Volume 43, Issue 8, 15 October 1965

Discreteness‐of‐Charge Adsorption Micropotentials. II. Single Imaging
View Description Hide DescriptionMethods are developed for calculating exact potentials and fields arising from discrete hexagonal arrays of monopoles imaged by a perfect conductor. They allow calculation of the potential and field at any point outside a plane electrode upon which such a hexagonal array of charges (of infinite extent) is adsorbed, and are applied to the problem of determining various electrical properties of such adsorbed arrays. We give particular attention to the micropotential and energy of adsorption for ionic adsorption in both electrolytic and gaseous systems and consider further application of the results to electron emission from metals. Detailed comparison with earlier approximate treatments is made, and when appropriate, errors appearing there are corrected. For example, we find that the rearrangement energy on incremental adsorption of ions does not vanish, as previously suggested, except for a nonphysical limit. Exact results are obtained for the rearrangement energy, and it is usually found to be a significant contribution to the total adsorption energy. Sufficient graphical and numerical results are given for all computed quantities so as to be of direct use to the worker in this general area.

Angular Correlation Functions: Application to Dielectric Absorption in Polar Fluids
View Description Hide DescriptionIn recent work on the theory of dielectric absorption, the complex dielectric constant has been expressed as a function of the correlation function of the electric moments of the system. Here, the equations are written down which relate this correlation function to the time‐dependent distribution functions for molecular orientation. Two model calculations of the singlet angular distribution function are discussed; and the pair‐wise distribution function is related to the singlet motion by an extension of the convolution approximation to rotational motion. The correlation function of the permanent dipole moments is obtained from these distribution functions, and the contribution of induced dipoles to the correlation function of the moments is briefly discussed. Finally, the theoretical expressions developed here are applied in a partial analysis of the dielectric absorption properties of liquid water.

Improved Free‐Volume Theory of Liquids. VI. Further Tests of the Worm Model for the 12–6 Potential
View Description Hide DescriptionThe worm theory of liquids has been tested over a rather broad range of physical conditions. Although the theory does represent a considerable improvement over the earlier cell theories, it generates accurate estimates of the thermodynamical functions and of the equation of state only for restricted ranges of temperature and pressure. The basis for this conclusion is provided by the comparisons of theory and experiment given in this and in a previous paper.

Improved Free‐Volume Theory of Liquids. VII. Radial Distribution Functions for the Worm Model
View Description Hide DescriptionRadial distribution functions have been calculated on the basis of the worm model. The calculations are specific to a system of molecules which interact according to the Lennard—Jones 12–6 potential. Comparisons are provided of our results with those of other theories and with Monte Carlo calculations and experiment. The agreement between theory and experiment (and/or Monte Carlo data) is found to be satisfactory only at very high pressures.

Study of the Properties of an Excess Electron in Liquid Helium. I. The Nature of the Electron—Helium Interactions
View Description Hide DescriptionIn this paper we present a theoretical study of the free and localized states of an excess electron in liquid helium. Electron—helium interactions are treated by the pseudopotential method, while multiple scattering effects on the properties of a quasifree electron in the dense fluid are treated using the Wigner—Seitz model. It is demonstrated that the plane‐wave state is not the lowest energy state for an excess electron in liquid helium and that fluid deformation leads to a localized state of lower energy. The large, repulsive helium‐atom pseudopotential coupled with the small heliumpolarization potential lead to electron localization which may be attributed entirely to short‐range repulsions. The following experimental observations are adequately interpreted by these results: (a) The energy barrier of liquid helium for electrons, (b) the density‐dependent transition from a delocalized state to a localized state of the excess electron, (c) the mobility of an excess electron in normal ^{4}He and in ^{3}He. Pressure and temperature effects on the electron bubble are also discussed. It is concluded that a pressure‐induced transition from the localized to the delocalized state of the excess electron will not occur in the fluid domain even at high pressures. Finally, we present some speculations concerning the optical properties of the excess‐electron center.

Study of the Properties of an Excess Electron in Liquid Helium. II. A Refined Description of Configuration Changes in the Liquid
View Description Hide DescriptionIn this paper we present a study of the structural changes in liquid helium in the vicinity of an excess electron. We have used the formal similarity between the pair distribution function of an N‐boson system, with the wavefunction expressed as the product of pair wavefunctions, and the pair distribution function of a classical fluid. The present model leads to an interfacial surface energy term which is in good agreement with the observed surface tension of liquid helium at 0°K. An important contribution to the bubble energy arises from the volume kinetic energy arising from the excess kinetic energy of the fluid atoms removed from the boundary layer. The bubble radius of 12.4 Å calculated herein is found to be in excellent agreement with the available experimental data.

On the Quantum‐Mechanical Fokker—Planck Equation
View Description Hide DescriptionTaking the Wigner function as the basic N‐body distribution function, quantum‐mechanical modifications to the Fokker—Planck equation are studied using the two different approaches introduced, respectively, by Kirkwood and Prigogine. In Sec. II the weak coupling limit is studied using the time‐smoothing technique first introduced by Kirkwood; the results of this section can perhaps form the basis of an approximate treatment of the transport phenomena of pure liquids. In Sec. III we treat the problem of the Brownian motion of a heavy particle employing the perturbation techniques developed by Prigogine and co‐workers; in this section it is found that the form of the quantum‐mechanical Fokker—Planck equation is identical to the classical equation, but that the friction coefficient contains quantum‐mechanical corrections. Finally, it is noted that the weak coupling result obtained by the time‐smoothing technique is identical to the approximation of small momentum transfer (Sec. III), if in the latter case the scattering cross section is represented in the Born approximation.

Additivity of Heats of Combustion, LCAO Resonance Energies, and Bond Orders of Conformal Sets of Conjugated Compounds
View Description Hide DescriptionThe notion of ``conformal sets'' of aromatic and other conjugated compounds is introduced. To a good approximation, it is found that the sum of the heats of combustion of the compounds in one set equals the sum for the compounds in the other set. Similar additivity occurs for their LCAO‐calculated total π‐electron energies, LCAO bond orders and free valences. Bond lengths and dissociation energies are also considered. Conformal sets are defined in terms of their number and quality of self‐returning random walks on the atoms of the conjugated compounds. Even‐electron nonalternant and alternant compounds are included, as are, in certain cases, radicals. Some insight into the additivity for LCAO properties is obtained by relating the coefficients in LCAO secular equations to random walks and using contour integral formulas connecting LCAO properties with secular determinants.

On the Theory of Chemiluminescent Electron‐Transfer Reactions
View Description Hide DescriptionA mechanism is described for chemiluminescent electron‐transfer reactions. It is shown that in the case of very exothermic homogeneous electron‐transfer reactions, the intersection of the potential‐energy surface of the reactants with that of electronically unexcited products occurs only at high energies. The rate of formation of unexcited products then becomes slow. A numerical estimate of this slowness is made using known homogeneous rate constants for ordinary electron‐exchange reactions. An intersection occurs at lower energies when one of the products of a highly exothermic electron transfer is electronically excited, thereby reducing the exothermicity. The product may emit light or subsequently form a state that does.
A rather different situation is shown to occur at electrodes: the system can now reduce the ``exothermicity'' by having the electron transfer into a high unoccupied level of the conduction band or from a low occupied level of the latter. The large width of the conduction band in metals permits much latitude in reducing the exothermicity thereby.
These results are compared with present experimental findings that chemiluminescent electron transfers occur in solution rather than on electrodesurfaces.

Dissociation and Isomerization of Vibrationally Excited Species. III
View Description Hide DescriptionThe equations of Part I for the specific and over‐all unimolecular reaction‐rate constants are extended slightly by including centrifugal effects in a more detailed way and by making explicit allowance for possible reaction‐path degeneracy (optically or geometrically isomeric paths). The expression for reaction‐path degeneracy can be applied to other types of reactions in discussions of statistical factors in reaction rates.

First and Second Triplets of Solid Benzene
View Description Hide DescriptionAbsorption spectra have been taken of the O_{2}‐perturbed first (^{3} B _{1u }) and second (^{3} E _{1u }) triplets of solid benzene at 4.2°K. Spectra of both C_{6}H_{6} and C_{6}D_{6} were obtained. The (0–0) bands of the first triplet occur at 29 674±25 cm^{−1} for C_{6}H_{6} and 29 851±25 cm^{−1} for C_{6}D_{6}. For the second triplet they lie at 36 560 cm^{−1}±50 for C_{6}H_{6} and 36 784±50 cm^{−1} for C_{6}D_{6}. The result for the first triplet of C_{6}H_{6} compares very favorably with Evans' gas‐phase O_{2}‐perturbed spectrum. It is also in satisfactory agreement with Nieman's accurate phosphorescence measurements on isotopic mixed crystals of benzene which place the C_{6}H_{6} (0, 0) band position in the crystal at 29 657.1 cm^{−1}. Many precautions were taken to eliminate the possibility of mis identification of the second triplet. The observation that the O_{2}‐enhanced first triplet and the O_{2}‐enhanced bands in the 36 600‐cm^{−1} region always appear together and with approximately the same relative intensities is considered to be the best evidence for the assignment. However, the rather broad structure obtained by the O_{2}‐perturbation technique does not allow all the uncertainties in the identification to be completely removed, nor does it allow a detailed study of this interesting state.
A detailed evaluation of the purity of the benzene is made, and a method is described for the preparation of material having ultrahigh spectroscopic purity. Crystals, up to 5 cm in length, of this very highly purified C_{6}H_{6} and C_{6}D_{6} were studied at 4.2°K to ascertain if the singlet—triplet absorptions could be seen in the absence of a perturbation. The long crystals showed some sharp and some broad (Δν≈150 cm^{−1}) absorptions starting at 36 947±50 cm^{−1} in C_{6}H_{6} and at 37 147±50 cm^{−1} in C_{6}D_{6}. The broad absorptions correlate reasonably well with the features assigned to the second triplet in the O_{2}‐perturbation experiments. The first triplet is too weak to be observed in the long‐crystal experiments. The position of the second triplet lies about 3000 cm^{−1} above that given by the Pariser—Parr calculation. This places the second triplet about nine‐tenths rather than half of the distance from the lowest triplet to the lowest excited singlet.

Transposition of the Theories Describing Superconducting Systems to Molecular Systems. Method of Biorbitals
View Description Hide DescriptionA new theory of the electronic structure of molecules is proposed, based on a wavefunction of the following type:This wavefunction is called the self‐consistent‐field biorbital wavefunction, and the factors ψ(i, j) are called biorbitals. This function is discussed, the corresponding energy expression is derived, and the variational equations leading to the best possible φ^{SCF–BI} are presented. This theory is the molecular version of Blatt's theory of superconducting systems.

Velocity Distribution of a Dilute Chemically Reacting Gas
View Description Hide DescriptionFrom the Wigner—Wilkins integral equation for the velocity distribution of a dilute chemically reacting gas which exchanges energy with a Maxwellian diluent gas, a simple integral equation for the high‐energy region is derived. For a constant reaction cross section σ_{ r }, the velocity distribution of the reactive molecules in the high‐energy region is n(x)=x^{p} exp(—x ^{2}), where , σ_{ s } is the scattering cross section, λ=(1—μ)/(1+μ), μ is the ratio of the reactive to diluent molecule mass, and x is the normalized velocity of the reactive molecule. A modified form of the Wigner—Wilkins equation is then used to extend these results into the thermal region. Small changes in the reaction cross section can produce significant changes in the velocity distribution; the perturbations of the Maxwellian distribution become even more pronounced when scattering is anisotropic in the forward direction.

United‐Atom Approximation Applied to Argonlike Molecules. II. PH_{2} ^{−}, PH_{3}, and PH_{4} ^{+}
View Description Hide DescriptionThe united‐atom treatment has been applied to the systems PH_{2} ^{−}, PH_{3}, and PH_{4} ^{+}. For PH_{3} and PH_{4} ^{+}, satisfactory values are obtained for the total molecular energy and ``breathing'' force constants when compared with experiment. The model gives theoretical bond lengths which are in good agreement with the observed values. Coherent x‐ray scattering factors are presented for PH_{3} and PH_{4} ^{+} along with a discussion of the molar diamagnetic susceptibilities. The value of the proton affinity derived here for PH_{3} is found to be in general agreement with the experimental results.
Except for the considerable improvement in the proton‐affinity calculation, the results obtained here are in keeping with the conclusions derived from our earlier work concerning the united‐atom study of H_{2}S.

Infrared Spectra of Anhydrous Rare‐Earth Nitrates from 4000–100 cm^{−1}
View Description Hide DescriptionThe infrared spectra of the anhydrous rare‐earth nitrates are presented in the region 4000–100 cm^{−1}. Strong coordination of the nitrate groups to the metal atom occurs. Possible different types of metal—oxygen bonding in these compounds is discussed.

Relationships between Polymer Structure and Glass Temperature. I. The n‐Alkanes
View Description Hide DescriptionThe constants in the empirical equation were evaluated from published viscosities η and densities d for 23 n‐alkanes. All three adjustable parameters A, B, and T _{0} were found to vary with chain length. The parameter T _{0} is predicted with good precision by the Gibbs—DiMarzio theory of the glass transition with a ``flex energy'' ε of 490.8 cal/mole. This value is in agreement with determinations by other methods.

Molecular Entanglement Theory of Flow Behavior in Amorphous Polymers
View Description Hide DescriptionAn approximate molecular theory of steady flow in amorphous polymers has been developed by considering the properties of a deforming entanglement network. The dynamics of entanglement formation between pairs of molecules and its influence on the density of entanglements during steady deformation were examined. From a simple model for these processes, apparent viscosity was calculated as a function of the shear rate. An approximate calculation of the normal‐stress differences was also made. The viscosity was found to be dependent on the rate of deformation, and at sufficiently high shear rates it approached proportionality with (shear rate)^{−¾}. The onset of non‐Newtonian behavior occurs in the same region of shear rates as that predicted by the coil‐distortion theory of steady flow. However, unlike the predictions of the latter theory, dynamic‐viscosity frequency curves and apparent‐viscosity shear‐rate curves should not necessarily superimpose, because the dispersion mechanisms are fundamentally different in the two cases. Comparison with experimental results reported in the literature showed generally satisfactory agreement in both the shape of the viscosity master curves and in the order of magnitude of the normal‐stress differences.

ESR Spectra and Structure of the Fluorinated Methyl Radicals
View Description Hide DescriptionThe ESR spectrum of the CF_{3} radical has been observed during the irradiation of liquid C_{2}F_{6}. The isotropic fluorine hyperfine splitting in this radical has a value of 144.75 G which is considerably larger than other known values for α fluorine atoms. Well‐resolved spectra of CF_{3}·, CHF_{2}·, and CH_{2}F·, as well as certain of the lines of the ^{13}C containing radicals, have also been observed in krypton and xenon matrices. The ^{13}CF_{3} and ^{13}CHF_{2} radicals have carbonhyperfine constants of 271.6 and 148.8 G showing that the unpaired electron is in an orbital with considerable s character and that these radicals are therefore non‐planar. The carbon and fluorine splittings in CF_{3}· are sufficiently large that third‐order correction terms are necessary in considering the line positions. As a result it is demonstrable that the carbon and fluorine hyperfine constants are of the same sign.

Electron Spin Resonance of Radical Cations Produced by the Oxidation of Aromatic Hydrocarbons with SbCl_{5}
View Description Hide DescriptionElectron spin resonance(ESR) studies have been made of radical cations prepared by the oxidation of 24 aromatic hydrocarbons with SbCl_{5} in CH_{2}Cl_{2}solvent. The transfer of reagents, reactions, and ESR measurements were carried out at temperatures below −78°C in the absence of air. The stability of the radicals in solution, and the resolution of ESRhyperfine structure, depended critically on temperature and on the concentrations of hydrocarbon and SbCl_{5}. The coupling constants for the ESR spectra of the radical cations of perylene, anthracene, 9,10‐dimethylanthracene, and naphthacene were essentially identical to those for the respective ions in H_{2}SO_{4}. The ESR spectra for the radical cations of pyrene, naphthalene, dibenzo‐(a,c)triphenylene,Δ^{9,9′}‐bifluorene, and tetraphenylethylene have also been reduced to coupling constants. The positive radical ion resulting from the oxidation of naphthalene is a symmetrical dimer with the unpaired electron divided equally between the two molecules. The generality of McConnell's relationship a _{H}=Q _{ρ} was examined by comparing experimental coupling constants with Hückel spin densities for available radical anion and cation data. A least‐squares fitting procedure yielded Q=28.6 for anions and Q=35.7 for cations. Quantitative comparisons were made between experimental coupling constants and those computed from the Colpa—Bolton and Giacometti—Nordio—Pavan theories. Both theories were found to account appropriately for the differences in magnitude between the coupling constants for aromatic negative and positive ions.

Calculated Bonding in the Pseudohalide Anions and Their Hydracids
View Description Hide DescriptionThe bond characters and energies of the nitrogen‐containing pseudohalide anions, CN^{−}, N_{3} ^{−}, NCO^{−}, NCS^{−}, NCSe^{−}, NCTe^{−}, and their common hydracids, as well as some related isoelectronic systems, have been calculated using an internally consistent procedure based on the utilization of ``unsymmetrical group orbitals'' (LCGO method). Coulomb integrals are related to the valence‐state electronegativities which are obtained by three different procedures. It is concluded that the results are more sensitive to the rate of change of the Coulomb integral with charge than to the value of the Coulomb integral itself. This LCGO—MO method greatly simplifies the calculations and gives results identical with the longer LCAO procedures which correlate well with the physical and chemical properties of the substances. The calculations account for the relative acid strengths and correlate quantitatively with the experimentally determined C–N stretching‐force constants and ^{14}N NMRchemical shifts.