The resurgence of interest in properties of molecules of icosahedral symmetry follows the
discovery of the C60 molecule. Due to the high symmetry of the icosahedron, almost all the
electronic and vibrational levels are highly degenerate, so that in dealing with properties of these
systems, the Jahn-Teller interaction must almost always be allowed for. This thesis primarily
explores the properties of the icosahedral G ⊗ (g ⊕ h] interaction and related subsystems in the
strong coupling régime. Mappings of the adiabatic potential energy surfaces facilitate an analysis
of the geometrical phase or Berry phase acquired by the quantal system on transportation around
adiabatic circuits in parameter space. The Berry phase information in conjunction with an
analysis of tunnelling, determines the properties of the ground state.
The use of Ham factors achieves a characterization of the various coupling régimes. However,
characterization of G and H Jahn-Teller systems, requires an extension of the standard definition
of these Ham factors. In such cases the extended matrix elements between and within vibronic
tunnelling sublevels cannot be ignored. A calculation of all the standard and extended matrix
elements is presented. A further introduction of a matrix of Ham factors facilitates the description
of H multiplicity within an H manifold.
Finally, two problems are investigated numerically; aimed at making some allusion towards
possible experimental manifestations of G ⊗(g(⊕)h). The first investigation considers the variations
in the eigensystem and Ham factors of G ⊗ (g ⊕ h), as a function of the coupling to the two
modes. The second investigation considers the structure in the optical absorption line shapes
for the transitions from orbital singlet to quartet, G, states in an icosahedral environment. The
quartet states are subject to both spin-orbit and linear Jahn-Teller interactions.</p