Volume 47, Issue 6, 15 September 1967

Microwave Spectral Linewidth of OCS Molecule Due to Molecular Collisions
View Description Hide DescriptionA double‐modulation microwave spectrograph has been used to determine the width of j 1→2 line of OCS molecule at different pressures due to collisions with the following perturbers: carbonyl sulfide, argon, oxygen, air, methyl bromide, and water vapor. Experimental linewidth parameters for OCS–OCS, OCS–Ar, OCS–O_{2}, and OCS–N_{2}molecular collisions are interpreted using Anderson—Tsao—Curnutte theory for the mixed‐interactions case. The molecular quadrupole moments of OCS, O_{2}, and N_{2} are found to be 2.0±2.0, 2.0±0.2, and 5.0±0.5 D·Å, respectively.

Compton Scattering Factors for Spherically Symmetric Free Atoms
View Description Hide DescriptionIncoherent scattering factors for all spherically symmetric free atoms have been computed from numerical SCF Hartree—Fock wavefunctions. The complete Waller—Hartree theory with all exchange terms has been used.

On the Evaluation of Rate Coefficients from Limited Cross‐Section Data
View Description Hide DescriptionAn analysis is presented of the error involved in calculating rate coefficients from cross‐section data in a limited energy range. For reactive cross sections of the form σ(E) = a(E—E ^{*})^{ n } exp[—b(E—E ^{*})], where E is the relative kinetic energy and E ^{*} is the threshold energy, it is shown that an upper bound on the error can readily be derived in terms of the parameters n, b, and E′—E ^{*}, where E′ is the largest kinetic energy for which σ(E) has been determined. For cross sections which can be represented by a polynomial of degree N in the range E ^{*} to E′, the m‐point Laguerre integration formula yields exact results for the rate coefficient for m≥½(N+2), and also provides the criterion for the required energy range E ^{*} to E′.

Polarizations of the Low‐Energy S_{n} ←S _{0} Transitions in 9,10‐Anthraquinone
View Description Hide DescriptionThe polarizations of the transitions to the lowest n—π^{*} and three π—π^{*} singlet states of 9,10‐anthraquinone have been measured relative to that of the phosphorescence by the method of photoselection. Assuming that the n—π^{*} and two π—π^{*}transition moments lie in the molecular plane and utilizing the previous result that the T _{1}←S _{0}transition moment is parallel to the carbonyl axes (Z), the n—π^{*} absorption is found to be polarized along Y and the first three π—π^{*}transitions along Y, Z, Y, respectively. Y is the axis in the molecular plane perpendicular to the carbonyl axis. The ordering of the π^{*} states is compared with the predictions of the two available SCF calculations and with the predictions of the Platt spectroscopic moment theory.

Vibration Frequencies of NO_{2} ^{−} and NO_{3} ^{−} Ions in KBr Crystals
View Description Hide DescriptionInfrared‐absorption spectra of doped KBr crystals at temperatures of approximately 8°K were measured in the wavelength range 2 to 15 μ. From the absorption spectra of crystals doped with NO_{2} ^{−} and NO_{3} ^{−} ions, and with ions enriched in the isotopes^{15}N and ^{18}O, frequencies of normal vibrations of six isotopic NO_{2} ^{−} molecules, and eight isotopic NO_{2} ^{−} molecules were determined. These frequency values were used to calculate a set of force constants for NO_{2} ^{−} and NO_{3} ^{−} ions in the KBr lattice, using the most general harmonic force field (GFF). In mdyn/Å units, values obtained for NO_{2} ^{−} were: f_{r} =7.50, f_{rr} =1.66, f _{ rα}=0.39, f _{α}=1.67, and for NO_{3} ^{−}: f_{r} =7.62, f_{rr} =1.43, (f _{α}—f _{αα})=1.09, (f _{ rα}—f _{ rα}′)=−0 .48, f _{γ}=1.506.
In crystals doped with divalent metal ions as well as NO_{2} ^{−} and NO_{3} ^{−} ions, new absorption bands appeared. These were perturbations of the vibrational frequencies of NO_{2} ^{−} and NO_{3} ^{−} ions, caused by association between divalent metal ions and NO_{2} ^{−} and NO_{3} ^{−} ions. In the case of NO_{3} ^{−}, the doubly degenerate asymmetric stretching band was split into two nondegenerate bands by the perturbation, which is thus almost certainly caused by a nearest‐neighbor association between metal and nitrate ions. While the other frequency shifts of nondegenerate vibrations can give no information about the symmetry of the surrounding lattice, they can also be explained by a nearest‐neighbor interaction between divalent metal and NO_{2} ^{−} and NO_{3} ions.

Microwave Spectra of Nitrogen‐Containing Molecules. II. Structure, Dipole Moment, Quadrupole Coupling Constants, and Barrier to Internal Rotation of N‐Methyl Ethyleneimine
View Description Hide DescriptionThe microwave spectrum of N‐methyl ethyleneimine has been studied in the region 8 to 28 GHz, and the following rotational constants have been obtained for the ground state:A=16443.15 MHz, B=7219.84 MHz, C=6152.69 MHz. Stark‐effect measurements led to dipole‐moment components of μ_{ a }=0.07±0.02 D μ_{ c }=1.234±0.015 D, and a total dipole moment of 1.236±0.02 D. Internal rotation splittings were observed in the v=1 torsional state and gave a barrier to internal rotation of 3462±35 cal/mole. The quadrupole coupling constants of ^{14}N in the inertial axis system were found to be: χ_{ aa }=3.35, χ_{ bb }=0.63, and χ_{ cc }=−3.98±0.02 MHz. The molecular structure has been described, and the previously mentioned results have been discussed with regard to other existing data. A tentative estimate has been made of the quadrupole coupling constants in the field‐gradient principal‐axis system.

Factor Group Splitting and Zeeman Splitting of the Lowest Triplet State of Phenazine
View Description Hide DescriptionThe T _{1}←S _{0} transition of the phenazine crystal is studied in polarized light, and in high magnetic fields, with the crystals at 4.2°K. A factor group splitting of 4 cm^{−1} is observed in the 0, 0 band. The two components have the same polarization—predominantly along the b axis. The polarization, relative intensity, and Zeemanspectra are used to assign the plus and minus factor group states in association with a crystal symmetry adapted spin—orbital coupling theory.
The lowest triplet state is assigned as ^{3} B _{2u } in agreement with theoretical predictions.

Study of Energy Transfer from Sb^{3+}, Bi^{3+}, Ce^{3+} to Sm^{3+}, Eu^{3+}, Tb^{3+}, Dy^{3+}
View Description Hide DescriptionEnergy transfer from the sensitizers (S) Sb^{3+}, Bi^{3+}, or Ce^{3+} to the activators (A) Sm^{3+}, Eu^{3+}, Tb^{3+}, or Dy^{3+} has been studied in four different host lattices, viz., YOCl, YBO_{3}, YAl_{3}B_{4}O_{12}, and YPO_{4}. The efficiency of the energy transfer in these systems varies markedly. This is the case even for a given combination of S and A in different host lattices. In most, but not all, cases SS energy transfer is also important. The SA energy transfer is probably nonradiative and occurs mainly by exchange interaction if the S emission band overlaps only 4f−4f absorption bands of A, but by electric multipole—multipole interaction, if the S emission band overlaps allowed absorption bands of A. Taking these ideas and the energy overlap of S emission and S absorption or S emission and A absorption into account, it is possible to explain the results in a qualitative way. If, however, the energy difference between the state S^{*} (excited) +A and the state consisting of the ions S^{+}+A^{−} is relatively small, no energy transfer by exchange is observed.

Theory of Molecular Polarizabilities
View Description Hide DescriptionBuckingham's theory of the interaction between a polarizable charge distribution and an external electric field is presented and extended in a unified way. Transformations of components of the molecular polarizability tensors under change of the coordinate origin are derived. Relationships between tensor components in systems of axial and spherical symmetry are given.

Corresponding Orbitals and the Nonorthogonality Problem in Molecular Quantum Mechanics
View Description Hide DescriptionGiven any two sets of spin orbitals a_{i} and b_{j}, there exist equivalent sets â_{i} and b̂_{j} such that their overlap matrix is diagonal, i.e., 〈â_{i}  b̂_{j} 〉=d̂_{ii} δ_{ ij }. This is the basis of the corresponding orbital transformation of Amos and Hall. Their transformation is shown to have widespread application to quantum chemistry. It leads to a simple generalization of the Slater—Condon rules for the expectation value of an operator between two determinantal wavefunctions when the spin orbitals of one function have no simple orthogonality relationship to those of the other function. In the case of single‐determinantal wavefunctions, use of the corresponding orbital transformation and the integral Hellmann—Feynman formula leads to a very simple expression for the energy difference associated with two similar configurations of a molecular system. Extensions to limited configuration interaction expansions are discussed. Given single‐determinantal wavefunctions for two related molecular systems, it is shown that the corresponding orbitals are those which are most nearly molecularly invariant in the sense of maximum overlap. A comparison of the Pitzer—Lipscomb wavefunctions for the staggered and eclipsed forms of ethane reveals that six of the nine corresponding orbitals have an overlap of no less than 0.999998 in the two configurations. Use of the corresponding orbital transformation overcomes various computational difficulties encountered with Löwdin's cofactor method for treating the nonorthogonality problem.

Viscosity of Entangling Polydisperse Polymers
View Description Hide DescriptionThis paper presents a theory of viscosity in steady shearing flow for bulk polymers and concentrated polymer solutions. The basis for the theory is the supposition that intermolecular chain entanglements control the magnitude of the viscosity and that the decrease in viscosity with increasing shear rate is caused by shear‐induced changes in the network of entanglements. It was found possible to represent the effect of entanglements by an additional term in the segmental friction coefficient, and to incorporate the effects of polymer concentration, molecular weight distribution, and shear rate in the final result. At low shear rates the viscosity reduces to η_{0}=(const) (φn̄_{x} )^{3.5} for highly entangled chains, where φ is the volume fraction of polymer and n̄_{x} is an average chain length slightly greater than the weight average. The form of the viscosity—shear rate master curve was found to depend on the chain‐length distribution of the polymer. Departures from Newtonian behavior occur at lower shear rates the broader the distribution, but at sufficiently high shear rates the behavior becomes similar for all distributions. The master curve for monodisperse polymers was in good agreement with measurements on solutions of narrow distribution polystyrene. The limiting power‐law exponent in was found to be 9/11 rather than ¾ as given by an earlier theory. The master curve calculated for most‐probable distributions (M̄_{w} /M̄_{n} =2) agreed moderately well with the empirical master curve of Bueche and Harding and with data on solutions of unfractionated polystyrene.

Geometry of Ozone and Azide Ion in Ground and Certain Excited States
View Description Hide DescriptionA series of ab initio SCF MO calculations for ground and various closed‐shell excited states of O_{3} and N_{3} ^{−} have been carried out with a view toward investigating relationships between the geometries of these states. The calculations give quantitative verification to many of the assumptions made in previous empirical theories dealing with this subject and enlarge the scope of former quantitative schemes to include more quantitative predictions relative to steepness of potential surfaces of molecular states. Thus it is shown that the calculated SCF total energy surfaces of both O_{3} and N_{3} ^{−} can be compared quite concisely in terms of a small number of differentiating orbital‐energy curves.

Intersystem Crossing in Gaseous Molecules
View Description Hide DescriptionThe theory of electronic relaxation in the solid phase is discussed in terms of gaseous molecules. Two limits are examined—the ``small‐molecule limit'' (α limit), where electronic relaxation cannot occur in the free molecule, and the ``big‐molecule limit'' (ω limit), where electronic relaxation at a rate virtually identical with that in the solid can take place in the absence of any external perturbation. An ``intermediate case'' (μ case) is described where no relaxation can occur in the completely free molecule, but where only extremely minute perturbations are required to induce such a process. Intersystem crossing in benzene is close to the borderline between the ω limit and the μ case. Although the concepts are somewhat inexact quantitatively because of the difficulty in estimating matrix elements and the vibrational density‐of‐state function, the theory does indicate that the zero‐point level of the ^{1} B _{2u } state could behave like a μ case, while the excited vibrational levels of this electronic state, those mainly reached in past experiments, may lie in the ω limit. Available experimental data support the idea that the higher vibrational levels of the ^{1} B _{2u } state of benzene very likely lie in the ω limit, but cannot shed light on the behavior of the zero‐point level.
Theoretical expressions for the rate of intersystem crossing, the fluorescence lifetime, and the fluorescence yield as functions of pressure are worked out. Theory shows that in both the α limit and in the μ case the absolute fluorescence yield approaches unity in the limit of zero pressure. For the ω limit, the absolute fluorescence yield can very well be less than unity even for a molecule in free space. Some new experiments are suggested by the theoretical conclusions.

Configuration of Isolated Polymer Molecules Adsorbed on Solid Surfaces Studied by Monte‐Carlo Computer Simulation
View Description Hide DescriptionThe configurations of adsorbed polymer molecules with excluded volume were simulated on a four‐choice simple cubic lattice using a computer. The average values of the fraction of segments on the surface, loops off the surface, normal distance of the end of the molecule from the surface, root mean square of the normal distance, maximum normal distance from the surface, and root‐mean‐square end‐to‐end distance were calculated for various lengths of the molecules and various attractive energies between segments and the surface. When these averages over the configurations are compared with previous results which do not account for the excluded‐volume effect, important differences are found.

Crystal Field in Rare‐Earth Trichlorides. I. Overlap and Exchange Effects in PrCl_{3}
View Description Hide DescriptionOverlap and exchange contributions to the crystal field of PrCl_{3} have been determined using an LCAO—MO model. The contributions to the parameters are found to be 8.5×greater than the magnitudes predicted by the simple point‐charge electrostatic model. However, the calculated values of these parameters are still 21% less than the experimental values and several reasons for this discrepancy are suggested.

Very High Pressure Effects upon the EPR Spectrum of Ruby
View Description Hide DescriptionThe variation in the electron paramagnetic resonance(EPR) spectrum of ruby as a function of pressure has been observed to beyond 70 kbar for a magnetic field orientation parallel to the crystalline c axis. The data can be interpreted in terms of the usual spin Hamiltonian:, where the spectroscopic splitting factor g‖ has the same value as at ambient pressures, but where the zero‐field splitting δ increases in a linear fashion from ∼0.38 to ∼0.43 cm^{−1}. The experimental apparatus has been described previously, although important improvements in the pressure seal and in the pressurecalibration have been made and are described in an Appendix.

Optical Anisotropy of Chain Molecules. Theory of Depolarization of Scattered Light with Application to n‐Alkanes
View Description Hide DescriptionDepolarization of light scattered by chain molecules is treated according to the valence‐optical scheme which rests on the assumption that the polarizabilitytensor α for the molecule may be formulated as the sum of contributions of individual bonds, or of constituent groups. The anisotropy of the polarizability is described by the tensor invariant γ^{2} (proportional to the sum of the squares of the elements of the traceless part of α), which determines the depolarization ratio. This invariant (γ^{2}) is averaged over all configurations of the chain molecule treated in the rotational isomeric‐state approximation, interdependence of rotations about neighboring bonds being taken into account. Beyond the stated assumptions, the treatment is exact; it is applicable to chains of any length, and of any specified structure. Illustrative calculations for the n‐alkanes are presented and compared with experimental depolarization results for liquids of this homologous series and for their solutions in solvents of low optical anisotropy.

On the Excess Helmholtz Function of Isotope Mixtures
View Description Hide DescriptionIt is observed that, if one represents the Helmholtz function of an isotopic mixture by that of a pseudosingle component system with mass , where the m _{α} are the masses of the components, and the x _{α} their mole fractions, a good approximation is obtained. The Helmholtz function of the pseudosingle component system is obtained via the quantal law of corresponding states. Plausible, though nonrigorous, theoretical arguments for the validity of this observation are presented.

Molecular Theory of Counterion Conductivity and Self‐Diffusion in Polyelectrolyte Solutions
View Description Hide DescriptionA previous theory of counterion self‐diffusion in a polyelectrolyte solution is extended to describe the counterion flow in the presence of an applied electric field. A term proportional to the velocity of the polyion is obtained in a natural way, with no assumptions concerning ``bound'' counterions. The previous expression for the counterion self‐diffusion coefficient is regained as a special case. The significance of the formal agreement with the ion binding model is discussed at some length.

Enthalpies of Mixing in Some Binary Alkaline‐Earth Chlorides
View Description Hide DescriptionThe integral enthalpies of mixing in the six binary systems formed by MgCl_{2}, CaCl_{2}, SrCl_{2}, and BaCl_{2} have been determined calorimetrically. The results are discussed in terms of the theories of Reiss, Katz, and Kleppa and of Davis and Rice, each suitably modified to take into account the higher charge on the cation. The molar enthalpies of mixing for the systems CaCl_{2}–SrCl_{2}, CaCl_{2}–BaCl_{2}, and SrCl_{2}–BaCl_{2} can be represented by the following approximate expression which is simply related to the corresponding expression for the binary alkali halides, previously derived by Hersh and Kleppa: Here X _{1} and X _{2} are the mole fractions of the two components; δ_{12}=(d _{1}—d _{2})/d _{1} d _{2}, where d _{1} and d _{2} are the sums of the ionic radii of anion and cation in the two salts, while Z=Z _{1} Z _{2}, the product of the charges of the anion and cation in the component salts (here Z=2). The term U _{0} ^{++} represents an estimate of the contribution to the enthalpy of mixing arising from the London dispersion interaction between next‐nearest‐neighbor cations. The quoted expression does not hold for the three binary systems involving MgCl_{2}, which are all much less exothermic and which exhibit considerable energetic asymmetry. In all cases the enthalpy of solution of MCl_{2}in MgCl_{2} is more endothermic than the opposite process. The results for these three systems are interpreted to support the view, originally advanced by Førland, that pure MgCl_{2} has a certain tendency to form covalently bonded Mg–Cl–Mg bridges. The breaking of these bridges gives rise to a significant endothermic contribution to the enthalpy of mixing, over and above the contributions arising from Coulombic and dispersion forces.