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From Mathematical Models to Quantum Chemistry in Cluster Science Copyright is owned by the Author of the thesis. Permission is given for a copy to be downloaded by an individual for the purpose of research and private study only. The thesis may not be reproduced elsewhere without the permission of the Author. From Mathematical Models to Quantum Chemistry in Cluster Science A thesis presented in partial fullment of the requirements for the degree of Doctor of Philosophy in Chemistry at Massey University, Albany New Zealand Lukas Trombach 2019 Abstract 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 ve, 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 struc- tures 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 struc- tures 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 ex- tended Lennard-Jones potential derived from coupled-cluster calculations for the xenon dimer, and, although the shape of the potential is not too dierent from a regular (6,12)-LJ potential, the number of minima increases substan- tially. 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 inuence. i Acknowledgements With the conclusion of this thesis another chapter of my life is coming to an end, and there are many people who supported me in one way or another along my path and I would like to express my sincere gratitude to all of them. First and foremost, I have to thank my supervisor Peter for all the support and assistance he gave me during the four years I spent as a member of his research group. The selection of research projects and the way he directed me towards the interesting questions that needed to be answered were both excellent. I would also like to thank Elke for taking on co-supervision and always helping me with any questions I had. My time in the Schwerdtfeger group has been lled with countless great memories, all due to the people that I shared our premises with. Lukas_0, Lukas_1 and Jayson were always available for a competitive round of table tennis and helped me tremendously with all kinds of scientic questions and problems. I want to thank Lukas_0, Jayson, Paul and Morten specically for being great oce mates and for all the interesting and fruitful discussions we had. Furthermore, thanks for all the conversations about “crypto” and stock markets to Antony and for all the discussions about writing a good PhD thesis and life in general to Odile. A special thanks goes out to Vesna for keeping the group up and running in the rst place. I am very grateful for all the moral and nancial support I received from my parents, my grandparents and my brother. The certainty that there was al- ways someone who would stand by me no matter what happened contributed immensely to the way I was able to approach this degree. Although, there were around 18,000 km and probably ten time zones separating us, Susi was always ready for a phone call when I needed it, and I’m very grateful to have her as a friend. I also wish to thank Emma and Victor for helping me to stay active and for being amazing friends and climbing buddies. And last but not least, I would like to thank Bali for making my last couple of months a little bit more special. Helping me to take my mind of all the work that was lying ahead of me made nishing this thesis just so much easier. iii Contents Abstract i Acknowledgements iii List of Publications vii List of Acronyms ix List of Figures xi List of Tables xvii 1 Introduction 1 1.1 Nanotechnology . .1 1.2 Cluster Science . .2 1.3 The Potential Energy Surface . .4 1.4 Outline . .6 I Theoretical Background9 2 Graph Theory 11 2.1 The Denition of a Graph . 11 2.2 Planar Graphs . 14 2.3 Graph Matching . 17 3 antum Chemistry 19 3.1 The Schrödinger Equation . 19 3.2 The Born-Oppenheimer Approximation . 20 3.3 The Hartree-Fock Approximation . 21 3.4 Density Functional Theory . 24 3.5 Periodic Boundary Conditions . 29 3.6 Basis Sets . 30 3.7 Description of Core Electrons . 32 4 Geometry Optimisation 35 4.1 General Considerations about Minima . 35 4.2 Properties of Optimisation Algorithms . 37 4.3 Quadratic Models . 38 4.4 Implementation for Two-Body Interaction Potentials . 43 4.5 Global Optimisation . 45 v 5 Interaction Potentials 51 5.1 Thermodynamic Considerations . 51 5.2 Lennard-Jones Potential . 54 5.3 Sticky-Hard-Sphere Potential . 57 II Methods 61 6 Program Package Spheres 63 6.1 Structural Optimisation and Analysis . 63 6.2 Graph-Theoretical Analysis . 67 6.3 Additional Functionalities . 68 6.4 Implementations in Detail . 69 III Results 73 7 Golden Dual Fullerenes 75 7.1 Introduction . 75 7.2 Topological Aspects . 76 7.3 Computational Details . 80 7.4 Structure and Stability . 81 7.5 Convergence Towards the Innite Structure . 89 7.6 Simulation of Photoelectron Spectra . 90 7.7 Conclusion . 92 8 From Sticky-Hard-Sphere to Lennard-Jones-Type Clusters 95 8.1 Introduction . 95 8.2 Computational Details . 98 8.3 Exploring the Limits of Lennard-Jones . 99 8.4 (6,12)-Lennard-Jones Clusters from Basin-Hopping . 105 8.5 Conclusion . 108 9 The Gregory-Newton Clusters 111 9.1 Introduction . 111 9.2 Computational Details . 113 9.3 The Gregory-Newton Problem for Soft Potentials . 113 9.4 Rigid Gregory-Newton Clusters and Corresponding Graphs . 115 9.5 Symmetry-Broken Lennard-Jones Gregory-Newton Clusters . 119 9.6 Adding a 14th Sphere . 122 9.7 Conclusion . 125 IV Appendix 127 A List of Gregory-Newton Contact Graphs 129 Bibliography 149 vi List of Publications The results presented in this thesis have been released previously in the fol- lowing publications: [1] L. Trombach, S. Rampino, L.-S. Wang, and P. Schwerdtfeger, “Hollow Gold Cages and Their Topological Relationship to Dual Fullerenes”, Chemistry–A European Journal 22, 8823 (2016). [2] L. Trombach, R. S. Hoy, D. J. Wales, and P. Schwerdtfeger, “From sticky-hard-sphere to Lennard-Jones-type clusters”, Physical Review E 97, 043309 (2018). [3] L. Trombach and P. Schwerdtfeger, “Gregory-Newton problem for kissing sticky spheres”, Physical Review E 98, 033311 (2018). vii List of Acronyms AFM atomic force microscope BFGS Broyden-Fletcher-Goldfarb-Shanno COS center-to-outer sphere DFP Davidon-Fletcher-Powell DFT density functional theory ECP eective core potential EDM Euclidean distance matrix eLJ extended Lennard-Jones fcc face-centered cubic FR Fletcher-Reeves GDF golden dual fullerene GEA gradient expansion approximation GGA generalised gradient approximation GN Gregory-Newton GNC Gregory-Newton cluster GTO Gaussian-type orbital hcp hexagonal-closed packed LCAO linear combination of atomic orbitals LDA local density approximation LJ Lennard-Jones MDS minimum distance sphere OMP Open Multi-Processing ix PAW projector-augmented wave PES potential energy surface RMS root mean square SCF self-consistent eld STM scanning tunneling microscope SHS sticky-hard-sphere STO Slater-type orbital x List of Figures 1.1 (a) STM image of a clean gold (100) surface showing atomic res- olution. The ridges are a result of surface reconstructions. The image is part of the public domain. (b) STM image of Xenon atoms arranged on a nickel (110) surface in a pattern resembling the IBM logo. Reprinted by permission from Springer Nature Cus- tomer Service Centre GmbH: Springer Nature, “Positioning Single Atoms with a Scanning Tunnelling Microscope”,[3] ©1990. .2 1.2 Buckminsterfullerene C60........................3 1.3 (a) Example of a two-dimensional potential energy landscape and (b) the same hypersurface represented with contour lines. Red lines mark the boundaries of the basins of attraction around the minima (blue dots) and transition states (green dots).
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