Volume 39, Issue 12, 15 December 1963

Unitary Operator Formalism of the Eigenvalue Problems
View Description Hide DescriptionUtilization of the parameters involved in the system under consideration leads to a unitary operator formalism for dealing with eigenvalue problems. The existence of a unitary linear operator which transfers a known function, being predetermined for a special choice of the parameter, to the general eigenfunction is demonstrated and its properties are discussed. The related hypervirial theorems for variational Hellmann—Feynman wavefunction are discussed. The scaling problem and virial theorem are treated in some detail by means of the unitary operator techniques.

On the Density—Intensity Calibration in Electron Diffraction Studies
View Description Hide DescriptionA general formulation for the dependence of the optical density on exposure in electron diffraction photographs is given, and an analytical function for the density—intensity calibration is expressed in terms of observable density data.

Electron Diffraction Study of the Molecular Structures of Sulfur Tetrafluoride (SF_{4}) and Thionyl Tetrafluoride (SOF_{4})
View Description Hide DescriptionThe molecular structures of SF_{4} and SOF_{4} in the gaseous state were determined by electron diffraction using sector‐microphotometer data. Both molecules were confirmed to have trigonal bipyramidal structures (C _{2v } symmetry), with two nonequivalent sets of F–S bond distances. The bond lengths, in terms of the center of gravity parameter, r_{g} , and the bond angles are as follows:
The polar F–S–F angles are measured along an arc which bisects the equatorial FSF angle.

Isotope Effects on Internal Frequencies in the Condensed Phase Resulting from Interactions with the Hindered Translations and Rotations. The Vapor Pressures of the Isotopic Ethylenes
View Description Hide DescriptionInteraction of the hindered translations and rotations with the internal vibrations in the condensed phase leads to isotope‐dependent shifts of the internal frequencies. This isotope dependence is a necessary consequence of the fact that the coordinates representing the translations and rotations of the molecule as a whole are, in general, isotope dependent. This external—internal interaction is investigated and the XH_{2} system is employed as a simple example to demonstrate the nature of the interaction. It is shown that the experimentally determined vapor pressures of the isotopic ethylenes may be rationalized when the external—internal interaction is taken into account. A force field based on a simple cell model for liquid ethylene is obtained which yields good agreement with the experimentally determined vapor‐pressure isotope effects. The force field, while not unique, fits (and is in fact partially based on) other available data on liquid ethylene.

Statistical Mechanics of Nonspherical Molecules
View Description Hide DescriptionA general approach to the calculation of the properties of classical fluids made up of nonspherical molecules is discussed. It is assumed that the total interaction energy is a sum of pairwise interactions which are arbitrary functions of the molecular separations and orientations. The pairwise interaction energy, and the two‐molecule correlation functions are written as series in the complete set of orthonormal functions of the molecular orientation angles, with coefficients which are functions of the distance of separation. Explicit expressions are given for the thermodynamic properties of these systems, and for the dielectric constant and rotational friction constant. Equations for the coefficients in the series for the pair correlation function are obtained by using a coupling parameter which determines the magnitude of the nonspherical part of the potential. Finally, it is shown that the number and kind of terms which appear in the series for the interaction energy and the pair correlation function are determined by the symmetry of the molecules. In this way, one can bring about a considerable reduction in the complexity inherent in the calculation of the ensemble average properties of molecules having arbitrary shapes.

Systematics of Vibrational Relaxation
View Description Hide DescriptionA large number of data points for the vibrational relaxation time (pτ_{ v } in atm sec) of simple systems have been logarithmically plotted vs (T°K)^{—⅓}. It appears that each system is well represented by a straight line, and that most of these straight lines when extended to higher temperatures intersect near the point [pτ_{ v }=10^{—8} atm sec, (T°K)^{—⅓}=0.03]. Systems with a small reduced mass μ are exceptions to such a simple convergence, and in an improved scheme, the location of the convergence point is dependent on the reduced mass. Such a presentation has lead to an empirical equation correlating available measurements of vibrational relaxation times:, where θ is the characteristic temperature of the oscillator in K deg. This equation reproduces the measured times within 50% for systems as diverse as N_{2}, I_{2}, and O_{2}–H_{2}. In the worst case thus far, O_{2}–Ar near 1000°K, it is off by a factor of 5.

Boltzmann Equation and Inverse Collisions. II. A Model for Association and Dissociation
View Description Hide DescriptionThe techniques of classical nonequilibrium statistical mechanics are applied to a gas which exhibits two distinct types of molecular interaction. The first allows chemical bonding to the extent of dimer formation; the second, impulsive in nature, provides a tractable mechanism for three‐body energy exchange. Hydrodynamicequations are derived for each of the two quasicomponents. A formal description of the reaction kinetics is developed which leads to a perturbation method for obtaining corrections to a theory based on an assumption of uniformly canonical distributions.

Statistics of Irreversible Termination in Homogeneous Anionic Polymerization
View Description Hide DescriptionThe effect of irreversible termination on the molecular weight distribution of a polymer prepared by homogeneous anionic polymerization is examined for cases in which the initiator is monofunctional and initiation is instantaneous. A general procedure for the calculation of the chain length distribution, as well as its moments, is presented. The procedure is then illustrated by an exact treatment of an isothermal batch polymerization for which we obtain the ratio r _{σ}(t) of the weight to number average molecular weights as a function of the fraction of monomer consumed, the initial ratio σ of the terminator concentration to that of the initiator, the ratio ρ of the rate constant for termination to the rate constant for propagation, and the average degree of polymerization at time t, 〈n ^{1}〉_{ t. }
When σ is much less than unity, and the limiting number average D.P. 〈n ^{1}〉_{∞} is very large, r _{σ}(t) approaches the following value as the reaction nears completion: r _{σ}(∞) = 1+σρ(2+ρ)^{—1}.

Separation of Gas Mixtures in Supersonic Jets
View Description Hide DescriptionThe supersonic jets resulting from the free expansion of nitrogen—hydrogen mixtures from small sonic nozzles were carefully studied by means of pressure and composition probing. The apparent concentration of heavy species in the core of the jet, originally observed by others, was confirmed. The mechanism by which this separation occurs is shown to be a consequence of the presence of the probe. The deceleration occasioned by the bow shock on the probe is suggested as the cause of the separation. The previously suggested mechanism of radial migration of light species from the jet axis does not seem to be applicable to the present results.

Survey of the Spectra of the Divalent Rare‐Earth Ions in Cubic Crystals
View Description Hide DescriptionThe rare‐earth ions may exist in the divalent state in suitable host crystals such as CaF_{2}. All of the trivalent ions from Ce to Yb (and probably also La) are reduced in situ to the divalent state in CaF_{2} by gamma irradiation. The spectra of most of these ions show that the ground and first few excited states derive from f^{n} configurations, but the weak absorption due to these is masked at higher energies by strong broad bands of the parity‐permitted f^{n} →f^{n}—1 d transitions. The excitation energy of these spectra have been calculated in a first approximation as the energy difference between the ``Hund rule,'' single‐determinant states of the configurations f^{n}—1 d and f^{n} . This procedure satisfactorily accounts for the remarkable variations in the excitation energy in passing from one ion to the next in the series with the exception of Gd^{++}, Ce^{++}, and Tb^{++}. Gd^{++} probably has f ^{7} d for its ground configuration, while Ce^{++} and Tb^{++} are borderline cases. The spectral structure probably arises chiefly from the crystal‐field splitting of the d orbital, since each ion in CaF_{2} has a similar spectrum, and the spectra change drastically in sites of other than cubic symmetry.

Smooth Curve Approximation to the Energy‐Level Distribution for Quantum Harmonic Oscillators
View Description Hide DescriptionIt is pointed out that the distribution of energy levels for a set of quantum oscillators may be expressed exactly as the inverse Laplace transform of a function closely related to the quantum partition function. This observation leads quite easily to a smooth curve approximation for the energy‐level distribution which can be expressed in terms of a generalized Bernoullipolynomial. A connection between this approximation and those given by Schlag and Sandsmark and by Whitten and Rabinovitch is established. A detailed comparison with Whitten and Rabinovitch's results is made for the special case of degenerate oscillators.

Investigation of Early Ionization Processes in Shocked Xenon
View Description Hide DescriptionMeasurements of ionization rates in noble gases have been of considerable interest, with much of the recent experimental work performed at very high temperatures with shock‐tube techniques. As a result, there have been several conflicting theories formulated to trace the history of ionization phenomena. A major difficulty has been the problem of precisely identifying the earliest stage of the ionization process. Since the probability of atom—atom collisions leading to ionization is small, another source of energy sufficient to promote the necessary early stage of the process must be provided. It appears that under conditions of relatively low temperatures, as in weak shock waves, the assumption of a photoexcitation mechanism of the gas is plausible. Biberman and Veklenko have suggested such a mechanism for the production of excited states of rare gases in shock waves. This mechanism has been used to explain the initial shock‐front ionization. The nature of the processes responsible for visible emission at the shock front, which has been designated as impurity radiation, appears to be a result of the transfer of energy from the excited rare‐gas atoms to the impurity molecule.
Spectral observations of the shock zone in xenon have been made in a carefully outgassed shock tube (10^{—6} mm Hg). The absorptionspectrum is typical of molecular fragments of carbon (C_{2}) which has been shown earlier to be localized in the region of 4500 to 4960 Å. Calcium or sodium, usual alkali impurities, were not observed in any measurement. The emission appeared to be predominantly from the xenon and possible carbon impurities.
Microwaveabsorptionmeasurements were made normal to the shock front. By use of a plasma model which assumes a linear varying electron density, microwave attenuation in the region of cutoff was measured to determine the rate of ionization and the plasma collision frequency. The observed ionization rate was of second order with respect to xenon pressure for the temperature range of 4000° to 9000°K. The observed activation energy was 1.60 eV±0.2. This was the approximate activation energy for the following reactions in xenon:The experimentally determined rate expression for ionization is given asRadiation at the shock front appears to be best described by the excitation of impurity or by the dissociation of Xe_{2} as suggested by Tanaka:

Analogies between Higher‐Order Analytical and Numerical Perturbation—Iteration Methods
View Description Hide DescriptionAn analogy is pointed out between Hirschfelder's perturbation—iteration method for the determination of eigenvalues and eigenfunctions of linear differential operators and Roothaan's numerical procedure applicable to finite matrices. An analytical procedure analogous to Ostrowski's third‐order matrix method is shown to lead to unmanageable differential equations.

Reaction between Nitric Oxide and Ozone in a Supersonic Nozzle
View Description Hide DescriptionThe rate of reaction between nitric oxide and ozone in a supersonic nozzle was measured in the temperature range 245–345°K by following the absorption of ultraviolet radiation by ozone at 2537 Å. The specific rate constant was found to beThe result is in good agreement with previous studies and confirms the bimolecular nature of the reaction. The experimental pre‐exponential factor is in satisfactory agreement with one calculated from transition state theory.

Dielectric Dispersion of a Polar—Nonpolar Liquid Mixture Near the Critical Solution Temperature
View Description Hide DescriptionThe dielectric dispersion of the mixture nitrobenzene and 2,2,4‐trimethylpentane has been measured at a single concentration over a temperature range from 40°C to the temperature of turbidity and at wavelengths of 575 m, 1.26, 3.22, and 10.0 cm. It was found that the dispersion can be characterized by a single relaxation time over most of the temperature range but that a distribution of relaxation times was necessary near the critical solution temperature. The distribution was also found to broaden as the temperature was lowered. The magnitude of the relaxation time was found to tend to a maximum value as the critical solution temperature was approached. The variations of the static dielectric constant and the density were also measured and found to exhibit the usual maxima. The data obtained are discussed in a qualitative manner in terms of cybotactic regions existing within the liquid and similarities between the dielectric and structural relaxation processes are pointed out.

Upper and Lower Bounds for Ground‐State Second‐Order Perturbation Energy
View Description Hide DescriptionThe first‐order perturbationequation, using the Dalgarno—Lewis formulation, is arranged in a form analogous to the Poisson equation for the electrostatic potential produced by a charge distribution in a medium of variable dielectric constant. Thus, the Thomson and Dirichlet variational principles of electrostatics can be used to obtain approximate solutions to the first‐order perturbationequation for systems in either the ground state or the lowest energy state of a given symmetry. The Thomson principle provides a useful lower bound to the second‐order perturbation energy. The Dirichlet principle is derivable from the Rayleigh—Ritz or Hylleraas principles and gives an upper bound to the second‐order energy. For excited states, the Sinanoğlu principle provides the upper bound. By optimizing the scaling of the trial perturbed wavefunction, the Hylleraas principle is presented in a somewhat improved form. As an example, the polarizability of atomic hydrogen is used to illustrate both the Thomson and Dirichlet principles and to place upper and lower bounds on the polarizability.

Polyhedral Clathrate Hydrates. V. Structure of the Tetra‐n‐butyl Ammonium Fluoride Hydrate
View Description Hide DescriptionThe structure of (n‐C_{4}H_{9})_{4}N^{+}F^{—}·32.8 H_{2}O is tetragonal, of space group P4_{2}/m with the unit‐cell dimensions a=23.52 Å and c=12.30 Å. The clathrate host structure is similar to that of the and consists of a tetragonal pseudo body‐centered arrangement of groups of five face‐sharing pentagonal dodecahedra interlinked by tetrakaidecahedra and pentakaidecahedra. Whereas in the benzoate there are four salt molecules per unit cell, in the fluoride there are five. This difference in the number of guests accommodated in essentially the same host structure is possible because the four tetrakaidecahedra occupied by the benzoate groups are available for the four alkyl groups of the additional cation. In the fluoride hydrate, four of the cations have the central N^{+} atom at the fourfold positions (j) and that of the fifth is disordered over the twofold position (f). Thus the structure has lower symmetry than the benzoate hydrate, which is P4_{2}/mnm by reason of the disorder of the four cations over eightfold positions.
The fluoride ions replace water molecules in the host lattice, with F^{—}···O distances of 2.80 to 2.87 Å forming an anionic host lattice. As in the other clathrate hydrate structures of this type, some of the pentagonal dodecahedra which are distorted by the presence of the cations are occupied by a guest species presumed to be additional water molecules.

Polyhedral Clathrate Hydrates. VI. Lattice Type and Ion Distribution in Some New Peralkyl Ammonium, Phosphonium, and Sulfonium Salt Hydrates
View Description Hide DescriptionThe crystal data and preparation are discussed for 22 new peralkylated ammonium, phosphonium, and sulfonium salt hydrate crystals. Although these structures have not been determined in detail, it is possible to classify them according to the host‐lattice type and to draw conclusions by analogy with known structures concerning the order and disorder in the distribution of the cations and anions with respect to the water lattice.

Forces in Molecules. III. Analysis of an Empirical Potential Function
View Description Hide DescriptionThe oscillating potential function reported in Part II of this series is analyzed in detail for all diatomic molecules for which sufficient experimental data exist. The potential function is calculated and compared to the experimental curve for several internuclear positions on either side of the equilibrium position. Criteria are given for determining the region of usefulness of this function. Some further discussion of the possible physical significance of the oscillating potential is also included.

Hyperfine Electron Spin Resonance Spectrum of the NH_{2} Free Radical
View Description Hide DescriptionThe unrestricted Hartree—Fock method in the LCAO—MO approximation is used to calculate the coupling constants for nitrogen and hydrogen in the NH_{2} free radical. Both Slater and self‐consistent field orbitals are used as basis functions. The effect of a spin annihilator after energy minimization, and the effects of ``freezing'' the nitrogen core are examined. The results are compared with those obtained in a recent configuration interaction calculation; and with the experimental results.