Deltahedra, orbitals

Aromaticity is the simplest way to explain the stability of unsaturated cyclic hydrocarbons with (4n + 2) electrons delocalized in the rr-orbitals perpendicular to the ring plane.1 Even though the introduction of the aromaticity concept in chemistry is quite old, its definition is still controversial. It is not surprising to find many attempts to define this term depending on different approaches to describe the electronic structure. In view of these problems of subjectivity, it is remarkable that aromaticity is useful to rationalize and understand the structure and reactivity of many organic molecules. As a result, the concept of aromaticity is truly a cornerstone in organic chemistry. In 1971, Wade proposed a similar concept to describe delocalized cr-bonding in closed-shell boron deltahedra.2-4 However, stability based on aromaticity had not been confirmed for any metallic moiety until Li et al. published their seminal paper entitled Observation of all-metal aromatic molecules, 5... 

The bipolar (e.g. bipyramids and bicapped antiprisms) and non-polar (e.g. D2d-dodecahedron and tricapped trigonal prism) deltahedral custers are best analysed in terms of the interactions between the two sets (polar and non-polar or equatorial and non equatorial) of symmetry-equivalent atoms which make up the cluster1573. In this way it has been shown that, although by symmetry there are no degenerate L /L" pairs and therefore no symmetry-induced departures from the (n + 1) rule, the frontier orbitals of bipolar deltahedral clusters consist of two parity matched U1 and L e pairs, giving rise to possible SEP counts of (n - 1), (n + 1) or (n + 3). In the case of the non-polar deltahedra the frontier orbitals are non-degenerate 1/ and L" orbitals, giving rise to possible SEP counts of n, (n + 1) or (n + 2). 

The availability of d orbitals on transition metal vertices leads to the possibility ofelectron-poor or hypoelectronic transition metal clusters with n vertices having less than 2 -I- 2 apparent, skeletal electrons. Such electron-poor clusters form deltahedra containing tetrahedral chambers, i.e., deltahedra with one or more vertices of degree 3 (Figure 9). The simplest examples of such deltahedra are the capped tetrahedra, of which the trigonal bipyramid (i.e., the monocapped tetrahedron) with five vertices is the smallest. The capped tetrahedra consist of a series of fused tetrahedral chambers with faces in common. An example of a cluster based on a bicapped tetrahedron is... 

The discovery of boron deltahedra in elemental boron and metal borides and later in polyhedral boranes generated an interest in computational studies on these structures as soon as suitable computational methods became available. The earliest computational work on boron deltahedra was the 1954 study by Longuet-Higgins and Roberts on the Be octahedra found in metal boride studies using the secular determinants obtained from linear combinations of atomic orbitals (LCAO). This work was followed shortly by a study of boron icosahedra which predicted the existence of a stable anionic icosahedral... 

To apply this method it is necessary to know the irreducible representations corresponding to for the core and external orbitals and for the surface orbitals for the deltahedra of interest (Table 3). The pure surface orbitals are starred in Table 3 these orbitals correspond to irreducible representations found in Tjt but not in Ta. 

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