Gerald E. Hite

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.

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”).

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.

Determinations of the πNσ term and s-wave scattering lengths☆

Abstract Determinations of the σ term and s-wave scattering lengths for pion-nucleon scattering are made using interior dispersion relations. The result for the σ term is obtained by extrapolating to the Cheng-Dashen point in only one variable. It is completely independent of the pion-nucleon coupling constant and insensitive to d and f waves in the resonance region. The use of interior dispersion relations allows a series of s-wave scattering length determinations to be made. It is observed that data at intermediate scattering angles gives more consistent results than data near the forward or backward directions, lending further support to the σ term calculation which utilizes only interior angle information.

The I = 0, s-wave ππ scattering length☆

Abstract Interior dispersion relations are applied to πN scattering amplitudes to extract the I = 0, s-wave ππ scattering length, a00. Two methods are used, the second of which also incorporates known ππ scattering phase shifts. We have used the πN amplitude analysis of Pietarinen and obtained values of a00: (0.28 ± 0.09)μ− and (0.25 ± 0.08)μ− for the two methods respectively, consistent with other recent determinations from Ke4 and πN → ππN data. The 1971 πN phase-shift analysis of Ahmehed and Lovelace was also used successfully in the first method to obtain the value a00: (0.15 ± 0.14)μ−, consistent with the above. The second method was found to be ineffective in the face of relatively unsmooth discrepancy functions obtained from the Ahmehed-Lovelace phase shifts.

Interacting dimers on a Sierpinski gasket

Abstract The different manners of embedding subgraphs of many disjoint edges in a parent graph arise in several different chemical and physical contexts. The problem of enumeration, or more generally of weighted summation, of such embeddings seems quite difficult to great quantitatively, at least for general extended graphs. The special extended parent graph treated here with the inclusion of interaction between dimers is that of a Sierpinski gasket with a fractal dimension of ln 3/ln 2. The manners of solution should extend to other fractal graphs with low ‘ramification degree’.