John J. Kozak

Analytic Approach to the Theory of Phase Transitions

An approximation to the first equation of the Kirkwood coupling parameter hierarchy and other model equations for the singlet distribution function are cast into the standard Hammerstein form of nonlinear integral equation. We give a criterion for the existence and uniqueness of solutions of this equation involving the first negative eigenvalue of the kernel, which allows us to establish temperatures and densities where the solution is unique. Multiple solutions of the nonlinear equation are associated with instability of the single phase and thus signal a phase transition. A necessary condition for the existence of other solutions of small norm is given by a bifurcation equation. These new solutions are associated with the freezing transition, and the periodic singlet density of the solid falls naturally out of the theory. The bifurcation equation can be related to the Kirkwood instability criterion, but, in contrast to this, predicts no transition for a system of hard rods when a model kernel is used. T...

Determination of the pair polarizability tensor for the Ne diatom

The pair polarizability tensor A for the neon diatom Ne2 has been determined as a function of the Ne–Ne separation using an extended Gaussian basis set and the finite perturbation coupled Hartree–Fock technique. Our results are compared with calculations reported previously on the Ne diatom (the earlier calculations were based on a density functional approach), and are found to differ significantly in the estimate of the anisotropy in the tensor A.

Electron reactions and electron transfer reactions catalyzed by micellar systems

The kinetics of the reaction of hydrated electrons with pyrene, pyrene butyric acid, and pyrene sulfonic acid (PSA) have been investigated in aqueous solutions of cetyltrimethylammonium bromide (CTAB). With all three solubilizates the formation of the electron adduct (P−) occurs very rapidly with rate constants ≳1011M−1⋅sec−1. These abnormally high rate constants are shown to be due to fast trapping of eaq− in the positive potential field of the micelle and subsequent efficient penetration of electrons into the micellar interior. A similar enhancement was observed for electron transfer reactions between CO−2 and solutes solubilized in or on the micelle. For example CO−2 readily transfers an electron to pyrene sulfonic acid on the surface of the micelle. This reaction does not occur in homogeneous solution but is catalyzed by the positive electrostatic surface potential. Addition of electrolyte drastically reduces the rate of eaq− and CO2− with solubilizates. The Debye−Huckel theory of electrolytes was inv...

Influence of geometry on light harvesting in dendrimeric systems

The exact analytic expression for the mean time to trapping (or mean walklength) for a particle (electron/exciton) performing a random walk on a finite dendrimer lattice with a trap at the center of the dendrimer was obtained. Exact analytic expressions have also been obtained for articulated/extended dendrimeric systems. The full dynamical behavior was determined for each case studied via numerical solution of the stochastic master equation, and the results obtained were shown to be a direct consequence of the structural properties of the dendrimeric system. These studies are linked to the behavior observed in experiments on light harvesting in dendrimeric supermolecules.

Study of the structure of molecular complexes. IX. The Hartree–Fock energy surface for the H2O–Li–F complex

A large number of geometrical configurations (250) are computed with a large Gaussian basis set in the Hartree–Fock approximation for the H2O–Li–F complex. The many‐dimensional potential energy surface has been sampled by keeping the molecule of water at a fixed position and by allowing the lithium and the fluorine to assume many positions in space. Because of the symmetry (C2v) of the water molecule, the 250 computations correspond to a sampling of about 600 configurations. The sampling includes a few highly repulsive configurations (up to about 300 kcal/mole in repulsion); the remaining points are either in the strongly attractive regions or in the weakly attractive regions of the surface. The stabilization energy of the complex reveals the existence of at least three possible structures: the Li–F–H2O structure (with C2v symmetry), with a stabilization energy (relative to H2O, F−, and Li+) of about −186 kcal/mole; a second Li–F–H2O structure with the fluorine forming a hydrogen bond (with one of the H–O...

Properties of solutions to the Yvon–Born–Green equation for the square‐well fluid

The Yvon−Born−Green equation under the superposition approximation was solved numerically for a system of molecules interacting via the square−well potential with σ2/σ1 = 1.85, for a temperature−density range bounded by 0 < ϑ < 0.55 and 0 < λ0 < 13. Solutions of the YBG equation throughout this range were found to be unique, and upon examination of the resulting thermodynamic properties, a gas−liquid coexistence region was located and evidence for a high−density, low temperature fluid transition was found. Relative to the range of ϑ − λ0 considered, the down−range properties of the pair−correlation function were used as a guide in carrying through a formal analysis of the YBG equation using basic theorems on the existence and uniqueness of solutions of nonlinear integral equations. Lower bounds on the limit of stability of the pure gas and liquid phase were identified and correlated with the thermodynamic behavior determined via numerical solution of the YBG equation.

Application of the Theory of Orlicz Spaces to Statistical Mechanics. I. Integral Equations

In this paper it is suggested that there may exist a fundamental relationship between the variables of thermodynamics, the operators associated with certain nonlinear integral equations of statistical mechanics, and the properties of a class of convex functions, called N functions, investigated by Krasnoselskii and Rutickii. In particular, it is pointed out that the most general theoretical framework within which all these problems can be studied is that provided by the theory of Orlicz spaces. In the first part of our study, presented here, it is shown that the existence of solutions to certain nonlinear integral equations, derived either from the BBKYG hierarchy or from the grand partition function using a variational approach, can be established with some generality. The relationship between our results and those obtained by Ruelle is discussed.

Theoretical investigation of fluorescence concentration quenching in two‐dimensional disordered systems. Application to chlorophyll a in monolayers of dioleylphosphatidylcholine

A master equation approach is used for investigating energy transfer and trapping in two‐dimensional disordered systems, where the traps are statistical pairs of pigment molecules closer than a critical distance Rc. Fluorescence decay curves are calculated over a range of concentrations as a function of Rc and the Forster transfer radius R0. The concentration dependence of the lifetimes is compared to the fluorescence self‐quenching data that Chauvet et al. obtained from real‐time measurements in monolayers of chlorophyll a and dioleylphosphatidylcholine (DOL). This dependence is found to be close to second order and for a choice of Rc =10 A the experimental data are fit if R0=78±2 A. This value is in close agreement with those found in the literature from depolarization measurements.

On the Relaxation to Quantum‐Statistical Equilibrium of the Wigner‐Weisskopf Atom in a One‐Dimensional Radiation Field. I. A Study of Spontaneous Emission

A theoretical study of the phenomenon of spontaneous emission has been carried out, using as a model the Wigner‐Weisskopf atom in a one‐dimensional radiation field. The calculation is performed within the framework of the Prigogine theory of nonequilibrium statistical mechanics. When the model is solved exactly to first order in the coupling parameter α and the evolution in time of the diagonal elements of the density matrix ρ is studied, it is found that the relaxation to equilibrium is characterized in part by a sequence of slowly damped oscillations. This result seems to be in agreement with the observation made by Zwanzig, namely, that exponential decay in time seems not to be universal, and may, in fact, be hidden behind some other kind of time dependence. An approximate theory is developed alongside the exact one, and corresponding terms in each treatment are compared numerically. It is found that, for small values of the coupling parameter α (α ≤ 0.1) and for sufficiently large values of τ, defined...

Relaxation to Quantum‐Statistical Equilibrium of the Wigner‐Weisskopf Atom in a One‐Dimensional Radiation Field. III. The Quantum‐Mechanical Solution

A study is undertaken to cast light on difficulties, which arose in the first two papers of this series, pertaining to the occurrence of negative probabilities in the weak‐coupling solution of the generalized Prigogine‐Resibois master equation for the model of the Wigner‐Weisskopf atom in a one‐dimensional radiation field. The Schrodinger equation is solved exactly for the model with the initial condition for spontaneous emission, and then the weak‐coupling approximations to the solution, both for an infinite and for a finite system, are derived as inverse Laplace transform integrals. An extensive analysis, theoretical and numerical, of these is undertaken, and comparison is made with the corresponding results based on the master equation. In particular, quantitative estimates of the Poincare recurrence times for finite systems are made. It is found that considerable differences exist between the statistical‐mechanical and quantum‐mechanical results, but that both manifest nonanalyticity in the coupling p...