Theoretical studies on cluster compounds
This Thesis describes some theoretical studies on ligated and bare clusters. Chapter 1 gives a review of the two theoretical models, Tensor Surface Harmonic Theory (TSH) and Jellium Model, accounting for the electronic structures of ligated and bare clusters. The Polyhedral Skeletal Electron Pair Theory (PSEPT), which correlates the structures and electron counts (total number of valence electrons) of main group and transition metal ligated clusters, is briefly described. A structural jellium model is developed in Chapter 2 which accounts for the electronic structures of clusters using a crystal-field perturbation. The zero-order potential we derive is of central-field form, depends on the geometry of the cluster, and has a well-defined relationship to the full nuclear-electron potential. Qualitative arguments suggest that this potential produces different energy level orderings for clusters with a nucleus with large positive charge at the centre of the cluster, enabling the spherical jellium model to be applied to alkali metal clusters seeded with magnesium and zinc. Analysis of the effects of the non-spherical perturbation on the spherical jellium shell structures leads to the conclusion that for a cluster with a closed shell electronic structure a high symmetry arrangement which is approximately or precisely close packed will be preferred. It also provides a basis for rationalising those structures, which have been predicted using ab initio calculations, of clusters with incomplete shell electronic configurations In Chapter 3, the geometric conclusions derived in the structural jellium model are developed in more detail. Alkali metal clusters with closed shell electronic configurations according to the jellium model adopt geometries of high symmetry and based on the Td , Oh and Ih point groups. For high nuclearity clusters alternative high symmetry structures can occur and those which are either the most close packed or spherical are predicted to be the most stable. When the jellium closed shell "magic numbers" coincides with one of these high symmetry structures then the cluster will be particularly stable. The group theoretical consequences of the Tensor Surface Harmonic Theory are developed in Chapter 4 for[ML2]n, [ML4]n and [ML5]n clusters where either the xz and yz or x2-y2 and xy components to Lπd and Lδd do not contribute equally to the bonding. The closed shell requirements for such clusters are defined and the orbital symmetry constraints pertaining to the interconversion of conformers of these clusters are described. In Chapter 5 Stone's Tensor Surface Harmonic methodology is applied to high nuclearity transition metal carbonyl cluster compounds with 13-44 metal atoms. Two limiting bonding situations are identified and represented in terms of general electron counting rules. If the radial bonding effects predominate the clusters are characterised by 12ns+Δi valence electrons, where Δi is the characteristic electron count of the interstitial moiety. If radial and tangential bonding effects are important then the total number of valence electrons is 12ns+2(ss+si-l), where ss and si are the number of skeletal bonding molecular orbitals associated with surface (ss) and interstitial (si) moieties. Chapter 6 develops a new theoretical framework to account for the bonding in the high nuclearity ligated clusters with columnar topologies. The wave functions of columnar metal clusters can be expressed as an expansion based on the particle on the cylinder problem. This bonding analysis is applied to clusters containing columns of triangles and squares. In Chapter 7 the origin of non-bonding orbitals in molecular compounds is reviewed and analysed using general quantum mechanical considerations. A combination of the pairing theorem and a group theoretical analysis leads to a definition of the number of the non-bonding molecular orbitals in co-ordination, polyene and cluster compounds. The non-bonding molecular orbitals have been generated by defining the nodal characteristics of the relevant orbitals and evaluating the solutions under the appropriate boundary conditions. The stereochemical role of nonbonding molecular orbitals in co-ordination compounds is also discussed.