Volume 53, Issue 2, 15 July 1970

Application of Many‐Body Perturbation Theory to the Hydrogen Molecule
View Description Hide DescriptionThe hydrogen molecule total energy,polarizabilities, and electron‐coupled nuclear spin‐spin Fermi interaction are studied by many‐body perturbation theory. Two potentials, the Hartree and Hartree–Fock, are considered. Through second order the energies are within several kilocalories per mole of the accurate nonrelativistic values. The Hartree–Fock expansion is reasonably accurate through first order for polarizabilities and gives very good agreement with experiment if a rather simple geometric approximation is used. For the spin–spin coupling constant, good agreement with experiment is obtained using the Hartree potential and considering some second‐ and higher‐order diagrams.

Theoretical Analysis of the Electronic Structure and Molecular Properties of the Alkali Halides. V. Potassium Chloride and Lithium Bromide
View Description Hide DescriptionThe results of recent self‐consistent field calculations on potassium chloride and lithium bromide are presented. Special emphasis is given to an analysis of molecular properties including nuclear quadrupole coupling constants, isotope shifts, spectroscopic constants, relativistic interactions, molecular force constants, and the electronic dipole moment.

Noniterative Solutions of Integral Equations for Scattering. IV. Preliminary Calculations for Coupled Open Channels and Coupled Eigenvalue Problems
View Description Hide DescriptionThe homogeneous integral solution formalism developed by Sams and Kouri is illustrated for coupled open channel scattering calculations and coupled eigenvalue calculations. This method for scattering calculations compares favorably, both in accuracy and in computation time required, with the Lester–Bernstein close coupling solution of the differential equations for scattering and the Johnson–Secrest amplitude density method. Preliminary results for coupled eigenvalueintegral equations are also encouraging. A simple technique for applying the quantization condition in searching for the eigenvalues of coupled integral equations is presented, and is employed in the calculations reported. Example results are given for 16, 23, and 30 coupled scattering channels and for 1, 2, and 3, coupled integral eigenvalueequations.

Rotation‐Vibration Energy Transfer in Collisions between and Ar and N_{2}
View Description Hide DescriptionCollisional relaxation of in high rotational levels of and 1 has been investigated with respect to transitions from rotational levels in to levels in . Initial nonequilibrium rotational distributions of in and 1 were produced by monochromatic photodissociation of H_{2}O with the radiation of a krypton resonance lamp at 1236 and 1165 Å. The effect of added foreign gases (Ar and N_{2}) on the population of individual levels in and 1 has been studied under steady‐state conditions by observing the emission intensities of individual lines in the (0, 0) and (1, 1) bands of the transition. The essential observation was made on the population of the rotational level in . The population of this level increased significantly in the presence of Ar and N_{2} beyond the initial population produced from H_{2}O alone. In comparison, the population of adjacent levels remained relatively unchanged or decreased when foreign gas was added. The effect on the level is attributed to the collisional transfer process where the energy difference, , between the two levels is small compared to the heat bath energy,. For this process and its reverse, a rate constant of the order of 10^{−11} cm^{3} molecule^{−1}·sec^{−1} has been derived. According to angular‐momentum conservation, the process involves, in the case of Ar as collision partner, an increase in the impact parameter which is estimated to be about 0.3 Å.

Approximate Evaluation of the Second‐Order Term in the Perturbation Theory of Fluids
View Description Hide DescriptionThe second‐order term in the free energy,, as given by the Barker–Henderson perturbation theory is considered for the square‐well and 6:12 potentials. The second‐order term is written in terms of integrals of the distribution functions of the unperturbed hard‐sphere system. If the superposition approximation is used for the third‐ and fourth‐order distribution functions, these integrals may be evaluated fairly easily and values for may be obtained. Comparison of these values with either the exact second‐order term Monte Carlo calculations of the exact second‐order term gives exact agreement for the lattice gas, fairly good agreement for the square‐well potential, and good agreement for the 6:12 potential.

Thermal Decomposition of N_{2}O_{4} and NO_{2} by Shock Waves
View Description Hide DescriptionA study of the reactions N_{2}O_{4} + M⇄2NO_{2} + M and 2NO_{2} + M⇄2NO + O_{2} + M was carried out using the shock‐wave method. For the first of these two reactions the rate equation was taken to be based on previous results. For argon as the inert gas , with a reactant mole fraction less than 0.1, the rate constant was found to be . Incorporated in these results was the new application of fully dispersed shock waves and these weak waves were found to produce more accurate results than the conventional use of partly dispersed shock waves. For the second reaction, the rate equation was taken to be , and with argon as the inert gas the dissociationrate constants were found to be and . An analytical description of the flow fields agreed well with the experimental values which were determined by a light absorption technique.

Infrared Spectra of Matrix‐Isolated Hydrogen Sulfide in Solid Nitrogen
View Description Hide DescriptionThe infrared spectra of H_{2}S in a series of solid nitrogen matrices at 20°K are recorded for the 2500–4000‐cm^{−1} region. From a curve‐resolution analysis of the SH stretching region, assignments of peaks are made to monomer, dimer, and polymer species. The H_{2}S dimer absorptions lead to a postulated open structure with a single hydrogen bond. An estimate based on frequency‐shift data indicates a hydrogen bond strength of roughly half that found in the matrix‐isolated water dimer.

On the Nature of the Crystal‐Field Approximation. IV. Generalization and Numerical Results
View Description Hide DescriptionA modified molecular orbital theory was developed by Herzfeld and Goldberg to show the effect of the neglect of the wavefunction overlap and the electron exchange on the results of the conventional crystalfield theory. It was applied by them to calculate the interaction of a hydrogen atom in the state and a hydrogen molecule in the ground state in two different simple nuclear configurations. In one case the atom nucleus was taken to lie on the perpendicular bisector of the molecular axis ( case), and in the other all the three nuclei were taken to lie on a line ( case). The method was generalized for the p^{n} ‐atom–H_{2}‐molecule system ( case) by Goldberg (. Numerical results were given for the p ^{1}‐atom–H_{2}‐molecule system ( and cases) by Berger and Herzfeld. In the present calculation this generalization is further extended to the p^{n} ‐atom–H_{2}‐molecule system ( case) with numerical results given for the and cases. It is shown that the generalized crystal‐field theory can account for some of the shortcomings of the conventional crystal‐field theory. In particular, calculation shows how the “center of gravity” rule and the inversion result are modified. It also shows how the degeneracy is lifted in a system consisting of a diatomic molecule and an atom in the configuration. The generalization of the Herzfeld–Goldberg method clearly demonstrates that the conventional crystal‐field theory cannot be considered to be a fundamentaltheory for the description of the wide range of phenomena that as a semiempirical method it has dealt with so well.

Excited‐State Variational Calculations with Orthogonality Constraint
View Description Hide DescriptionOpen‐shell pi‐electron calculations have been performed on several singlet excited states of pyridine, pyrazine, pyrimidine, pyridazine, and naphthalene using a recently developed variational technique. Spectral results obtained via the new method are compared with results based on virtual orbitals and experiment. Configuration interaction was found to be necessary for adequate state representation, and the results of the new method are generally better than virtual orbital results, but the CI calculation over separately minimized configurations is very laborious. This new method has some computational advantages in certain applications, but it cannot be recommended for spectral calculations since acceptable spectral results can be obtained more easily using the Pariser–Parr–Pople scheme.

New Combining Rule for Intermolecular Distances in Intermolecular Potential Functions
View Description Hide DescriptionA geometric mean combining rule is proposed for intermolecular distances in potential functions of the Mie–Lennard‐Jones or Buckingham types. It is shown to have considerable advantages over the Lorentz rule of additive diameters.

Nuclear Quadrupole Resonance in Hexamethylenetetramine: Modulation and Reorientational Broadening
View Description Hide DescriptionThe temperature dependences of the frequency and linewidth of the nuclear quadrupole resonance absorption in hexamethylenetetramine have been measured in the range between 297–380°K using a Zeeman modulated spectrometer. The results are reported and interpreted both in terms of the molecular reorientation that is known to take place at these temperatures and the modulation broadening expected from this type of instrument.

Mode Expansion in Equilibrium Statistical Mechanics. I. General Theory and Application to the Classical Electron Gas
View Description Hide DescriptionA new expansion for the Helmholtz free energy of a classical system is presented. The potential energy of the system is assumed to be composed of two parts: a “reference system” potential energy and a perturbation potential, which is the sum of two‐particle potentials. The two‐particle perturbation potential energy is assumed to have a Fourier transform. Collective variables, which are the Fourier transforms of the single‐particle density, are introduced, and the canonical ensemble partition function is expressed as an infinite series. The first term in the result for the Helmholtz free energy is the reference system free energy. The second is a mean field term, and the third is the random phase approximation. Subsequent terms involve correlations among the collective variables in the reference system. The general results are applied to the special case in which the reference system is the ideal gas. A density and temperature dependent renormalized potential arises from the analysis in a straightforward way. When the results are specialized to the case of the Coulomb potential, agreement with the ionic cluster theory is obtained, and the renormalized potential is the usual Debye–Hückel potential. Further applications of the technique are mentioned.

Equation of State for Liquids. Calculation of the Shock Temperature of Carbon Tetrachloride, Nitromethane, and Water in the 100‐kbar Region
View Description Hide DescriptionThe model for calculating shock temperature in liquids is presented as an extension of the Walsh–Christian model for metals. The model is based on an analysis showing shock temperature to be more sensitive to variations in than in , and it takes account of the temperature dependence of . Measured shock temperatures for carbon tetrachloride are compared with calculated values as a test of the constant and models. The constant model overestimates shock temperature and is inappropriate to polyatomic liquids. The agreement obtained with the model suggests that it will be valuable for calculating more realistic values of temperature in shock initiation studies of liquids in the neighborhood of 100 kbar.

Heat Capacity of Dilute Polymer Solutions
View Description Hide DescriptionFlory's treatment of the excluded volume integral for pairs of segments of polymer chains is extended to include temperature dependence of the enthalpy and entropy of mixing. The heat capacity term thereby introduced is evaluated from the known critical solution temperatures for polyisobutylene–benzene and polystyrene–cyclohexane. The ratio at infinite dilution is calculated to be −3.1 and −4.1, respectively. A less accurate estimate of this quantity for polystyrene–decalin is employed in an analysis of data obtained by Berry. It is shown that partial draining effects are not as large as previously deduced, and that the results for this system are rationally interpreted by inclusion of temperature dependent thermodynamic functions for the mixture.

Phase Separation and the Critical Index for Liquid–Liquid Coexistence in the Sodium–Ammonia System
View Description Hide DescriptionA precise determination of the liquid–liquid coexistence curve has been made for solutions of sodium in liquid ammonia. The critical index , defined by the mole fraction vs temperature dependence, , undergoes a rather abrupt change from 0.502 ± 0.010 to 0.34 ± 0.02 when approaches to within 1.8°C of the consolute temperature. The validity range of Landau molecular field theory is used to deduce an interaction cluster extending over at least 140 ammoniated sodium centers.

Analytical Approach to the Shock Compressibility of 18 Cubic‐Lattice Metals
View Description Hide DescriptionAn average value of 11 / 6 is taken to represent the Grüneisen constant of these eighteen metals: Li, Na, K, Rb, Cs, Fe, Mo,Ta, W, Al, Co, Ni,Cu, Pd, Ag, Pt, Au, and Pb. An analytical model is formulated for the interpretation of shock‐wave behavior of the_e metals. The model is shown to be as heuristic as the classical model of gas dynamics, and the adiabatic indices and compression ratios of both models at shock infinity may now be arranged as orderly spectra and (4, 5, 6, 7), respectively. The investigation seeks to emphasize the interrelation between shock and isentropic compressibilities, and it also provides favorable evidence for the use of Slater formula to evaluate at high pressure.

Electron Spin Resonance of Gamma‐Irradiated Single Crystal of Dihydrouracil
View Description Hide DescriptionFree radicals in gamma‐irradiated dihydrouracil have been identified as –C_{(5)}H_{2}–C_{(6)}H–. Two sets of resonance lines due to two magnetically distinct molecules were observed. From the direction cosines of the α‐proton coupling tensor it has been concluded that the N_{(6)} atom, originally significantly out of the plane defined by the atoms N_{(1)}, C_{(2)}, N_{(3)}, and C_{(4)}, comes to that plane after the abstraction of a hydrogen atom from C_{(6)}. The nonequivalence of the β‐proton couplings proves that the C_{(5)} atom remains out of the plane even in the radical form. Calculated bondenergies for the four C–H bonds indicate that the H_{(6′)} hydrogen atom should be the most easily removed. That is in agreement with the observations.

Isoelectronic Molecules. II. First‐ and Second‐Order Physical Properties
View Description Hide DescriptionIt has been shown that first‐ and second‐order physical properties of isoelectronic molecules are interrelated in a simple way. This follows from the fact that the corresponding first‐ and second‐order operators are homogeneous functions of position and nuclear charge. The results are rigorously true for noninteracting electrons and can be extended to include interelectronic effects as well.

Consistent Force Field Calculations. II. Crystal Structures, Sublimation Energies, Molecular and Lattice Vibrations, Molecular Conformations, and Enthalpies of Alkanes
View Description Hide DescriptionA set of energy functions of internal coordinates and interatomic distances is used for calculating simultaneously and consistently many properties of alkanes. Comparison between calculated and experimental data is used as a means for systematically selecting energy functions and determining their parameters. Properties calculated are: unit‐cell parameters; heat of sublimation; molecular and lattice vibrations;thermal expansion of the unit cell—all these for n‐hexane and n‐octane crystals. Enthalpy differences—for gauche–trans butane and for axial–equatorial methyl‐cyclohexane. Excess enthalpies for the series cyclopentane through cyclodedcane and molecular conformations and vibrations for a selected number of cyclo‐ and n‐alkanes molecules. Some of the results and conclusions from these calculations are: molecular vibrations in the crystal are in part higher than in the gas, and the corresponding enthalpy difference contributes to the heat of sublimation equally as much as the lattice vibrations. Anharmonicity of lattice and molecular vibrations contributes significantly to the calculated unit‐cell parameters of crystals of n‐hexane and n‐octane at 0°K; thermal expansion is largest in the a _{1} direction. The Lennard‐Jones and Coulomb (12–6–1) potential is inadequate to account simultaneously for intra‐ and intermolecular interactions; the 9–6–1 potential and the exp‐6–1 potential are equally preferable. The intermolecular interactions exihibit an anisotropy not fully represented by functions of interatomic distances;anisotropy is the source of systematic discrepancies between the calculated and measured unit‐cell dimensions.

High‐Accuracy Wavefunctions and Matrix Elements for Vibration–Rotation States of Diatomic Molecules Using the Dunham Potential
View Description Hide DescriptionAn iterative procedure for deriving accurate analytic wavefunctions for the Dunham molecular oscillator is described, and the wavefunctions so obtained are tabulated for the 10 lowest vibrational states. These wavefunctions are more accurate than those which are available for simpler models (e.g., the simple harmonic oscillator with an added cubic term or the Morse oscillator) and, in fact, can be credited with accuracies at least as great as those associated with the Rydberg–Klein–Rees numerical methods. The inclusion of rotational motion is also described. Finally, a method for the rapid derivation of analytic expressions for off‐diagonal radial matrix elements between vibration–rotation states is detailed. This method makes use of only certain parts of the analytic wavefunctions, relaying on a recursion relation to complete the development.