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The possibility to study quantum interference phenomena at ambient conditions is an appealing feature of molecular electronics. By connecting two porphyrins in a cofacial cyclophane, we create an attractive platform for mechanically controlling electric transport through the intramolecular extent of π-orbital overlap of the porphyrins facing each other and through the angle of xanthene bridges with regard to the porphyrin planes. We analyze theoretically the evolution of molecular configurations in the pulling process and the corresponding changes in electric conduction by combining density functional theory (DFT) with Landauer scattering theory of phase-coherent elastic transport. Predicted conductances during the stretching process show order of magnitude variations caused by two robust destructive quantum interference features that span through the whole electronic gap between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO). Mechanically-controlled break junction (MCBJ) experiments at room temperature verify the mechanosensitive response of the molecular junctions. During the continuous stretching of the molecule, they show conductance variations of up to 1.5 orders of magnitude over single breaking events. Uncommon triple- and quadruple-frequency responses are observed in periodic electrode modulation experiments with amplitudes of up to 10 Å. This further confirms the theoretically predicted double transmission dips caused by the spatial and energetic rearrangement of molecular orbitals, with contributions from both through-space and through-bond transport.

This thesis envisions the analysis of mechanosensitive porphyrin cyclophanes embedded into single-molecule junctions in close collaboration with physicists from the Dulić and van der Zant groups. Meticulously designed molecular structures are synthesized to potentially allow the elucidation of fundamental structure-property relations using the mechanically controlled break junction (MCBJ) technique. In the first chapter of this thesis, the use of seven closely related porphyrin derivatives, combined with an unsupervised cluster algorithm to identify certain classes of conductance from the data collected by the MCBJ technique, which allowed the elucidation of three competitive transport paths as published as Edge Article in Chemical Science (2019) is presented. In the second chapter of this thesis, our interests are geared towards the design and synthesis of porphyrin cyclophanes, which would allow observing mechanosensitivity when implemented into a two-terminal MCBJ set-up. The first generation of cyclophanes was designed and the unanticipated product was electrically characterized as published as Article in The Journal of Organic Chemistry (2020). In a second design, we achieved the successful synthesis of the target structure, which was presented as Article in The Journal of Organic Chemistry (2020) combined with preliminary transport studies. In the last chapter, the preliminary results of detailed experimental and theoretical distance manipulation studies of this cyclophane are shown. The concept of single-molecule transport studies, a guide for the different platforms and techniques as well as transport mechanisms for direct electrical measurements and an overview about porphyrins as building blocks in such experiments is given in the introduction of this thesis, which was published as Review Article in Nanoscale (2021).

A method for the transformation of organic thioacetates, a widely used functionality for the preparation of self-assembled monolayers on gold surfaces, into unsymmetrical disulfides is reported. Disulfides are readily immobilized on gold in contrast to thioacetates, which usually require a deprotection step prior to bonding to the metal surface. The potential of the method for the controlled preparation of unsymmetrical disulfides has been demonstrated with model compounds comprising several thioacetates, which were readily converted into the corresponding unsymmetrical disulfides.

Tropical geometry connects the fields of algebraic and polyhedral geometry. This connection has been used to discover much simpler proofs of fundamental theorems in algebraic geometry, including the Brill-Noether theorem. Tropical geometry has also found applications outside of pure mathematics, in areas as diverse as phylogenetic models and auction theory. This dissertation seeks to answer the question of when the minors of a symmetric matrix form a tropical basis. The first chapter introduces the relevant ideas and concepts from tropical geometry and tropical linear algebra. The second chapter introduces different notions of rank for symmetric tropical matrices. The third chapter is devoted to proving all the cases, outside symmetric tropical rank three, where the minors of a symmetric matrix form a tropical basis. The fourth chapter deals with symmetric tropical rank three. We prove that the 4 × 4 minors of an n×n symmetric matrix form a tropical basis if n ≤ 5, but not if n ≥ 13. The question for 5 < n < 13 remains open. The fifth chapter is devoted to when the minors of a symmetric matrix do not form a tropical basis. We prove the r × r minors of an n × n symmetric matrix do not form a tropical basis when 4 < r < n. We also prove that, when the minors of a matrix (general or symmetric) define a tropical variety and tropical prevariety that are different, then, with one exception, the two sets differ in dimension. The exception is the 4 × 4 minors of a symmetric matrix, where the question is still unresolved. The sixth chapter explores tropical conics. A correspondence between a property of the symmetric matrix of a quadric and the dual complex of that quadric is demonstrated for conics, and proposed for all quadrics. The seventh chapter reviews the results and proposes possible questions for further study. The first appendix is devoted to correcting a proof in a paper cited by this dissertation. The second appendix is a transcript of the Maple worksheets used to perform the computer calculations from the fifth chapter.