The localization spread and polarizability of rings and periodic chains

The localization spread gives a criterion to decide between metallic and insulating behavior of a material. It is defined as the second moment cumulant of the many-body position operator, divided by the number of electrons. Different operators are used for systems treated with open or periodic boundary conditions. In particular, in the case of periodic systems, we use the complex position definition, which was already used in similar contexts for the treatment of both classical and quantum situations. In this study, we show that the localization spread evaluated on a finite ring system of radius R with open boundary conditions leads, in the large R limit, to the same formula derived by Resta and co-workers [C. Sgiarovello, M. Peressi, and R. Resta, Phys. Rev. B 64, 115202 (2001)] for 1D systems with periodic Born-von Kármán boundary conditions. A second formula, alternative to Resta’s, is also given based on the sum-over-state formalism, allowing for an interesting generalization to polarizability and other similar quantities

The topology of fullerenes

Fullerenes are carbon molecules that form polyhedral cages. Their bond structures are exactly the planar cubic graphs that have only pentagon and hexagon faces. Strikingly, a number of chemical properties of a fullerene can be derived from its graph structure. A rich mathematics of cubic planar graphs and fullerene graphs has grown since they were studied by Goldberg, Coxeter, and others in the early 20th century, and many mathematical properties of fullerenes have found simple and beautiful solutions. Yet many interesting chemical and mathematical problems in the field remain open. In this paper, we present a general overview of recent topological and graph theoretical developments in fullerene research over the past two decades, describing both solved and open problems. WIREs Comput Mol Sci 2015, 5:96–145. doi: 10.1002/wcms.1207 Conflict of interest: The authors have declared no conflicts of interest for this article. For further resources related to this article, please visit the WIREs website

Topological Ring-Currents in Condensed Benzenoid Hydrocarbons

Topological ring-currents are defined as being pi-electron ring-current intensities in condensed, benzenoid hydrocarbons that (i) are calculated by the simplest Hückel-London-Pople- McWeeny method, (ii) are based on a molecular geometry of regular hexagons of carbon atoms, and (iii) are expressed as a ratio to the corresponding ring-current intensity calculated, by the same method, for benzene. Once a particular benzenoid hydrocarbon has been specified, such topological ring-currents are predetermined and do not further depend on any subjective (or other) parameters; they are, therefore, purely graph-theoretical indices, reliant solely on knowledge of a vertex-adjacency matrix for the graph representing the connectivity of the carbon atoms in the benzenoid molecule under study. For convenient reference, tables of all known topological ring-current intensities – some published, and others so-far unpublished – are presented for future evaluation and possible comparison with other graph-theoretical indices that characterise the individual rings of a condensed, benzenoid hydrocarbon

Synthesis and Study of the Electrochemical and Optoelectronic Properties of π-Conjugated Poly-P-Phenylene-Based Molecular Wires

Organic photovoltaics will play an important role in supplementing the energy needs of the twenty first century in a most cost-effective way to convert the solar energy into usable forms of energy. One of the bottlenecks in promoting widespread use of photovoltaic devices for solar energy storage is the inefficiency of the devices, which in part arises due to the inefficient charge separation and long-range charge transport. Design and synthesis of efficient charge-transfer materials would require one to first identify and establish the structural features necessary in a given molecular wire, which may promote effective charge transfer to long distances. Accordingly, herein, we describe the syntheses and study of the electrochemical and optoelectronic properties a number of different series of poly-p-phenylene based oligomers in order to probe the mechanism and extent of hole delocalization in molecular wires containing a large number of p-phenylenes. In order to develop an understanding of hole delocalization in π-conjugated oligomers/polymers, we have synthesized and systematically studied well-defined series of oligomers of π-conjugated poly-p-phenylene based molecular wires with varying inter-ring dihedral angles ( ) between the monomer units (e.g. RPPn, ~ 33º; oligofluorenes, ~ 37º and 0º; or planarized angular polyfluorenes, ~ 0º) together with the effect of end capping groups, such as alkyl (iAPP) and alkoxy (ROPP). The experimental evaluation of redox and optical properties of various molecular wires studied herein showed that the HOMO density distribution extends over the entire chain as expected based on the observed linear cos[π/(n + 1)] trend albeit gravitating toward the center of the molecule, whereas the hole distribution, which determines optoelectronic properties of the oligomer cation radicals, is found to be limited to 7-8 phenylene units. The study of poly-p-phenylene oligomers with electron-donating end-capping groups (i.e. alkyl- or alkoxy-groups) showed that the hole distribution is strongly impacted by end-capping groups and it gravitates from the central position to the end of the oligomeric chain while the distribution of HOMO density remains in the middle of the π-conjugated wires. Such migration of the hole toward the end of the chain due to the electron-donating alkoxy groups has significant experimental consequences, i.e. the optical and redox properties saturated at five phenylene units in case of alkoxy-capped poly-p-phenylene wires. These results were further reconciled by DFT calculations and by a recently developed multistate parabolic model