Near omni-conductors and insulators: Alternant hydrocarbons in the SSP model of ballistic conduction

Abstract

Within the SSP (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 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.

Molecular graphs and molecular conduction: the d-omni-conductors

“…It can be proved that the 81 conceivable classes reduce to exactly 14. 25 This treatment suggests a generalisation for near-omni nonbipartite graphs.…”

“…The SSP (source-and-sink-potential) model was introduced by Ernzerhof et al [1][2][3][4][5][6][7][8][9][10][11][12][13][14] as a simple but effective description of ballistic molecular conduction. In its graph theoretical (Hückel) incarnation, [15][16][17][18][19][20][21][22][23][24][25] it predicts transmission as a function of energy for a two-wire device from an expression involving a functional of four characteristic polynomials: those of the molecular graph and three subgraphs. In the context of π systems, a chemical (molecular) graph is one that is connected and has a maximum degree of 3 or less.…”

“…The 27 categories reduce to exactly 13 that are realisable by graphs. 25 There are families of chemical graphs for which omni-conducting behaviour is not possible. For them we can define the notion of a near-omni conductor, by making a well defined partition of the set of devices arising from a molecular graph.…”

“…The colours correspond to the starred and unstarred atoms of the alternant hydrocarbon. 28 Bipartite molecular graphs cannot have full omniconduction, for well understood mathematical reasons, but we can make a near-omni 25 classification by separating distinct devices into inter and intra types: in an intra device the two connection vertices belong to the same partite set; in an inter device they belong to different partite sets. For ipso devices, we can ignore the colour of the connection vertex.…”

The conductivity of superimposed key-graphs with a common one-dimensional adjacency nullspace

“…Different three letter acronyms are proposed in[6,7] to classify classes of molecular graphs as conductors or insulators with respect to the graph-theoretical distance between two connecting vertices of the graph across which there is a small bias voltage.…”