Cluster orbitals

As mentioned earlier, one of the features of these molecules is that usually there are fewer electron pairs than close contacts (bonds) in the molecule. We derive a very general model for these systems by considering three types of orbital at each deltahedral vertex, assembled by considering the one s and three p orbitals at each center. Initially, then we focus on the main group examples 22.1-22.3 and 22.11-22.14. There is an orbital, 22.15, pointing away from each vertex of the 

Derivation of orbital diagrams, built from the radial orbitals of the fragments, for the tetrahedron and the trigonal bipyramid, by capping a trigonal planar unit with one or two new fragments, respectively. Notice that the tetrahedron Is a nido trigonal bipyramid. 

---

In particular the last observation is easily understood by the molecular orbital picture given above. The 7z>orbitals of the monomers are required for a stabilization of the otherwise anti-bonding cluster orbitals of t2 symmetry which must accept six electrons. If this mixing is prevented because these orbitals adopt the electron density of the ligands, e.g., 7z>electrons of the side-on coordinated cyclopentadienyl groups, their contribution to the cluster stability is minimized or in particular cases the formation of clusters does not occur at all. Thus, the substituents attached terminally to the clusters strongly influence their stability by the different donor or acceptor capabilities. A further effect may result from the different steric demand of the substituents which will be discussed below. 

The Nb3 binary halides with seven electrons available for metal-metal bonding within the triangle are electron poor clusters. A simplified MO scheme suggests that the HOMO and LUMO possess metal d character.645 One unpaired electron is expected to be present. The existence of a low-lying metal cluster orbital is also indicated by the electronic spectra. 

FIGURE 10. (a) (He I) PE spectrum of l,2-dimethyl-l,2-disila-cZoso-dodecaborane(12) ((H3CSi)2 BioHio) with Koopmans assignment by MNDO eigenvalues as well as cluster orbital diagrams for the three radical cation states of lowest energy (b) comparison of MNDO charge distributions for both 1,2-carborane as well as 1,2-bis(silamethyl)-c/oso-dodecaborane(l 2)... 

Let us look at bare clusters from a different point of view. What would a naked [Be]2- octahedral cluster be tike It has 20 eve, 6 short of the requirement. This number of electrons is sufficient to serve as the required seven sep and provide three external lone pairs however, three of the out-pointing external cluster orbitals would be empty and a structural rearrangement would be required to create a significant gap between MO 10 and the tiu set 11-13 (left side of Figure 2.21). Hence, in the same way that BH3 is only found as base adducts, so too octahedral [B6]2- would be expected to be found coordinated to bases. 

How strong is the tendency to fill an empty external cluster orbital The linked cluster pair shown in Figure 2.22 is instructive. Oxidation of [B10H10]2- results in the loss of two electrons and a proton (effectively H- is lost). The shape remains the same so the 11 sep associated with cluster bonding are unaffected. Hence, an external outward-pointing empty orbital is generated. Dimerization to form [B2oHi8]2-arises by formation of two three-center-two-electron bonds between the clusters. 

We now categorize the basis atomic orbitals in any, roughly spherical, cluster according to the number of nodal planes they possess that contain the radius vector. Hence, s atomic orbitals are classified as cr-type cluster orbitals, and so are p and d 2 orbitals, where the local axes at each vertex are chosen with the X and y directions tangential to the surface of the sphere so that z points outwards along the radius vector. Radially directed hybrids, such as sp are also a-type under this classification. Examples will be illustrated below. 7r-type cluster orbitals have one intrinsic nodal plane containing the radius vector, for example, p , Py, and dy in the same axis system. A y and Aj 2 y2 functions contain two such nodal planes, and are known as 5 orbitals. ... 

A set of a-type atomic orbitals transform amongst themselves under point group operations like scalar quantities. Hence, to consttuct descent-in-symmetry-type LCAO cluster orbitals, we simply follow the same reasoning as for the linear and cychc polyenes, above, and define (unnormalized) cr-type cluster orbitals as... 

Figure 1 The (uimormalized) (T-cluster orbitals of an Octahedral cluster, such as Lie for which the basis functions are 2s orbitals. The Mght and dark shading indicates the phase the atoms are represented as small filled black circles. (This figure was produced using Mathematica 2.0 Wolfram Research fnc., 1990.)...

Hence, odd n cluster orbitals may be obtained from their even partners by rotating the p orbital components through 90° about the radius vector. These cluster orbitals may be written as... 

Transition metal clusters also have Axy and atomic orbitals, which are classified as 5-type in TSH theory. To represent the transformation properties of these orbitals, we use second derivatives of the spherical harmonics, that is, tensor spherical harmonics - hence the name of the theory. As for the vector surface harmonics, there are again both odd and even 5 cluster orbitals, denoted by L and L, respectively. Usually, both sets are completely filled in transition metal clusters, and we will not consider their properties in any detail in this review. However, the cases of partial occupation are important and have been described in previous articles. ... 

For a cluster with n vertices, the cluster orbital transformation involves n functions. For deltahedral geometries, it is generally not necessary to use the cluster orbitals from the set with L -F 1 until aU the 2L + 1 members of the set based upon the spherical harmonics 4>) have been used. However,... 

In radially bonded clusters, the bonding cluster orbital energy levels lie lower when the number of nearest neighbors per atom is maximized, and this usually corresponds to close-packed high-symmetry structures. However, in clusters that do not correspond to completely filled, P and so on, shells, it may be more favorable for the cluster to distort and open up a significant HOMO-LUMO gap. The... 

---