Deltahedra

In a deltahedral ML compound, the linear combinations of the n ligand orbitals are built up in such a way that the n lowest L quantum states are always used before higher L 

However, the crystallographically independent Li- - -Li distances of the individual tetramers are similar within the estimated standard deviations (esd). They decrease from 256 pm in [MeLi]4 (1) to 253 pm in [EtLi]4 (2) and 241 pm in [t-BuLi]4 (3). Interestingly, the Li—C bond lengths are almost invariant at 226 2 pm and close to the mean Li—C bond distance of 230 pm from the CCDC. In addition, [EtLi]4 and [t-BuLi]4 display relatively short Li- - -C distances. In the latter they are only 10 pm longer than the Li—Ca bonds (Table 1). 

FIGURE 6. Solid-state structures of the basic [RLi]4 tetramers [MeLiL (1), [EtLiL (2) and [t-BuLi]4 (3) 

TABLE 1. Donor-base-free tetrameric bthium organics 

Due to the missing binding carbanions, the related Li- - -Li distances in the unoccupied Lis triangles are considerably longer (294-318 pm Table 2) than those in the tetrahedra 

FIGURE 8. Solid-state structures of donor-base-free [RLiJg hexamers 

---

Fig. 2. Idealized deltahedra and deltahedral fragments for closo, nido and arachno boranes and heteroboranes. From left to right the vertical columns give generic closo, nido, and arachno frameworks bridge hydrogens and BH2 groups are not shown, but when appropriate they are placed around the open...

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. 

Wade stated some further rules for open clusters that are interpreted as deltahedra with missing vertices. They are of special importance for boranes ... 

For convex deltahedra (polyhedra for which all edges are triangular), =—/, and... 

Figure 9.36 Five typical PBU/C(D-polyhedra), which Bernal [180-182] allocated DRP of monospheres (they are sometimes called the Bernal holes) (a) tetrahedron, (b) octahedron, (c) trigonal prism, (d) Archimedian s antiprism, and (e) dodecahrdron. It is northway that not all these D-polyhedrons are simple, but all are deltahedra, thus the relation 9.60a (see Section 9.7.1) is applicable for them.

Fig. 1-2. The most nearly spherical deltahedra found in the boranes B H 2 (6 < n < 12) and isoelectronic carboranes.

Balakrishnarajan and Jemmis [32, 33] have very recently extended the Wade-Mingos rules from isolated borane deltahedra to fused borane ("conjuncto ) delta -hedra. They arrive at the requirement of n I m skeletal electron pairs corresponding to 2n + 2m skeletal electrons for such fused deltahedra having n total vertices and m individual deltahedra. Note that for a single deltahedron (i.e., m = 1) the Jemmis 2n + 2m rule reduces to the Wade-Mingos 2n I 2 rule. 

The graph-theoretical 4N + 2 Hiickel rule analogy with the aromaticity of two-dimensional polygons requires that N = 0 in all the three-dimensional deltahedra. The Jemmis-Schleyer interstitial electron rule [55], originally introduced for nido half-sandwich species, also relates the 4N + 2 Hiickel rule to the delocalized deltahedra directly In this treatment, N is typically 1. 

Apparently Hypoelectronic Deltahedra in Bare Clusters of Indium and Thallium Polyhedra with Flattened Vertices... 

Mingos rules. This hypoelectronicity or electron poverty (fewer than the Wade-Mingos 2n + 2 skeletal electrons) in the bare metal cluster anions Enz (z < n + 2) leads to deltahedra not only different from those in the deltahedral boranes but also different from those in hypoelectronic metal carbonyl clusters of metals such as osmium. 

The shapes of the deltahedra in the apparently hypoelectronic clusters of the heavier group 13 metals bear an interesting relationship to their electron counts. All such hypoelectronic deltahedra can be derived formally from the standard closo-borane structures (Figure 1-2) by flattening one or more vertices towards... 

Organometallic Deltahedral Clusters of the Heavier Croup 13 Metals and More Complicated Structures Derived from Deltahedra... 

Fig. 1-9. Flattening one to three vertices in 9-, 10-, and 11-vertex deltahedra to give the deltahedra found in apparently hypoelectronic group 13 metal clusters. Vertices of degrees 3, 4, and 6 are indicated by , , and respectively. Vertices of degree 5 are unmarked.

Closo, nido, arachno, hypho deltahedral clusters based on n main group vertex atoms are characterized by An + 2, An + 4, An + 6 and An + 8 total valence electrons respectively. The corresponding optimum VEC for closo and nido deltahedra... 

Figure 4.26. Optimum values, in accordance with Wade s rules and as summarized by Miller et al. (2002), of the valence electron concentration VEC (total number of electrons divided by the cluster atom number) as a function of the number of cluster atoms for closo and nido deltahedra. On the left the values computed for the main group elements and on the right those relevant to the transition metals.

The structures, relative stabilities, and relative Lowry-Bronsted acidities of carboranes and boranes as well as related anions, Lewis base adducts, and heteroelement analogs are rationalized primarily on the basis of rudimentary coordination numbers. The principal factors, in decreasing order of importance, are (a) the various deltahedra and deltahedral fragments, (b) the placement of bridge and endohydrogens, (c) the placement of carbon and other heteroelements, and d) the resulting coordination number of boron. 

In 1971, a note 164) was published favoring the hypothesis that the carboranes, boranes, their isoelectronic anions, Lewis base adducts, and heteroatom-substituted analogs should be viewed as constructed about the vertices of either the most spherical series of triangular-faceted polyhedra (deltahedra) found to be characteristic of the dicarba-cZoao-carboranes (Fig. 1) or, with one lone exception, fragments of the series of deltahedra produced by the successive removal of the highest coordinated vertices that sequentially define the nido and arachno classes. This position was in conflict with the then prevalent shibboleth that all nido and arachno compounds [except B5H9 (I-N5)] had or would prove to have icosahedral fragment structures. 

The rules in overall decreasing order of importance essentially state that the ideal structures for carboranes will be based on most spherical deltahedra (rule 1) the BE hydrogens will tend to be placed in the lowest possible coordination environments (rule 2) when elements to the right of boron in the periodic table are incorporated into the deltahedron or deltahedral fragment, they will tend to preempt low-coordination sites (e.g., carbon) or, if electron-deficient, high coordination sites (rule 3) and, lastly, boron will eschew seven-coordinate BH or six-coordinate... 

As the initial paper (164) was concerned only with boron and carbon, those cases wherein the skeletal atoms involved would be more content in different coordination sites and where, therefore, different deltahedra (with differently coordinated vertices) would be preferred were not considered. However, Wade has deduced that (CO)3CrC6H8 is a nido compound (four-coordinate carbon and nine-coordinate chromium)... 

When the systems under consideration have additional electrons available for skeletal bonding, fewer bonds are required and the more open, nido and arachno deltahedra fragments describe their structures instead of closed deltahedra. 

---