Volume 38, Issue 10, 15 May 1963

Theory of the Effect of Temperature on the Electron Diffraction Patterns of Diatomic Molecules
View Description Hide DescriptionThe effect of temperature on the electron diffraction pattern of a diatomic molecule is considered from the standpoint of the simple kinematicscattering theory utilizing a quartic vibrational potential. The potential is obtained by an expansion of about its minimum value r _{0}. The second‐order wavefunction for the nth vibrational and Jth rotational state of the system has been obtained, and expressions for the electron diffraction quantities r_{g}, l_{e} ^{2}, and M (s) have been computed. General results for the quantity M (s) utilizing the approximate eigenfunctions of the complete Morse potential and incorporating an approximate treatment of the effect of centrifugal stretching are also presented. Explicit expressions for M (s) for the first three vibrational states as derived by this treatment are given. Appropriate sums over all the vibrational and rotational states have been carried out to obtain the temperature dependence for the above quantities. Estimates of the effect of temperature on the parameters r_{g} and l_{e} ^{2} at 300° and 1500°K for representative diatomic molecules are given.

Structure of Formaldoxime
View Description Hide DescriptionThe microwave spectra of cis‐CHDNOH, trans‐CHDNOH, C^{13}H_{2}NOH, CH_{2}N^{15}OH, and CH_{2}NO^{18}H were assigned. Together with the results of previous work on the normal species and CH_{2}NOD, the data give the following planar structure for formaldoxime: bond lengths in Å; CH_{ cis }=1.085, CH_{ trans }=1.086, C=N=1.276, NO=1.408, OH=0.956; bond angles; H_{ cis }CN 121°46′, H_{ trans }CN 115°33′, CNO 110°12′, NOH 102°41′.
The most significant feature of the structure is the difference between cis‐ and trans‐HCN bond angles. This ``tilt'' of the methylene group appears to be too large to be attributed to steric repulsion. The C=N and N–O bond lengths are consistent with the Schomaker—Stevenson relation if allowance for resonance is made.

Stark Effect and Dipole Moment of Methyl Fluoride
View Description Hide DescriptionHigh‐precision Stark effectmeasurements have been made on the J=1→2 transition of methyl fluoride in the shorter millimeter wave region. A parallel‐plate Stark cell employing silver‐coated glass plates held apart by optically ground quartz spacers was employed. Millimeter wave power was fed between the plates by two sets of horns, one of which allowed for measurement of the ΔM=0 transitions and the other for measurement of the ΔM=±1 transitions. The dc voltage across the Stark plates was measured with a standard potentiometer. Both the first‐ and second‐order effects were measured. The resulting value for the dipole moment of CH_{3}F is μ=1.8555±0.0015 D for the ground vibrational state.

Semiempirical SCF‐LCAO—MO Treatment of Furan
View Description Hide DescriptionThe self‐consistent field—molecular orbital method including configuration interaction is utilized for the study of the π‐electronic structure of the five‐membered heterocyclic molecule furan. The calculation proceeds through the density matrix formalism of McWeeny. A judicious choice of the two empirical core parameters for oxygen, along with the semiempirical approximations of Pariser and Parr, leads to a satisfactory description of the ionization potential, electronic spectra, and chemical reactivity of furan.

Forces in Molecules. II. A Differential Equation for the Potential‐Energy Function
View Description Hide DescriptionA second‐order, linear, homogeneous, differential equation for the diatomic molecular potential‐energy function, U_{n} (R), is derived. Several aspects of this equation are studied in detail. The solutions of an approximate form of the equation suggest an empirical function that slowly oscillates in the complete interval 0≤R≤∞. This function is fit about the optimum minimum for 23 molecules. An approximate differential equation is derived for the function g_{ij} (R)≡U_{i} (R)—U_{j} (R), where U_{i} and U_{j} are any two potential‐energy curves for different electronic states. This equation is valid in an interval for which g_{ij} is small—that is, for near degeneracies. An application to two states of H_{2} ^{+} is given.

Heat of Formation of Methylene
View Description Hide DescriptionRecent theoretical estimates of the heat of formation of methylene prompted the reanalysis of the electron impact data of Langer, Hipple, and Stevenson on CH_{4}, CH_{3}, and CH_{2}. The analysis indicates that the potentials are not wholly self‐consistent but consideration of only CH_{2} ^{+} and CH^{+} ions leads to self‐consistent reactions. These yield a most probable value of 95±5 kcal mole^{—1} for the heat of formation of methylene, in excellent agreement with the theoretical estimates.

Spin—Spin Interaction in Methylene
View Description Hide DescriptionCalculations of the spin—spin interaction for methylene have been carried out. It is predicted that a resonance line should be observed at 3.3×10^{10} cps in the ESRspectrum of the solid system. It is shown that the splitting of the rotational lines in the electronic spectrum of methylene should be observable with an instrument with a resolving power of ∼0.5 cm^{—1} in the ultraviolet region of the spectrum.

Kinetic Theory of Nonspherical Molecules. V
View Description Hide DescriptionA generalized Boltzmann equation for fluids composed of molecules with arbitrary internal degrees of freedom and which interact according to classical mechanics through arbitrary, noncentral pair‐additive forces is developed using techniques similar to those used by Hollinger and Curtiss. Several special cases are considered: (1) molecules which interact only through impulsive forces of repulsion, (2) molecules for which the only internal degrees of freedom are rotational degrees, and (3) a combination of both special cases. The third special case leads to a Boltzmann equation appropriate to a fluid composed of rigid nonspherical molecules, and provides a firm basis upon which to develop a formal kinetic theory. As a special example we consider a gas made up of rigid, convex, nonspherical molecules with symmetric‐top mass distributions. Rigorous equations of hydrodynamics are presented for such a fluid. It is shown that the kinetic theory of a dilute multicomponent system of these molecules is formally the same as that for the loaded‐sphere system recently studied by Dahler and Sather. Finally, it is shown how the methods developed by these authors can be applied to the problems of solving the Boltzmann equation and of calculating the transport coefficients for the nonspherical molecular species considered in the present paper.

Kinetic Theory of Loaded Spheres. I
View Description Hide DescriptionA rigorous formalism is developed for the theory of transport in fluids composed of loaded‐sphere molecules. From this theory formulas are derived for the fluxes of mass, linear momentum, internal or spin angular momentum, and energy. The existence of inverse collisions is established and used to prove a weak form of the Boltzmann H theorem. Finally, we present a kinetic theory for dilute multicomponent mixtures of loaded spheres and derive explicit formulas for the phenomenological gas transport coefficients. Since the only critical assumption imbedded within this treatment is that of molecular chaos, it appears that our theory for loaded spheres shares the ``rigor'' of the Chapman—Enskog theory for structureless molecular species. Detailed numerical calculations of transport coefficients will be communicated in a subsequent paper of this series.

Molecular Orbitals for H_{2} Based on Central Potentials. Average Hamiltonian and Correlation Energy
View Description Hide DescriptionThe exact Hamiltonian for H_{2} is averaged over trial functions composed of molecular orbitals which are defined by a model suggested by the authors [J. Chem. Phys. 33, 1803 (1960)]. The three trial functions studied are (a) the single configuration type, (b) a two‐configuration type including the σ_{ u } orbital defined by the model, and (c) a function composed of a product of the lowest σ_{ g } orbitals multiplied by a correlation function including one term having σ_{ u } symmetry. The results from the single configuration function indicate that the lowest σ_{ g } molecular orbital is a good approximation to an SCF orbital. The configuration interaction (CI) calculation indicates that a CI expansion based on these orbitals will converge very slowly to an eigenfunction of H_{2}. Adding a one‐term correlation function gave significant improvement, and the construction of a more complicated correlation function, by means of the variation principle, would undoubtedly lead to a good approximation to the exact eigenfunction. Rather than proceed along these lines, analysis of the various average values contributing to the total energy and the corresponding values predicted by the model suggests a different approach. It is found that, at the equilibrium internuclear distance, potential energy terms computed from the model agree with the exact values much better than average values based on the three trial functions, while the kinetic energy is much more poorly represented. This indicates that calculation of the total energy from the potential energy by the virial theorem would give better results than using the less exact kinetic energy term to construct the total energy and, at the same time, provide a means of calculating the total energy without knowledge of the correlation function. A constraint on the average of the exact Hamiltonian is introduced to provide a rigorous foundation for the above observation, and an appropriately modified virial equation is derived. An approximate solution of this equation is investigated. It is found that a complete solution is not possible without knowledge of the exact energy at one point.

Correlation Diagram for H_{2}. Oscillator Strengths
View Description Hide DescriptionThe eigenvalues of a one‐electron Hamiltonian, based on a model of the hydrogen molecule proposed earlier by the authors [J. Chem. Phys. 33, 1803 (1960)], have been calculated for several states as a function of internuclear distance. These results are shown as a correlation diagram which is complete for all states correlating with separated atoms having principal quantum numbers up to three. This correlation diagram provides a semiquantitative justification for the model. A number of physical implications of the diagram are discussed. Several members of the set of one‐electron eigenfunctions of the model Hamiltonian have been used to calculate oscillator strengths for single electron excitations. The values 0.28, 0.556 were found for the oscillator strengths corresponding to the Lyman and sum of the Lyman and Werner bands, where the best available estimates for H_{2} are 0.27, 0.65, respectively. A number of other oscillator strengths, for which there is no available information for comparison, have also been calculated.

One‐Electron Potential Function for H_{2}
View Description Hide DescriptionA one‐electron potential function is shown to exist such that the exact energy of the hydrogen molecule can be calculated with only a knowledge of the solution to the one‐electron Schrödinger equation, its eigenvalue and an average, over this solution, of the potential energy of one electron in the mean field of the remaining electron. Hence, given this potential function in advance, the exact energy of H_{2} can be calculated from the solution of a one‐electron problem. A formula for this potential function is obtained in terms of the exact eigenfunction for H_{2}. The formula can only be applied to accurately calculate the potential function when the hydrogen eigenfunction is known, and hence the formula is useful primarily to demonstrate the existence of the potential. On the basis of the Heitler—London wavefunction, an approximate numerical comparison between this potential function and that suggested by Slater is given. An empirical formula for the potential function is suggested, and an undetermined parameter is obtained using an approximate eigenfunction.

Effect of Penetrating Radiation on the Production of Persistent Internal Polarization in Electret‐Forming Materials
View Description Hide DescriptionThe electret state was produced in Teflon and carnauba wax by simultaneous action of penetrating radiation and an electric field. Polymethylmethacrylate, polyvinylacetate, polystyrene, polyethylene, and nylon did not form electrets under these conditions. The shapes of isothermal decay curves and thermal depolarization ``glow curves'' were used to calculate activation energies for the depolarization process in Teflon. The average activation energy as calculated from the ``initial rise'' of thermodepolarization curves was 0.93 eV. The persistent heterocharge of carnauba wax and of certain polymers may be caused by the orientation of dipolar units composed of a positive molecular ion and a partially solvated electron.

Molecular Reorientation in Liquids. I. Distribution Functions and Friction Constants
View Description Hide DescriptionThe orientational motion of rigid molecules in liquids obeying classical mechanics is considered. The equations of motion of these systems are Euler's equations. It is assumed that the torque on a molecule can be represented by a friction term proportional to angular velocity plus a randomly fluctuatingtorque. It is then shown that the equations of motion for a spherical top have solutions of the form of the solutions of the Langevin equation. The components of the diagonalized rotational friction constant tensor are given as time integrals of the autocorrelation functions of the components of the angular velocity and an approximate expression is obtained which gives the rotational friction constants in terms of ensemble averages of the angular derivatives of the intermolecular potential function. A differential equation is derived for the time‐dependent distribution function for the Eulerian angle displacements. The solutions of this equation can be written in terms of the solutions of the Schrödinger equation for the asymmetric top multiplied by exponentially decaying functions of time. Molecules possessing elements of symmetry in their interaction potential functions are considered, and it is shown that the components of the rotational frictiontensor approach zero in these systems and that the rotational motion over relatively long time intervals is then determined by the inertial terms in the equation of motion. The application of these results to the theory of rotational nuclear magnetic and dielectric relaxation in liquids is briefly discussed.

Molecular Reorientation in Liquids. II. Angular Autocorrelation Functions
View Description Hide DescriptionThe normalized angular autocorrelation functions which appear in the theories of rotational nuclear magnetic and dielectric relaxation are considered in the light of the results presented in Paper I of this series. The time‐dependent spherical harmonics in the equations for the autocorrelation functions are written as functions of the molecular Eulerian angular displacements. The time dependence of the autocorrelation functions is calculated using the distribution functions obtained in I, and it is shown that the results are strongly dependent upon the magnitude of the components of the diagonalized rotational frictiontensor. If all components are large and unequal, five relaxation times appear in the autocorrelation function for nuclear relaxation and three times appear in the function for dielectric relaxation. If one or more of the components are small, the autocorrelation functions are found to be dependent upon the inertial terms in the molecular equations of motion. Autocorrelation functions are evaluated in the case that all the components of the rotational frictiontensor of a spherical top are zero, and an approximate equation for the autocorrelation function is suggested for systems in which dynamical coherence is the predominant factor in the reorientational motion. These equations are Gaussian functions of time, with time constants which depend only upon the inertial properties of the molecule. In principle, the results of this paper, together with the results of I, allow one to compute rotational relaxation behavior in liquids from molecular interaction potential functions.

Nuclear Spin Relaxation in Liquids. Spheroidal Molecules
View Description Hide DescriptionNuclear spin—lattice relaxation times have been measured as a function of temperature for a number of liquid hydrocarbons. These data, together with other available measurements on rigid, spheroidal molecules, are compared with the Bloembergen, Purcell, and Pound (B.P.P.) theory of spin—lattice relaxation in liquids, and it is shown that the B.P.P. calculation of the rotational contribution to the relaxation time gives a value which are much shorter than the total experimental relaxation times. It is then assumed that the time dependence of the rotational angular autocorrelation functions of these molecules is dominated by dynamical coherence, rather than by frictional forces as assumed in the B.P.P. theory. If one calculates the net spin—lattice relaxation timeT _{1} by summing the translational contribution calculated from the B.P.P. theory and the rotational contribution calculated from an approximate autocorrelation function valid for small but nonzero friction constants, one obtains values for T _{1} which are in quantitative agreement with the data for nonpolar, spheroidal molecules. Furthermore, when the two calculations are compared with the data for polar spheroidal molecules, it is seen that frictional forces are the predominant factor in the rotational motion of these systems, but the values obtained for the rotational friction constant from the Stokes—Einstein equation are in error by a considerable amount.

Infrared Spectrum of Chlorodifluoramine, ClNF_{2}
View Description Hide DescriptionThe infrared spectrum of chlorodifluoramine, ClNF_{2}, has been examined, and all but one fundamental frequency assigned. ClNF_{2} is readily identified by three intense absorptions at 930, 854, and 697 cm^{—1} corresponding to symmetric and asymmetric N–F stretching, and N–Cl stretching, respectively.

Crystal Structures of KCuCl_{3} and NH_{4}CuCl_{3}
View Description Hide DescriptionThe crystal structures of the isomorphous garnet‐red copper salts, KCuCl_{3} and NH_{4}CuCl_{3}, have been determined. Each compound contains discrete, planar, Cu_{2}Cl_{6} ^{=} dimers. These dimers are stacked above each other along the crystallographic a axis. The cation exhibits a ninefold coordination.
KCuCl_{3} is antiferromagnetic below 30°K. Both KCuCl_{3} and NH_{4}CuCl_{3} are pleochroic with maximum visible absorption when the electric vector is parallel with the Cu–Cu vector of the dimer.

Theory of the Dielectric Constant of a Nonpolar Fluid
View Description Hide DescriptionTaking the dipole interactions of molecules into consideration, a statistical theory of the dielectric constant of a nonpolar fluid is presented. It is assumed that the molecules have a constant polarizability and no higher‐order multipole moments. The deviation of the Clausius—Mossotti formula from experiments is expressed in a density series. It is found that coefficient B in the generalized Clausius—Mossotti Eq. (5.1) has a relation to coefficient A. B is determined explicitly as a function of temperature for isotropic spherical molecules with hard‐sphere potentials, square‐well potentials, and Lennard‐Jones potentials. The results are compared with the data of Johnston, Oudemans, and Cole on inert gases.

New Method for Determining Semiempirical One‐Center Coulomb Repulsion Integrals
View Description Hide DescriptionUtilizing the ``core—peel'' approximation for atoms and assuming that correlation errors in orbital calculations of potential energy are independent of orbital exponents, a model is devised for assessing the ``true'' repulsion energy associated with a two‐electron wavefunction (geminal). For two‐electron atoms, results corresponding to the integral (1s1s:1s1s), but with correlation, conform to experiment within 0.01 eV. Values are also obtained for the integrals corresponding to (2p2p:2p2p) in boron to fluorine in various valence states.