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In Hückel theory, the bond number is the sum of the orders of the π bonds incident on a given carbon center. From the work of Coulson and his school, it has been believed for over 70 years that the bond number has a maximum of 3 and that this bound is realized by exactly one conjugated framework, that of the trimethylenemethane radical. Search of published literature and archived correspondence failed to find any formal proof of these two statements. Here, we provide a new formula for bond number that leads to an easily checked proof of both. The bond number of graphene is 1.574 597… (90.9% of the mathematical limit), and this value appears to act as a separator for the classes of metallic and semiconducting single-walled nanotubes, as defined within Hückel theory.

Within the source-and-sink-potential model, a complete characterisation is obtained for the conduction behaviour of alternant π-conjugated hydrocarbons (conjugated hydrocarbons without odd cycles). In this model, an omni-conductor has a molecular graph that conducts at the Fermi level irrespective of the choice of connection vertices. Likewise, an omni-insulator is a molecular graph that fails to conduct for any choice of connections. We give a comprehensive classification of possible combinations of omni-conducting and omni-insulating behaviour for molecular graphs, ranked by nullity (number of non-bonding orbitals). Alternant hydrocarbons are those that have bipartite molecular graphs; they cannot be full omni-conductors or full omni-insulators but may conduct or insulate within well-defined subsets of vertices (unsaturated carbon centres). This leads to the definition of "near omni-conductors" and "near omni-insulators." Of 81 conceivable classes of conduction behaviour for alternants, only 14 are realisable. Of these, nine are realised by more than one chemical graph. For example, conduction of all Kekulean benzenoids (nanographenes) is described by just two classes. In particular, the catafused benzenoids (benzenoids in which no carbon atom belongs to three hexagons) conduct when connected to leads via one starred and one unstarred atom, and otherwise insulate, corresponding to conduction type CII in the near-omni classification scheme.

This paper shows how to include Pauli (exclusion principle) effects within a treatment of ballistic molecular conduction that uses the tight-binding Hückel Hamiltonian and the source-sink-potential (SSP) method. We take into account the many-electron ground-state of the molecule and show that we can discuss ballistic conduction for a specific molecular device in terms of four structural polynomials. In the standard one-electron picture, these are characteristic polynomials of vertex-deleted graphs, with spectral representations in terms of molecular-orbital eigenvectors and eigenvalues. In a more realistic many-electron picture, the spectral representation of each polynomial is retained but projected into the manifold of unoccupied spin-orbitals. Crucially, this projection preserves interlacing properties. With this simple reformulation, selection rules for device transmission, expressions for overall transmission, and partition of transmission into bond currents can all be mapped onto the formalism previously developed. Inclusion of Pauli spin blockade, in the absence of external perturbations, has a generic effect (suppression of transmission at energies below the Fermi level) and specific effects at anti-bonding energies, which can be understood using our previous classification of inert and active shells. The theory predicts the intriguing phenomenon of Pauli perfect reflection whereby, once a critical electron count is reached, some electronic states of devices can give total reflection of electrons at all energies.

We re-derive the tight-binding source-sink potential (SSP) equations for ballistic conduction through conjugated molecular structures in a form that avoids singularities. This enables derivation of new results for families of molecular devices in terms of eigenvectors and eigenvalues of the adjacency matrix of the molecular graph. In particular, we define the transmission of electrons through individual molecular orbitals (MO) and through MO shells. We make explicit the behaviour of the total current and individual MO and shell currents at molecular eigenvalues. A rich variety of behaviour is found. A SSP device has specific insulation or conduction at an eigenvalue of the molecular graph (a root of the characteristic polynomial) according to the multiplicities of that value in the spectra of four defined device polynomials. Conduction near eigenvalues is dominated by the transmission curves of nearby shells. A shell may be inert or active. An inert shell does not conduct at any energy, not even at its own eigenvalue. Conduction may occur at the eigenvalue of an inert shell, but is then carried entirely by other shells. If a shell is active, it carries all conduction at its own eigenvalue. For bipartite molecular graphs (alternant molecules), orbital conduction properties are governed by a pairing theorem. Inertness of shells for families such as chains and rings is predicted by selection rules based on node counting and degeneracy.

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.

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.

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.

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.

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.

Calculations on the novel argon insertion compounds C6F6Ar(n), n = 1-6, where the argon atoms are inserted into the C-F bonds in perfluorobenzene, suggest that all possible species, with one to six inserted argon atoms, occupy minima on their respective potential energy surfaces. Ring-current plots using the ipsocentric model indicate that there is no disruption of the aromatic pi system upon argon insertion, and consequently all insertion compounds are aromatic according to the magnetic criterion. The barrier height for decomposition of the single-insertion compound, C6F6Ar, into C6F6+Ar is 19.5 kcal/mol at HF6-311G** and 29.5 kcal/mol at B3LYP6-311G**, suggesting that, although clearly thermodynamically unstable, argon-perfluorobenzene insertion compounds may be stable kinetically. Preliminary calculations indicate that other rare gas-perfluorobenzene insertion compounds may also be metastable. Both C6F6Ne and C6F6He are predicted to occupy minima on their respective potential energy surfaces.