dimensional lattice on the basis of fivefold symmetry, various distortions, translations and voids occur in the actual crystal structures, (c) The distortion angle 0, which varies from 0° to 25°, for various boron atoms in crystalline boron and metal borides.

Figure 6.2 Basal plane of a-rhombohedral boron showing close-packed arrangement of B 2 icosahedra. The B-B distances within each icosahedron vary regularly between 173-179 pm. Dotted lines show the 3-centre bonds between the 6 equatorial boron atoms in each icosahedron to 6 other icosahedra in the same sheet at 202.5 pm. The sheet-s are slacked so that each icosahedron is bonded by six 2-centre B-B bonds at 171 pm (directed rhombohedral ly, 3 above and 3 below the icosahedron). B12 units in the layer above are centred over 1 and those in the layer below are centred under 2.

The thermodynamically most stable polymorph of boron is the /3-rhombohedral modification which has a much more complex structure with 105 B atoms in the unit cell (no 1014.5 pm, a 65.28°). The basic unit can be thought of as a central Bn icosahedron surrounded by an icosahedron of icosahedra this can be visualized as 12 of the B7 units in Fig. 6.1b arranged so that the apex atoms form the central Bn surrounded by 12 radially disposed pentagonal dishes to give the Bg4 unit shown in Fig. 6.3a. The 12 half-icosahedra are then completed by means of 2 complicated Bjo subunits per unit cell,... 

The structures of boron-rich borides (e.g. MB4, MBfi, MBio, MB12, MBe6) are even more effectively dominated by inter-B bonding, and the structures comprise three-dimensional networks of B atoms and clusters in which the metal atoms occupy specific voids or otherwise vacant sites. The structures are often exceedingly complicated (for the reasons given in Section 6.2.2) for example, the cubic unit cell of YB e has ao 2344 pm and contains 1584 B and 24 Y atoms the basic structural unit is the 13-icosahedron unit of 156 B atoms found in -rhombohedral B (p. 142) there are 8 such units (1248 B) in the unit cell and the remaining 336 B atoms are statistically distributed in channels formed by the packing of the 13-icosahedron units. 

The stabilizing influence of small amounts of B (M/B > 0.25) in the voids of the metal host lattice varies with the mode of filling (partial or complete) of the interstitial, mostly O, sites and whether the compounds develop from the binary-intermetallic host lattice. The structures of B-rich compounds (M/B < 4) are mainly determined by the formation of regular, covalent B polyhedra (O, icosahedron) and the connections between them (B frame structures). Typical metal (M) borides therefore are found within a characteristic ratio of metal to boron 0.125 < M/B < 4. 

Assuming perfect stoichiometric structures, the stabilization of the boron frameworks of MB2, MB4, MBg, MBj2 and elemental B requires the addition of two electrons from each metal atom. Whatever the Bj2 unit, icosahedron or cubooctahe-dron, 26 electrons are required for internal bonding and 12 for external bonding. Since the 12 B possesses only 36 electrons, the metal must supply two electrons to each Bi2 group. The results for YB,2 are consistent with this model measurements indicate that one electron per Y is delocalized in the conduction band. ... 

In a-B12 the icosahedra are arranged as in a cubic closest-packing of spheres (Fig. 11.16). In one layer of icosahedra every icosahedron is surrounded by six other icosahedra that are linked by three-center two-electron bonds. Every boron atom involved contributes an average of electrons to these bonds, which amounts to -6 = 4 electrons per icosahedron. Every icosahedron is surrounded additionally by six icosahedra of the two adjacent layers, to which it is bonded by normal B-B bonds this requires 6 electrons per icosahedron. In total, this adds up exactly to the above-mentioned 10 electrons for the inter-icosahedron bonds. 

The dodecahydrododecaborate anion, B H 2-, is termed unique with considerable justification. This ion and its perhalo derivatives, e.g., Bi2C1i22-, are the most symmetrical molecular aggregates known. The boron atoms occupy the vertices of a regular icosahedron and each is bonded terminally to a hydrogen atom all boron atoms are environmentally equivalent.7,8 This anion is the only known example of the 7 symmetry group.8 General spectral, physical, and chemical properties of Bx2Hi22-are detailed in a paper by Muetterties et al.9... 

One well-known derivative of the B12H122 ion is the carborane, B10C2H12. Note that this species is neutral because each carbon atom has one more electron than does a boron atom. Because the carbon atoms can be located in any two positions in an icosahedron, there are three isomers of B10C2H12 that differ in the location of the two carbon atoms in the structure. These isomers have the structures shown in Figure 13.6. 

Even though qualitative bonding descriptions of metal atom clusters up to six or seven atoms can be derived and in some cases correlated with structural detail, it is clear that most structures observed for higher clusters cannot be treated thus. Nor do the structures observed correlate with those observed for borane derivatives with the same number of vertices. Much of borane chemistry is dominated by the tendency to form structures derived from the icosahedron found in elemental boron. However, elemental transition metals possess either a close-packed or body-centered cubic arrangement. In this connection, one can find the vast majority of metal polyhedra in carbonyl cluster compounds within close-packed geometries, particularly hexagonal close-packing. 

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