Volume 55, Issue 9, 01 November 1971

Ising Chain with Several Phase Transitions
View Description Hide DescriptionThe one‐dimensional spin‐½ Ising model with very long‐range ferromagnetic interaction and first and second neighbor antiferromagnetic interactions is solved exactly in a direct magnetic field. A study of the finite temperature behavior confirms the existence of up to four first‐order phase transitions ending in classical critical points. We find a confluence of three critical lines at a ``tricritical'' point with exponents , and . We find also a confluence of four critical lines at a ``tetracritical'' point with exponents , and . We show the existence of triple lines and ``quadruple'' lines, along which three or four phases are simultaneously in contact with each other, respectively. For some special values of the interaction strengths, five phases are in contact with each other at a ``quintuple'' point. We remark that our ``multiple'' lines and ``multicritical'' points are not ``true'' multiple lines and multicritical points but that they would be so in a somewhat extended model. This model satisfies a generalized form of the ordinary phase rule of Gibbs. In distinction to various models considered recently, our model does not exhibit a singular phase‐boundary diameter.

Properties of Self‐Avoiding Walks Not Constrained to Lattices
View Description Hide DescriptionA study of self‐avoiding walks not confined to a lattice was performed. By use of the method of direct enumeration, walks of 14 steps were generated. Two different models were studied. Configurational properties, such as the mean square end‐to‐end distance, the mean‐square radius of gyration, etc., plus hydrodynamic as well as phase transitionproperties were obtained. They are discussed in terms of the theories incorporating the excluded volume effect. Some of the results were applied to the DNA molecule.

Molecular Vibrations in a Crystal Lattice: The Normal Coordinate Treatment of Cs_{2}LiCo(CN)_{6}
View Description Hide DescriptionA normal coordinate analysis including lattice modes has been carried out for Cs_{2}LiCo(CN)_{6}. The majority of the changes in frequency found for the normal modes of Cs_{2}LiCo(CN)_{6} with respect to the isolated ion can be reproduced without changing the intramolecular potential constants. However, the internal valence CN force constant for Cs_{2}LiCo(CN)_{6} is shown to be higher than that of . The low frequency modes are characterized by extreme mixing of intra‐ and intermolecular vibrations.

Photoelectron Emission Spectroscopy of Solvated Electron Solutions
View Description Hide DescriptionInstrumentation and methodology were developed for photoelectron emission (PEE) spectroscopy of sodium solutions in hexamethyl phosphoric triamide (HMPA) for irradiation from 1 to 3.7 eV. PEE currents were corrected by the Thomson equation for back scattering in the gas (nitrogen at 200–800 torr) above the solution. The Schottky effect for PEE was smaller than for thermionic emission and decreased with increasing photon energy. PEE spectra exhibited two steps corresponding, respectively, to emission by solvated electron species and metal species. The corresponding thresholds, obtained from Einstein plots, are approximately 1.4 and 2.6 eV. The first threshold is compared with the energy of the 1s state for a simple cavity model. Quantum yields were low, e.g., 10^{−5} emitted electron per incident photon. Thermionic and PEE currents are correlated.

Photoelectron Emission Spectroscopy of Solutions: An Interpretation
View Description Hide DescriptionThe process of photoelectron emission (PEE) by solutions is decomposed into two steps: generation of electrons in unbound states in solution by irradiation; escape of unbound electrons into the gas above the solution. The transition rate for the first step is expressed in terms of the photon energy and the energy for transition to the continuum. The complete absorption band for bound‐unbound transitions in solution is calculated. Experimental PEE currents for different emitters (solvated electrons in liquid ammonia or hexamethyl phosphoric triamide, ferrocyanide in water) vary linearly with photon energy after normalization to a constant transition rate. This result is interpreted by assuming that unbound electrons undergo a random walk in solution and lose their kinetic energy at a rate inversely proportional to their speed. The Schottky effect on PEE is also interpreted.

Development of the Bond‐Charge Model for Vibrating Diatomic Molecules
View Description Hide DescriptionThe bond‐charge model of a vibrating diatomic molecule previously described by Borkman, Simons, and Parr [J. Chem. Phys. 49, 1055 (1968); 50, 58 (1969)] is embedded in an exact theory of molecular vibrations near equilibrium, and shown to be a natural first approximation to the exact description. The model, based upon the Fues potential, is made exact by letting the quantities W _{1} and W _{2} depend on , with the electronic potential energy and kinetic energy, respectively, still having the forms and −W _{0}+W _{2}/R ^{2}. It is shown that, with no loss of accuracy, one may take , which establishes the previous parameterization of W _{1} and W _{2} in terms of a bond charge q. A potential function of the form is generated from a ``multipole expansion'' of the electronic potential energy, and a model is given which includes interactions between atomic dipoles at the nuclei and bond charges. The atomic dipoles are related to the number of valence electrons of an atom, accounting for the periodic‐table column dependence of force constant relations recently pointed out by Calder and Ruedenberg. The model is shown to possess the properties in accord with recent studies on the nature of charge densities and bond orders.

Use of the Nonsymmetrical Wavefunction. II. Dynamic Polarizability of Molecular Hydrogen
View Description Hide DescriptionThe nonsymmetrical (NS) wavefunction is employed to calculate the real‐ and imaginary‐frequency dynamic polarizabilities of the hydrogen molecule, and double perturbation theory allows one to calculate them through first order in electron correlation within the same theoretical framework. From the imaginary‐frequency polarizability, the London coefficients for the dispersion interactions of and are also obtained. The results in the shielding and geometric approximations are compared with other theoretical and experimental values, and it is found that the use of the NS wavefunction can be extended to time‐dependent problems with reasonable accuracy and much ease of computation.

Absorption Coefficients of Helium between 599 and 610 Å; Transition Moment for He_{2}
View Description Hide DescriptionAbsorption coefficients of helium have been measured in the region of the continuum‐bound transition of He_{2} (599–610 Å) for temperatures of 77, 200, and 298°K. The absorption coefficientk is proportional to the square of the helium density n ^{2}. The quantity k/n ^{2}, ranging between 10^{−39} and 10^{−42} cm^{5}, is in excellent agreement with the continuum‐bound absorption coefficients of He_{2} calculated by Sando and Dalgarno, supporting a value of 0.20 a.u. for the electronic transition moment of He_{2} between 2 and 3 Å.

Structure of Dielectric Fluids. I. The Two‐Particle Distribution Function of Polar Fluids
View Description Hide DescriptionA molecular fluid of identical molecules with a rigid dipole moment in an arbitrarily shaped volume is considered. The volume may or may not be embedded in a dielectric continuum. It is shown that when an arbitrary external field is applied, the constitutive relation between the local polarization and the local macroscopic electric field is valid under some completely acceptable restrictions. In the establishment of this relation a crucial role is played by a long‐range part of the two‐particle correlation function, for which an explicit expression is obtained. One term in this long‐range part is explicitly dependent on the shape of the sample volume and on the surroundings. The resulting dielectric constant of the molecular fluid can be formally expressed in only the local interactions of the molecules and is thus independent of the surroundings and the shape of the sample.

Calculated Spectroscopic Parameters and the Intermediate Spin‐Orbit Coupling Diagram in the Interpretation of Spectra
View Description Hide DescriptionAb initio relativistic SCF Dirac‐Fock and Dirac‐Slater calculations of the free‐ion Coulomb interaction parameter, F _{2}, and the spin‐orbit coupling parameter, ζ, have been made for all known oxidation states of actinides,thorium through nobelium. These calculated parameters, when normalized to the experimental spectroscopic parameters, were used to initiate least‐squares fits to nine experimental electronic energy levels of anhydrous KAmO_{2}CO_{3}, and to 15 levels of aqueous in 1M . Correlation with 35 lower levels of the f ^{4} intermediate spin‐orbit coupling diagram allowed term assignments to be made to the experimental free‐ion levels. Some differences in term assignments were noted between the solution and solid state spectra, but substantial agreement was found between the spectra taken in 1M measured here and some earlier spectra taken in 1M HClO_{4}. The ``best'' values for the spectroscopic parameters of 5f ^{4} Am (V) were: , and . The predicted trends for these parameters for the heavy actinides is discussed.

3d Wavefunctions in ^{57}Fe Mössbauer Spectroscopy
View Description Hide DescriptionIn a remarkable article, Ganiel has shown that ^{57}Fe quadrupole splitting due to a single 3d electron is constant, for any real wavefunction with real coefficients spanning either the T _{2g } or E_{g} subspaces. We examine here the effects on V_{zz} and η and show that these vary in complex fashion. Hence, the sensitive effects of structure and bonding on the efg tensor can be buried in a constant quadrupole splitting for this case, and the values of V_{zz} and η must be extracted by magnetic perturbation or other methods if the nature of the 3dwavefunctions is to be determined.

Viscous Liquids and the Glass Transition. III. Secondary Relaxations in Aliphatic Alcohols and Other Nonrigid Molecules
View Description Hide DescriptionThe dielectricpermittivity and the dielectric loss factor of 5‐methyl‐3, 4‐methyl‐3 and 3‐methyl‐3‐heptanol, n‐ and iso‐butanol, 1,2‐propanediol, dimethyl and diethyl phthallate, and 3‐methylpentane have been measured from 50 to 10^{5} Hz and from −196 to about 20°C above their respective glass transition temperatures. The glass transition temperature T_{g} of these substances, several more isomeric octanols, and 1‐phenyl‐1‐propanol have been measured by differential thermal analysis. All substances except for 3‐methylpentane and iso‐butanol show either a well‐defined secondary relaxation peak in tanδ, or a clear indication of the presence of a secondary relaxation below their . Arrhenius plots for the α‐relaxation process of the isomeric octanols are linear with an activation energy of 16–18 kcal/mole, while for other substances they are nonlinear with the activation energy changing from 30 to 70 kcal/mole. The Arrhenius plots for the secondary relaxations are linear and have an activation energy of 4–8 kcal/mole. It is pointed out that the presence of a spectrum of relaxation times in liquids near T_{g} is not necessarily concomitant with non‐Arrhenius behavior. It is concluded that the presence of secondary relaxations should be considered as a characteristic property of the liquid in or near the glassy state, and do not require specific intramolecular mechanisms for their existance.

Condensation of Hydrogen on Tungsten
View Description Hide DescriptionThe condensation kinetics of hydrogen on the (110), (111), and (100) planes of tungsten are examined as functions of crystal temperature, coverage, and population of the various binding states. Measurements were carried out simultaneously on all three crystals in the same ultrahigh‐vacuum system in order to accurately compare sticking coefficients and amounts adsorbed. There are large variations in the initial sticking coefficients s _{0}; at 78°K these vary between 0.26 on the (111) plane to 0.07 on the (110) plane. This implies that the electronic transition involved in bond formation depends sensitively on the geometric and electronic structures of the substrates. Also s _{0} is observed to be almost independent of substrate temperature for all planes, showing that phonon assisted processes do not determine the energy transfer. The coverage dependence of s shows considerable variation between states. In all cases a weakly adsorbed precursor state is involved, but the kinetics of conversion to the tightly bound states are quite different for different planes and states. Models are developed to explain condensation into multiple states for sequential and simultaneous population of the states on a plane, and these are compared with the experimental results.

Unit Compressibility Slopes and a New Virial Coefficient Interrelation
View Description Hide DescriptionThe slopes of isochors and isotherms as they cross the unit compressibility line are found to constitute a corresponding property for many fluids over a wide range of densities extending from dilute gases to compressed liquids. This correspondence is represented by a simple function, and its consequences for equations of state are examined. In particular, a new interrelation among the virial coefficients is derived. The interrelation is tested for the Lennard‐Jones 12,6 intermolecular potential, and the agreement permits an estimation of this potential's previously unknown sixth virial coefficient as a function of temperature.

Shear Viscosity of 3‐Methylpentane—Nitroethane near the Critical Point
View Description Hide DescriptionShear viscosity coefficients for a mixture of 3‐methylpentane—nitroethane of critical composition were determined over the temperature range by an improved capillary viscometer. The data were analyzed by alternate procedures both of which give strong support to the view that the viscosity singularity is indeed very weak over a wide temperature range.

Correlation Studies on . I. The Wavefunctions
View Description Hide DescriptionFour configuration interaction calculations on the molecule—ion are reported. Two Gaussian lobe basis sets are used and calculations are made with only doubly substituted configurations and also with both doubly and singly substituted configurations. The best calculation gives an energy of −1.34050 hartree at a fixed internuclear distance of 1.6406 bohr. The wavefunctions are analyzed in terms of natural orbitals and compared with each other and with some wavefunctions from the literature.

Correlation Studies on . II. Electron Densities and Expectation Values
View Description Hide DescriptionEffects of correlation on one‐electron distribution and other properties are calculated for the molecule ion. Effects due to singly substituted and doubly substituted configurations are essentially additive for this closed shell species. The effect of doubly substituted configurations is to shift charge from the molecular center to nuclear regions, mainly outside the molecule. This is rationalized in terms of the many‐electron theory of Sinanoğlu. The singly substituted configurations cause a slight electronic contraction similar to that caused by initial atomic—molecular formation. Neglect of singly substituted configurations is not justified when one is seeking improvement of properties beyond the HF‐SCF level.

Angular Variation of Intensities of Forbidden Lines in in Parallel and Perpendicular Configurations
View Description Hide DescriptionThe angular variation of intensities of electronic forbidden and nuclear forbidden transitions in in parallel and perpendicular configurations was investigated. The intensity expressions for these transitions, assuming that the direction of the rf magnetic field makes an arbitrary angle with the direction of the static magnetic field, were derived. It is found that there is a good agreement between the theoretical and experimental results. From the comparison of the spectra in the parallel and perpendicular configurations for and , it is concluded that D, A, and B have the same sign.

Low‐Magnitude, Pure‐Primary, Hydrogen Kinetic Isotope Effects
View Description Hide DescriptionIt is shown theoretically that low‐magnitude, primary, hydrogen kinetic isotope effects may be characterized by the same types of temperature‐dependence irregularities that have been previously shown to be expected fairly commonly for secondary hydrogen isotope effects and primary heavy‐atom isotope effects. Upper limits are given for the magnitudes of pure‐primary and mixed‐primary—secondary hydrogen kinetic isotope effects possibly associated with such temperature‐dependence ``anomalies.''

Exact Solution of the Mean Spherical Model for Fluids of Hard Spheres with Permanent Electric Dipole Moments
View Description Hide DescriptionThe mean spherical model is solved in closed form for a fluid of hard spheres with permanent electric dipole moments. Both the pair distribution functiong(12) and the direct correlation functionc(12) consist of a spherically symmetric term and two other terms with different dependences on the orientations of the two dipole moments. The spherically symmetric part is the solution of the Percus‐Yevick equation for hard spheres. The angle‐dependent terms satisfy two coupled integral equations, which can be decoupled by appropriate changes of the dependent variables. The solutions are expressed in terms of the solution of the Percus‐Yevick equation for hard spheres for both positive and negative densities. The effect on g(12) of the finite size of the sample is calculated for the case of a sphere. The correction term in g(12) is found to be of order 1/V, where V is the volume of the sample. It is a function not only of the relative distance vector of the two molecules, but also of their positions in the sample. The contribution to the polarization is, nevertheless, constant throughout the sample, in agreement with classical electrostatics. The dielectric constant ε, calculated by Kirkwood's formula, is obtained in closed form. It is a function of a single variable which does not contain the hard sphere diameter.