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Volume 29, Issue 2, 01 August 1958

Statistical Computation of Reaction Probabilities
View Description Hide DescriptionCalculations have been carried out on a high‐speed electronic digital computer to determine statistically the transition probability for a simple chemical reaction. Using the London‐Eyring‐Polanyi form for the potential energy of the colinear system, H+H_{2}, we have solved the classical equations of motion for the system under a great variety of conditions. Collisions of a hydrogen atom and a hydrogen molecule, with and without initial molecular vibration, were studied as a function of the total kinetic energy. By carrying out numerous calculations for a sufficiently large number of different initial conditions, we obtained an average behavior of the system, which we believe provides a satisfactory approximation to the quantum mechanical average. It was found that the reaction probability was greatest when the energy was just slightly above the activation energy. Moreover, it was observed that, if the molecule was initially vibrating, only a part of the energy of vibration was available for reducing the kinetic energy of translation required for activation. It was also found that the conversion of energy between vibration and translation was relatively improbable.

Quantum Statistical Theory of Plasmas and Liquid Metals
View Description Hide DescriptionA Debye‐Hückel‐type theory is described for an assembly of completely ionized atoms, the nuclei being treated classically and the electrons by the Thomas‐Fermi method. The thermodynamic functions are derived by considering the Debye charging process, and the virial theorem is shown to hold. Numerical results are given for hydrogen and iron near normal solid densities, and are probably accurate only at high temperatures (kT>5 ev for hydrogen and kT>100 ev for iron). At these temperatures, the results do not differ greatly from those of the ordinary Thomas‐Fermi theory of the atom except for the additional contributions of the nuclei.

On the Approximate Calculation of Eigenvalues with Special Reference to the Helium Atom
View Description Hide DescriptionA method is developed for treating the error in the energy of a variation function as a perturbation, the variation function serving to describe the ``unperturbed state.'' An effective potential is defined from the variation function, and comparison of this with the correct potential for the helium atom gives useful information about the behavior of the variation functions as the electronic separation approaches zero. In the second section, iterative methods of improving a variational wave function without increasing the number of parameters are discussed and the conditions under which they will be applicable are examined.

Application of Schwarz's Inequality to Perturbation Theory
View Description Hide DescriptionIt is shown by use of the variation theorem, that under fairly general conditions the energy given to an odd order of approximation by perturbation theory is an upper bound to the correct energy. Under less general conditions it is shown that for a positive definite perturbation the energy given by an even‐order approximation is a lower bound and for a negative definitive perturbation it is again an upper bound. Some applications are discussed.

Radial Distribution Functions in Binary Gas Mixtures
View Description Hide DescriptionThe equations of Born and Green are generalized to the two component case. An approximate solution is formally obtained by a linearization procedure and is found to contain a singularity which is tentatively identified with the gas‐liquid branch point of the radial distribution function. Temperature‐density relations are found in the critical region by roughly locating this singular point. The results lend weight to the hypothesis that the singularity corresponds to the branch point.

Reactions of Gaseous Molecule Ions with Gaseous Molecules. IV. Experimental Method and Results
View Description Hide DescriptionThis paper is concerned with a detailed description of quantitative studies of certain secondary reactions involving simple molecules and ions that occur in the ion source of a mass spectrometer. It is found that the phenomenological reaction cross sections for reactions of the typeare independent of the temperature of the gas in the ion source, and vary inversely with the square root of the product of the ion repeller field strength and the reduced mass of the reactant ion‐reactant molecule pair. For the reactant ion (P ^{+}) a rare gas, hydrogen, nitrogen, or carbon monoxide, and the resultant molecule (R) hydrogen, the cross sections for formation of the secondary ion, S^{+}=PH^{+}, are found to be one or two orders of magnitude greater than the ordinary, kinetic theory collision cross sections for similar atoms and molecules.

Reactions of Gaseous Molecule Ions with Gaseous Molecules. V. Theory
View Description Hide DescriptionIon‐molecule reactions of the sort observed as secondary reactions in mass spectrometers have been treated by the methods of the modern kinetic theory; that is, the rate of reaction is expressed in terms of the velocity distribution functions of the reactants and the cross section for the reaction. The cross section, which is calculated by means of the properties of the classical collision orbits, is found to have an inverse square root dependence on energy. The ion distribution function, which is far from Maxwellian, is found by means of an explicit solution of the Boltzmann equation. A simple relation is given which relates the mass spectrometric data to the specific rate of the same reaction under thermal conditions. For the simpler molecules, this rate may be calculated completely a priori, with excellent agreement with experiment.

Evaluation of Two‐Center Exchange Integrals
View Description Hide DescriptionThe evaluation of two‐center exchange integrals for any homonuclear molecules using Slater‐type wave functions is reduced to the solution of Poisson's equation. An expression is obtained for the exchange integrals in terms of integrals which can be evaluated analytically. In particular, all cases which arise from 2s and 2pwave functions are discussed in detail and the auxiliary integrals evaluated. The pertinent results are collected in tables, and the final result for a particular exchange integral is given explicitly.

Theory of Dilute Polymer Solution. II. Osmotic Pressure and Frictional Properties
View Description Hide DescriptionThe osmotic second virial coefficient of flexible polymer solutions is calculated on the basis of the pearl‐necklace model with continuous medium approximation. Proper account is taken not only of the intermolecular interaction but also of the intramolecular interaction of segments or the so‐called ``excluded volume effect.'' Evaluation is carried out to the triple contact approximation. The results show that the intramolecular interaction has a marked influence on the molecular weight dependence of the coefficient.
Then, theories of the intrinsic viscosity and frictional coefficient are developed on the same model as above, following the Kirkwood and Riseman scheme. The results show that, if α is the linear expansion factor of the polymer coil, the intrinsic viscosity is asymptotically proportional to α^{2} for the free‐draining case and α^{2.43} for the impermeable case. This implies that even in the limit for impermeable molecule, the hydrodynamic radius of the polymer coil is not proportional to its statistical radius.
Finally a thorough comparison of the present theories with experimental data is made, taking solutions of polystyrene in cyclohexane as an example. The results prove that the theories are very fitted for structural interpretation of experimental data.

Electric Field Gradients in Ionic Crystals. I. Nuclear Quadrupole Coupling Constants
View Description Hide DescriptionIn ionic crystals and metals the coupling constant of a quadrupolar nucleus can often be written in the simplified form eQ(Δε)(1—γ_{∞}). The polarization factors 1—γ_{∞} were calculated in previous work by T. P. Das, E. G. Wikner, and R. Bersohn. In the present work the factors Δε are calculated in the form of lattice sums. Under favorable circumstances a good estimate of the nuclear quadrupole moment Q can be made. When Q is known by other means, information about the distribution of charges in crystals may be obtained. It is shown, for example, that Al_{2}O_{3} is essentially ionic. The agreement between experimental and observed coupling constants proves that the field gradients in ionic crystals and metals is of long‐range origin in contrast to molecular crystals.

Intramolecular Twisting Effects in Substituted Benzenes. II. Ground‐State Properties
View Description Hide DescriptionThe effects on ground‐state properties of twisting a substituent group about the substituent‐ring bond in substituted benzenes are discussed from the viewpoint of semiempirical MO theory. The ground‐state properties are discussed with reference to a parameter θ, which generally increases as the substituent is twisted. The substituent‐ring bond order varies approximately as cosθ, and the following vary approximately as cos^{2}θ: the resonance energy, charge densities, ring C–C bond orders and the π‐electronic dipole moment. The θ dependence of the total dipole moment is discussed. Numerical applications to N,N‐dimethylaniline and related molecules are described, including a detailed treatment of the dipole moments of N,N‐dimethylaniline and some of its ortho‐substituted derivatives. A brief discussion of the valence state of the dimethylamino group is included.

Magnetic Susceptibility of Ferrous Ammonium Tutton Salt at Liquid Helium Temperatures
View Description Hide DescriptionThe magnetic susceptibility of ferrous ammonium sulfate hexahydrate has been measured between 0.95 and 4.2°K by both the Faraday and the ballistic mutual inductance bridge methods. At 4.2°K the susceptibility is (1.45±0.05)×10^{—3} emu/g, close to the value given by the high temperature Curie‐Weiss law, 9.71×10^{—3}/(T+3.2). Below 2°K the susceptibility is constant at (1.85±0.05)×10^{—3} emu/g. The results are shown to be compatible with available specific heat data in the low temperature region, provided that the lowest level is taken to be a singlet and the constant susceptibility is ascribed to temperature‐independent paramagnetism of the lowest level.

Relative Intensities of Microwave Absorption Lines
View Description Hide DescriptionA method is described for measuring the relative intensities of microwave absorption lines up to 80 mc. apart to within a few percent, using a Stark‐modulated resonant cavity absorption cell of novel design. Barriers to internal rotation in CH_{3}CHO and CH_{3}CH_{2}F have been obtained by measuring the relative intensities of torsional satellites. The resulting barrier heights are: 1103±60 cal for CH_{3}CHO, 3310±210 cal. for CH_{3}CH_{2}F, in good agreement with the values 1162±30 cal for CH_{3}CHO, and 3306±100 cal for CH_{3}CH_{2}F obtained from splitting measurements by Kilb, Lin, and Wilson, and by Herschbach, respectively.
Correction factors are given for errors in intensity and splitting measurements due to small doublet splittings.

Generalized United‐Atom Theory of Molecules. I. Spherical Contributions in the sσ Wave Functions for H_{2} ^{+}
View Description Hide DescriptionThe generalized united‐atom view of molecules suggests that electronic states of symmetric molecules may be characterized by means of ``atomic'' quantum numbers.
A test for the purity of the quantum number l in sσ states in H_{2} ^{+} is facilitated by the Tibbs‐Wannier theory of a Coulomb field with cutoff. Comparison shows that 96% of the ground‐state electronic energy is due to contributions of spherical functions.

Generalized United‐Atom Theory. II. A One‐Center Model for Cyclic Polyenes
View Description Hide DescriptionIn cyclic polyenes the quantum numbers of Hückel and Platt are related to the atomic magnetic quantum number m_{l} . A one‐center treatment shows that, within the tubular assumption, the lowest one‐electron wave functions have roughly the same (ρ, z) dependent part and are distinguished chiefly by the dependence on the azimuthal angle [open phi]. The one‐electron level scheme to first approximation is identical to that of Platt's electron‐gas theory. Reintroduction of the lower molecular symmetry mixes states of different m_{l} and splits the m_{l} =±(2N+1) degeneracy. The result is suggestive of the LCAO‐MO theory, although the latter can be shown to be basically different. A semiempirical united‐atom theory can be formulated in close parallel to the LCMO theory of Coulson, Longuet‐Higgins and Dewar.

Determination of the Binding Energy of He_{2} ^{+} from Ion Scattering Data
View Description Hide DescriptionData on the elastic and the charge exchangescattering of beams of He^{+} ions in He gas between 4 and 400 ev are analyzed to determine precisely the interaction energies for the lowest ^{2}Σ_{ u } and ^{2}Σ_{ g } states of He_{2} ^{+}. A Morse function is used to represent the energy for the attractive ^{2}Σ_{ u } state and an exponential is chosen for the repulsive ^{2}Σ_{ g } state. An attempt is made to determine the three disposable parameters strictly from the scattering data, but does not lead to a unique result. However, use of the spectroscopic value of the equilibrium distance for the ^{2}Σ_{ u } state permits a unique determination, leading to interaction energies valid from about 0.9 A to 3.8 A. The energies are in good agreement with available quantum‐mechanical calculations and give a dissociation energy of 2.16 ev for the ^{2}Σ_{ u } state. The cross‐section calculations are given in reduced units in a form readily applicable to the calculation of binding energies for similar systems as data may become available.

Studies in Nonequilibrium Rate Processes. IV. The Rotational and Vibrational Relaxation of a System of Rotating Oscillators
View Description Hide DescriptionIn continuation of previous papers in this series a theoretical study has been made of the rotational and vibrational relaxation of a system of rotating oscillators, represented by a rigid rotator‐harmonic oscillator model, in their interaction with a constant temperature heat bath. The relevant relaxation equations for this system have been derived and both numerical solutions, computed with an IBM 704 electronic computer, and approximate analytical solutions have been obtained for the mode and rate of change of the rotational‐vibrational level population with time. The results show that, owing to the large order of magnitude difference in the efficiency of translational‐rotational and translational‐vibrational energy transfer, there is very little coupling between the vibrational and rotational relaxation and the system of rotating oscillators relaxes essentially as if composed of two independent and noninteracting systems of harmonic oscillators and rigid rotators. A calculation of the rotational relaxation time,t(relax), using the adiabatic transition probabilities given by Brout and by Takayanagi and the concomitant selection rules ΔJ=±1 (for heteronuclear molecules) gives values for t(relax) which are greater by a factor of several orders of magnitude (∼10^{2}) than the experimental relaxation times found from shock wave studies. This indicates clearly that rotational relaxation does not take place via a stepwise ``ladder process'' involving only transitions between neighboring rotational levels.

Mass Spectra of Some Simple Esters and Their Interpretation by Quasi‐Equilibrium Theory
View Description Hide DescriptionMass spectra and appearance potential data are reported for normal and isotopically labeled methyl formate, methyl acetate, and ethyl formate, and reaction mechanisms are proposed for the production of the spectra. The data are analyzed from the standpoint of the quasi‐equilibrium theory of Eyring and co‐workers. For the data from 70‐volt ionizing electrons, frequency factors can be adjusted so that the calculated patterns agree well with the experimental data. These frequency factors seem qualitatively satisfactory although there are some difficulties with the values for some of the rearrangements and hydrogen abstraction processes. However the theory is clearly inadequate at lower ionizing voltages. The probable explanation is that the electronic states are in narrow bands which do not overlap, so that at low voltages a portion of the ionic fragments are produced by direction dissociation, without the redistribution of energy which is required by the quasi‐equilibrium theory.

Microwave Spectrum, Internal Barrier, Structure, Equilibrium Configuration, and Dipole Moment of Methyl Monofluorosilane
View Description Hide DescriptionThe microwave spectrum of methyl monofluorosilane has been investigated in the region 18 to 38 kmc. Several transitions of the isotopic species CH_{3}SiH_{2}F and CH_{3}SiD_{2}F are split into doublets because of coupling of internal and over‐all rotation. These splittings give a barrier to internal rotation of 1559±30 cal/mole. The form of the potential barrier was assumed to be . A few transitions belonging to the first excited torsional state were observed and assigned. The splittings in the excited state are consistent with the barrier as determined from ground state observations.
Rotational constants of CH_{3}SiH_{2}F, CH_{3}SiD_{2}F, C^{13}H_{3}SiH_{2}F, CD_{3}SiH_{2}F, and CH_{3}SiDHF have been determined and were used to calculate the structure of the molecule.
The structural parameters are:
A study of CH_{2}DSiH_{2}F proves conclusively that methyl monofluorosilane in its equilibrium configuration has the methyl group staggered with respect to the SiH_{2}F group.
The dipole moment was found to be 1.71D.

Organic Photoconductors. IV. Sign of the Charge Carrier
View Description Hide DescriptionIf the energies of carriers in the conduction band and valence band of a dye can be determined with respect to the corresponding levels in cadmium sulfide, then the sign and magnitude of the photo‐emf developed at a junction between dye and cadmium sulfide can often be interpreted as indicating a transfer of carriers between conduction bands, conduction hence being due primarily to electrons; or as a transfer between valence bands, conduction thus being due to holes. In this way the majority carrier in several cationic dyes is shown to be electrons; in erythrosin, an anionic dye, holes; and in methyl chlorophyllide‐a, a non‐ionic dye, probably electrons.