From mathematical models to quantum chemistry in cluster science : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Chemistry at Massey University, Albany, New Zealand
The structures and stabilities of hollow gold clusters are investigated by means of density functional theory (DFT) as topological duals of carbon fullerenes. Fullerenes can be constructed by taking a graphene sheet and wrapping it around a sphere, which requires the introduction of exactly 12 pentagons. In the dual case, a (111) face-centred cubic (fcc) gold sheet can be deformed in the same way, introducing 12 vertices of degree five, to create hollow gold nano-cages. This one-to-one relationship follows trivially from Euler’s polyhedral formula and there are as many golden dual fullerene isomers as there are carbon fullerenes. Photoelectron spectra of the clusters are simulated and compared to experimental results to investigate the possibility of detecting other dual fullerene isomers. The stability of the hollow gold cages is compared to compact structures and a clear energy convergence towards the (111) fcc sheet of gold is observed.
The relationship between the Lennard-Jones (LJ) and sticky-hard-sphere (SHS) potential is investigated by means of geometry optimisations starting from the SHS clusters. It is shown that the number of non-isomorphic structures resulting from this procedure depends strongly on the exponents of the LJ potential. Not all LJ minima, that have been discovered in previous work, can be retrieved this way and the mapping from the SHS to the LJ structures is therefore non-injective and non-surjective. The number of missing structures is small and they correspond to energetically unfavourable minima on the energy landscape. The optimisations are also carried out for an extended Lennard-Jones potential derived from coupled-cluster calculations for the xenon dimer, and, although the shape of the potential is not too different from a regular (6,12)-LJ potential, the number of minima increases substantially.
Gregory-Newton clusters, which are clusters where 12 spheres surround and touch a central sphere, are obtained from the complete set of SHS clusters. All 737 structures result in an icosahedron, when optimised with a (6,12)-LJ potential. Furthermore, the contact graphs, consisting only of atoms from the outer shell of the clusters, are all edge-induced sub-graphs of the icosahedral graph. For higher LJ exponents the symmetry of the potential energy surface breaks away from the icosahedral motif towards the SHS landscape, which does not support a perfect icosahedron for energetic reasons. This symmetry breaking is mainly governed by the shape of the potential in the repulsive region, with the long-range attractive region having little influence.