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.</p