Volume 55, Issue 4, 15 August 1971

Mode Expansion in Equilibrium Statistical Mechanics. III. Optimized Convergence and Application to Ionic Solution Theory
View Description Hide DescriptionThe mode expansion is an infinite series for the Helmholtz free energy of a system of classical particles with spherically symmetric interparticle potentials. A systematic procedure is presented for making the mode expansion converge as quickly as possible for particles whose potential includes a strong repulsion. This optimization procedure is physically related to the excluded volume effects produced by the repulsive forces. When the mode expansion is truncated after the one‐mode term (the random phase approximation), the optimized result is related to the spherical modelintegral equation. The optimized mode expansion is applied to the restricted primitive model of aqueous 1–1 and 2–2 electrolyte solutions and compared with the results obtained from Monte Carlo calculations for salt concentrations up to . The comparison indicates that in this concentration range, the optimized mode expansion converges fairly rapidly. Truncation of the expansion for the 1–1 electrolyte after the two‐mode term yields values for the Helmholtz free energy, osmotic coefficient, and activity coefficients which are in almost perfect agreement with the Monte Carlo calculations up to . Accurate values of the excess internal energy are also obtained. For 2–2 electrolytes, the convergence of the optimized mode expansion is not as rapid as that for 1–1 electrolytes. Even so, when truncated after the two‐mode term, the optimized mode expansion for the 2–2 electrolyte predicts configurational energies which differ by at most 6% from the Monte Carlo results for concentrations between and .

Matrix‐Isolation Raman and Infrared Spectra, Molecular Structure, and Force Constants for XeO_{3}F_{2}
View Description Hide DescriptionBoth the Raman and the infrared spectra of low‐temperature matrix‐isolated molecules of XeO_{3}F_{2} are reported. symmetry for the molecules is indicated. The observed fundamental frequencies in cm^{−1} are . Force constants are reported in terms of the orbital valency force field assumption.

Reaction of Hydrogen Atoms with C_{2}H_{4} and C_{2}D_{4}
View Description Hide DescriptionA study of the photolysis of HI in the presence of C_{2}H_{4} and C_{2}D_{4} was carried out at low conversions, in the presence of large pressures of moderator, and at various temperatures. In the experiments with C_{2}D_{4} the formation of small percentage of HD was detected and it was found that the HD/H_{2} ratio provides a very sensitive test of the presence of hot reactions. The observed rate of hydrogen formation was used to evaluate the ratio of rate constants for the following reactions: The Arrhenius parameters for were found to be and, using the previously determined value for , we find . This value is based actually on the value of for H+I_{2} as determined by Sullivan and extrapolated to our conditions. C_{2}D_{4} has been found to react about 10% faster than C_{2}H_{4}.

Influence of Molecular Geometry, Orientation, and Dynamics on Angular Correlation Patterns from Rotationally Labeled Macromolecules
View Description Hide DescriptionIn order to determine the sensitivity of gamma‐ray angular correlation patterns from solute macromolecules labeled with rotational tracers such as ^{111m }Cd, a theoretical study was made of the behavior expected under certain conditions. A nucleus of spin , acted upon by an axially symmetric electric field gradient, and bound to a rodlike macromolecule, was considered. Under static conditions (no molecular rotation), the time‐dependent correlation pattern is quite sensitive to molecular orientation and, for oriented molecules, to the angle between the axis of the field‐gradient tensor and the molecular axis. A general equation and results for selected geometric configurations are given. When molecular rotation is allowed, a classical model is applicable if the rotation is sufficiently slow. This model is used to calculate relaxation curves for several geometrical configurations under the condition that the macromolecules rotate about their long axes. These curves are shown to have considerable diagnostic value. Finally, the applicability of rotational tracers in the light of these results is discussed.

Collinear Collisions of an Atom and a Morse Oscillator: An Approximate Semiclassical Approach
View Description Hide DescriptionThe semiclassical treatment by Heidrich, Wilson, and Rapp of atom–harmonic oscillator collinear collisions with exponential and Morse repulsive potentials, is extended to include collisions, with the same potentials, between an atom and a Morse oscillator. The classical equations of motion for the Morse oscillator system lead to a differential equation which can be solved in the same form as the harmonic oscillator system by an iterative method. It is found that for vibrational transitions between low lying levels and for relative translational energies on the order of several oscillator quanta only one or two iterations are necessary for convergence to within a few percent. The results are compared with a calculation for harmonic oscillator systems.

Numerical Calculation of Vibrational Relaxation and Dissociation for a Quantum Anharmonic Oscillator
View Description Hide DescriptionThe vibrational relaxation and dissociation of gaseous H_{2} highly diluted in an argon heat bath have been studied numerically at temperatures between 2500 and 15 000°K. Collisional transition probabilities between the vibrational levels of H_{2} were calculated by the method of Schwartz, Slawsky, and Herzfeld in a form modified for a Morse oscillator. The master equation was solved by numerical integration to give the time dependence of the vibrational level populations and of dissociated molecules in a simulated shock‐wave experiment. The absolute values of the calculated rate coefficients for dissociation are in good agreement with experiment, but the predicted vibrational relaxation times are about a factor of 35 shorter than those determined experimentally. At all temperatures, the upper vibrational levels in the dissociating gas are substantially depleted below their equilibrium populations. The Arrhenius activation energy for dissociation is 102.5 kcal mole^{−1}, slightly lower than the dissociation energy of 103.2 kcal but higher than the experimentally determined activation energies by about 10 kcal. An incubation period of about 1 to 1.5 vibrational relaxation times precedes the establishment of approximately steady dissociation. The computed incubation behavior is compared with that predicted from the diffusion theory model of Brau, Keck, and Carrier. The vibrational relaxation behavior of this anharmonic oscillator molecule has been examined. In the absence of dissociation, the relaxation time as defined by the Bethe–Teller equation differs from its SHO value by less than 10%, and is found to show variations of less than 10% with time. The present results are also used to examine some approximations made in theories of coupled vibration and dissociation.

Crystal Field Splitting of the Energy Levels of Nd^{3+} and Er^{3+} in CaWO_{4}
View Description Hide DescriptionCrystal field parameters appropriate for symmetry were fit to data on the level splittings of Nd^{3+}:CaWO_{4}. The symmetry assignments of the sublevels of now seem firmly established. The “missing” sublevel in was selected by comparing to existing Nd^{3+}:PbMoO_{4} data. Spectral data which were used to interpret the and sublevel structure are presented. The total rms deviation for the term without g‐value fitting was 9.4 cm^{−1}. When g‐value fitting was included, the rms deviation was 10.7 cm^{−1}. The total rms deviation for the was 7.2 cm^{−1}.

Highly Conducting Ion‐Radical Salts of Tetrathiotetracene
View Description Hide DescriptionThe electrical resistivity and thermoelectric power of the organic semiconductor tetrathiotetracene (TTT) and its chloride, bromide, iodide, and thiocyanate salts are reported as a function of temperature. The ion–radical salts, of nearly 1:1 stoichiometry, were discovered to be highly conductingorganic semiconductors and have resistivity values at room temperature ranging from 2.3 × 10^{3} Ω·cm for TTT–chloride to 0.71 Ω·cm for TTT–iodide. The sign of the Seebeck coefficient of TTT–chloride corresponds to an n‐type semiconductor, while for TTT and the other salts it indicates p‐type behavior. Infrared absorption spectra of the solids, uv–visible spectra of solutions, preliminary x‐ray powder patterns and density measurements are correlated with electrical properties. The influence of constitution of the ion–radical salts on the electrical resistivity and possible mechanisms of conduction are discussed.

Generalized Boltzmann Equation for Molecules with Internal States
View Description Hide DescriptionThe general form of the Boltzmann collision operator is discussed. It is argued that this should be the natural transition (super‐) operator for a binary collision that is defined analogous to the ordinary transition operator in collision theory. Properties of the collision term are discussed, in particular pointing out its general nonhermiticity and nondefiniteness as well as its behavior under parity, rotations, and time reversal. In general, there is no theorem.

ESR Studies of Cu, Ag, and Au Atoms Isolated in Rare‐Gas Matrices
View Description Hide DescriptionElectron spin resonancespectra were observed on Cu, Ag, and Au atoms isolated in Ne, Ar, Kr, and Xe matrices at ∼ 4°K. The hyperfine coupling constants and the values were determined and examined for the matrix effect. With xenon matrices a superhyperfine structure with magnetic xenon nuclei was partially resolved, and by means of computer simulation, it was shown that these atoms are substitutionally incorporated within the Xe lattice. Also an evidence for atom‐vacancy pairing was observed for Cu in Ne matrix.

NMR Relaxation in Cholesterol and Cholesteric Liquid Crystals
View Description Hide DescriptionThe protonnuclear spin lattice relaxation has been measured in cholesterol and its esters (n‐propionate, n‐decanoate, myristate, and oleate) in the solid, mesophase, and liquid regions using conventional pulsed NMR techniques at a frequency of 30 MHz. All solid phases and the mesophases of all the esters except the oleate exhibited relaxation which could be fit within experimental error with a single exponential. Nonexponential relaxation, which was analyzed assuming two relaxation times, was observed in all liquid phases and throughout the mesophase of the oleate ester. Relaxation in the mesophases of all the esters had very little temperature dependence and this, plus its behavior at the phase transitions, indicated that the relaxation mechanism might be similar to that in nematic liquid crystals. It was not possible to determine the primary source of the nonexponential relaxation behavior but it did not require the presence of an ester group.

Entropy Change and Fluctuation in Reaction System
View Description Hide DescriptionThe irreversible entropy change due to a chemical reaction near equilibrium in a closed system is expressed by , where is the Boltzmann constant, and is the degree of advancement. The reciprocal of the factor , according to the stochastic approach to chemical kinetics, is the fluctuation in the number of molecules of reacting species at chemical equilibrium. It is similarly shown that the factor , which Eigen and de Maeyer introduced in the thermodynamicanalysis of chemical relaxation, is equivalent to the fluctuation. The interrelations between chemical affinity, equilibrium reaction rate, and relaxation time are also discussed through these relations to the fluctuation.

F‐Aggregate Centers in Sodium Chloride. I
View Description Hide DescriptionThermal aggregation of electron excess color centers in NaCl, x irradiated at 80°K, was studied by observation of changes in optical absorption in pulse annealing experiments. An initial 80 h x irradiation introduced large concentrations of and positive hole centers. As annealing temperatures were raised (80 → 235°K) recombination of thermally released positive holes with electrons in centers produced centers. Thermal aggregation of centers occurred above 240°K when centers become mobile. centers form and disappear and centers increase rapidly in number near 265°K as a result of the reactions and . centers form by the reaction below 320°K and disappear slowly by the reactions and , above 330°K. Optical absorption band maxima at 80°K are: M^{+}, 1.22 eV (1015 nm); M_{1}, 1.74 eV (713 nm); R^{+}, 1.80 eV (690 nm); R_{1}, 2.28 eV (543 nm) and R_{2}, 2.07 eV (600 nm). N_{1} and N_{2} band systems were observed but were not studied in any detail. centers also were formed by direct ionization of centers by 300‐nm light. The concentration, however, was low because of a counterbalancing of photoionization and electron capture processes. centers were formed in x‐irradiated NaCl by excitation with light absorbed in the R_{2} band. They could be transformed back to centers by x‐irradiation or center bleaching.

Thermodynamics of Microcrystallites and Its Relation to Nucleation Theory
View Description Hide DescriptionWe develop a generalized Einstein model for calculating the thermodynamic properties of small crystalline clusters of atoms. The validity of the model is established through the molecular dynamics calculations by Dickey and Paskin and the numerical normal modeanalysis by Burton for certain cluster packings. The rapid computational procedure and low computer storage requirements for our model calculations enable us to easily compute thermodynamic quantities for cluster configurations and sizes beyond those previously studied. Polyhedron cluster packings and “approximately spherical” cluster packings are used to generate clusters of various configurations. The Einstein theory is used to study the Helmholtz free energy, vibrational energy, and entropy for the various cluster configurations and for cluster sizes up to ∼ 1000 atoms. The model calculations are compared with the predictions based on the capillarity approximation used in nucleation theory. The simplicity of the method enables one to calculate the thermodynamic quantities for other types of systems.

Two Modes of Diffusion in Liquids
View Description Hide DescriptionWe propose that in the case of diffusion in liquids two modes of diffusion are present. The first mode, a flow mechanism, resulting from the movement of the solvent molecules, is the only one available to large diffusants having molecular weights at least an order of magnitude larger than that of the solvent. Smaller diffusants can use this mode but also move by activated diffusion within the lattice of the solvent matrix. The form of the variation of the activation energy for diffusion with size of the diffusant is discussed. Semiempirical calculations of diffusion coefficients and activation energies for several (small to medium‐sized) diffusants in water are performed and the results are compared with available data.

Time Correlation Functions for a Molecular Gas: Magnetic Field Effects
View Description Hide DescriptionTime correlation functions appropriate for the study of thermal conduction and viscous dissipation are considered for a molecular gas in the presence of a static magnetic field (Senftleben–Beenakker effect). The classical Boltzmann equation is used in the formulation although the generalization to the quantum mechanical case is straightforward. The results are illustrated by an explicit calculation for diamagnetic molecules within the framework of a simple model used by Kagan and Maksimov. The time correlation functions which correspond to the nondiagonal components of the thermal conductivity and viscositytensors show the most interesting structure. The results are discussed and further extensions of the present calculations are outlined.

Liquid Mixture Excess Properties and Gas Solubilities by the Hard‐Sphere Model
View Description Hide DescriptionThermodynamic properties of binary liquid mixtures containing simple species are calculated with the hard‐sphere model originally proposed by Longuet‐Higgins and Widom. No liquid mixture data is used. The only mixture parameter employed is , the deviation parameter for the geometric‐mean mixing rule for intermolecular energy parameters. Very encouraging agreement with experiment is observed for both close‐boiling systems and for mixtures containing a supercritical solute. A calculational scheme for mixtures containing the quantum gases hydrogen and helium is also developed with the same model.

Raman Scattering Studies of Liquids and Glasses. I. Liquid and Supercooled Liquid Glycerol
View Description Hide DescriptionThe Raman scattering technique was used to investigate the low‐lying intermolecular modes in liquid and vitreous glycerol from 65–353°K. It was found that while the integrated intensity of the polarized spectrum decreases with decreasing temperature, the depolarization ratio remains more or less constant (≅ 0.68). A large decrease in the scattering intensity occurs near the glass transition temperature. Considering that a normal mode of vibration is responsible for both the ir absorption and the Raman scattering, we have derived an expression which relates the absorption coefficient to the scattering power spectrum. The derivation was obtained with a similar technique used by Kubo and Tomita in their theory of magnetic resonance absorption and relaxation. The derived expression was used to calculate the function (the absorption coefficient divided by the frequency ) from the measured Raman spectrum. It was shown that the so obtained curves can be fitted to within experimental uncertainty by a simple damped‐harmonic oscillator function with a temperature‐dependent damping constant. Except for a transition region at 260–280°K, the temperature behavior of the damping constant derived from the depolarized Raman spectrum can be described in terms of two activation energies. For the temperature region above 285°K, the activation energy corresponds to the hydrogen bondenergy, and for the region below 280°K the activation energy is considered to arise from the anharmonicity in the hydrogen bond potential. The reason for the temperature insensitive Raman scattering cross section in vitreous materials is also given.

Effect of Pressure on the Electronic Structure of Phthalocyanine and Iron Phthalocyanine Derivatives
View Description Hide DescriptionThe electronic structures of phthalocyanine, ferrous phthalocyanine, and the adducts with pyridine, 3‐picoline, 4‐picoline, and piperidine have been studied to 175 kbar pressure using optical absorption and Mössbauer resonance. One observes shifts of the optical absorption peaks to lower energy and changes in relative intensity of peaks which can be explained in terms of changes in the configuration interaction. Fe(II) in ferrous phthalocyanine is in an intermediate spin configuration at all pressures. In the axially coordinated adducts the Fe(II) is low spin at 1 atm. For the pyridine and picoline adducts this is caused by the increased backbonding to the axial ligands in addition to the phthalocyanine and by repulsion due to bonding. These factors both tend to separate the orbitals from the orbitals. In the piperidine adduct only the latter factor is operative. With increasing pressure the low spin compounds convert partially to a new spin state, believed to be intermediate spin. The picolines show the most conversion, the pyridine adduct less, and the piperidine complex still less. The general explanation lies in the decrease in backbonding at high pressure caused by the tendency for occupation of the ligand orbitals by ligand electrons. A similar effect has been observed in ferrocyanides and in phenanthroline complexes. The differences among the various adducts can be explained in terms of differences in the degree of backbonding and bonding.

Effect of Pressure on the Electronic Structure of Protoporphyrin IX, Hemiporphyrins, and Related Compounds
View Description Hide DescriptionStudies have been made on the effect of pressure to 170 kbar on the electronic structure of protoprophyrin IX, several hemiporphyrins, and related compounds using optical absorption and Mössbauer resonance. The optical absorption peaks which correspond to transitions shifted to lower energy and decreased in intensity with increasing pressure. This could be explained in terms of changes in the configuration interaction involving a shift of the electrons to the outer portions of the porphyrin ring. Imidazole protohemichrome is a low spin ferric compound. At room temperature and pressures above about 45–50 kbar it reduced apparently to an intermediate spin (or mixed spin) ferrous state. At 110° it reduced at moderate pressure to a low spin ferrous state, but with increasing pressure the Fe(II) transformed to intermediate spin. Hemin and hematin are high spin ferric compounds. With increasing pressure they reduced apparently to intermediate spin (or mixed spin) ferrous compounds. The amount of reduction with pressure was consistent with the location and shift of the observable metal to ligand‐charge transfer bonds. Two ferric compounds with square pyramidal symmetry like hemin and hematin but with planar neighbors of oxygen or sulfur instead of nitrogen were studied. Both reduced with increasing pressure, in one case with a probable change of spin state. All reductions observed were reversible.