Thomas G. Schmalz

Compact self-avoiding circuits on two-dimensional lattices

Close-packed self-avoiding walks and circuits, as models for condensed polymer phases, are studied on the square-planar and honeycomb lattices. Exact solutions for strips from these lattices are obtained via transfer matrix methods. Extrapolations are made for the leading asymptotic terms in the count of compact conformations on the square-planar lattice. The leading asymptotic term for each lattice is bounded from below, and it is noted that boundary effects can be important.

Dimer coverings and Kekulé structures on honeycomb lattice strips

The problem of covering every site of a subsection of the honeycomb lattice with disjoint edges is considered. It is pointed out that a type of long-range order associated to such coverings can occur, so that different phases can arise as a consequence of the subsections boundaries. These features are quantitatively investigated via a new analytic solution for a class of strips of arbitrary widths, arbitrary lengths, and arbitrary long-range-order values. Relations to work on the dimer covering problem of statistical mechanics and especially to the resonance theory of benzenoid hydrocarbons are noted.

Generalizations of the Stone-Wales rearrangement for cage compounds, including fullerenes

Abstract Generalizations of the Stone-Wales (pyracylene) rearrangement are proposed for fullerenes and other cage compounds. These involve various numbers of pairs of hexagons fused via a zig-zag sequence of edges, ending in two faces which are polygons of any ring size. This generalized rearrangement may lead either to automerization, or more often to rearrangements changing one isomer of a cage into another. Applications to fullerenes with 5- and 6-membered rings are discussed, as well as other types of cages with larger and smaller rings.

Wavefunction comparisons for the valence-bond model for conjugated π-networks

Approximate ground-state wavefunctions for valence-bond (or Heisenberg) models are obtained both within Néel-state-based and within Kekulé-state-based resonance-theoretic approaches. Comparisons are made between these and other general approaches, with particular emphasis on organic π-network systems. Attention is drawn to the manner in which the quality of the different approximation schemes changes with variations in structural characteristics of the system. It is suggested that resonance-theoretic ideas are most appropriate for (aromatic benzenoid) systems with low coordination number, whereas Néel-state based ideas are most appropriate for (3-dimensional) structures with higher coordination number (and little “frustration”).

Kekulé count and algebraic structure count for unbranched alternant cata-fusenes☆

Abstract Algorithms for making Kekule-structure and algebraic-structure enumerations for unbranched catacondensed hydrocarbons with even-membered rings are described. A pictorial presentation of the algorithms is obtained and is shown to reduce to an earlier well-known recursion (of Gordon and Davison) applicable to polyhex chains.

Extended ?-networks with multiple spin-pairing phases: resonance-theory calculations on poly-polyphenanthrenes

The poly-polyphenanthrene family of extended π-network strips with members ranging from polyacetylene to graphite is considered in terms of the locally correlated valence-bond or Heisenberg Hamiltonian. Resonance theory wavefunctions which provide a variational upper bound to the ground state energy are developed in a graph-theoretic formalism extendable to more general localized wavefunction cluster expansions. The graph-theoretic formalism facilitates the use of general transfer matrix techniques, which are especially powerful in application to quasi-one-dimensional systems such as are illustratively treated here. It is argued that these strips exhibit states of different long-range spin-pairing orderings. Novel properties associated with these different resulting phases are briefly indicated, including the possibilities of solitonic excitations and the reactivity at the ends of the strips. The qualitative arguments are supported by numerical calculations for strips up to width 8.

REACTION GRAPHS FOR REARRANGEMENTS OF PENTAGONAL-BIPYRAMIDAL COMPLEXES

Abstract The reaction graphs for pentagonal-bipyramidal structures have 504 vertices when enantiomerism is considered, and 252 vertices when it is ignored. Smaller subgraphs are presented in detail, obtained from the above integral reaction graphs by contracting two pairs of opposite edges in four-membered circuits into a single edge. Based on the symmetries of these smaller graphs with 24 and 12 vertices, respectively, considerations on the two integral reaction graphs are presented. Chemically relevant species are discussed.