Conduction of molecular electronic devices: qualitative insights through atom-atom polarizabilities.

The atom-atom polarizability and the transmission probability at the Fermi level, as obtained through the source-and-sink-potential method for every possible configuration of contacts simultaneously, are compared for polycyclic aromatic compounds. This comparison leads to the conjecture that a positive atom-atom polarizability is a necessary condition for transmission to take place in alternant hydrocarbons without non-bonding orbitals and that the relative transmission probability for different configurations of the contacts can be predicted by analyzing the corresponding atom-atom polarizability. A theoretical link between the two considered properties is derived, leading to a mathematical explanation for the observed trends for transmission based on the atom-atom polarizability.

Omni-conducting and omni-insulating molecules.

The source and sink potential model is used to predict the existence of omni-conductors (and omni-insulators): molecular conjugated π systems that respectively support ballistic conduction or show insulation at the Fermi level, irrespective of the centres chosen as connections. Distinct, ipso, and strong omni-conductors/omni-insulators show Fermi-level conduction/insulation for all distinct pairs of connections, for all connections via a single centre, and for both, respectively. The class of conduction behaviour depends critically on the number of non-bonding orbitals (NBO) of the molecular system (corresponding to the nullity of the graph). Distinct omni-conductors have at most one NBO; distinct omni-insulators have at least two NBO; strong omni-insulators do not exist for any number of NBO. Distinct omni-conductors with a single NBO are all also strong and correspond exactly to the class of graphs known as nut graphs. Families of conjugated hydrocarbons corresponding to chemical graphs with predicted omni-conducting/insulating behaviour are identified. For example, most fullerenes are predicted to be strong omni-conductors.

Symmetry-extended counting rules for periodic frameworks.

A symmetry-adapted version of the Maxwell rule appropriate to periodic bar-and-joint frameworks is obtained, and is further extended to body-and-joint systems. The treatment deals with bodies and forces that are replicated in every unit cell, and uses the point group isomorphic to the factor group of the space group of the framework. Explicit expressions are found for the numbers and symmetries of detectable mechanisms and states of self-stress in terms of the numbers and symmetries of framework components. This approach allows detection and characterization of mechanisms and states of self-stress in microscopic and macroscopic materials and meta-materials. Illustrative examples are described. The notion of local isostaticity of periodic frameworks is extended to include point-group symmetry.

Capturing the elusive aromaticity of bicalicene.

The ring-current aromaticity of the bicalicene molecule arises, in spite of the 16 π carbon perimeter, from strong local diatropic circulations on the two pentagonal rings, as shown by current-density maps computed at the ipsocentric RHF/6-311G** and DFT/6-311G** levels of theory. Conjugated-circuit models cannot capture this pattern of circulation as it arises from 'ionic' contributions in a valence-bond picture. Canonical molecular-orbital analysis reveals a cancellation of paratropic and diatropic frontier-orbital contributions, which explains the difficulties that Hückel-based models have in producing qualitatively correct current-density maps for this molecule. Other measures of aromaticity reflect, to different extents, the dominance of the 'tetraionic' contribution to the aromaticity of this species.

Aromatization of fulvene by complexation with lithium.

At the B3LYP/6-311++G(d,p) level, approach of a lithium atom to a face of the fulvene molecule leads to formation of a complex with binding energy 41 kcal/mol and significant ion-pair character. The fulvene moiety gains a delocalized aromatic cyclic π system, documented by the geometry-based aromaticity index HOMA, and a strong diatropic ring current, visualized by ipsocentric calculation of the π current-density, which leads to an "aromatic" NICS value of -11 ppm.

A selection rule for molecular conduction.

Conditions for transmission of a pi-conjugated molecular conductor are derived within the source and sink potential approach in terms of numbers of nonbonding levels of four graphs: The molecular graph G and the three vertex-deleted subgraphs obtained by removing one or both contact vertices. For all bipartite and most nonbipartite G, counting nonbonding levels gives a simple necessary and sufficient condition for conduction at the Fermi level. The exceptional case is where G is nonbipartite and all four graphs have the same number of nonbonding levels; then, an auxiliary requirement involving tail coefficients of the four characteristic polynomials must also be checked.

Fragment analysis of single-molecule conduction.

In the tight-binding source and sink potential model of transmission in single-molecule pi-conjugated conductors, vanishing of the opacity polynomial defines a necessary condition for zero conductance at a given energy. Theorems are given for calculating opacity polynomials of composite devices in terms of opacity and characteristic polynomials of the subunits. These relations rationalize the positions and shapes of zeros in transmission curves for devices consisting of molecules with side chains or of units assembled in series and take an especially simple form for polymeric molecules with identical repeat units.

Conduction in graphenes.

It is shown that, within the tight-binding approximation, Fermi-level ballistic conduction for a perimeter-connected graphene fragment follows a simple selection rule: the zero eigenvalues of the molecular graph and of its subgraph minus both contact vertices must be equal in number, as must those of the two subgraphs with single contact vertices deleted. In chemical terms, the new rule therefore involves counting nonbonding orbitals of four molecules. The rule is initially derived within the source and sink potential scattering framework, but has equivalent forms that unify the molecular-orbital and valence-bond approaches to conduction. It is shown that the new selection rule can be cast in terms of Kekule counts, bond orders, and frontier-orbital coefficients. In particular, for a Kekulean graphene, conduction pathways are shown to be ranked in efficiency by a (nonmonotonic) function of Pauling bond order between the contact vertices. Frontier-orbital analysis of conduction approximates this function. For a monoradical graphene, the analogous function is shown to depend on Pauling spin densities at contact vertices.

Experimental and theoretical characterization of the hexaazidophosphate(V) ion.

(PPN)[P(N(3))(6)] (2) was synthesized by the metathesis of Na[P(N(3))(6)] (1) and (PPN)N(3) (PPN(+) = {(Ph(3)P)(2)N}(+)), allowing for the isolation and full characterization of a stable hexaazidophosphate(V) salt by (31)P and (14)N NMR, UV absorption, IR and Raman spectroscopy, elemental and thermal analyses, X-ray diffraction, and Hartree-Fock and density functional theory calculations. The colorless single crystals of 2 are triclinic, space group P1, a = 9.6296(9), b = 9.8158(9), c = 10.1414(10) A, alpha = 92.635(5) degrees , beta = 93.437(5) degrees , gamma = 92.105(4) degrees , and Z = 1. The [P(N(3))(6)](-) ion of 2 is isolated in the solid state and adopts S(6) symmetry both in the crystal and in solution. Thermogravimetric analysis and differential scanning calorimetry measurements reveal a surprising thermal stability of 2 (T(dec) ca. 200 degrees C). No friction sensitivity was encountered.

Stacked-ring aromaticity: an orbital model.

Visualization of induced current density using the ipsocentric CHF/CTOCD-DZ/6-31G** approach gives a direct demonstration of the literature proposal of reversal of [4n]annulene antiaromaticity on stacking cyclooctatetraene (COT) rings into a superphane. Through-space interactions lead to a closed-shell in which paratropicity of planar COT units is quenched, and layered diatropic currents arise from magnetic response of two pairs of frontier orbitals. A general orbital model rationalizes the differences in current between stacked aromatic and antiaromatic rings.

Full spectral decomposition of ring currents.

Within the ipsocentric method for calculation of molecular magnetic response, projection of perturbed orbitals onto the virtual orbital space allows partition of induced current density into contributions from individual virtual excitations between occupied and unoccupied orbitals, enabling detailed assignment of the origin of currents in, e.g., benzene, cyclooctatetraene, borazine, coronene, and corannulene. Whereas delocalized currents in benzene and planar cyclooctatetraene are described by transitions within the valence space, localized currents in the borazine pi system involve excitations outside the valence space.

Current density, chemical shifts and aromaticity.

The intimate link between chemical shifts and magnetic criteria for aromaticity prompts a search for detailed understanding of patterns of current density induced in pi systems by external magnetic fields. Conceptual and practical advantages of calculation of current densities with a specific method of distribution of origin of vector potential, the ipsocentric choice, where the induced current density at each point is calculated with that point as origin, are outlined. This choice leads uniquely to canonical molecular orbital contributions that are free of unphysical occupied-occupied mixing. Characteristic magnetic response of delocalized pi systems is then effectively restricted to the activity of a small number of frontier electrons, governed by simple symmetry and node-counting rules, and readily visualized in current density maps. Localized orbitals for sigma systems can also be used, again eliminating occupied-occupied mixing. For integrated properties (magnetizability and nuclear shieldings), the ipsocentric method gives, in a well-defined sense, the orbital contributions that are best for purposes of interpretation. The general theory is illustrated by maps for a set of annelated pentalenes; the known benzopentalene (1) and 1,2:4,5-dibenzopentalene (2), the still unknown isomer of 2, 1,2:5,6-dibenzopentalene (3), and cyclopent[b,c]acenaphthylene (4), an unknown isomer of pyracylene, all of which consist of fusions of formally aromatic and anti-aromatic pi-conjugated systems.

Zigzags, railroads, and knots in fullerenes.

Two connections between fullerene structures and alternating knots are established. Knots may appear in two ways: from zigzags, i.e., circuits (possibly self-intersecting) of edges running alternately left and right at successive vertices, and from railroads, i.e., circuits (possibly self-intersecting) of edge-sharing hexagonal faces, such that the shared edges occur in opposite pairs. A z-knot fullerene has only a single zigzag, doubly covering all edges: in the range investigated (n /= 38, all chiral, belonging to groups C(1), C(2), C(3), D(3), or D(5). An r-knot fullerene has a railroad corresponding to the projection of a nontrivial knot: examples are found for C(52) (trefoil), C(54) (figure-of-eight or Flemish knot), and, with isolated pentagons, at C(96), C(104), C(108), C(112), C(114). Statistics on the occurrence of z-knots and of z-vectors of various kinds, z-uniform, z-transitive, and z-balanced, are presented for trivalent polyhedra, general fullerenes, and isolated-pentagon fullerenes, along with examples with self-intersecting railroads and r-knots. In a subset of z-knot fullerenes, so-called minimal knots, the unique zigzag defines a specific Kekulé structure in which double bonds lie on lines of longitude and single bonds on lines of latitude of the approximate sphere defined by the polyhedron vertices.

Addition patterns in carbon allotropes: independence numbers and d-codes in the Klein and related graphs.

The problem of predicting stoichiometries and patterns of chemical addition to a carbon framework, subject solely to the restriction that each addend excludes neighboring sites up to some distance d, is equivalent to determination of d-codes of a graph, and for d = 2 to determination of maximum independent sets. Sizes, symmetries, and numbers of d-codes are found for the all-heptagon Klein graph (prototype for "plumber's nightmare" carbon) and for three related graphs. The independence number of the Klein graph is 23, which increases to 24 for a related, but sterically relaxed, all-heptagon network with the same number of vertices and modified adjacencies. Expansion of the Klein graph and its relaxed analogue by insertion of hexagonal faces to form leapfrog graphs also allows all heptagons to achieve their maximum of 3 addends. Consideration of the pi system that is the complement of the addition pattern imposes a closed-shell requirement on the adjacency spectrum, which typically reduces the size of acceptable independent sets. The closed-shell independence numbers of the Klein graph and its relaxed analogue are 18 and 20, respectively.

A catalogue of growth transformations of fullerene polyhedra.

Carbon insertion or extrusion mechanisms transforming one fullerene to another are presented as patch replacements on the fullerene surface. A systematic catalogue is constructed for the topologically distinct local insertion/extrusion transformations of fullerenes, classified by patch boundary and pentagon content. All pairs of patches with the same boundary but different numbers of atoms, i.e., growth patches, containing up to five pentagons, with an upper limit for the boundary length that depends on the number of pentagons, are listed. New transformations and infinite series of transformations are identified.

A catalogue of isomerization transformations of fullerene polyhedra.

Representation of isomerization and carbon insertion or extrusion mechanisms as patch replacements on a fullerene surface allows construction of a catalogue of topologically distinct local transformations of fullerenes, classified by patch boundary and pentagon content. All isomerization patches and isomerization pairs containing up to five pentagons and with an upper limit for the boundary length depending on the number of pentagons are listed. Several infinite series of transformations are identified.

Version of zones and zigzag structure in icosahedral fullerenes and icosadeltahedra.

A circuit of faces in a polyhedron is called a zone if each face is attached to its two neighbors by opposite edges. (For odd-sized faces, each edge has a left and a right opposite partner.) Zones are called alternating if, when odd faces (if any) are encountered, left and right opposite edges are chosen alternately. Zigzag (Petrie) circuits in cubic (= trivalent) polyhedra correspond to alternating zones in their deltahedral duals. With these definitions, a full analysis of the zone and zigzag structure is made for icosahedral centrosymmetric fullerenes and their duals. The zone structure provides hypercube embeddings of these classes of polyhedra which preserve all graph distances (subject to a scale factor of 2) up to a limit that depends on the vertex count. These embeddings may have applications in nomenclature, atom/vertex numbering schemes, and in calculation of distance invariants for this subclass of highly symmetric fullerenes and their deltahedral duals.

Cahn-Ingold-Prelog descriptors of absolute configuration for carbon cages.

A simplified procedure is described for assigning Cahn-Ingold-Prelog descriptors to stereocentres in spheroalkanes (the CnHn molecules, with n even, based on trivalent, polyhedral carbon frameworks, a class which subsumes the fulleranes). By extension, similar descriptors can be found for the atoms of fullerenes and related carbon-only molecules. Assignment maps are given for chiral fullerenes C28 C76, C78, C84 and C140, and for a number of spheroalkanes. Cases of breakdown of the simple procedure for triangle-rich spheroalkane molecular graphs are discussed.

Addition patterns, codes and contact graphs for fullerene derivatives.

The mathematical concept of the d-code and its associated contact graph give a model for sterically constrained addition patterns in fullerene derivatives C60Xm and C70Xm. In combination with simple electronic arguments, the stoichiometries, symmetries, and location of addends can be predicted, yielding a small number of candidates for further study. For example, sterically and optimal solutions C60Xm with pairwise separation of d bonds between addends are found at m(d) = 24(2), 12(3), 6(4,5), 2(6 to 9). The solution for C60X24 is unique, and the model selects 12 candidates for C60X12 from a starting set of 11661527060 possibilities.