Tangential orbitals

Fig. 3. 2-4. Radial (bottom) and tangential (top) bonding molecular orbitals for cyclopropane [12e] (left) and hexa-borane(6) dianion (right) illustrating the planar and three dimensional a aromaticity, respectively. Tangential orbitals of [B6H6]2 are triply degenerate.

An important consequence of the nonutilization of tangential orbitals is that platinum clusters often do not obey the normal electron counting rules and appear to be electron deficient (19,21,29,58,75,76). Electron counts are usually intermediate between those found in normal transition metal clusters (58-68) and those observed in gold clusters (58,78), but no satisfactory general electron counting theory has been developed for Pt-containing clusters. In small Pt clusters constructed from PtL2 units, theoretical studies have shown that the total electron count depends on the relative orientation of the... 

Numerous triosmium-platinum clusters have been prepared, many of them derived from Os3Pt(/i-H)2(CO)10(PR3) (1) (40,40a). Although, as discussed in Section III, the participation in cluster bonding of the tangential orbitals of nonconical PtL units may be difficult to predict a priori, the observed tetrahedral skeleton of 1 (40,40a) and nonplanar coordination geometry at the Pt atom suggest that the Pt atom should behave as an 18-electron center. The 58-CVE count therefore implies that the cluster is unsaturated, and this is consistent with EHMO studies (88,98) and the high chemical reactivity of 1. The crystal structures of la (40,40a) and the 58-CVE derivatives 21 (108) and 22 (118) all show unusually short Os(/ -H)Os distances [2.789(1),... 

PROBLEM 2.4.1. Given that the average earth-moon distance is Rem = 3.844 x 108 m and that the moon s revolution around the earth is 27.3 days (from which its tangential orbital velocity is vm = 1.0186 x 103 m s 1), compute the mass of the earth. 

Answer. The eve count for [Al77R2o]2 is 253 so it is an odd electron cluster. Considering the inner two clusters with full radial and tangential bonding (the smaller one with an interstitial A1 atom) yields a count of 50 + 178 = 228. If the outer AIR fragments are attached by radial bonding alone the total count is 228 + 20 x 2 = 268 which is 15 more than observed and perhaps 14 more than a probable even electron species. Hence, one would have to assume seven tangential orbitals empty. 

Other hybridization schemes could be used as long as they result in one radially directed and two tangential orbitals... 

The nitrogen 2 s and 2p valence AO s conveniently oriented for interaction, are shown in Fig. 5. This orientation naturally divides the valence AO s into three orbital sets, namely, the 2s set, the 2 px set and the 2 py, 2 pz set. The latter two sets we refer to as radial and tangential p orbitals, respectively. The radial orbitals point toward the molecular center and the tangential orbitals are perpendicular to the radial orbitals. Let us consider these sets separately and imagine various GO s placed one by one between the nitrogen atoms. For each GO we try to find that combination of nitrogen orbitals which has the same symmetry, until we have as many symmetry orbitals (SO s) as there are nitrogen valence orbitals in the set. 

Fig. 14. Examples of high nuclearity transition metal clusters, with interstitial metal atoms, where the surface tangential orbitals make a partial contribution to the metal-metal bonding...

It is therefore, apparent that, despite the fact that the vertices of a 3-connected polyhedron may all lie on the surface of a single sphere, the arrangement of these vertices represents a poor coverage of the sphere s surface. Thus, these polyhedra do not emulate the sphere sufficiently well for their skeletal MO s to be good approximations to the particle on a sphere solutions. In particular, the low number of close atom-atom contacts (i.e. cluster edges) is insufficient to enable a distinction to be made, in all cases, of the bonding characteristics of the even (L") and odd parity (L31) tangential orbitals. 

Quinn has derived simplified forms of Eqs. (36) and (37) for axial point groups, where the pe and p9 components of the tangential p orbitals are seperable. The tangential orbitals are merely formed by taking the L° MO s with a basis set of pe and p orbitals, rather than radial orbitals. The p8 orbitals have the same local symmetry as Tz, while the p orbitals have the local symmetry of TRz154) ... 

When there is only one tangential orbital as in the M(CO)4 group of [V(CO)4(AuPPh3)6] or Mn(CO)4(AuPPh3)3 the elliptical VAu6 or planar, rhombic MnAu3 structure obviously leads to a more effective overlap. 

The tangential orbitals are, in principle, no more complex to build up. Starting again with the arachno trigonal bipyramid and octahedron we need the tangential... 

FIGURE 22.4. The tangential orbitals of the square and triangle. These break down into two types. Those parallel (li) and those perpcnclieular (i) to the figure axis of the molecule. 

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