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Volume 70, Issue 6, 15 March 1979

The variational method and the stochastic–Liouville equation. I. A finite element solution to the CIDN(E)P problema)
View Description Hide DescriptionA variational formulation is developed for the stochastic–Liouville equation (SLE). It is shown how this formulation may be used as a general basis for the study of numerical and approximate methods of solution of the SLE. The finite element method is developed for the approximate solution of the spin–density matrix elements using the variational formulation. The method is illustrated by employing it to obtain a compact computer‐oriented solution to the (high‐field) chemically‐induced spin polarization problem. This solution is both more efficient as well as more accurate than the previous treatment by Pedersen and Freed using finite difference methods. Various features of finite element and finite difference methods are compared from the viewpoint of this solution. The great flexibility of finite element methods for solution of the SLE is discussed.

Optical absorption spectra of localized electrons generated at 1.6 K in polar matrices: Evidence for presolvated electrons
View Description Hide DescriptionGlassy samples of ethanol, methanol +5 mol % water, 10 M and 18 M NaOH in H_{2}O, 5 M K_{2}CO_{3} and 7 M NaClO_{4} in D_{2}O were x irradiated at 1.6 K. After irradiation the optical absorptionspectra of the localized electronsgenerated were recorded at 1.6 K. Red shifts relative to spectra after irradiation at 4.2 K were observed in all matrices indicating stabilization of some presolvated electrons characterized by a nonequilibrium first solvation shell. Particularly in deuterated matrices, where observation further into the infrared is possible, a long infrared tail assigned to presolvated electrons is detected at 1.6 K; these infrared tails are not stabilized at 4.2 K. Analysis of the spectral shifts after irradiation at 77, 4.2, and 1.6 K indicates that an increasingly larger population of presolvated electrons is stabilized as the temperature is lowered and that a parallelism for electron solvation exists in all the matrices studied including ethanol. In the methanol and aqueous matrices, electron solvation is much more efficient than in ethanol, and the predominant population of localized electrons is largely solvated even at 1.6 K. The spectral shifts in 18 M NaOH glasses are also consistently interpreted in terms of presolvated electron stabilization and do not require postulation of (Na^{+}–e^{−}) pair stabilization as suggested in earlier work at 77 K.

Hydrodynamic theory for vibrational dephasing in liquids
View Description Hide DescriptionA hydrodynamic model is presented for the dephasing of molecular vibrations in liquids; the effects of vibrational anharmonicity are included. Agreement with coherent picosecond pulse experiments and isotropic Raman line shape data is good for lower viscosityliquids, and the qualitative temperature and density dependence of the dephasing rate is predicted correctly. The final result shows a surprising formal similarity to the isolated binary collision model developed previously.

EPR studies of paramagnetic rhodium centers in LiH and LiD
View Description Hide DescriptionEPR spectra have been obtained for two independent paramagnetic Rh centers in UV‐irradiated single crystals of LiH and LiD. The spin Hamiltonians for both species indicate S=1/2 and are assigned to Rh^{2+} ions with low‐spin 4d ^{7} electron configurations. Analysis of the anisotropicg‐factor for the tetragonal center shows the unpaired electron is the Rh 4d _{ x2−y2} orbital. The unpaired electron occupies the 4d _{3z } ^{2} _{−r } ^{2} orbital in the orthorhombic center. The g factor, hyperfine constants, and protium superhyperfine constants for the Rh centers in LiH and LiD are compared with similar parameters for Rh^{2+} ions in several halide and oxide crystals. Although the isotropic protium hyperfine values indicate significant ‖1s≳−‖4d≳ orbital interactions, the total covalency in the Rh–H(D) bonds is rather small since the hydrogen ‖2p≳ orbitals cannot participate. The displacements of H ligands in the Rh centers have been determined from the direct‐dipolar contributions to the superhyperfine tensor.

Role of pseudomolecules in vibrational relaxation in solidsa)
View Description Hide DescriptionA quantum‐mechanical theory of the participation of pseudolocal modes in vibrational energy relaxation (VER) of molecules in solids is developed. A simple closed expression for the VER rate constant is derived in terms of the microscopic properties of the system. This expression is used to explore the role of local translational modes in a one‐dimensional model. The results of model calculations are consistent with experimental findings for the system Cl_{2} isolated in rare‐gas matrices.

Comments on the magnitude and sign of the quasifree electron energy in the rare‐gas solids as inferred from ESR studies of Frenkel impurity states
View Description Hide DescriptionWe point out that the magnetic properties of tightly‐bound H atom impurities in dense, nonpolar solids are sensitive not only to local contributions to an effective potential, but also to electron– (bulk) medium interactions which dictate the nature of the ground state of an excess electron in these materials.

Ultraviolet preresonance Raman scattering of benzene derivatives. I. Excitation profiles for fundamentals
View Description Hide DescriptionThe excitation frequency dependence of the Raman scattered intensities of several fundamentals of three benzene derivatives (toluene, m‐xylene, and fluorobenzene) has been measured in the ultraviolet spectral region approaching the lowest lying absorption band in each of these molecules. As in benzene, the most preresonance active fundamentals are the C–H out‐of‐plane bending modes. The scattering cross section of these modes is found to result from the vibronic coupling of the ^{1} B _{ a,b } states (∼190 nm) and weak nearby out‐of‐plane polarized transitions. Most of the preresonance scattered intensity for the other modes reported here (the ring‐breathing, ring‐substituent, and internal methyl vibrations) have Raman active electronic origins in the far UV. With excitation very close to resonance with the lowest lying ^{1} L _{ b } transition, this source of scattering cross section becomes conspicuous for those modes seen prominently in this transition (the ring breathing and ring substituent vibrations). In addition the ring‐substituent fundamentals are found to derive significant scattering cross section from the region of the ^{1} B _{ a,b } transitions.

Ultraviolet preresonance Raman scattering of benzene derivatives. II. Interference effects in the excitation profiles of the vibronically active fundamentals
View Description Hide DescriptionRaman resonance and antiresonance behavior is observed in the ultraviolet preresonance excitation region for the scattering of those vibrations known to be vibronically active in the ^{1} L _{ a } and ^{1} L _{ b } electronic transitions of toluene, m‐xylene, and fluorobenzene. These effects are found to be due to the constructive and destructive interferences between scattering contributions derived from the vibronic coupling of the low lying ππ transitions and transitions in the far‐UV region. This behavior is seen to reflect the varying sign of the allowed ^{1} L _{ b } and ^{1} L _{ a } transition moment among these derivatives. Calculations are performed which show how the allowed transition moment in m‐xylene is of opposite sign to that of these two monosubstituted benzenes.

Laser photoelectron, photodetachment, and photodestruction spectra of O^{−} _{3}
View Description Hide DescriptionFixed frequency laser photoelectron spectrometry and variable frequency laser photodetachment and photodestruction spectroscopy of the ozonide ion, O^{−} _{3}, have been accomplished. The electron affinity of ozone is measured to be EA(O_{3}) =2.1028(25) eV, in good agreement with previous measurements of less accuracy. Progressions in the spectra are analyzed to yield the symmetric stretching frequency and the bending frequency of the ozonide ion to be 982(30) and 550(50) cm^{−1}, respectively. While no evidence is found for a long lived excited electronic state of O^{−} _{3}, an excited electronic state of neutral ozone is found roughly 0.7–1.1 eV above the ground state. Models for the dissociation of O_{3} ^{−} are examined to explain why the photoelectron and photodetachmentspectra fail to show a strong progression in the symmetric bending vibrational mode. Attempts to measure the electron affinity of CO^{−} _{3} were unsuccessful. Limits placed by this attempt and our EA(O_{3}) value are invoked in a discussion of some recent disagreements in the literature on the thermochemistry of CO^{−} _{3} and O^{−} _{3}.

Electron–molecule scattering in momentum space
View Description Hide DescriptionWe examine the Fourier transform of the Schrödinger equation for electron–molecule scattering, treated as potential scattering from a multicenter distribution of charged fixed in space. When the angle ϑ between R↘,the internuclear vector of a diatomic target, and q↘, the momentum transfer, is held fixed during the collision, then the directions of incidence and scattering are fixed relative to R↘. The process is then described as having a dynamical dependence on the magnitude of q↘, q, from which the scattering angle is determined, and a parametric dependence on q↘′s direction relative to R↘. This approximation is used routinely at high energies in the calculation of the Born amplitude. Fixed–nuclei coordinate–space studies suggest that this approximation can be extended to low energies, provided the amplitude is taken from the solution of the integral equation of momentum space rather than from its inhomogeneity, proportional to the Born amplitude. To obtain this solution, we use the above constraint in the inhomogeneity, and we adopt a similar constraint in the evaluation of the kernel of the integral term. That is, we constrain R↘ to be in the same direction relative to q↘′, a virtual momentum transfer belonging to the kernel, as it is to q↘. In this approximation, the process is described as having a dynamical dependence on two magnitudes, q and q′, and a parametric dependence on the single angle ϑ_{ R }. This causes the equations for the ’’radial’’ or k space partial waves to be diagonal for each value of the orientation angle ϑ_{ R }. Calculations are performed for the e, H_{2} scattering in the static approximation, and cross sections averaged over ϑ_{ R } are shown to be in good agreement with cross sections calculated by use of coupled spherical and coupled spheroidal partial wave theories. The angular distribution in the static approximation is also calculated at an incident energy close to 7 eV, where exchange is relatively unimportant. This result is in reasonably good agreement with that of R matrix theory in the static–exchange approximation. The extension of the theory to treat exchange is formulated and discussed. Also its extension to treat more complicated molecular targets is discussed.

Spin densitites in triplet benzophenone
View Description Hide DescriptionThe lowest photoexcited triplet state of benzophenone in single crystals of 4,4′‐dibromodiphenyl ether has been studied by optically detected ENDOR. The hyperfine coupling tensors for eight protons and carbonyl–^{13}C have been determined by analyzing the angular dependence of the ENDOR spectra. A point–dipole model is presented to account for the anisotropichyperfinetensor elements based solely on spin densitites and molecular geometry. A set of spin densities and a ring–twist angle have been found which, together, best reproduce the observed dipole interaction tensors as calculated from our model. Data from zero‐field ODMR experiments on ^{17}O–benzophenone performed by S. Yamauchi and D. W. Pratt at the University of Pittsburgh enabled us to remove the correlation between n and π spin densities on oxygen. A final fit of all data shows the n electron to be localized on oxygen and the π electron to be delocalized onto the rings.

Fluorocarbons as oxygen carriers. I. An NMR study of oxygen solutions in hexafluorobenzene
View Description Hide DescriptionRelaxation times of ^{13}C and ^{19}F (or ^{1}H) nuclei of hexafluorobenzene (or benzene) solutions of oxygen at 25° C are measured at several frequencies using variable concentrations of oxygen (up to 0.09 M under 6 atm). The influence of paramagnetic oxygen on neighboring solvent nuclei can be explained by dipolar electronic spin–nuclear spin interactions modulated by translation diffusion in the case of C_{6}H_{6}solutions only. The results for C_{6}F_{6}solutions suggest a discontinuous diffusion through the C_{6}F_{6} liquid ’’lattice’’ with a short residence time (≲10^{−12} sec) in privileged positions of the lattice.

Raman spectrum of phase III of solid CH_{4} in the lattice and intramolecular regions
View Description Hide DescriptionTwo samples of solid CH_{4} were grown under high pressures. The pressures of these samples at 10 K place them in phases III and IV, respectively. The Raman spectra of these samples in the lattice and intramolecular regions have been observed. These spectra provide no evidence of a III–IV phase transition. The lattice lines are assigned to translational or librational modes on the basis of their observed mode Grüneisen parameters. The translational modes are then assigned on the basis of a three‐site structure recently proposed for phase III. The intramolecular bands of CH_{4} in phase III contain broad and sharp lines. The broad lines are assigned to lattice sidebands based on the volume dependence of their frequencies. The sharp lines are then assigned to the pure vibrational modes of three types of molecules in phase III.

On the Raman spectrum of argon dimersa)
View Description Hide DescriptionThe pressure broadening of the rotational Raman transitions of argon dimers is calculated, using the semiclassical method. Simple models to compute the line broadening of noble gas dimers are put forward. The predictions of these simple models are compared with the results of the full semiclassical calculation. The optimal experimental conditions to resolve the Ar_{2}Raman spectrum are discussed.

Mechanisms of energy transfer in the deuterium fluoride systema)
View Description Hide DescriptionA three‐dimensional trajectory study has been employed to determine rate coefficients as a function of temperature for the important energy‐transfer processes that occur in DF(v _{1})+DF(v _{2}) collisions. From this study, it was predicted that the v→v energy‐transfer processes occur by means of Δv=±1 transitions and that the rate coefficients for the v→v processes DF(v _{1}=1)+DF(v _{2}) →DF(v′_{1}=0)+DF(v′_{2}=v _{2}+1) with v _{2}±1 through 5, respectively, decrease with increasing vibrational quantum number v. The computed rate coefficients for the v→v processes are k (v _{1}=1, v _{2}=1; v′_{1}=0, v′_{2}=2) =1.3×10^{13} cm^{3}/mole sec and k (v _{1}=1, v _{2}; v′_{1}=0, v′_{2}=v _{2}+1) =1.61^{1−}Hv ^{ 2 } k (1,1;0,2) at 300° K. These v→v processes correspond to near‐resonant vibration‐to‐vibration (v→v) intermolecular energy transfer. The v→R energy‐transfer processes occur by converting multiple quanta of vibrational energy of a vibrationally excited DF molecule into rotational energy of the same molecule. This process is nonresonant v→R intramolecular energy transfer. These multiquantum v→R transitions provide more ways to distribute the vibrational energy of the vibrationally excited DF molecule into rotational energy and thereby populate its high rotational states. The high rotational quantum states are relaxed slowly by R→ (R′,T) processes. A rotational nonequilibrium model is used to calculate quenching rate coefficients for vibrational relaxation of DF(v _{1}=1) by DF(v _{2}=0). The results are in good agreement with available experimental data.

Two classification systems for smectic phases, based on symmetry and order
View Description Hide DescriptionOn the basis of evidence in the literature, it is concluded that the basic molecular packing underlying all smecticstructures is a herringbone arrangement of the molecular planes with an orthogonal lattice of the molecular centers. Starting from this basic packing, two sets of five orthorhombic and three monoclinic smectic layer structures are derived which can all be classified with crystallographic space‐group symbols. One set is for centrosymmetric ’’molecules,’’ the other for noncentrosymmetric ’’molecules.’’ A second classification of the same set of structures is developed on the basis of the symmetry of the molecular positions. This classification is equally specific but more easily visualized, and it leads to simpler classification symbols. With an additional symbol to indicate order or disorder in the stacking of the smectic layers, a set of 14 possible smecticstructures is obtained (each structure allowing centrosymmetric as well as noncentrosymmetric ’’molecules’’), and all known layered smectic phase types are classified within this system, each phase type assigned to a different structure. The structure of the smecticB phase appears to be different than usually assumed, and the so far not understood smecticF phase has been assigned a specific structure.

High temperature thermodynamics of the solutions of hydrogen in palladium‐silver alloys
View Description Hide DescriptionThe thermodynamic properties of solutions of hydrogen in palladium‐silver alloys with silver contents ranging from 25 to 50% have been determined by a calorimetric‐equilibrium method at 555 and 700 K. The results are compared with corresponding data recently reported for hydrogen in Pd and in Pd_{0.9}Ag_{0.1}. (1) For the alloy Pd_{0.75}Ag_{0.25} the thermodynamicmeasurements give no evidence for an order–disorder transition in the considered temperature range. (2) Comparisons between the solutions of hydrogen and deuterium in the same alloy and at the same temperature allow calculations to be made of the vibrational frequencies of hydrogen. The calculated values of ?_{H} show a very significant increase with increasing silver content. (3) The partial excess entropies of hydrogen in dilute solutions in palladium‐silver alloys are r e d u c e d significantly compared to the values in pure palladium. It is probable that these changes, in large measure, are caused by a nonrandom distribution of the hydrogen atoms among the available interstitial sites. However, other possible interpretations also are discussed.

Extended accuracy for analytic model potentials for diatomic molecules
View Description Hide DescriptionFormulas have recently been published which permit the determination of very accurate perturbed‐Morse‐oscillator (PMO) potentials from spectroscopic data. Many of the formulas were presented without a mathematically rigorous derivation. The present work contains such derivations, based on the use of Lagrange series techniques. The derivations are also modified to provide equivalent formulas for two other analytic model potentials, the Dunham and Simons–Parr–Finlan potentials. In addition, general relationships between parameters of the three models are presented, in terms of which one can directly obtain parameters of one model from those of another; the formulas are applied to determine parameters for n?20 for the other two models from published PMO parameters for CO. The suitability and appropriateness of the various models are discussed, especially in relation to problems of convergence and errors due to truncation and roundoff.

Application of field theoretical methods to the calculation of spectroscopic band shapes of molecules in strongly interacting solvents. IV. Microscopic interpretation of the extended rotational diffusion model
View Description Hide DescriptionMethods of field theory have been used to derive e x a c t formal expressions for the frequency Fourier transforms of spectroscopic memory functions. In particular these functions for free‐ and environment‐interacting rotors have been obtained. The nature of the parameter τ_{ J } that appears in the extended diffusionmodels has been identified as being the imaginary part of the self energy. All theoretical methods based upon the memory function techniques must use approximations to the exact formal memory functions derived in this paper. In those regions in which hydrodynamic treatments are no longer valid, the parameter τ_{ J } must appear as a frequency dependent quantity; we have demonstrated this frequency dependence. Finally by using the finite temperature Green’s function method we have derived two expressions for τ_{ J }, one of which is of second order in the solute–solvent interaction potential and the other is a far more exact infinite order result. Both of these results show temperature dependence.

Electric dipole moment of SO_{2} in ground and excited vibrational states
View Description Hide DescriptionRadiofrequency electric resonancespectra of SO_{2} molecular beams have been observed for a number of rotational levels of the (000), (100), (010), (001), (200), (110), (200), and (011) vibrational states. Heated supersonic nozzle beams of SO_{2} in argon were found to give enhanced signal‐to‐noise ratios. Stark effectmeasurements for the vibrational ground state give μ_{000}=1.63305(4) D and polarizability anisotropies of α_{ a a }−α_{ b b }=1.62(40) Å^{3} and α_{ a a }−α_{ c c }=0.1(7) Å^{3}. The dipole moment data for excited vibrational states can be expressed as μ=1.62673+0.00017(v _{1}+1/2) − 0.00684(v _{2}+1/2) + 0.01943(v _{3}+1/2) − 0.00002(v _{1}+1/2)^{2} + 0.00002(v _{1}+1/2) (v_{2}+1/2) − 0.00003(v _{2}+1/2)^{2} − 0.00012(v _{2}+1/2) (v _{3}+1/2). By combining the Stark effect data with infrared intensity measurements, the electric dipole moment function for SO_{2} through second derivative terms has been determined.