Volume 52, Issue 7, 01 April 1970

Numerical Solutions of the Percus–Yevick Equation for the Lennard‐Jones (6–12) and Hard‐Sphere Potentials
View Description Hide DescriptionNumerical solutions of the Percus–Yevick equation for the Lennard‐Jones (6–12) potential are presented largely for high densities and the liquid region in a form particularly useful to workers who have access only to small computers which cannot be used for solving the PY equation. The solutions, given in a form suitable for calculating either the radial distribution functions or direct correlation functions, are deposited with the National Auxiliary Publications Service (Document No. NAPS‐00720). A table of densities and temperatures at which the PY solutions are available is given. Energies and pressures, including the compressibility pressure for temperatures below the critical, are also tabulated and comparisons are made with molecular dynamics results. Solutions for the hard‐sphere potential are also presented completing the tables of Throop and Bearman [J. Chem. Phys. 43, 2408 (1965)] for large intermolecular separations.

Variational Perturbation Theory Study of Some Excited States of Two‐Electron Atoms
View Description Hide DescriptionThe Hylleraas–Scherr–Knight variational perturbation method has been applied to the two‐electron states and . The calculations have been carried through to 20th‐order wavefunctions, with 96‐, 126‐, and 162‐term expansions of the Hylleraas type, thus yielding estimates of the energy through the 41st order. The perturbation energy coefficients are tabulated through the 21st order for the states , and , and the corresponding nonrelativistic energies are presented for the first 10 elements of the helium isoelectronic sequence, with the exception of H^{−} to which the perturbation expansions do not apply. Where comparison is possible, our results are in fair agreement with those obtained from conventional variational calculations. A survey of the variational perturbation procedure is also given, presenting the basic formulas and phrasing two important theorems.

Linear Hard‐Sphere Gas: Variational Eigenvalue Spectra of the Energy Kernel
View Description Hide DescriptionThe relaxation problem for the linear hard‐sphere gas is considered using a Rayleigh–Ritz expansion of the energy kernel for transitions in a test‐particle/heat‐bath system at variable mass ratio . The basis sets used are the “exact” eigenfunctions for the problem in the Rayleigh limit and the required expansion matrices can be obtained to order at least 20 × 20 by algebraic procedures. Good, converged eigenvalues are obtained in the discrete region so long as these are not within about 10% of the continuum threshold . Where possible, unconverged “pseudoeigenvalues” in the continuum region are also determined and their use to “represent” the continuum in the initial‐value problem is discussed. A number of applications are considered. We obtain the relaxation of distribution functions and mean energies and investigate the energy‐autocorrelation function for equilibrium fluctuations, as a function of mass ratio. The latter proves to be very nearly Gaussian for all in the range . The results also allow a detailed evaluation of the range of validity of the Rayleigh–Fokker–Planck (RFP) equation in energy space. For processes dominated by the RFP approximation is adequate from about , but with progressive deterioration of the predicted higher eigenvalues. Where comparable the results agree with those of Shapiro and Corngold [Phys. Rev. 137, A1686 (1965)] obtained for the neutron thermalization problem with somewhat different methods and scope.

Exact Treatment of the Long‐Range Dipole–Dipole Interaction between Two Helium Atoms
View Description Hide DescriptionWe present a new expression for the long‐range dipole–dipole interaction coefficient in terms of a polarizabilitylike response function. Using certain results of Schwartz, we are able to evaluate this coefficient in the case of the interaction between two helium atoms and, by careful error analysis, to set accurate error bounds to this quantity. We obtain .

Atomic and Molecular Calculations with the Pseudopotential Method. VI. Two‐Valence‐Electron Energy Values for Doubly Excited States
View Description Hide DescriptionThe pseudopotential method is applied to a number of doubly excited states of Be, Mg, Al^{+}, Si^{2+}, Ca, and Zn. Most of these excited states are in the autoionizing region. Besides the pseudopotential approximation, the further approximation of “freezing” one of the two valence electrons is used. The fixed valence electron is selected to be the state of the one‐valence‐electron system which is the limit for the particular series of states in the two‐valence‐electron system, e.g., for the Be series the fixed valence electron is set at the value of the Be^{+} state. Comparison is made with the observed values and other calculated values with good agreement.

Study of the H^{+} and OH^{−} Hydrated Ions by the CNDO/2 Method
View Description Hide DescriptionThe ions obtained by hydration of H^{+} and OH^{−}, e.g., H_{3}O^{+}, H_{5}O_{2}, H_{7}O_{3}, H_{9}O_{4} ^{+}, H_{3}O_{2} ^{−}, H_{5}O_{3} ^{−}, and H_{7}O_{4} ^{−} have been studied theoretically using the CNDO/2 method. The results indicate chain structures are generally preferred albiet weakly over cyclic structures by the total energy criterion and in the case of positive ions over proton centered structures. Binding energies are found to be in reasonable agreement with experimental data. Charge distributions and individual bond energy effects have been examined in detail and are generally consistent with the notion that protons serve as bridging centers for delocalization which remain strongly positive throughout both series to shield the fragment oxygens from each other.

Heat Capacity of Silver Iodide. IV. Heat Content of Quenched Stoichiometric Material
View Description Hide DescriptionExperiments have been performed to measure the heat content of stoichiometric silver iodide, water quenched from the high‐temperature phase to room temperature. The data shows the retention of large amounts of energy during quenching, and this is shown to be consistent with a model of lattice disorder which was developed to explain anomalous thermal properties of annealedmaterial.

Heat Capacity of Silver Iodide. V. Heat Content of Nonstoichiometric Material
View Description Hide DescriptionA program of study of the cation disorder in α‐AgI is completed by a series of heat content experiments on quenched samples which contain small excesses of silver. The results presented are analyzed in terms of lattice disorder, and a model of the behavior of dopedmaterial is developed. This model shows that despite significant differences in thermal properties, the cooperative disordering of the cations is very similar in stoichiometric and dopedmaterial. It is proposed that deviation from stoichiometry causes an elevation of the critical temperature for disordering.

Electron Paramagnetic Resonance Studies of a Viscous Nematic Liquid Crystal
View Description Hide DescriptionWe have studied the order as a function of temperature in the liquid crystal bis(4′‐n‐octyloxybenzal)‐2‐chloro‐1,4‐phenylenediamine using vanadyl acetylacetonate as a paramagnetic probe. The order found in the upper part of the mesophase (105–179°C) was normal. However, in the lower part of the mesophase (40–105°C) the order found was much higher than ordinarily observed, approaching complete alignment near the melting point. In addition, the EPR lines assumed an unusual shape in this lower temperature region. The observations were accounted for by assuming the liquid crystal was too viscous to allow free tumbling of the probe molecules, and by developing a new model in which a spatial rather than a temporal average was performed. Supporting evidence for the validity of the assumption was obtained experimentally by demonstrating that the liquid crystal had an unusually long response time for reorientation in the magnetic field.

High‐Resolution He i and He ii Photoelectron Spectra of Xenon Difluoride
View Description Hide DescriptionThe vertical I.P.'s of the first eight ionizations in the XeF_{2} molecule have been determined, and they compare favorably with the results of Gaussian‐type orbital calculations. The first two ionic states of XeF_{2} are the ^{2}Π_{3/2} (12.42 eV) and ^{2}Π_{1/2} (12.89 eV) spin–orbit components formed by ionization from the highest‐filled orbital . As judged by the vibronic structure of the transitions, the two upper states are linear, with Xe–F distances very much like those of the ground state. The fifth (15.60 eV) and sixth (16.00 eV) I.P.'s similarly involve the ejection of an electron from the orbitals, and thus they are separated by the action of spin–orbit coupling. Substitution of He ii excitation (40.82 eV) for the He i excitation (21.21 eV) results in both the additional ionization of electrons from the molecular orbital (22.5 eV), and a drastic change in the relative intensities of certain peaks in the lower I.P. region. Rydberg bands in the optical spectrum of XeF_{2} can be correlated with those of the photoelectron spectrum, yielding assignments in essential agreement with those given by earlier workers. The work with the He ii resonance line is described in some detail because a description of its use in photoelectron spectroscopy has not previously appeared in the literature.

Theoretical Interpretation of the Electron Scattering Spectrum of CO_{2}
View Description Hide DescriptionA theoretical study of processes observed in electron–CO_{2}scattering experiments is presented. Potential curves for CO_{2} ^{−} are calculated to determine which scattering processes proceed through formation of short‐lived negative molecule–ion compound states. The four dissociative attachment processes occurring below 20‐eV incident energy are shown to result from four different electronic states of CO_{2} ^{−}. Vibrational excitation of CO_{2} in the 3.0–4.5‐eV incident energy range is shown to proceed through compound‐state formation. The calculated energies of the compound states are shown to agree with the absorption spectra of CO_{2} ^{−} trapped in crystalline solids. The inelastic scattering below 3.0 eV is shown to be nonresonant and of a direct nature. For this process the leading term in the angular dependence of the cross section is determined by dipole selection rules, which leads to strong forward peaking. For small‐angle scattering the cross section is shown to be about 10^{−16 − }10^{−17} cm^{2} in magnitude. Further, the discussion on direct vibrational excitation cross sections will generally be valid for other nonpolar molecules with infrared‐active vibrational modes.

Phosphorescence of Phenylcarboxylic Acids and Their Salts
View Description Hide DescriptionThe phosphorescence spectra and lifetimes of a series of carboxylic acids, phenyl–(CH_{2})_{ n }–COOH, and their salts have been measured. It is shown, contrary to accepted attitudes, that the lowest triplet state of benzoic acid is not an excitation localized on the benzene ring but involves considerable delocalization onto the carboxyl group. The lowest triplet energy of the carboxyl group and its radiative lifetime are estimated as 29.7 kK and 4 sec, respectively.

Luminescence of Systems Containing Hydroxyl Ion
View Description Hide DescriptionA fluorescence of alkali halide single crystalsdoped with the corresponding alkali hydroxide has been observed and is attributed to the emission of a photoproduced hydroxyl radical. A phosphorescence of systems containing the hydroxide ion has also been noted at and is attributed to a emission of the OH radical. Photoprocesses in systems containing the OH^{−} ion, as well as in aqueous systems containing certain salts and acids, are discussed in a qualitative way.

Energy States and Intermolecular Interactions in Molecular Aggregates: Resonance Pair Spectra of Crystalline Naphthalene
View Description Hide DescriptionVery high‐resolution optical spectra reveal fine structure in the and electronic transitions of naphthalene isotopic mixed crystals, C_{10}H_{8} as a guest in a C_{10}D_{8} host. Extensive experimental data allow the spectral lines to be characterized and assigned. Some of the structure corresponds to the resonance splitting arising from neighboring pairs of guest molecules in the host lattice. In the Frenkel tight‐binding approximation, this structure gives directly the intermolecular excitation transfer matrix elements responsible for the exciton mobilities and the energy band structures of both pure and mixed molecular crystals. For the state the largest interchange equivalent interactions are approximately + 15.3 cm^{−1} and + 3 cm^{−1} while the largest translation equivalent interactions are approximately + 3.7, − 5.1, − 7.9, and − 3.3 cm^{−1}. For the state the largest interchange equivalent interaction is 1.25 cm^{−1}.

Hydrogen Bonding of H_{2} ^{+} to the He Atom: Energy Surface for Linear HeH_{2} ^{+}
View Description Hide DescriptionUsing pseudonatural orbitals as a basis for the superposition of configurations (SOC), we have calculated the adiabatic potential energy surfaces for linear HeHH^{+} and HHeH^{+}. Except for the different basis set of Gaussian functions used, these calculations were carried out identically to one reported earlier for linear H_{3}. It is found that HeHH^{+} has a potential energy minimum ∼ 5 kcal deep for , and This “hydrogen bonding” effect is given by the Hartree–Fock results and is little changed by SOC. HHeH^{+} is found to have a high potential energy barrier of ∼ 25 kcal for the symmetric complex,

Raman Scattering in Crystalline Polyethylene
View Description Hide DescriptionThe Raman scattering in crystalline polyethylene, deuterated polyethylene, and n‐paraffins has been investigated as a function of temperature. Crystal‐field splitting of the methylene bending mode observed at room temperature increases at − 160°C in both polyethylene and deuterated polyethylene. No splitting of the skeletal stretching mode and the methylene twisting modes in polyethylene is observed at room temperature, but the two components of the space group fundamental are resolved at − 160°C. Similarly, splitting of the methylene wagging mode in deuterated polyethylene is observed at − 160°C but not at room temperature. The observed crystal‐field splittings are discussed in terms of splittings calculated from repulsive forces between hydrogen atoms on neighboring chains in the polyethylene crystal.

Two‐Electron Fractional Parentage Coefficients for the Configurations l^{n}
View Description Hide DescriptionAn equation for computing the two‐electron functional parentage coefficients of the configurations in terms of the one‐electron fractional parentage coefficients of and is derived. Abbreviated tables of the two‐electron coefficients of the lowest lying terms of the and configurations are given. The orthogonality, charge conjugation, and symmetry properties of the two‐electron fractional parentage coefficients are discussed briefly. The use of the coefficients in calculating matrix elements of two‐particle operators is discussed, and the appropriate equations are given.

Nuclear Overhauser Effects and ^{13}C Relaxation Times in ^{13}C–{H} Double Resonance Spectra
View Description Hide DescriptionThe effect of proton decoupling upon carbon‐13 magnetic resonancespectra is treated in detail with the Solomon formulation for multiply irradiated spin systems. The factors affecting the nuclear Overhauser enhancement are discussed in terms of competition between the dipole–dipole relaxation mechanism essential to the Overhauser effect and other relaxation processes. Using a density matrix formulation, it is exhibited that the maximum Overhauser effect (achieved when the dipolar mechanism dominates) is independent of the number of hydrogen atoms interacting with a relaxing carbon‐13 nucleus. On the other hand, if the molecular tumbling motion is isotropic and the dipole–dipole mechanism dominates the relaxation process, then to a first approximation can be shown to depend inversely upon the number of directly bonded hydrogen atoms. Expressions for treating an AMX three‐spin system are given also, and the effect of the third spin is discussed. The highly symmetric adamantane molecule provides an excellent test case for the validity of the theoretical conclusions as it is an isotropic tumbler because of the rigidity and its tetrahedral symmetry. Furthermore, it has two nonequivalent carbon positions having different numbers of hydrogen atoms. The experimental Overhauser enhancement was found to be the same for both carbons and the for these two carbons, measured with the adiabatic rapid passage technique, reflected the two‐to‐one factor expected for CH and CH_{2} groups, respectively. Minor deviations are explained with small contributions from vicinal hydrogens. Having exhibited the validity of the theoretical treatment for adamantane, it becomes evident that exceptions may be interpreted in terms of deviations from isotropic tumbling motion. As most molecules lack the symmetry to be isotropic in rotational diffusional processes, this experimental technique offers an important way for studying anisotropic molecular motion in liquids.

Semiempirical VB Calculation of the (H_{2}I_{2}) Interaction Potential
View Description Hide DescriptionA simple, nonionic valence‐bond formalism is employed to obtain the interaction potential energy for the (H_{2}I_{2}) system. The system is treated as one consisting of four electrons moving in the potential of two nonpolarizable I^{+} cores and two protons. All three‐ and four‐center integrals are omitted to obtain the London expression for the system potential. All remaining Coulomb and exchange integrals are evaluated semiempirically in terms of ground state and lowest triplet state potentials by utilizing the Heitler–London expression for the energy of the diatomic systems. The result is a closed‐form analytical expression for the system potential energy as a function of the six interparticle distances. Potential‐energy contour maps are presented for the and distorted planar trapezoidal conformations, symmetric and nonsymmetric linear arrangements, and a planar “staggered” conformation. The three‐body (H_{2}I) potential‐energy surface is also investigated and contour maps for the 90° and linear surfaces are given.

Electric Dipole Moment of SiO and GeO
View Description Hide DescriptionThe molecular beam electric resonance spectra of SiO and GeO have been measured. The electric dipole moments (in Debye) of SiO and GeO in the lower vibrational states are: The difference between observed and calculated dipole moment is quite similar in CO and SiO.