Volume 34, Issue 6, 01 June 1961

Effect of Inert‐Gas Moderators on the (n, γ) Activated Reaction of I^{128} with CH_{4}
View Description Hide DescriptionIn the presence of a large excess of gaseous methane, 54.4±0.5% of I^{128} formed by (n, γ) activation was found to become stabilized in organic combination. The effects of inert‐gas additives in moderating the reaction of I^{128} with CH_{4} were determined in an effort to ascertain the mechanism. The data, extrapolated to zero mole‐fraction methane, indicate that xenon is capable of reducing the organic I^{128} to 11% whereas neon, argon, and krypton each reduce it to only about 36%. These data suggest that of the 54.4% organic I^{128}, about 18.4% forms as a result of hot I^{128}reactions, 11% as a result of excited iodine atoms or I^{+} ions in the ^{3} P _{2}, ^{3} P _{1}, and/or ^{3} P _{0} states, and 25% as a result of reactions of I^{+}(^{1} D _{2}) ions.

Quantum Mechanics of Mobile Electrons in Conjugated Bond Systems. I. General Analysis in the Tight‐Binding Formulation
View Description Hide DescriptionThe quantum‐mechanical treatment is carried through for a set of electrons in a homonuclear conjugated bond system of arbitrary size, including electronic interaction and including all overlap effects between neighbors. All framework contributions are obtained by explicit integration over the framework Hamiltonian, including the effect of nonconjugated neighbor atoms and differentiating between different types of conjugated atoms (joint, nonjoints, etc.). Expressions are given for the ground‐state energy, ionization potential,electron affinity, electronegativity, and for the configuration interaction matrix for the calculation of excited states, assuming singly excited configurations. The results take simple forms permitting instructive interpretations. The partial additivity of one‐electron binding‐energy contributions, obtained as eigenvalues of topological molecular orbitals, and the approximate validity of the ``neglect of differential overlap'' is proved.

Quantum Mechanics of Mobile Electrons in Conjugated Bond Systems. II. Augmented Tight‐Binding Formulation
View Description Hide DescriptionThe analysis of homonuclear conjugated systems, which has been given within the limits of the tight‐binding approximation in the first paper of this series, is extended to include the interactions between non‐neighbor atoms and the variation in the interactions between neighbor atoms. Both kinds of interactions are included as perturbation effects on the tight‐binding approximation. As a consequence, the formalism developed for the latter is not being complicated and the interpretations remain unchanged. Application to benzene and naphthalene illustrates the approach.

Quantum Mechanics of Mobile Electrons in Conjugated Bond Systems. III. Topological Matrix as Generatrix of Bond Orders
View Description Hide DescriptionIt is shown that in homonuclear conjugated systems the various bond orders and similar quantities can be written as matrix functions of the topological incidence matrix. This entails the existence of a number of useful general relations between these various quantities. The relations include as special cases: Coulson and Rushbrooke's theorem on charge orders in alternants; G. G. Hall's theorem on bond orders in alternants; McWeeny's theorem on the formal charges; Ham and Ruedenberg's correlation between Coulson and Mulliken bond orders for neighbors in alternants; closely related is Ham‐Ruedenberg‐Platt's relation between valence‐bond bond orders and molecular orbital theory. A number of new relations between bond orders are derived and discussed. The generalization from alternants to nonalternants is given particular attention.

Quantum Mechanics of Mobile Electrons in Conjugated Bond Systems. IV. Integral Formulas
View Description Hide DescriptionFormulas are established for all carbon and hydrogen penetration integrals occurring in the tight‐binding approximation of the theory of mobile electrons. The orbital exponents of the penetrated shielding orbitals may differ from the orbital exponent of the penetrating 2pπ electrons. A simple correlation between kinetic energy integral and overlap integral is found.

Quantum Mechanics of Mobile Electrons in Conjugated Bond Systems. V. Empirical Determination of Integrals between Carbon Atomic Orbitals from Experimental Data on Benzene
View Description Hide DescriptionIt is shown that the experimental information given in the ultraviolet spectrum of benzene uniquely determines the two‐center Coulomb integrals between the 2pπ atomic orbitals of carbon as a function of the internuclear distance. The calculations are carried out including all terms involving neighbor overlap. For the many‐center electron‐interaction integrals, the Mulliken approximation and the London approximation are both considered with little difference in the results. Some peculiar properties of the empirically determined distance dependence are compared with the theoretical behavior of two‐center Coulomb integrals. The empirical values for the resonance integral and the Coulomb integral are also found. The investigation differs from similar work of Pariser by the inclusion of overlap effects.

Quantum Mechanics of Mobile Electrons in Conjugated Bond Systems. VI. Theoretical Evaluation of Energy Contributions
View Description Hide DescriptionThe general formulas of the first paper in the series [K. Ruedenberg, J. Chem. Phys. 34, 1861 (1961)] are used to analyze the energy contributions of the molecular framework for a π‐electron system in detail, without use of additional assumptions. The valence‐state potential of carbon, used in the framework potential, and the 2pπ atomic orbitals are determined by a minimum principle. The Coulomb integral and resonance integral are evaluated from their constituent integrals in the tight‐binding approximation. The agreement with the empirical values [see K. Ruedenberg, J. Chem. Phys. 34, 1878 (1961)] is unsatisfactory for the Coulomb integral but very good for the resonance integral. The approximate proportionality between energy matrix elements and overlap integrals is proven for a variation of the internuclear distance between 1.26 and 1.54 A.

Energies of Excited Electronic States as Calculated with the Zero Differential Overlap Approximation
View Description Hide DescriptionIt is shown that electronic excitation energies, calculated using the zero differential overlap approximation, can always be expressed in terms of the difference between two‐electron Coulombic integrals. These differences can then be used as empirical parameters for the calculation of excitation energies.

Vaporization of Iridium and Rhodium
View Description Hide DescriptionThe vaporization of iridium and rhodium have been studied by Knudsen effusion and Langmuir evaporation techniques. The vapor pressure of iridium over the temperature range of from 2100° to 2600°K is represented by the equation: logp_{mm} =10.46— (33980/T), and the vapor pressure of rhodium over the temperature range of from 2050° to 2200°K by the equation: logp_{mm} =10.28— (28300/T). Third‐law analyses of the data yield the following heats of vaporization:iridium, ΔH _{298}=158.4±0.5 kcal/mole; rhodium, ΔH _{298}=132.8±0.3 kcal/mole. Estimated boiling points for iridium and rhodium are 4800° and 3980°K, respectively.

Potential Curves for Doubly Positive Diatomic Ions
View Description Hide DescriptionAn integral form of the quantum‐mechanical virial theorem, applicable to ionic as well as neutral systems, is established. This theorem is used to derive an approximate expression for the potential curve for a doubly positive diatomic ion in terms of the corresponding curve for a related neutral molecule. The appearance potentials of a number of these ions are calculated and compared with values obtained from electron‐impact measurements. Satisfactory agreement is found, but a crucial test of the theory is vitiated by uncertainties in the experimental value of either the appearance potential of the doubly charged ion or the dissociation energy of the corresponding neutral molecule. The accuracy of the approximations underlying the theory is estimated by a series of calculations on two‐electron systems.

Effect of Configuration Interactions on the Dissociation Energy and Hyperfine Structure Constants of the NO Molecule
View Description Hide DescriptionBy means of a 23‐dimensional configuration interaction calculation the energy and eigenfunction of the ground ^{2}II state are determined resulting in dissociation energy of 2.2 ev. The magnetic hyperfine and nuclear quadrupole coupling constants are calculated from this molecular function. The contact term  ψ(0) ^{2} is obtained as 0.71×10^{—24} cm^{—3} which agrees well with experiment. The values ofare not much affected by the configuration interactions.

Theory of Vibrational Relaxation in Liquids
View Description Hide DescriptionA new formulation of the theory of vibrational relaxation, based on Zener's semiclassical approximation, is presented here. The relaxation rate is shown to be proportional to the spectral density of the force exerted on the oscillator by its environment. The isolated binary collision theory is derived, but only with the condition that the collision frequency is much smaller than the oscillator frequency. This requirement is not satisfied in a liquid; we conclude that Litovitz's application of the isolated binary collision theory to liquids is not justified. A possible relation between vibrational relaxation and the self‐diffusion coefficient in a liquid is discussed.

Experimental Study of the Shape of the F Band Absorption in NaCl
View Description Hide DescriptionThe possibility of extending the authors' study of the optical absorption band associated with F centers in KCl to KI, KBr, and NaCl is explored. Due to the relative sizes and positions of the K and F bands in KI and KBr, one cannot resolve the two. The situation in NaCl is unclear; hence, a detailed study has been made. A unique F center can be formed in this crystal by various techniques. The optical bleaching properties depend on the means of production. Smakula's equation applies to this center, and it seems to be Pekarian. Our ``best equation'' for the width at half‐height H (in ev), has the form H ^{2}=0.075 coth (105.5/θ) — 0.01; where θ is the temperature. This study supports the general theory [J. J. Markham, Revs. Modern Phys. 31, 956 (1959)] although all the experimental problems have not been resolved.

Shock Tube Determination of Dissociation Rates of Oxygen
View Description Hide DescriptionThe rate of dissociation of oxygen in Xe–O_{2} mixtures was measured over a temperature range of 3000°K to 6000°K. An x‐ray densitometer was used to measure the density during the dissociation process behind a shock wave. It was possible to match the experimental data with theoretical density profiles over a wide range of compositions and initial conditions. The reactions considered werewhere M can be Xe, O_{2}, or O. Considering these species as third bodies, the deduced recombination rates in cc^{2} mole^{—2} sec^{—1} were 4.7×10^{17} T ^{—1}, 1.6×10^{18} T ^{—1}, and 4.8×10^{18} T ^{—1}, respectively. The third‐body efficiencies of O_{2} and O relative to Xe are 3 and 10. Experimental conditions were such that an accurate measurement of the exponent of the temperature could not be made. However, since the data showed it to be within the limits of —½ and —2, a value of —1.0 was arbitrarily chosen. The agreement between results reported here and previous work demonstrates the potential utility of this method for kinetic studies of other reactions.

Effect of Pressure on the Resistance of Iodine and Selenium
View Description Hide DescriptionThe effect of pressure from 60 to over 400 kbar has been measured on the resistance of selenium and iodine. Selenium exhibits a very rapid drop in resistance between 60 and 128 kbar; at 128 kbar it shows a discontinuous drop. At higher pressures its behavior is apparently metallic. Iodine shows a rapid drop in resistance from 60 kbar to the region of 225–255 kbar where there is relatively abrupt change of slope. At higher pressures the change in resistance with pressure is much smaller. It is interesting to note that the optical energy gap of selenium extrapolates to zero at about 130 kbar, while the optical gap for iodine extrapolates to zero at 240 kbar.

A Calculation of the Potential Energy Curves for Some Electronic States of Carbon Monoxide
View Description Hide DescriptionThe LCAO‐MO SCF orbitals for the ^{1}Σ^{+}ground state of carbon monoxide, constructed from 1s, 2s, and 2p atomic functions, have been calculated for 11 internuclear distances between 1.5 and 4.0 a.u. Using the unoccupied SCF orbitals a number of lower excited states have been calculated. The equilibrium distances, force constants, and ν_{00} energies have been calculated from the curves of E vs R for these various states. As these calculated curves do not separate at infinity into the atoms in their proper spectroscopic states, configuration interaction has been carried out with functions which cross the computed curves and separate into the atoms in their ^{3} P states. The lower states of carbon monoxide seem to be divided into three classes: (a) those which are reasonably well represented by our functions, namely, X ^{1}Σ^{+}, e ^{3}Σ^{—}, a′ ^{3}Σ^{+}, A ^{1}Π, and a ^{3}Π; (b) those which are not at all well represented by our functions, namely, b ^{3}Σ^{+}, B ^{1}Σ^{+}, and C ^{1}Σ^{+}; (c) those states for which the experimental assignment may be in error, namely, d ^{3}Π and F ^{1}Π.

Isotopic Separation Factor for the System Potassium Amalgam—Aqueous Potassium Hydroxide
View Description Hide DescriptionThe isotopic fractionation of potassium between potassium amalgam and aqueous potassium hydroxide was measured at room temperature. The single‐stage separation factor was 1.006±0.002 (95% C. I.).

Paramagnetic Susceptibility of Polycrystalline Praseodymium Metal
View Description Hide DescriptionThe paramagnetic susceptibility of polycrystallinepraseodymium metal has been measured between 300 and 1500°K. The measurements are in good agreement with the predictions of the Van Vleck theory of paramagnetism based on a localized f‐electron model of noninteracting Pr^{3+} ions. By making a reasonable correction for the contribution of conduction electrons to the paramagnetism, a best fit to the experimental results is obtained by using a screening constant σ=34.

On the Theory of Helix—Coil Transition in Polypeptides
View Description Hide DescriptionThe evaluation of the configurational partition function of a polypeptide molecule, with the internal rotation angles as variables, leads to an improved treatment of the phenomenon of helix‐coil transition in polypeptide molecules. The conditional probabilities of occurrence of helical and coiled states of the peptide units are obtained in the form of a 3×3 matrix. The order of this matrix is the lowest possible for the model employed, and is derived by a logical procedure which serves to eliminate redundancies in the enumeration of states. The eigenvalues of this matrix yield the various molecular averages as functions of the degree of polymerization, temperature, and molecular constants. Explicit formulas are given for the degree of intramolecular hydrogen bonding, average number of helical sequences, and the distribution of their lengths, as well as the number average and the weight average of these lengths.

Theory of Solutions. III. Thermodynamics of Aggregation or Polymerization
View Description Hide DescriptionPolymeric and colloidal systems are usually polydisperse, containing possibly hundreds or thousands or more, of subspecies differing only in degree of polymerizationn. The required thermodynamic approach is that appropriate to a highly multicomponent system. Let P(n) be the fraction of polymer molecules of size n. The distribution functionP(n) will depend on one or more parameters, for example, the mean n̄ and standard deviation σ in the case of a Gaussian distribution. The thermodynamic problem becomes tractable if we replace, as composition variables, the numbers of molecules of all the polymer subspecies by the total number of polymer moleculesN and the parameters of P(n). The new set of variables may be only two, three, etc., in number. Our main interest in this paper is to investigate how the thermodynamic functions of the system change when the distribution functionP(n) changes, as, for example, in a kinetic study in which P(n) evolves sufficiently slowly with time or in thermodynamic studies on different samples of the same polymer.