Volume 51, Issue 12, 15 December 1969

Collision Dynamics and the Statistical Theories of Chemical Reactions. I. Average Cross Section from Transition‐State Theory
View Description Hide DescriptionBy recasting the transition‐state‐theory formula for the reaction rate of a general bimolecular reaction and making use of Laplace transform techniques, an expression for the average reaction cross section as a function of the total system energy is derived; the result for three‐dimensional collisions have been given previously by Marcus. For the collinear model, a reaction probability replaces the cross section. Cross sections and probabilities for a number of special cases are given. It is shown that under certain conditions, improper high‐energy limits result. The relation of the present conclusions to unimolecular rate theory is indicated.

Theory of Reactive Scattering. I. Homogeneous Integral Solution Formalism for the Rearrangement τ‐Operator Integral Equation
View Description Hide DescriptionThe integral equation for the rearrangement operator is used as the basis for a discussion of reactive scattering. The concept of the amplitude density introduced by Johnson and Secrest is extended to reactive collisions by means of the operator. The homogeneous integral solution method of Sams and Kouri is used to develop a noniterative solution to the integral equation satisfied by the reactive‐scattering amplitude density. This method is characterized by very stable behavior and is capable of quite accurate solutions with a relatively large step size. We consider the formulation of the method first for a general reactive collision. Both open and closed channel contributions to the reactive scattering amplitude density are considered. We employ the Feshbach projection operator formalism to treat the effects of complex formation due to virtual excitation of internal degrees of freedom. After discussing the general reactive collision, we next deal with the specific problem of atom–diatom reactions. Notation and techniques developed by Miller are used to treat problems associated with coordinates. Unlike Miller's discussion of reactive scattering, the present approach does not require inclusion of square integrable functions since the integral equation for the operator (and the equation for the amplitude density) already takes explicit account of the effects described by such terms. Further, the present approach allows one to easily study the effects of vibrational and/or rotational excitation on the reaction rate since the initial state appears explicitly as an inhomogeneity in the integral equation for the reactive scattering amplitude density. Finally, we discuss the manner in which the matrix may be obtained from the reactive scattering amplitude density. It results that the reactive matrix may be expressed in terms of the nonreactive matrix.

Observations on ESR Linewidths and Concentration Measurements of Gas‐Phase Radicals
View Description Hide DescriptionThe factors affecting linewidths of gas‐phase atoms and free radicals as observed under various conditions with the usual type of ESR spectrometer are reviewed. The significance of the linewidth in the problem of determining relative concentrations of labile species from their ESR signal heights is emphasized, which is a matter of great importance in chemical kinetic applications. For all species (except S‐state atoms) in dilute concentration, the true unsaturated linewidth is simply determined by the total pressure, and several experimental examples of this are given. Relative concentrations of such species at constant pressure may thus be easily monitored by their signal heights. For S‐state atoms (ground‐state H and N being the most important examples) in a nonparamagnetic diluent, the true linewidth depends on the atom concentration via spin‐exchange collisions, so that more caution is necessary in monitoring them. The practical considerations for handling this problem are discussed. The H–H spin‐exchange cross section was measured to be 18 Å^{2}, which is shown to be in good agreement with the theoretical value as estimated by the method of Wittke and Dicke. The D–D cross section was found to be the same within experimental error. Other cross sections evaluated from linewidth measurements are (in angstroms squared) . A significant dependence of on the state of Cl was observed. The factors affecting the spin–lattice relaxation time are also noted, and it is shown that for H atoms the controlling step is probably diffusion to the walls of the flow tube. Values of in the apparatus employed were about 50 × 10^{−6}, and the difference between this and the much longer value reported in the literature for N atoms is discussed.

Paramagnetic Anisotropy and Two‐Dimensional Antiferromagnetism in Copper Formate Tetrahydrate
View Description Hide DescriptionThe paramagnetic anisotropies of the single crystals of copper formate tetrahydrate have been measured between 300 and about 80°K. The distinct layer structure parallel to the (001) plane make this crystal a good example of a two‐dimensional antiferromagnetic lattice. While the average moment is lower than the normal value of , the crystal shows in this temperature range a molecular anisotropy which is typical of normal copper compounds. Although the molecular moment in the plane of the layer is only about compared to an almost normal value of normal to it, the temperature variation of the principal molecular susceptibilities show that both and obey a Curie–Weiss law with almost the same Weiss constant, and . This result is in agreement with the theory of a two‐dimensional antiferromagnetic lattice, but disproves the earlier belief that the temperature variation of and should be quite different, in this case obeying a Curie law and obeying a Curie–Weiss law with a large Weiss constant. The temperature variation of and within this temperature range is quantitatively explained using the high temperature series expansion solution for a two‐dimensional Heisenberg antiferromagnet.

Polyatomic SCF Calculations Utilizing Anisotropic Basis Sets of Slater‐type Orbitals
View Description Hide DescriptionFully optimized SCF wavefunctions using minimum Slater‐type basis sets have been obtained for BH_{3}, NH_{3}, C_{2}H_{2}, C_{2}H_{4}, HCN, and H_{2}CO. Calculations employing anisotropic minimum basis sets are reported for NH_{3}, H_{2}O, HCN, and H_{2}CO. The lower SCF energies of the calculations for anisotropic sets are primarily due to a reduction in electron–electron repulsion. Improvements in calculated dipole moments occur when independent p‐orbital components are used to describe lone pairs and covalent bonds. A geometry‐optimized calculation for H_{2}O is also reported.

Dependence of Generalized Oscillator Strengths of Ethylene on Momentum Transfer
View Description Hide DescriptionGeneralized oscillator strengths are calculated as a function of scattering angle or , the momentum transfer, for seven electronic transitions in ethylene in order to characterize each type of transition. The valence transitions and , and the Rydberg transitions and are studied. Characteristic minima in for the Rydberg transitions appear. This behavior is attributed to the presence or absence of nodes in the molecular orbitals. It is suggested that this phenomenon be used as a probe of the type of molecular orbitals which are involved in a particular transition as well as the location of optically forbidden transitions. Qualitative agreement is obtained with experimental results for all but the transition, which does display a minimum. This is probably due to configurational mixing with Hartree–Fock Rydberg states.

Minima in Generalized Oscillator Strengths: C_{2}H_{4}
View Description Hide DescriptionA characteristic of electron impact excitation of low‐lying Rydberg states is a minimum in the generalized oscillator strength as a function of , the momentum transfer. This characteristic is used as a probe of the Rydberg character of four transitions in C_{2}H_{4}. Three, the , and transitions, have long been identified as Rydberg and are found to exhibit the characteristic minimum. A fourth transition is normally termed a valence excitation, and a theoretical calculation using Hartree–Fock molecular orbitals had predicted no minimum. For an energy loss of 8.0 eV which is identified with the valence transition a definite minimum is observed. Speculation on the source of this anomaly centers on a type of valence–Rydberg configurational mixing which can occur in many molecules. It is suggested that the presence or absence of minima in generalized oscillator strength curves be used to probe this aspect of the character of the excited state.

Isotopic Volume Shifts in the Inert Gas Solids at 0°K
View Description Hide DescriptionRecent perturbative calculations of isotopic shifts in Ne are extended to include these shifts for the other inert gas solids.

Differential Cross Sections for the Electron‐Impact Excitation of Vibrational Levels in Molecules
View Description Hide DescriptionThe angular dependence of scatteredelectrons which have excited a particular vibrational level of the ground electronic state of a molecule is investigated. General expressions are developed for the differential electron‐scattering cross sections for the excitation of vibrational levels in terms of derivatives of the ground‐state electron density and derivatives of the total electric dipole and quadrupole moments. Results for the small‐angle limit are obtained, and the errors in the theory and their possible effect on the small‐angle scattering are discussed. It is shown that contributions to the scattering from overtone and combination frequencies increase in comparison to contributions from the infrared allowed fundamental transitions with increasing scattering angle. The relationship between the electronic contribution to the large‐angle scattering and certain vibrational corrections to the vibrationally averaged three‐dimensional molecular electron density is pointed out.

Consecutive Metastable Peaks in Mass Spectra
View Description Hide DescriptionThe observation of consecutive metastable peaks in mass spectra has prompted a calculation for the abundance of these species. Expressions for fractional abundance as a function of primary and secondary rates and residence times are derived for all species of normal and metastable ions. Primary and secondary rates for a toluene consecutive dissociation were calculated using the method of Vestal, Wahrhaftig, and Johnston, and the breakdown curves were determined. By assuming a constant energy‐transfer function the normalized areas of the calculated curves are found to compare very favorably with the mass spectrum obtained in this laboratory with a double‐focusing mass spectrometer.

Nuclear Magnetic Resonance and Magnetic Susceptibility of Pr_{3}Al_{11}, Nd_{3}Al_{11}, and EuAl_{4}
View Description Hide DescriptionThe Knight shifts of the ^{27}Al NMR in Pr_{3}Al_{11}, Nd_{3}Al_{11}, and EuAl_{4} have been measured between 86 and 300°K, and have been found to be different for inequivalent Al sites. The results are discussed in terms of the Ruderman–Kittel–Kasuya–Yosida theory. The magnetic susceptibilities have been measured in the temperature region from 300 down to 4.2°K, and found to deviate from Curie–Weiss laws only for temperatures lower than 40°K.

Infrared Spectra of Crystalline Cyclopentane and Cyclopentane‐d _{10}
View Description Hide DescriptionInfrared spectra of crystalline cyclopentane and cyclopentane‐d _{10} have been measured at different temperatures between 4000 and 200 cm^{−1}. Spectroscopic evidence shows that form I (− 120°C) and form II (− 140°C) have the same degree of disorder as the liquid. In the ordered crystal form III (− 180°C) the molecule has a rigid bent conformation. By the mixed‐crystal technique the nature of the observed splittings has been established. This has permitted us to show that the site symmetry of the molecule in form III is , and packing concepts together with the spectroscopic data led to the choice of as the possible space group of the lowest‐temperature form.

Hartree–Fock Approximation to the Autoionizing States of Helium
View Description Hide DescriptionAs a first step in estimating the utility of the Hartree–Fock method for calculating positions and widths of autoionizing levels of atoms and molecules, an investigation of the helium problem has been initiated and is reported here. Wavefunctions and energies for the ten lowest states are calculated by numerically integrating the Hartree–Fock equation for solutions corresponding to energy levels above the ground state. Except for all these states are autoionizing, and each is probably the lowest bound state for each closed channel for . The energies of the states are found to be between the nth and (n − 1)th thresholds for but for the calculated levels fall below the (n − 1)th threshold. The method of numerical integration used is described, and the wavefunctions are reported in terms of an expansion in a basis determined by a least‐squares curve‐fitting method.

Collision‐Induced Absorption Spectrum of Gaseous Oxygen at Low Temperatures and Pressures. II. The Simultaneous Transitions and
View Description Hide DescriptionThe collision‐induced absorption spectra have been measured at room temperature and at 87°K for bands in the and simultaneous electronic systems for oxygen. The binary absorption coefficients were found to increase with decreasing temperature for . The band shapes for this system suggest that the Hamiltonian which is responsible for intensity borrowing depends on the angular orientation of the O_{2} molecules in the collision pair since selection rules are needed to account for the bandwidth. The relative intensity of the (1–0) and (0–0) bands indicates that the interaction Hamiltonian is also strongly modulated by the vibrational coordinates of O_{2}. The frequency shift of this simultaneous transition indicates that the intermolecular distance parameter for is 3% larger than for . The unusual band shape for the band is interpreted in terms of an exiton interaction for the combination. Although bound state (O_{2})_{2} molecules undoubtedly exist at low temperatures these data provide no unambiguous spectroscopic evidence of their presence.

NMR Study of Internal Motions in Certain Iron Group Fluosilicates
View Description Hide DescriptionNMR study of ferrous fluosilicate hexahydrate indicated the presence of motion of both proton and fluorine nuclei. Only a single narrow line was observed for protons for any arbitrary orientation of a single crystal with respect to the applied magnetic field. This can be interpreted in terms of a phase‐correlated flip motion of the interproton vectors between two disordered orientations or in terms of a hindered rotation of the Fe(H_{2}O)_{6} octahedron about the fourfold axes, together with the flip motion. The fluorine second moment indicated that the SiF_{6} octahedron also is undergoing reorientation. The temperature variation of the powderlinewidth showed a transition around 195°K and led to rather low values for the potential barriers hindering the motions. No significant temperature variation of the linewidth was observed for hexahydrated cobalt fluosilicate in the temperature range between 90°K and room temperature. Similar observations in a powder sample of tetrahydrated copper fluosilicate also showed the presence of internal motions. The linewidth transition in this case took place at about 220°K and was found to be rather abrupt. The potential barrier for the motion was found to be relatively high.

Note on the Perturbation Equation of State of Barker and Henderson
View Description Hide DescriptionPerturbation theory of equation of state due to Barker and Henderson is reformulated. This new formulation makes it possible to calculate the Helmholtz free energy of a fluid system analytically by perturbation relations of Barker and Henderson, without use of any approximation or need of any numerical table for the hard‐sphere reference system other than the original Percus–Yevick approximations. The results are compared with the calculation of Barker and Henderson, and it is shown that the two agree with each other at all the temperatures which are compared, while the present method produces compressibilities slightly closer to the experimental and machine‐calculated data. The results of the present method, based on average Percus–Yevick hard‐sphere compressibilities, are also compared with the result of other theories of equation of state of simple fluids, molecular dynamics, and Monte Carlo calculations.

Dielectric Relaxation in Nitroalkanes
View Description Hide DescriptionPermittivity and dielectric loss ( and ) of four nitroalkanes have been measured at five frequencies at 3, 15, 30, 45, and 60°C. The values of and have been used to plot Cole–Cole diagrams. The relaxation time and the distribution parameter have been calculated from these arc plots. It has been observed that the relaxation time increases with the increase in the number of carbon atoms and decreases with increase in temperature. The dielectric relaxation and the viscous flow phenomena have been compared using the theory of chemical rate processes.

Crossed‐Beam Study of the Reactive Asymmetry of Oriented Methyl Iodide Molecules with Rubidium
View Description Hide DescriptionA crossed molecular‐beam study of the reactive asymmetry of oriented methyl iodide molecules reacting with rubidium is described. The velocity‐selected molecular beam of CH_{3}I is focused by an inhomogeneous electric six‐pole field; the CH_{3}I molecules are then oriented with respect to the incoming atomic beam of Rb by means of a homogeneous, weak, electric field.Measurements of the intensity of the RbI product have been made at various laboratory scattering angles as a function of the relative alignment of the CH_{3}I with respect to the incident relative velocity vector. A computer analysis of the data has shown that the differential reactive scattering cross section (in the backward center‐of‐mass direction) for the “favorable” CH_{3}I alignment exceeds that for the “unfavorable” alignment by a factor of 4 or more. A simple hardsphere model, approximating methyl iodide as a partially reactive sphere, accounts for many of the important features of the observations.

Dipolar Coupling and Molecular Vibrations in Crystals. III. Polarizabilities of Molecular Anions and the Internal Field in Some Rhombohedral Crystals
View Description Hide DescriptionUsing the formalism and the dipole coupling sums developed in an earlier paper the equations relating the optical refractive indices to ionic polarizabilities are given for certain classes of rhombohedral crystals. Anion polarizabilities are calculated from measured refractive indices and previously calculated cation polarizabilities for several series of crystals. Finally the validity of the widely used Lorentz–Lorenz internal field in these systems is examined.

Photodissociation of O_{2} and N_{2} in the Far Vacuum uv and Production of Excited O and N Atoms
View Description Hide DescriptionPhotodissociation of O_{2} and N_{2} yielding highly excited O and N atoms has been investigated in the region from 1000 to 450 Å by observing the fluorescence of fragments in the region between 1500 and 1100 Å. Fluorescence intensities were measured as a function of the incident wavelength. The following dissociation processes have been identified: The threshold wavelengths of these processes coincide closely with the corresponding dissociation energies. The fluorescence intensities of the 1305‐ and 1200‐Å resonance lines of O and N, respectively, were calibrated and absolute cross sections for the production of and have been determined. Depending on the incident wavelength, cross sections of up to 4 × 10^{−18} and 5 × 10^{−19} cm^{2} are found for the and production, respectively.