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Volume 68, Issue 6, 15 March 1978

Polarization propagator calculations of frequency‐dependent polarizabilities, Verdet constants, and energy weighted sum rules
View Description Hide DescriptionExpressions for the frequency‐dependent polarizability, the Verdet constant, and the energy sum rules have been derived consistent through third order in the electronic repulsion using an analytical polarization propagator method. In this approach, the second order optical properties are obtained directly from the propagator without calculating the individual excitation energies and transition moments which appear in the sum‐over‐states procedures.

Frequency‐dependent polarizabilities and Verdet constants for He, Be, CO, and FH
View Description Hide DescriptionFrequency dependent polarizabilities and Verdet constants for He, Be, CO, and FH have been calculated within a first (coupled Hartree–Fock) and second order polarization propagator approach. Except for regions close to excitation thresholds, small differences are found between frequency‐dependent polarizabilities calculated in the two orders. The improvements in the polarizability near an excitation threshold are caused by the better description of the excitation spectrum in the second order theory. The trends in the Verdet constants are similar to those found for the polarizability except that the improvements in the second order approach are substantial also away from an excitation threshold (up to 60% of the difference between the coupled Hartree–Fock and the experimental result).

An intermolecular potential for (NH_{3})_{2}
View Description Hide DescriptionModel interatomic potential energy functions are presented for the ammonia dimer (NH_{3})_{2} which have the correct Coulombic and dispersion interactions for asymptotically large intermolecular separations. Existing data for the second virial coefficient of the gas together with the known structure and binding energy of the solid are used to constrain the parameters of the short range interatomic interactions. Finally, new experimental results for the differential collision cross section of NH_{3}+NH_{3} are used to test the proposed potential energy functions.

Calculation of dynamically induced electronic transitions of matrix‐isolated atomic nitrogen
View Description Hide DescriptionThe parity‐forbidden transitions ^{2} P→^{2} D and ^{2} D→^{4} S within the ground configuration (2p)^{3} of atomic nitrogen have been observed as phonon‐ and vibron‐induced emission bands in luminescencespectra of solid nitrogen and have been attributed to dipole radiation induced by dynamic crystal fields of odd parity with respect to the substitutionally trapped atom in the N_{2} lattice. We confirm this interpretation by an a p r i o r i calculation of the absolute transition probabilities and band shapes, using known atomic, molecular, and crystal properties of nitrogen and no adjustable parameters. The induced dipole moments are calculated from the vibrationally modulated quadrupole fields of the host molecules at the site of the polarizable guest atom. The dynamics of the doped lattice are treated by a normal‐coordinate analysis of the ’’supermolecule’’ consisting of the atom and its 18 nearest radially relaxed N_{2} neighbors suspended in the unperturbed surrounding N_{2} lattice. The calculated dipole‐weighted densities of phonon states and the resulting total transition probabilities agree well with observations. The corresponding emissions in rare‐gas matrices are represented via superpositions of empirical pair overlap dipole moments.

The coupled states approximation for scattering of two diatoms
View Description Hide DescriptionThe coupled states (CS) approximation is developed in detail for the general case of two colliding diatomic molecules. The high energy limit of the exact Lippmann–Schwinger equation is used to obtain the CS equations so that the sufficiency conditions of Kouri, Heil, and Shimoni apply. In addition, care is taken to ensure correct treatment of parity in the CS, as well as correct labeling of the CS by an effective orbital angular momentum. The analysis follows that given by Shimoni and Kouri for atom–diatom collisions where the coupled rotor angular momentumj _{12} and projection λ_{12} replace the single diatom angular momentumj and projection λ. The result is an expression for the differential scattering amplitude which is a generalization of the highly successful McGuire–Kouri differential scattering amplitude for atom–diatom collisions. Also, the opacity function is found to be a generalization of the Clebsch–Gordon weight atom–diatom expression of Shimoni and Kouri. The diatom–diatom CS body frame T matrix T ^{ J }(j _{1}′j _{2}′j _{12}′λ_{12}′‖ j _{1} j _{2} j _{12}λ_{12}) is also found to be nondiagonal in λ′_{12}, λ_{12}, just as in the atom–diatom case. The parity and identical molecule interchange symmetries are also considered in detail in both the exact close coupling and CS approximations. Symmetrized expressions for all relevant quantities are obtained, along with the symmetrized coupled equations one must solve. The properly labeled and symmetrized CS equations have not been derived before this present work. The present correctly labeled CS theory is tested computationally by applications to three different diatom–diatom potentials. First we carry out calculations for para–para, ortho–ortho, and ortho–para H_{2}–H_{2}collisions using the experimental potential of Farrar and Lee (as modified by Zarur and Rabitz). Our results are compared with CC results due to Green. In addition, we have derived and tested the statistical and truncated coupled states approximations for this system. Next, we carried out calculations for ortho–para H_{2}–H_{2}scattering using the configuration interactionsurface of Ostlund. Calculations at seven energies from 2000 to 14 000 cm^{−1} are reported; the CS results are compared to CC results. The third system studied is H_{2}–HCl, using a potential adapted from an HCl–He electron gas surface by Green. The results are again compared with CC studies due to Green. The results are quite encouraging and indicate that for systems (and specific transitions) satisfying the validity conditions of Kouri, Heil, and Shimoni, the CS is quantiative just as in the atom–diatom case. Furthermore, we expect the accuracy to improve as the energy of the collision increases.

Surface‐induced NMR line splittings and augmented relaxation rates in water
View Description Hide DescriptionNuclei in liquid water adsorbed onto substrates in layered silicates or ordered macromolecules have been shown to exhibit nuclear magnetic resonance line splittings. In aqueous proteinsolutions they also exhibit a dispersion in their longitudinal relaxation rate at low frequencies. Furthermore, the reduced line splittings and relaxation rates have been found to be the same for ^{1}H, ^{2}H, and ^{17}O and independent of the nature of the substrate. We show that these results are consistent with a surface‐induced anisotropy in the orientational probability distribution of the water molecules. Agreement between this theory and the experimental results is possible only if the values of the H–O–H angle and the coefficients for the field gradient anisotropies at the ^{2}H and ^{17}O nuclei are close to those found in the solid phase. The theory is consistent with a model for the water surface in which all of the molecules are statistically equivalent and rotationally mobile under a weak anisotropic constraint.

An analysis of the B ^{1}Π_{ u }–X ^{1}∑^{+} _{ g } band system of Na_{2}
View Description Hide DescriptionA critical analysis of the B ^{1}Π_{ u }–X ^{1}Σ^{+} _{ g } band system of the Na_{2} molecule has been made. The body of data on which the present analysis is based is much more extensive than that used in any previous analysis. The data for the analysis were obtained by a laser‐induced fluorescence technique in which many collisionally induced satellitefluorescence series were observed in addition to the directly excited series. We have found the coefficients in a Dunham expansion that replicates the frequencies of 12 591 lines that originate in 1297 different V′, J′ levels with an rms deviation between the observed and calculated frequencies of 0.011 cm^{−1}. The frequencies of the lines that appear in the present analysis are poorly replicated by use of the Dunham coefficients found by earlier workers. The RKR potential curves, for both the B and X states and the Franck–Condon factors for transitions between the two states, have been found. In contrast to previous analyses the theoretical centrifugal distortion coefficients, calculated from the rotationless RKR potential, agree closely with those directly found in the present analysis. The dissociation energies have been found to be 5988 cm^{−1} for the X state and 3116 cm^{−1} for the B state. The maximum of the potential barrier of the B state lies about 474 cm^{−1} above the ^{2} P _{3/2}+^{2} S _{1/2} levels to which the B state dissociates.

Reactions of N^{+} _{4} with O_{2}, CO_{2}, H_{2}, and D_{2} and mobilities of N^{+} _{4} in nitrogen
View Description Hide DescriptionThe rate constants for the reaction of N^{+} _{4} with O_{2}, CO_{2}, H_{2}, and D_{2} have been studied as a function of mean relative kinetic energy in a flow‐drift tube using a nitrogen buffer gas. The N^{+} _{2}/N^{+} _{4} equilibrium, which would otherwise deleteriously effect such a study, was suppressed by adding a trace of H_{2} as a purge gas that preferentially reacted with N^{+} _{2}. The reaction rates of N^{+} _{4} with CO_{2} and O_{2} are found to be fast and independent of energy from 0.04 to 0.16 eV. The reactions of N^{+} _{4} with H_{2} and D_{2}, however, have very small rates at room temperature, but the rates increase dramatically with increasing energy, an effect that is attributed to a sensitivity of the reactions to N^{+} _{4} vibrational excitation. The reaction of N^{+} _{4} with H_{2} was studied as a function of temperature and an activation energy obtained. The mobility of N^{+} _{4} in N_{2} was extended to higher E/N values than could be done in earlier studies since the effects of the N^{+} _{2}/N^{+} _{4} equilibrium were suppressed in the present investigation.

Effect of surface processes on hydrogen and nitrogen permeation. I. Adsorption
View Description Hide DescriptionThe permeation of hydrogen and nitrogen through a metal wall is analyzed for the case where surfaceadsorption is rate limiting in addition to volume diffusion. By applying equilibrium thermodynamics plus a minimum of kinetic theory, an expression is derived for steady state permeation. The expression reduces to the familiar Richardson equation in the limit of negligible surface resistance. In addition, unsteady permeation with slow adsorption is examined by solving appropriate finite difference equations. The results accurately predict anomalies commonly encountered in permeation and diffusion data at low temperatures and pressures, e.g., the p ^{ n } anomaly where n≳1/2, curvature or droop in Arrhenius plots, and anomalously low diffusivities. The analysis is useful not only in understanding the influence of surface effects on permeation, but also in applying permeation membranes to the study of surface phenomena.

Power law frequency dependent dielectric function and nonanalyticity properties
View Description Hide DescriptionExperiment yields for many materials a dielectric loss function χ^{″} (ω) having the behavior χ^{″} (ω) ∝ω^{ n−1}(n<1) over a substantial range of frequencies. Writing the conductivity σ (ω) =ωχ^{″} (ω) in the form σ (ω) ∝F_{0} ^{∞}ω^{2}/1+(ω^{2}/ν^{2})] g (ν) dν, where g (ν) is the relaxation frequency spectrum, we consider two examples in detail: (a) a relaxing center, at which relaxation is assisted by arrival of defects by diffusion; (b) hopping electronic conduction, with variable range, and thermally assisted. Example (a) is characterized (and dominated) at high frequencies by g (ν) ∝ν^{−5/2}, and for ν≳ν_{ D }, a characteristic Debye relaxation frequency, σ (ω) ∝ω^{1/2} results. Example (b) is dominated by low frequency modes, and for ν<ν_{ D } the behavior is like ω^{0.8} over a limited frequency range. If one allows in more detail for a distribution of activation energies, then it is argued that σ∝ω^{ n } where n<0.8. Some specific experimental results are briefly referred to.

Relativistic scattered‐wave theory. II. Normalization and symmetrization
View Description Hide DescriptionFormalisms for normalization and symmetrization of one‐electron Diracscattered‐wavewavefunctions are presented. The normalization integral consists of one‐dimensional radial integrals for the spherical regions and an analytic expression for the intersphere region. Symmetrization drastically reduces the size of the secular matrix to be solved. Examples for planar Pb_{2}Se_{2} and tetrahedral Pd_{4} are discussed.

Infrared spectral moments and mean squared torques of OCS mixed with noble gases
View Description Hide DescriptionThe spectral moments of OCS mixed with rare gases are measured for the ν_{1} and ν_{2} vibrations at various pressures in the 5–300 bar range. Hot bands, isotopic bands, and rotation–vibration interaction are taken into account for the two vibrations. The corrected experimental values of the second moments are in excellent agreement with the theoretical ones. The mean squared torques are therefore deduced, and an estimation of the experimental error is made. We obtain the same values of the mean squared torques from the ν_{1} and ν_{2} vibrations and they are found to be a linear function of density. An interpretation in terms of the intermolecular potential is given.

Vibrations of potassium and cesium thiocyanates crystals. III. Long‐wave lattice modes
View Description Hide DescriptionIn this paper, a dynamical calculation is reported for the orthorhombic crystals of potassium and cesium thiocyanates. A general description of the model used and theoretical results are presented. Although the model covers arbitrary wave vectors, and is applicable to both internal and external modes, numerical work is presently restricted to external long‐wave vibrations. The vibrational frequencies theoretically calculated and experimentally observed show reasonable agreement. Rotatory and translatory modes belonging to the same symmetry species are found to be strongly coupled. Some erroneous assignments made previously are rectified.

Absorption and circular dichroism spectra of the c i s‐ and t r a n s‐butadiene chromophores α‐ and β‐phellandrene
View Description Hide DescriptionThe absorption and circular dichroismspectra of gas‐phase (−) ‐α‐phellandrene and (−) ‐β‐phellandrene have been measured in the spectral region 300–135 nm. Solution spectra of these compounds were measured in perfluoro‐n‐hexane to 160 nm. The spectra were interpreted in terms of the excited electronic states of the c i s‐ and t r a n s‐butadiene chromophores. The overlapping valence and Rydberg states were unambiguously differentiated by comparison of the vapor and solution phase spectra. Comparison between absorption and CD spectra made possible the symmetry assignments of magnetic dipole allowed (as opposed to electric dipole allowed) excited states of butadiene. For both molecular systems the first observed excited singlet state corresponds to ^{1} B _{ u } for the t r a n s chromophore and to ^{1} B _{2} for the c i s chromophore. No evidence was found for a ’’lowest‐lying’’ singlet state of A _{ g } or A _{1} symmetry.

Theory of the spin Hamiltonian of CIO_{2} in single crystals of silver chlorate
View Description Hide DescriptionThe EPR of ClO_{2} in gamma irradiatedsingle crystals of AgClO_{3} was recently observed and the g and Atensors appeared rotated with respect to each other by approximately 6° about one of the molecular axis of ClO_{2}. This effect was studied by considering a crystal field that breaks the C _{2v } symmetry of ClO_{2}. The crystal field was estimated from a point charge model that includes the Ag^{+} and ClO_{−3} ions that are the nearest to the ClO_{2} fragment. This model gives a good qualitative agreement with the experimental results: It predicts the same axis and direction of rotation that were observed and a value of the angle of rotation between 0.8° and 4.4° (depending on the position of the ClO_{2} in the crystal lattice). In this model the gtensor is practically unaffected by the crystal fields and the orientation of the ClO_{2} fragment is consequently determined by the principal axis of this tensor.

Equation of state of fluid n‐D_{2} from P–V–T and ultrasonic velocity measurements to 20 kbar
View Description Hide DescriptionThe molar volume V and ultrasonic velocityu of fluid n‐D_{2} were measured simultaneously in a piston‐cylinder apparatus over the range 75<T<300 K and 2<P<20 kbar. The data were fitted to an equation of state of the Benedict type which reproduced the 1404 experimental data sets of V and u within an average deviation of 0.4%. Values of the thermodynamic properties of n‐D_{2} were calculated and compared with those from a parallel study of n‐H_{2} in an attempt to treat isotopic effects.

Polarized photochromism in solid solutions
View Description Hide DescriptionReversible photochromic transitions are in principle analogous to fluorescence. By this analogy reversible photochromic processes in a rigid transparent medium should display a characteristicanisotropyspectrum with respect to the polarization of the excitation and the monitoring lights. The polarized photochromismspectrum can help to elucidate the chemical structure of the phototransient and to assign the orientation of its transition dipoles. The following report presents and discusses polarizationspectra of physical and chemical photochromic processes, measured by a novel steady‐state method.

A potential surface for the two low‐frequency out‐of‐plane modes of s‐tetrazine
View Description Hide DescriptionPreviously reported data for s‐tetrazine‐d _{0} and ‐d _{2} have been used to determine a potential energy surface for the low‐frequency out‐of‐plane ring modes. For s‐tetrazine‐d _{0} this potential function in reduced coordinates is V (q _{α},q _{τ}) = (251.8/2) q ^{2} _{α}+(337.5/2) q ^{2} _{τ}+1.62q ^{4} _{α} −5.18q ^{2} _{α} q ^{2} _{τ}, where V is in cm^{−1} and q _{α} and q _{τ} are dimensionless ring‐puckering and ring‐twisting coordinates. Based on the limited frequency data available for the d _{2} compound the following potential function was determined and used to predict the Q‐branch structure for the low‐frequency mode: V (q _{α},q _{τ}) = (225.6/2) q ^{2} _{α}+(337.5/2) q ^{2} _{τ} +1.14q ^{4} _{α}−4.30q ^{2} _{α} q ^{2} _{τ}.

Rotational selection rules for nonradiative processes
View Description Hide DescriptionRecent experiments indicate that nonradiative transitions not only differ for various single vibronic levels but are also strongly dependent on the rotational states involved. Thus the rotational selection rules should be considered for these rotational transitions. Rotational matrix elements and selection rules are presented for internal conversion between two triplet or two singlet states and for intersystem crossing between a singlet and a triplet for polyatomic molecules in Hund’s case b. The selection rules are, in general, different for each process and depend on the nature of the angular momentum coupling as well as the interaction operator.

The adiabatic correction for nonlinear triatomic molecules: Techniques and calculations
View Description Hide DescriptionIn the Born–Oppenheimer (BO) approximation the isotopic‐mass‐independent BO electronic energy is the potential energy surface for nuclear motion. The first correction to the BO approximation is obtained by adding the isotopic‐mass‐dependent adiabatic correction C to this potential energy surface. The adiabatic correction for a nonlinear triatomic molecule in a nondegenerate electronic state is expressed in terms of one‐electron integrals over atomic basis functions. Both single determinant molecular orbital BO electronic wavefunctions and configuration interactionwavefunctions are discussed. The adiabatic correction is calculated for H_{2}O and H^{+} _{3} and their isotopic variants at the respective equilibrium internuclear geometries. Theses adiabatic corrections are combined with those previously calculated for diatomic molecules to obtain correction factors to equilibrium constants for H/D isotopic exchange reactions calculated within the framework of the BO approximation.