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Volume 57, Issue 7, 01 October 1972

Theoretical Studies of Collision‐Induced Transitions between Molecular Rotational Energy States
View Description Hide DescriptionA method is presented to calculate the rates of collision‐induced transitions on the basis of collision‐broadening theory. The rates are calculated for important collisional transitions responsible for the change in intensity ΔI/I of the signal transition observed in double resonance experiments involving H_{2}CO, HDCO, H_{2}CCO, HCN, and DCN. The values of ΔI/I derived from the calculated transition rates agree satisfactorily with the measured ΔI/I.

Moment Theory Bounds for the Second‐Order Optical Properties of Atoms and Molecules
View Description Hide DescriptionA variety of oscillator‐strength‐distribution frequency moments are employed in conjunction with theorems from the theory of moments to construct bounded estimates for the dynamic polarizabilities and related optical properties of atoms and diatomic molecules. This technique circumvents the explicit construction of an accurate approximation to the dipole oscillator‐strength distribution appearing in the Kramers‐Heisenberg formula, and relates the dispersive optical properties of a system to the simpler frequency moments of its excitation spectrum. The bounds obtained follow from the theory of Tchebycheff systems of functions, although the Stieltjes nature of the polarizability insures that various other related formalisms, including the theory of Padé approximants, Gaussian quadratures, variational principles, and linear programming methods, can also be employed in their construction. Illustrative applications of the moment technique are given for the ground states of atoms and diatomic molecules employing oscillator‐strength sum‐rule moments, which can be determined from the ground statewavefunction of the system, and negative‐even‐integer (Cauchy) moments, determined from both theoretical and semi‐empirical procedures. Highly accurate bounds for the refractivities, Verdet coefficients, and Rayleigh scattering cross sections of atomic and molecular hydrogen and helium are obtained using a small number of sum‐rule moments and the static value of the polarizability. The sum‐rule moment bounds obtained for the optical properties of the heavier inert gases, which exhibit more structured and complex dipole excitation spectra, are somewhat weaker than those for the smaller systems, although the mean values of the bounds are generally in good agreement with experiment. The long wavelength Faraday rotation data for neon appears to be somewhat anomalous, however. Theoretical and semiempirical Cauchy moments for the polarizability components of diatomic hydrogen, nitrogen, and oxygen are employed in the construction of Cauchy moment bounds. The theoretical results for molecular hydrogen, in the time‐dependent Hartree‐Fock approximation, are in good agreement with previous theoretical determinations of the dispersion in hydrogen, and the semiempirical results aid in the interpretation of uv Rayleigh scattering measurements for molecular hydrogen and nitrogen. Comparisons of the Cauchy moment bounds for the molecular Verdet coefficients with measured Faraday rotation data shows that all three diatomic gases satisfy the ``modified'' Becquerel formula. The ease with which bounds for the second‐order optical properties of atoms and molecules can be constructed when the required frequency moments are available suggests that the moment technique should be one of continuing interest, with extensions of the approach to additional linear and nonlinear optical susceptibilities expected to be similarly rewarding.

Optimized Cluster Expansions for Classical Fluids. III. Applications to Ionic Solutions and Simple Liquids
View Description Hide DescriptionIn a previous paper we have derived a number of simple approximations for the thermodynamic and structural properties of liquids. Here we test these approximations by applying them to models for ionic solutions and atomic liquids and comparing the results with those obtained from Monte Carlo and molecular dynamics simulations. In particular, the pair‐correlation functions for 1–1 primitive model aqueous ionic solutions are calculated using the EXP approximation. The internal energy for 2–2 salts is obtained from the approximation. The pair‐correlation function for the Lennard‐Jones fluid is calculated with the EXP approximation. In all cases the results obtained agree closely with those obtained from computer experiments. Thus, the approximation for the free energy and the EXP approximation for the pair correlation function provide a theory that is both very accurate and applicable to a wide variety of classical fluids.

Infrared Spectrum of Matrix‐Isolated Nitrous Oxide
View Description Hide DescriptionNitrous oxide, generated by the in situ thermal decomposition of hydroxylamine hydrochloride or ammonium nitrate, was isolated in some pressed alkali halide matrices. Its three infrared active fundamentals appeared at room temperature as single peaks and without the gas phase rotational structures. Cooling of the host matrix caused these bands to sharpen and to shift to higher frequencies. At any given temperature, however, the peak frequencies were essentially independent of the host matrix. The temperature dependence of the most intense ν_{3} fundamental was examined in detail, and the results indicated that there were two dominant trapping sites for N_{2}O in pressed KCl, KBr, and KI matrices. Furthermore, the relative concentrations of N_{2}O occupying these sites changed reversibly as the temperature was varied from about 90°K, our lower limit, to over 200°K. From these concentration changes, it was deduced that the transfer of a N_{2}O molecule from the low temperature trapping site to the high temperature site was an endothermic process by and was accompanied by a large increase in entropy of in all three potassium halide matrices.

Low Temperature Phase Transition in TlN_{3}. Infrared Absorption and Raman Scattering Study
View Description Hide DescriptionThe long wavelength modes of tetragonal (Phase I) TlN_{3} in the frequency range 30–2500 cm^{−1}, have been observed and assigned. The low frequency spectrum becomes fairly complex at low temperatures due to an orthorhombic distortion of the Phase I lattice. The possibility of a non‐centrosymmetric structure is indicated by an apparent breakdown of the rule of mutual exclusion for some of the lattice frequencies. The phase transition has been found to involve a distortion of the unit cell at followed by a distortion of the azide ions at The latter temperature is close to the value observed by differential thermal analysis. The low frequency Raman spectrum has been measured at different low temperatures and the temperature dependence of the frequencies are discussed in terms of microscopic approaches to the phase transition problem.

Correlation Effects and Molecular Tumbling in NMR Studies of Solid β‐(CH_{3})_{4}Si
View Description Hide DescriptionThe β phase of high purity samples (99.9% min) of tetramethylsilane (TMS), (CH_{3})_{4}Si, was investigated between 77°K and its melting point (174°K) employing static and rotating frame proton spin‐lattice relaxation (T _{1} and T _{1p }, respectively) measurements. Minima in the T _{1} curve near 90°K and in T _{1p } at 171°K correspond to methyl group reorientation and over‐all molecular tumbling, respectively. Activation energies and inverse frequency factors for both motions are derived and discussed. Deviations from exponential behavior in the longitudinal relaxation were noted below 140°K and are inferred to be due to a correlation in the relative motion of the protons within each methyl group. The exponentiality of the spin‐lattice relaxation above ∼140°K is associated with the onset of over‐all molecular tumbling motions. It is suggested that the presence of exponential proton magnetization decays in solids containing methyl groups may sometimes serve as an indication of the presence of reorientations of the molecular frame to which the methyl groups are attached. The possibility that the existence of exponential and nonexponential decays may be used by itself to gain information on interactions and motions in solids, has to be further investigated.

Variable Trap Problem in Isotopic Mixed Crystals of s‐Triazine
View Description Hide DescriptionTheoretical treatments of isotopic mixed crystals, in general, have assumed that the intermolecular interactions are equal and that the problem can be reduced to a consideration of the free molecule excitation energies. It is shown that for the isotopic s‐triazines the intermolecular interactions are different and that now it is necessary to consider a trap depth which is both concentration and distribution dependent. This has necessitated a modification of the existing theories. Experimental results have been obtained over the entire concentration range for the s‐triazine_{ d‐0}/s‐triazine_{ d‐3} mixed crystal system, and the results have been qualitatively explained in terms of a concentration dependent trap depth.

Kinetic Energy Release in Unimolecular Ionic Reactions. Thermochemical Aspects
View Description Hide DescriptionThe kinetic energy (T) released in the unimolecular reactions of metastable gaseous ions has been compared with literature values obtained by several methods for the same reactions occurring in the ion source. The internal energy difference between the long‐lived and the short‐lived reactant ions has a relatively small effect upon the kinetic energy release when rearrangement and elimination reactions are studied, large kinetic energy releases being observed by both methods. Simple cleavages, which are typically accompanied by far smaller energy releases, show large relative variations when the time‐scale of the reaction is changed with the values measured in the field‐free region being much lower (e.g., 0.002 eV vs 0.18 eV in the case of H loss from propane). These and other results provide strong evidence that metastable reactions occur from ions whose average internal energies are only slightly in excess of the activation energy for the process in question. The consequences of this result for studies on energy partitioning by mass spectrometry and for improving thermochemical data on ions are stressed and illustrated. Franklin has emphasized the importance of T values, extrapolated to ions of threshold internal energy, as a means of correcting for the excess energy of the activated complex and so improving the quality of thermochemicalmeasurements obtained by mass spectrometry. Since the metastable technique is directly applicable to ions of approximately threshold energy, no extrapolation is necessary, and a rapid and simple method of obtaining improved thermochemical data is therefore available. It is further suggested that these results should also provide a basis for correcting measured appearance potentials for the excess energy term arising from the presence of a reverse activation energy.

Quantum and Anharmonic Effects on the Temperature and Mass Dependence of Rate Processes in Solids
View Description Hide DescriptionThe contributions to the temperature and mass dependence by quantum statistical effects in the harmonic approximation are shown to give rise to terms proportional to T ^{−2n } and m^{−n} , respectively, where It is shown, explicitly, that anharmonic terms make no contribution to the mass dependence in the classical case, except in second order. An analysis of the temperature dependence of the pre‐exponential part of the diffusivity, for one atomic mechanism, including anharmonic effects, reveals, for Cu, that deviations from a constant value occur only at high temperatures in the classical case. On the other hand, quantum effects cause a positive curvature at low temperatures. The very small variation of D _{0} with temperature provides a proof that Arrhenius‐type behavior is nearly correct in the classical case despite anharmonic contributions to D _{0}.

Electron Paramagnetic Resonance of Gd^{3+} in Single Crystals of Various Rare‐Earth Ethyl Sulfates at Room Temperature and at 77°K
View Description Hide DescriptionElectron paramagnetic resonance studies of Gd^{3+} in single crystals of (where Ln^{3+} denotes La^{3+}, Pr^{3+}, Nd^{3+}, Sm^{3+}, Tb^{3+}, Dy^{3+}, Ho^{3+}, Er^{3+}, Yb^{3+}, and Lu^{3+}) have been carried out at X band at room temperature and at 77°K. The spin Hamiltonian parameters and and have been evaluated at both temperatures for each host except terbium ethyl sulfate at 77°K. The absorption linewidths for each transition have been measured at both temperatures. The present results are compared with previous results. A principal conclusion is that the magnitude of for Gd^{3+} in the various rare‐earth ethyl sulfates is more nearly a linear function of the ionic radius of the trivalent host ion than of its atomic number.

Interaction of Sound with Gas Phase Reactions
View Description Hide DescriptionReaction and sound propagation are coupled in ideal gas phase reactions which have a difference in the sums of stoichiometric coefficients of reactants and products, or which have a pressure‐dependent rate coefficient. Numerical and approximate analytical (WKB‐type) solutions are obtained for the hydro‐dynamic equations which take into account this coupling and which describe the spatial and temporal variations in density, velocity and concentration for an isothermal system far from equilibrium. Specific results are given for a unimolecular reaction of the type , where sound is shown to be amplified in both high and low pressure regions, with an accompanying change in frequency; the system is unstable to small perturbations. Possible applications include a new method for the determination of rate coefficients by acoustic means.

Monte Carlo Calculation of ESR Line Shapes in the Slow Motional Region
View Description Hide DescriptionThe ESR line shape in the slow modulation region is calculated from the Kubo‐Anderson theory by using a Monte Carlo technique for averaging the relaxation function. The problem of spurious peaks is solved by the means of a window technique. It is explicitly shown how to sample the relaxation function. The method is applied to axially symmetric and asymmetric secular gtensor and pseudosecular terms modulated by isotropic rotational diffusion. Other interactions, e.g., nonsecular terms, as well as other types of modulation may be included.

Existence of the Dielectric Constant in Rigid‐Dipole Fluids: The Direct Correlation Function
View Description Hide DescriptionThe question of whether the dielectric constant ε exists (is well defined) for a finite fluid system of rigid dipolar molecules is reconsidered and reformulated. It is found that this question can most simply be expressed in terms of the behavior of the position‐ and orientation‐dependent directcorrelation functionc(r_{1}, ω_{1}; r_{2}, ω_{2}). It is shown that ε exists if c satisfies the following two conditions: (a) for , where φ is the dipole‐dipole potential and σ is a length which is large microscopically but small macroscopically. (b) c(r_{1}, ω_{1}; r_{2}, ω_{2}) is of the form for , where e(ω) is the unit vector with orientation ω. An explicit (and new) expression for ε in terms of c is automatically obtained; its applicability is ensured if the above conditions are satisfied. These results lend new intuition and insight into the question of the existence of ε, and suggest a promising approach for future investigations of this question.

Rotational Diffusion Model with a Variable Collision Distribution
View Description Hide DescriptionStarting with an m‐diffusion model a matrix description is given of the rotational motion of a dipole molecule undergoing frequent collisions. This treatment gives rise to an analytical expression for the dipole correlation function and for the angular momentumcorrelation function in which a limited number of parameters from the model appear. It is argued that the collision distribution which determines the rotational diffusion process need not necessarily be a Poisson distribution. In liquids with strong interactions the distribution is governed by the frequency distribution of the medium. This leads to the inclusion of a librational motion in the rotational diffusionmodel. A comparison of simulations with different collision distributions and experimental data is given.

Model for the Repulsive Elastic Scattering of Atoms by Solid Surfaces
View Description Hide DescriptionA model is developed, enabling the intensities of diffracted beams to be calculated in closed form, starting from a pairwise (Yukawa‐type) interaction law between a gas atom and a solid atom. The treatment is more satisfying than most previous treatments, which calculate the diffracted intensities as functions of empirical parameters describing the over‐all gas‐solid interaction, without reference to the microscopic (pairwise) interaction. The model is illustrated by an application to the He–LiF system.

Dissociative Attachment of Thermal Electrons to N_{2}O and Subsequent Electron Detachment
View Description Hide DescriptionThe attachment of electrons to nitrous oxide has been studied, following pulse ionization, by a microwaveconductivity technique. For pure N_{2}O, and N_{2}O diluted with N_{2}, extrapolation of the experimentally observed attachment rate constant to zero pressure gives an upper limit of for the rate constant for two‐body attachment at 298°K. When C_{2}H_{6} or C_{2}H_{8} are used as diluents a higher two‐body rate constant of is found which is believed to represent the rate constant of the dissociative attachmentreaction. The much lower value for pure N_{2}O or mixtures is explained by the regeneration of electrons, via the reactions and , which largely nullifies the primary attachment step. The presence of a hydrocarbon prevents electron regeneration by irreversibly reacting with O^{−} and allows observation of the uncomplicated rate constant for dissociative attachment. The rate constant for dissociative attachment increases markedly with increasing temperature and in the range studied, 278 to 355°K, has an activation energy of . Confirming evidence for the proposed mechanism is obtained from the effect, on the attachment rate constant, of H_{2} added to mixtures. The H_{2} reacts with O^{−} by the associative detachment reaction to nullify the primary attachment step and to reduce the rate constant by an appropriate amount. In the steady‐state radiolysis of N_{2}O, the yields of the products N_{2} and NO are found to increase markedly with decreasing pressure below 50 torr. These increases in yields may be quantitatively related to the chain reaction resulting from dissociative attachment followed by electron regeneration.

Lattice Dynamics of M_{2}XY_{6} Hexabromometallates: Cs_{2}UBr_{6}
View Description Hide DescriptionA seven parameter model based on a modified Urey‐Bradley potential with long range Coulomb terms is used to describe the lattice dynamics of M_{2}XY_{6} compounds. Optical data are used to evaluate the force constants for the case of Cs_{2}UBr_{6}. The model is to be used as a starting point for a comprehensive calculation of the vibronic structure of Cs_{2}UBr_{6}.

Raman Spectra of Matrix Isolated Molecules: Application to the Force Field for CCl_{4}
View Description Hide DescriptionSmall isotopic frequency shift information, if precisely determined, provides an effective constraint on intramolecular force fields. The Raman spectrum of the ν_{4} mode of matrix isolated CCl_{4} exhibits a splitting pattern which is interpreted in terms of the chlorine isotope effect. These data are employed in assigning approximate limits to the crucial stretch‐bend interaction force constant (F _{34}). This force field is compared to potential functions derived by other investigators in which F _{34} is fixed by mean square amplitude information and relative Raman intensities. The Raman matrix spectra of CCl_{4} were recorded with a general purpose cryostat which was designed for use with a Cary Model 81 spectrophotometer equipped with an argon ion lasersource.Spectra were obtained from samples with dilutions of CCl_{4} in argon in mole ratios of 1:400 to 1:600.

Natural Bifurcation Coordinates for Three‐Dimensional Chemical Reactions
View Description Hide DescriptionThree‐dimensional atom‐diatom collisions which can lead to two possible rearrangement products are considered under the restraint of near‐linear intermediates. A coordinate system (natural bifurcation coordinates) is introduced to provide for smooth internal rotation of the system in the region of reaction path bifurcation; thus systems are allowed to ``wobble'' from one reaction path to the other. For both two‐ and three‐dimensional reactions, the Hamiltonian operator is derived from the classical kinetic energy.

Hindered Rotor States for the Planar Reaction
View Description Hide DescriptionA numerical procedure for solving the rotational differential equation for the planar reaction in natural difurcation coordinates is described. The procedure has the advantage that it may be applied uniformly over the entire range of the reaction coordinate. The symmetry of the differential equation is utilized to simplify the form of the wavefunctions, and their asymptotic behavior provides a state labeling criterion. The conditions under which any pair of energy levels may become either identically or nearly degenerate is investigated. It is shown that as the reactants approach the transition state (s=O), the rotational wavefunctions smoothly pass through forms characteristic of free rotation, hindered rotation, and finally of a bending vibration. At s=O, equal probability density is shown to be highly localized at the entrance of each rearrangement channel.