L. K. Runnels

Exact Finite Method of Lattice Statistics. I. Square and Triangular Lattice Gases of Hard Molecules

A general, feasible approach is presented for the evaluation of the statistical thermodynamics of interacting lattice gases. Exact solutions are obtained for lattice systems of infinite length and increasing finite width, using the matrix method which treats all densities on an equivalent basis. Through the application of symmetry reduction and the use of an electronic computer to perform logical as well as arithmetical operations, widths of up to 24 sites for two‐dimensional lattices can be handled. For examples studied, rapid convergence is obtained away from transition regions and in the vicinity of phase transitions the behavior appears to be a sufficiently regular function of the width to allow meaningful extrapolation to systems of infinite width. Two problems of two‐dimensional lattice gases are solved as illustrations of the technique: the square and triangular lattice gases with infinite repulsive interactions preventing the simultaneous occupancy of adjacent lattice sites (excluded‐volume effect...

Diffusion and Relaxation Phenomena in Ice

The underlying mechanisms of several rate processes in ice are examined through cross comparisons of the processes with each other and with experimental observations. The assumption that the migration of orientational defects (Bjerrum faults) is the common origin of dielectric and elastic relaxation leads to a predicted ratio of dielectric‐to‐elastic relaxation time of 32, in close agreement with experiment. The conclusion that a separate process is responsible for diffusion is based on a comparison of diffusion and dielectric relaxation data. The faster diffusive motion controls the thermal equilibration of the proton spins as well as the magnetic resonance linewidths; an interstitial migration appears to be the mostlikely diffusion mechanism.

Nature of the Rigid‐Rod Mesophase

The thermodynamic phases of the rigid‐rod gas are re‐examined. Introduced in its present form by Zwanzig, the model postulates elongated molecules of parallelepiped shape with only three allowed orientations (mutually perpendicular). We have added to the virial series the cluster integrals represented by eight‐point bicolored graphs. It is demonstrated that the Pade approximants generated by the truncated cluster expansion provide a much more stable sequence of parameters characterizing the isotropic–anisotropic phase transition. It is similarly shown that there are no stable phases predicted for which the orientational distribution is not axially symmetric. The relative volume change accompanying the transition is still predicted to be larger than that observed in the isotropic–nematic liquid crystal transition.

Exact Finite Method of Lattice Statistics. II. Honeycomb‐Lattice Gas of Hard Molecules

Thermodynamic properties have been obtained using the matrix method for a two‐dimensional honeycomb‐lattice gas of hard molecules which prevent simultaneous occupancy of nearest‐neighbor sites. Exact results have been obtained for cylinders of infinite length and finite circumference from 6 to 18 sites. Uniform trends with increasing circumference lend confidence to extrapolations used to infer the behavior of systems of infinite width. We find a great similarity with the corresponding square‐lattice gas and a second‐order phase transition at activity z=7.92±0.08, reduced pressure p/kT=1.12±0.05, and relative density ρ/ρ0=0.845±0.02. The compressibility appears to be infinite at the transition.

Exact Finite Method of Lattice Statistics. IV. Square Well Lattice Gas

The temperature‐dependent phase equilibrium of simple classical molecules has been studied, using a model based on the two‐dimensional square lattice. The intermolecular potential includes a hard core extending to the first‐neighbor distance and a finite interaction (attractive or repulsive) at the second‐neighbor distance. The transfer matrix method of calculation is used for lattices of infinite length and finite circumference up to 16 sites. At all temperatures studied there is a single phase transition, which is of first order for attractive interactions at sufficiently low temperatures. The model best describes the equilibrium between solid and gas at low temperatures or the equilibrium between solid and supercritical fluid at high temperatures. The deficiencies in the model which exclude a liquidlike phase are discussed.

Model for Correlated Molecular Rotation

We report studies of models designed to emphasize the thermodynamic consequences of coupled molecular rotation in condensed phases. The coupling arises from hard molecular cores of nonspherical shape, here represented by flat coplanar squares in a one‐dimensional array. Various forms are considered for the orientation‐independent part of the interaction, from none to harmonic to delta function (or lattice model); the first and third are shown to represent the high‐ and low‐temperature limits of the second, treated classically. The lattice model gives rise to a particular sort of first‐order phase transition, the significance of which is clarified by the temperature dependence of the properties of the harmonic model.

Classical rotators on a linear lattice

Exact thermodynamic properties are computed for a model of rotating, coplanar molecules with centers defining a regular one‐dimensional array. The interaction included is that between nearest neighbors, represented by a truncated expansion in Legendre functions. The problem reduces to an integral eigenvalue equation, and consequently the free energy must depend analytically on temperature.

Exact Finite Method of Lattice Statistics. V. The Thermodynamic Phases of a Triangular Lattice Gas

We present phase diagrams showing the stable phases of a two‐dimensional lattice gas. The molecules reside on the triangular lattice and have hard cores which exclude other molecules from the first‐ and second‐neighbor positions. An interaction w (positive or negative) is postulated for molecules separated by the third‐neighbor distance, and no interaction is experienced at still greater separation. For attractive third‐neighbor interactions (negative w) the phase diagram possesses only two regions: solid and fluid. The transition between them is first order at all temperatures, but more strongly first order at low temperatures. For positive w (soft repulsions), the phase diagram is topologically similar to that of helium. In addition to a solid region and a gaseous region, there are areas identified as a “normal liquid” and an “ordered liquid,” the latter being the stable phase as T → 0 at moderate pressures.

One‐Center Molecular Orbitals for HeH++ and HeH+ with Variable Origin

The feasibility of using a variable expansion point in one‐center calculations is investigated for the ground states of HeH++ and HeH+ with very limited basis sets of Slater‐type atomic orbitals. The energy improvement resulting from variation of the expansion point appears to be economical only if the wavefunction is of the simplest possible form, 1s or (1s1s). For these functions the maximum effect of varying the origin occurs in the vicinity of 0.4 a.u., where the improvement in the electronic energy is somewhat less than 2%.

Pair Distribution Function for a Gas of Hard Rods

The pair distribution function is approximated to third order in the density expansion for a gas of long, thin rods. Zwanzigs model is used in which the molecules are rectangular parallelepipeds of length l and square cross section d2. The long molecular axes can point in only three mutually perpendicular directions. The distribution function coefficients are calculated exactly as functions of relative position and orientation in the limit l→ ∞, d→0, l2d→1. The potential of mean force is found to be positive at low density for perpendicular orientations and negative at low density for parallel orientations. It is proved that in the above limit, parallel orientations do not contribute to the pressure.