Volume 55, Issue 2, 15 July 1971

Normal Coordinates: General Theory, Redundant Coordinates, and General Analysis Using Electronic Computers
View Description Hide DescriptionA system of doing normal coordinate analysis is described which is better suited for use with electronic computers than are current methods. Mass‐weighted Cartesian coordinates are used, and rotation and translation are not separated until the matrix diagonalization procedure. The matrix is generated numerically in a simple and accurate manner. Only one transformation is involved. The transformation which diagonalizes is simply the transformation from Cartesian to normal coordinates. The problem of redundant coordinates does not exist in this proposed method, but the advantage of using linear potential constants in strained and constrained systems is retained.

Radial Electron–Electron Distributions and the Coulomb Hole for Be
View Description Hide DescriptionIndividual radial electron–electron distribution functions,, are defined in terms of the eigen‐functions (the natural geminals, NG's) of the spin‐free 2‐matrix, . The total radial electron–electron distribution function,, is defined in terms of . It is demonstrated that the structure of individual two‐particle eigenfunctions can easily be determined from the corresponding . Distributions are obtained for the ground state of the Be atom by numerical Fourier transformation of x‐ray intensity values. The natural spin geminal (NSG) analysis by Barnett and Shull of Weiss' configuration interaction (CI) wavefunction is used for the determination of correlated , while the 2‐matrix analysis of Clementi's Restricted Hartree–Fock (RHF) function is employed in the evaluation of independent particle model (IPM) . The correlated and RHF are compared with that obtained from a two‐configuration wavefunction of Watson, which takes into account the near degeneracy of and orbitals. By subtracting the RHF from the Weiss , a description of the Coulomb hole function, , is obtained for the Be ground state. A measure of the Fermi correlation may be obtained with constructed from those and RHF geminals which involve the spatially symmetric or the spatially antisymmetric combination of and orbitals.

Polarized Vibrational Spectra of Single Crystal Sodium Tetrafluoroborate
View Description Hide DescriptionThe polarized Raman spectra of an oriented single crystal of NaBF_{4} were measured at 298 and 77°K in the region above 50 cm^{−1}. The polarized infrared transmission and reflection spectra of single crystal NaBF_{4} were also measured at room temperature in the region above 300 cm^{−1}. From the polarized vibrational spectra, the intramolecular modes were assigned, and estimates were made of the magnitude of the static field effects in crystalline NaBF_{4}.

Polarized Vibrational Spectra of Potassium Chlorate
View Description Hide DescriptionThe Raman spectra of single crystal KClO_{3} were measured in the region above 30 cm^{−1} at 298 and 77°K. Single crystalinfrared spectra in the internal mode region were measured by reflection at 298°K. Infrared spectra of the polycrystalline material also were measured at 77°K. Assignments for all of the crystal vibrations that are derived from correlation field splitting of the (internal) ClO_{3} ^{−} vibrations were made from the results of the polarizationmeasurements. Longitudinal optical mode frequencies for the and components obtained from polarized (infrared) specular reflection measurements on polycrystalline KClO_{3} are also reported. The bands due to the Raman active lattice modes were observed in the region below 200 cm^{−1} and were tentatively assigned to librational or to translatory‐type motions. The 2‐cm^{−1} splitting observed for the mode of the ^{35}Cl^{18}O^{16}O_{2} and ^{37}Cl^{18}O^{16}O_{2}isotopes in the low‐temperature Raman spectra was assigned to an orientational effect in the lattice.

Perturbation Theory of Product Hamiltonians through Fourth Order
View Description Hide DescriptionExpressions are derived through fourth order in a representation for the eigenvalues and off‐diagonal elements of Hamiltonians expressible as sums of products of operators from two Hilbert spaces ( and ), . The zeroth order Hamiltonian is assumed separable, and the eigenvalue differences in the space are assumed to be an order of magnitude larger than the eigenvalue differences in the space. The method involves successive contact transformations chosen to yield results in terms of matrix elements in the space and operators in the space. The technique allows for the exclusion of interactions between resonant states in the space for subsequent numerical diagonalization.

Semiclassical Transition Probabilities for Rotationally Inelastic Scattering
View Description Hide DescriptionDegeneracy averaged partial cross sections for rotationally inelastic scattering are calculated using semiclassical methods. Assuming a classical linear trajectory simple expressions are derived for the sudden limit, the classical limit, and first‐order limit. A single uniform approximation is obtained which converges to all three limits. The partial inelastic cross section is given by , where and are reduced parameters, is the Clebsch–Gordan coefficient, and is a set of integrals depending on the potential form; the 's are calculated for the ion–dipole and anisotropic van der Waals potentials and in the sudden limit for the dipole–dipole potential.

Rotational Barriers in Hydrogen Peroxide
View Description Hide DescriptionThe rotational barriers in hydrogen peroxide are investigated using LC(Hartree–Fock) AO MO SCF calculations. Systematic improvement of the basis set produces a definite trans barrier, demonstrating that this barrier, like all those previously studied, is calculable within the Hartree–Fock approximation. Allowance for geometry adjustment, particularly OOH angle variation, leads to a cis barrier much closer to agreement with experiment than that obtained assuming rigid rotation. The theoretical and experimental inplications of these findings are discussed.

Low Frequency Motions in Barium Azide Crystals
View Description Hide DescriptionA reinvestigation of the k ∼ 0 infrared and Raman active lattice modes of BaN_{6} crystals is reported. The symmetry species of the Raman active modes were determined unequivocally by oriented single crystal laser excited Raman spectroscopy. A new band at 40 cm^{−1} was resolved and a number of symmetry reassignments were made. The results are in essential agreement with the space group for the crystal. The transmission far infrared spectra of polycrystalline BaN_{6} and SrN_{6} were measured between 20 to 300 cm^{−1} and compared with the reflection spectrum of BaN_{6} reported in the previous study.

Picosecond Pulse Radiolysis. III. Reaction Rates and Reduction in Yields of Hydrated Electrons
View Description Hide DescriptionRate constants for reactions of the hydrated electron with a wide range of compounds, including simple salts and amino acids, have been measured in the time region 20–350 psec. In nearly all cases the second‐order rate constant for reaction with the hydrated electron did not change in the concentration range studied. However, the measuredrate constant was in general different to that obtained in dilute solutions, and this was attributed to incomplete formation of the ionic atmosphere around the hydrated electron before reaction. Apart from the hydronium ion, H_{aq} ^{+}, nearly all compounds decreased the initial hydrated electron yield: in all cases this decrease showed an exponential dependence on concentration. Compounds most efficient at decreasing the hydrated electron yield were cystine and cadmium salts, in both cases 0.39 mole/liter being needed to reduce the initial yield to 37%. The ability to reduce the initial yield showed no direct correlation with the corresponding hydrated electron rates. This and other evidence suggest that radiation‐produced electrons are reacting before solvation.

Hydrogen Dibromide Radical: Infrared Detection through the Matrix Isolation Technique
View Description Hide DescriptionWhen hydrogen bromide–bromine–argon mixtures are passed through a glow discharge and the products are condensed at 20°K, prominent absorptions appear at 727.4, 892.1, and 1053.0 cm^{−1}. With deuterium, these features shift to 496.1, 666, and 833 cm^{−1}. Both deuterium and bromine isotope shifts show that these features can be assigned to the BrHBr (BrDBr) free radical in a linear, symmetric structure with and . The evidence that these bands are not due to HBr_{2} ^{−} is reviewed. The vibrational potential function for includes a large quartic term and it suggests that the hydrogen bond in HBr_{2} is comparable in strength to that in HBr_{2} ^{−}.

Collisional Deactivation of
View Description Hide DescriptionThis paper reports determinations of collisional deactivation rate constants for by minor atmospheric constituents and other gases. The benzene–oxygen photochemical system was used to produce , and relative concentrations were measured by monitoring the intensity of its emission at 1.27 μ. The rate constants for the reaction were 4.53 × 10^{−18}, 5.60 × 10^{−18}, and (0.5–1.0) × 10^{−19} (cm^{3} molecule^{−1}·sec^{−1}) for M=H_{2}, H_{2}O, and N_{2}O, respectively. Upper limits were obtained for deactivation by He, Ar, CO_{2}, and SF_{6} of 8.0 × 10^{−21}, 8.3 × 10^{−21}, 1.5 × 10^{−20}, and 1.2 × 10^{−20}, respectively. The temperature dependence of the deactivation of by oxygen has been investigated. The rate constant can be expressed in the form , where and is the temperature in degrees Kelvin.

Vibrational Distribution Functions in Anharmonic Oscillators
View Description Hide DescriptionA model has been developed to calculate the steady‐state vibrational distribution of an anharmonic oscillator under conditions of thermal nonequilibrium. The model includes vibration–vibration and vibration–translation collisional processes as well as radiative decay. Numerical calculations are shown to be in reasonably good agreement with experimental data obtained in electrically excited N_{2}–CO mixing systems. It is shown that radiative decay may be the dominant loss mechanism in low‐pressure N_{2}–CO experiments.

Trajectory Surface Hopping Approach to Nonadiabatic Molecular Collisions: The Reaction of H^{+} with D_{2}
View Description Hide DescriptionAn extension of the classical trajectory approach is proposed that may be useful in treating many types of nonadiabaticmolecular collisions. Nuclei are assumed to move classically on a single potential energy surface until an avoided surface crossing or other region of large nonadiabatic coupling is reached. At such points the trajectory is split into two branches, each of which follows a different potential surface. The validity of this model as applied to the HD_{2} ^{+} system is assessed by numerical integration of the appropriate semiclassical equations. A 3d “trajectory surface hopping” treatment of the reaction of H^{+} with D_{2} at a collision energy of 4 eV is reported. The excellent agreement with experiment is an encouraging indication of the potential usefulness of this approach.

Reptation of a Polymer Chain in the Presence of Fixed Obstacles
View Description Hide DescriptionWe discuss possible motions for one polymer molecule P (of mass ) performing wormlike displacements inside a strongly cross‐linked polymericgel G. The topological requirement that P cannot intersect any of the chains of G is taken into account by a rigorous procedure: The only motions allowed for the chain are associated with the displacement of certain “defects” along the chain. The main conclusions derived from this model are the following:
(a) There are two characteristic times for the chain motion: One of them is the equilibration time for the defect concentration, and is proportional to . The other time is the time required for complete renewal of the chain conformation, and is proportional to .
(b) The over‐all mobility and diffusion coefficients of the chain P are proportional to .
(c) At times the mean square displacement of one monomer of P increases only like .
These results may also turn out to be useful for the (more difficult) problem of entanglement effects in unlinked molten polymers.

Vapor Phase Homogeneous Nucleation and the Thermodynamic Properties of Small Clusters of Argon Atoms
View Description Hide DescriptionThe steady state rate of homogeneous nucleation in a vapor of Lennard‐Jones spheres has been calculated using statistical mechanical techniques to compute the equilibrium concentrations of clusters in the vapor. All degrees of freedom of the clusters are explicitly considered. The harmonic approximation is used in the calculation of vibrational contributions to the cluster partition functions and the rigid body approximation is used in the calculation of rotational contributions. The Gibbs free energy of cluster formation as a function of cluster size is calculated and its dependence on temperature is examined. Size effects on the vibrational free energy are found to be mainly energetic rather than entropic. The rate of nucleation as a function of pressure is calculated at four temperatures and found to have a behavior similar to what would be expected from “liquid drop” model calculations.

Neutron Diffraction Study of Ice Polymorphs under Helium Pressure
View Description Hide DescriptionNeutron diffraction studies of several of the D_{2}O icepolymorphs were made with helium pressures to 3.5 kbar. Volume compressions were obtained for ices Ih, Ic, and IX at 2.1 to 2.8 kbar. The equilibrium phase boundaries between ices I–II and I–III in the presence of helium are shifted toward higher pressure values. The crystal structures of ices Ih, Ic, II, and IX at approximately 2.5 kbar are the same as the structures found for these ice forms at atmospheric pressure. A tetragonal structure is indicated for ice III with lattice parameters at − 23°C and 2.7 kbar of and . The ratio for ice IX has been found to be 1.000 ± 0.003 both at 1 bar and at 2.8 kbar.

Anisotropic Intermolecular Force Effects in Spectra of H_{2}– and D_{2}–Rare‐Gas Complexes
View Description Hide DescriptionSpectra of H_{2}–Ar, H_{2}–Kr, and H_{2}–Xe Van der Waals complexes, accompanying the , and transitions of the pressure‐induced fundamental absorption band of hydrogen, have been studied in a path length of 165 m at temperatures in the range 85–158°K. At the low total gas densities used, 1–2 amagat, the lifetimes of the complexes were sufficiently long to show a great deal of new detail in the spectra. The and branches (, where is the angular momentum of the complex) accompanying the overlap‐induced transitions, could be analyzed on a nonrigid rotation model to give Lennard‐Jones and parameters for the molecular pairs. The spectra accompanying the quadrupole‐induced , and transitions show and as well as and branches, and many of the lines are split by the anisotropy of the intermolecular forces. The splitting is qualitatively in agreement with a model with loose coupling between J, the angular momentum of the H_{2} molecule, and 1. Well‐resolved spectra of D_{2}–Ar, D_{2}–Kr, and D_{2}–Xe complexes accompanying the transition of deuterium were also obtained; these show more bound states than the H_{2}–rare‐gas spectra and anisotropic interaction effects are evident. The H_{2}–N_{2} complex shows a spectrum with a diffuse structure not previously observed.

Deuterium Quadrupole Coupling in Formyl Fluoride
View Description Hide DescriptionThe 1_{01}–0_{00} rotational transition in HFCO and DFCO was recorded with a beam maser spectrometer. Hyperfine structure due to the deuterium quadrupole coupling was observed for DFCO. Analysis of the spectrum yielded the deuterium quadrupole coupling strength along the D–C bond direction of . This rotational transition had not been reported previously for DFCO. A comparison of deuterium quadrupole coupling strengths with force constants along the H–C bond is made according to the theory of Salem.

On One‐Dimensional Model for Exchange Forces
View Description Hide DescriptionA one‐dimensional Schrödinger equation with the potential is considered. Asymptotic expansions for eigenvalues for large are presented. It is shown that the quantity obtained by means of the LCAO approximation exhibits incorrect behavior in the limit of large . Comparisons of asymptotic formulae with results of numerical calculation are given.

Theoretical Electronic Transition Probabilities in Diatomic Molecules II. 13‐Electron Sequence
View Description Hide DescriptionHartree–Fock (HF) electronic dipole‐momentum and dipole‐length transition moments, and , respectively, are presented as a function of the internuclear distance for the first negative and Meinel systems. They are compared with “experimental” and curves obtained by using relative band transition probability and upper‐state lifetime measurements recently reported in the literature. Vibrational averages of both the theoretical and “experimental” transition moments are computed and used to calculate some relative band oscillator strengths . The HF value of 0.1370 for the first negative system differs from experiment by a factor of 5.6. For the Meinel system the HF value of 1.065 × 10^{−2} is too large by a factor of 3.3 or 7.3 depending on whether the lifetime data of O'Neil and Davidson or that of Hollstein et al. are used to determine the experimental value. The HF transition‐moment calculations at a single value of are also reported for the and systems of C_{2} ^{−} and for the and systems of CN, CO^{+}, BO, BF^{+}, and BeF. Electronic absorption oscillator strengths are presented and compared with experiment where possible.