Volume 52, Issue 10, 15 May 1970

Variational Approach to the Dielectric Constant of a Polarizable Medium
View Description Hide DescriptionThe Kirkwood theory of the dielectric constant for a nonpolar fluid is shown to imply a negative electrostatic self‐energy for certain configurations of point dipoles. It is shown that this unphysical behavior can be eliminated by introducing a hard core of radius larger than or equal to the cube root of the molecular polarizability. The introduction of such a core also permits the derivation of variational upper and lower bounds on the dielectric constant. In particular, the Clausius–Mossotti formula can be shown to give too low a value for the dielectric constant of a polarizable sphere fluid. The relationship of these bounds to those developed elsewhere for the bulk properties of random two‐phase media is discussed, as is the effect of dipole fluctuations.

Soft‐Sphere Equation of State
View Description Hide DescriptionThe pressure and entropy for soft‐sphere particles interacting with an inverse twelfth‐power potential are determined using the Monte Carlo method. The solid‐phase entropy is calculated in two ways: by integrating the single‐occupancy equation of state from the low density limit to solid densities, and by using solid‐phase Monte Carlo pressures to evaluate the anharmonic corrections to the lattice‐dynamics high‐density limit. The two methods agree, and the entropy is used to locate the melting transition. The computed results are compared with the predictions of the virial series, lattice dynamics,perturbation theories, and cell models. For the fluid phase, perturbation theory is very accurate up to two‐thirds of the freezing density. For the solid phase, a correlated cell model predicts pressures very close to the Monte Carlo results.

Anisotropies and Absolute Signs of the Indirect Spin–Spin Coupling Constants in ^{13}CH_{3}F
View Description Hide DescriptionProton and ^{19}F magnetic resonance experiments on ^{13}CH_{3}F partially oriented in the liquid crystalsolventp,p′‐di‐n‐hexyloxyazoxybenzene show conclusively that , , and , with the signs absolutely determined. Assuming that there is no change of molecular geometry between the gas phase and the nematic solution, and neglecting solventeffects,isotope effects and corrections due to vibrational averaging, the experimental results suggest unexpectedly large anisotropies of the spin couplings, , (measured with respect to the molecular symmetry axis), whereas has negligible anisotropy. The anisotropies in are examined with respect to their sensitivities upon the appropriate average molecular geometries used in their determination, and the effects of solvent are also discussed. The accuracy of the above anisotropies cannot be determined without a knowledge of the complete vibrational potential of methylfluoride. However, it can be concluded that indirect spin coupling anisotropies do exist for directly bonded nuclei, and that at least one of them is sizeable, of the order of magnitude of the isotropic .

Rotational Structure of a Novel Raman Effect in Quantized Symmetric‐ and Spherical‐Top Molecules
View Description Hide DescriptionBy angular momentum techniques, the theoretical line intensities of a higher‐order odd‐parityRaman effect have been derived. This effect may be used as an alternative mechanism to study the silent modes forbidden in the infrared spectra, and in the conventional lower‐order Raman effect which is of evenparity. It can be compared with the recently discovered (three‐photon) hyper‐Radman effect which is also of oddparity and contrasted with the recently studied electronic Raman effect using the antisymmetric scatteringtensor which is of even parity: It has different selection rules (Secs. IV, V) and angular characteristics (Sec. III and Appendix B). It is Raman scattering in which one photon is in an electric dipole mode, whereas the other photon is in a magnetic dipole (or electric quadrupole) mode. The scatteringtensor is of odd parity. The intensities along the perpendicular direction of observation of the scattering by symmetric‐top molecules in their transition from the state and by spherical‐top molecules in their transition from the state to have been derived (Sec. II and Appendix A). Depolarization ratios for linearly polarized and for unpolarized incident light may be computed from these intensities. Similar intensities for scattering of circularly polarized light into forward as well as into arbitrary observation directions were derived (Sec. III and Appendix B). These allow the computation of the reversal coefficients. In the limit when both photons are in the electric dipole mode the results will agree with those for the conventional even‐parityRaman effect studied by Placzek and by Teller. The total intensity for the scattering of unpolarized light into an arbitrary direction and the integrated intensity for scattering into all directions have been obtained (Appendix B). Explicit forms of the new scatteringtensors, which are bipolar harmonics, in Cartesian coordinates are given (Sec. IV and Appendix C). Their similarity to, as well as difference from, spherical harmonics are shown. Special features are pointed out for the tensors. Their transformation under the point group, to which a symmetric‐top molecule may belong, and under the point group, to which a spherical‐top molecule may belong, have been derived and tabulated (Tables I and II). The scatteringtensor may connect different electronic states of oppositeparities in an electronic Raman effect, or may give rise to a rotational and vibrational Raman effect with different selection rules (Sec. V) from the conventional Raman effect, and thereby may serve as a supplement for the study of molecular structure.

High‐Temperature Evaporation and Thermodynamic Properties of Cm_{2}O_{3}
View Description Hide DescriptionCurium oxide has been shown to vaporize congruently as Cm_{2}O_{3}. Effusion measurements have been carried out, using tungsten and molybdenum cells, over a temperature range from 1800 to 2600°K and a pressure range from 10^{−9} atm to 0.5 mm. The linear least‐squares fit to the total volatility, calculated as Cm_{2}O_{3}, yields the equation . The corresponding and were computed to be 133.0 ± 2.7 kcal/mole of gas and 33.5 ± 1.2 cal/deg·mole of gas, respectively. The predominant vaporization process is thought to be the formation of CmO(g) and O(g), based on estimates of the heat of formation of Cm_{2}O_{3}(s), comparison of experimental and estimated entropies of vaporization, consistency between second‐ and third‐law heats of vaporization, and trends in the vaporization behavior of lighter actinide oxides. The heat of vaporization at 0°K would then be 429 ± 8 kcal/mole of Cm_{2}O_{3}.

Mercury Transition in Solid Rare Gases
View Description Hide DescriptionIt is shown that the gross shifts in the spectrum of the Hgtransition of Hg in solid‐rare gases can be correlated by considering the difference in the interaction potential between the lattice and the Hg atom in the and states, providing the distortion of the lattice caused by the Hg atom is taken into account. In Ar, Kr, and Xe the spectrum can consist of three separate components. It is quantitatively shown that one of the components arises from relatively isolated Hg atoms while the other two components result from Hg atoms having nearest neighbor Hg atoms.

Molecular Beams of Macroions. II
View Description Hide DescriptionUsing the electrospray technique and nozzle–skimmer system previously described, the influence of polystyrene concentration in the sprayed solution on the magnitude and substructure of the macroion current in the molecular beam has been investigated. Other variables studied include polarity of the beam, needle–nozzle and nozzle–skimmer distances, shape of spray, etc. Attempts to produce isolated macroions in the beam by carrier gas atomization were not successful; operation at reduced pressures and elevated temperatures was not found to be advantageous. In the case of positive macroions the current–repeller‐voltage curves had less substructure than in the case of negative beams, and the first stopping potential was more dependent upon the solution concentration, especially when using a relatively large nozzle–skimmer distance. For molecular weights of the polystyrene from 600 to 97 200 the relative macroion positive charge was roughly proportional to the surface area of the macroions, assuming the latter to be spheres and as calculated from the first measured stopping potential. In the case of negative beams the current at zero repeller voltage and at a solution concentration of 2.2 × 10^{−6} Mw was proportional to the third power of the molecular weight at the lower solute molecular weights, as is predicted for a diffusionalcharging process.

Exciton Wave Packet Localization on an Impurity
View Description Hide DescriptionThe time‐dependent localization of an exciton wave packet on a single impurity is calculated exactly for an infinite one‐dimensional crystal with nearest‐neighbor coupling. The problem is solved by a new algebraic operator method, and the resultant expression evaluated by computer. The dependence of the localization probability on the impurity strength and the distance of the initial absorption site from the impurity is discussed.

Constrained Variation Method in Molecular Quantum Mechanics. III. The Use of Multiple Constraints and the Hellman–Feynman Theorem
View Description Hide DescriptionThe result of (a) constraining the forces on the nuclei in the lithium hydride molecule to their theoretical values and (b) satisfying the virial theorem by a constrained variation technique are presented. A method for the imposition of multiple constraints was developed, and criteria for the reliability of wavefunctions with regards to the possibility of satisfying theoretical constraints are discussed.

Self‐Consistent Molecular Orbital Methods. VI. Energy Optimized Gaussian Atomic Orbitals
View Description Hide DescriptionMinimal basis atomic orbitals expressed as sums of Gaussian functions are presented for hydrogen and for the first row atoms boron to fluorine. The expansion coefficients and Gaussian exponents are determined by minimizing the total calculated energy of the atomic ground state. For expansion lengths of up to six Gaussians, two sets of atomic orbitals are reported. In the first set, which we describe as unconstrained, different Gaussian exponents are used for the and atomic orbitals. In the second set, the and atomic orbitals are constrained to share the same Gaussian exponents. It is shown that this constraint, which produces a significant gain in computational speed in molecular calculations, does not seriously reduce the quality of the atomic orbitals for given . A comparison of the contracted sets presented here with previous studies on uncontracted basis sets for the first row atoms, shows that the uncontracted Gaussian exponents are a poor approximation to those of the contracted functions.

Interpretation of Diamond and Graphite Compressibility Data Using Molecular Force Constants
View Description Hide DescriptionThe diamond and in‐plane graphite compressibility data of Lynch and Drickamer are analyzed in terms of simple bond‐compression and bond‐compression, buckling, and puckering models, respectively, using carbon–carbon stretching and out‐of‐plane displacement force constants from molecular studies. To the highest experimental pressures of over 300 kbar it is found that the quotient of potential energy of bond compression and macroscopic work of compression, for the same values of the lattice parameter ratio , remains essentially constant for diamond at 0.7 indicating that bond compression is the dominant mechanism for storing energy of compression. Similar quotients for graphite at corresponding values of vary from 1.4 at the lowest pressures to 0.6 at the highest for the in‐plane bond compression, from 100–4 for the out‐of‐plane buckling, and from 300–13 for the out‐of‐plane puckering with the last two modes assuming a fixed C–C in‐plane bond distance. It is suggested that in graphite bond compression is the primary mechanism for absorbing intraplanar energy of compression with out‐of‐plane buckling and puckering much less important until quite high pressures are reached, in contrast to earlier views. This would seem to lead to a partial understanding of the difficulties present in synthesizing diamond from graphite.

Distorted Wave Born Series for Rotational Inelastic Scattering
View Description Hide DescriptionThe distorted wave Born series is applied to the problem of rotational inelastic scattering of an atom by a diatomic molecule. The formulation is made in the total angular momentum representation, and for the special but important case of an exponential interaction a simple second‐order expression for the degeneracy averaged cross section is derived. Numerical results for two intermolecular potentials compare favorably with exact multistate calculations. The analytic results (1) show that differences between exact and DW results do not vanish near threshold, (2) show the applicability of the method to molecules heavier than H_{2}, (3) provide a method of assessing the relative order of contributions from various internal states, (4) provide a method of estimating higher‐order terms in the DWB series. The application of the second‐order DWB series to problems with different forms of potentials of interaction requires only a single numerical integration.

One‐Dimensional Random Lattice Systems Including DNA
View Description Hide DescriptionA first‐order difference equation is derived in the ratio of successive partition functions along the lattice for one‐dimensional lattice systems with a nonconstant 2 × 2 transfer matrix. In the case of DNA, the distribution functions of these ratios are easy to generate. Numerical values for the expected free energy, obtained using unsophisticated computer techniques, are reported. They agree with those given by Lehman et al. to within 2% computational accuracy. The random Ising model is also considered and the DNAgenerating equations shown to apply in a special case.

New Nuclear Quadrupole Resonance Frequencies at ^{14}N in Saturated Cyclic Amines
View Description Hide DescriptionThe values of the nuclear quadrupole coupling constants of ^{14}N and the asymmetry parameters in a series of saturated six‐, five‐, four‐, and three‐membered cyclic amines and of their N‐methyl derivatives have been measured at various temperatures, starting from liquid nitrogen temperature. Several resonance lines which could not be observed at 77°K, were detected at higher temperatures. By applying the Townes and Dailey theory to the experimental results, the electron populations of the atomic orbitals used by the nitrogen atom for the bonds with the carbon and hydrogen atoms have been evaluated. The values of the valence angles of nitrogen in the various kinds of molecules have also been evaluated. A particularly interesting result has been obtained for the four‐ and three‐membered cyclic amines.

Transport Mechanics in Systems of Orientable Particles. II. Kinetic Theory of Orientation Specific Transport for Hard‐Core Models
View Description Hide DescriptionThe role of orientation and polarization in the transport phenomena of hard‐core polyatomic gases is developed in detail from the standpoint of the Boltzmann equation. A rigorous orientation specific Chapman and Enskog scheme is used to establish linear constitutive relations and anisotropictransport coefficients for orientation‐specific translational and rotational diffusion and for the orientation‐summed fluxes of mass, momentum, and heat. Calculations are reported for the effect of polarization upon the shear viscosity and thermal conductivity tensors of loaded spheres. The anisotropic expressions for rotational and translational thermal diffusion, shear diffusion, and concentration diffusion coefficient tensors are determined for a dilute system of loaded spherocylinders in a bath of rigid spheres. With the use of time reversal invariance, a microscopic reciprocal theorem is established and employed to prove Onsager reciprocal relations among the transport coefficient tensors.

Construction of Tetrahedral Harmonics
View Description Hide DescriptionExplicit, detailed, and convenient algorithms have been developed for constructing tetrahedral harmonics for arbitrary J. These functions are expressed as linear combinations of spherical harmonics and also of symmetric top functions. A projection‐operator technique was used to produce projected functions which transform according to the rows of the irreducible matrix representations of . In general the set of projected functions so constructed may be linearly dependent. The idempotence of the projection operators was invoked to show that all the required orthonormal tetrahedral harmonics can be produced by the diagonalization of the projection‐operator matrices. The elements of these matrices are given explicitly for arbitrary J. In an Appendix, alternative procedures, including the Gram–Schmidt method, for the orthogonalization of the projected functions are treated briefly, with special emphasis on the use of these matrix elements. The “symmetry‐adapted functions” involve the , and a recursion relation is presented which facilitates an accurate and rapid calculation of these constants. An important new sum rule useful in calculations of integrated intensities is derived using the idempotence property. Various physical applications are discussed.

Effect of Internal Viscosity on the Deformation of a Linear Macromolecule in a Sheared Solution
View Description Hide DescriptionThe deformation of macromolecules in a solution subjected to time‐dependent shear flow is described by means of a modified Rouse model. Into every segment of this model an internal viscosity force is introduced which is proportional to the rate at which the end‐to‐end distance of the segment is changed, and acts in the direction of the tie line between the end points. No a priori assumptions have been made on the motion of the segments themselves. An analytical solution for the distribution function of the segment lengths can now no longer be derived. At low values of the internal viscosity, an approximate solution can be obtained by means of a perturbationcalculus. The effect of internal viscosity on the moments of the distribution function and, hence, on the average dimensions of the macromolecules, can be calculated. It appears that with constant shear flow at high rates of shear, as well as with oscillatory shear at high frequencies, an increase of the internal viscosity results in a decrease of the average deformation of the molecules. The flowbirefringence at high shear rates decreases as the internal viscosity increases, whereas the optical extinction angle is hardly influenced by it. In the literature there is no unanimity on the relation between the deformation of the molecules and the shear stress; in the present study we use the formula of Kirkwood and Riseman. In the case of constant shear flow, the apparent viscosity of the solution decreases with increasing rates of shear. With oscillatory shear flow the internal viscosity leads to a finite limiting value of the dynamic viscosity at high frequencies, and this value should increase when a constant shear rate is superimposed on the oscillatory shear rate. This theoretical prediction conflicts with experimental data.

Solubility of Condensed Substances in Dense Gases and the Effect on PVT Properties
View Description Hide DescriptionAn analytic relation is derived in terms of the molecular interactions for the density dependence of the solubility of a liquid or solid in a dense gas using a modified van der Waals equation. It is shown that at high gas densities the solubility is sharply reduced. It is also shown that the effect of the solubility enhancement on high‐density PVT measurements is smaller than previously thought. The analysis is applied to the case where the condensed material is liquidmercury. Solubility‐vs‐density isotherms and the effect of the solubility on PVT measurements are obtained at 50, 150, 250, and 400°C for the following gaseous species: He, H_{2}, Ne, Ar, Kr, Xe, N_{2}, NH_{3}, H_{2}O. The need for accurate experimental data at high gas densities to test further the essential features of the analysis is discussed.

Ion–Molecule Reactions Involving N_{2} ^{+}, N^{+}, O_{2} ^{+}, and O^{+} Ions from 300°K to ∼1 eV
View Description Hide DescriptionSeveral ion–molecule reactions of ionospheric interest have been studied in a drift‐tube–mass‐spectrometer apparatus for ions of mean energy from thermal energies to ∼ 1 eV. The measured rate coefficients in cubic centimeters/second are N^{+}+O_{2}: 5 × 10^{−10} from 0.039–0.9 eV; N_{2} ^{+}+O_{2}: 6 × 10^{−11} at 0.039 eV, decreasing to ∼ 8 × 10^{−12} at 0.9 eV; O_{2} ^{+}+NO: 3 × 10^{−10} from 0.039–1.6 eV; O^{+}+CO_{2}: 1 × 10^{−9} at 0.039 eV, decreasing to 5 × 10^{−10} cm^{3}/sec at 1.3 eV. The relevance of the first three reactions for the Earth's ionosphere and of the last reaction for the Martian atmosphere is discussed briefly.

Theory of the Hydrogen Bond: Ab Initio Calculations on Hydrogen Fluoride Dimer and the Mixed Water–Hydrogen Fluoride Dimer
View Description Hide DescriptionHigh‐accuracy molecular orbital calculations have been carried out on different geometries of the hydrogen fluoride dimer and the mixed water–hydrogen fluoride dimer. A zigzag (near linear) structure is predicted for the hydrogen fluoride dimer with a dimerization energy in reasonable agreement with experiment. One geometry of the mixed water–hydrogen fluoride dimer has a very large stabilization energy (10 kcal/mole), and a microwave experiment is proposed to determine its exact structure. Changes in molecular properties and charge distribution upon dimer formation are calculated and a dimer rotational barrier determined.