Showing papers in "Journal of Chemical Physics in 1969"

Equation of State for Nonattracting Rigid Spheres

Abstract: A new equation of state for rigid spheres has been developed from an analysis of the reduced virial series. Comparisons with existing equations show that the new formula possesses superior ability to describe rigid‐sphere behavior.

Self‐Consistent Molecular‐Orbital Methods. I. Use of Gaussian Expansions of Slater‐Type Atomic Orbitals

Abstract: Least‐squares representations of Slater‐type atomic orbitals as a sum of Gaussian‐type orbitals are presented. These have the special feature that common Gaussian exponents are shared between Slater‐type 2s and 2p functions. Use of these atomic orbitals in self‐consistent molecular‐orbital calculations is shown to lead to values of atomization energies, atomic populations, and electric dipole moments which converge rapidly (with increasing size of Gaussian expansion) to the values appropriate for pure Slater‐type orbitals. The ζ exponents (or scale factors) for the atomic orbitals which are optimized for a number of molecules are also shown to be nearly independent of the number of Gaussian functions. A standard set of ζ values for use in molecular calculations is suggested on the basis of this study and is shown to be adequate for the calculation of total and atomization energies, but less appropriate for studies of charge distribution.

Limiting Laws and Counterion Condensation in Polyelectrolyte Solutions I. Colligative Properties

Abstract: Formulas are derived for the osmotic coefficient, the Donnan salt‐exclusion factor, and the mobile‐ion activity coefficients in a polyelectrolyte solution with or without added sample salt. The formulas, which contain no adjustable parameters, are based on the (theoretical) observation by several workers that counterions will “condense” on the polyion until the charge density on the polyion is reduced below a certain critical value. The uncondensed mobile ions are treated in the Debye–Huckel approximation. In a restricted sense, the formulas are “limiting laws,” and this aspect is discussed at length. Detailed comparison with experimental data in the literature is given; agreement of the theory with experiment is usually found to be quantitative.

Oxygen isotope fractionation in divalent metal carbonates

Abstract: Equilibrium fractionation factors for the distribution of 18O between alkaline‐earth carbonates and water have been measured over the temperature range 0–500°C. The fractionation factors α can be represented by the equationsCaCO3–H2O, 1000 lnα = 2.78(106 T−2)− 3.39,SrCO3–H2O, 1000 lnα = 2.69(106 T−2)− 3.74,BaCO3–H2O, 1000 lnα = 2.57(106 T−2)− 4.73.Measurements on MnCO3, CdCO3, and PbCO3 were made at isolated temperatures. A statistical‐mechanical calculation of the isotopic partition function ratios gives reasonably good agreement with experiment. Both cationic size and mass are important in isotopic fractionation, the former predominantly in its effect on the internal vibrations of the anion, the latter in its effect on the lattice vibrations.

Viscous Liquids and the Glass Transition: A Potential Energy Barrier Picture

Abstract: Recent attempts have been made to assess the relative merits of the free volume and entropy theories of viscous flow in glass‐forming liquids by accurate measurement of viscosity over wide temperature ranges, and subsequent comparison with the equations derived from these theories. In the author's view, this effort is misguided. The theories are crude and qualitative, and such tests are too stringent. It is better to make qualitative or semiquantitative comparison of a wide variety of physical phenomena; judged by this criterion, the entropy theory appears more successful. It is conjectured that further progress can be made by accepting the crude, naive character of any model we are likely to find tractable for the foreseeable future, and recognizing both the values and limitations of such models. A picture of the flow process in viscous liquids is proposed, in an attempt to answer certain questions about the molecular steps in flow either answered unsuccessfully or ignored by present theories, in the hop...

Variational Treatment of Hydrodynamic Interaction in Polymers

Abstract: It has recently been pointed out by several authors that the Kirkwood–Riseman diffusion tensor normally used to treat hydrodynamic interactions in polymer molecules is not necessarily positive definite, an unphysical behavior which results from the neglect of short‐range contributions to the interaction between chain segments. The problem can be reformulated as a minimization of the rate at which energy is dissipated by the motion of the suspending fluid, and in this way we have obtained an approximate diffusion tensor which (1) is positive definite for all configurations of the polymer molecule, (2) approaches the Kirkwood–Riseman diffusion tensor at large separations between the interacting segments, and (3) can be written as the true diffusion tensor plus a positive definite correction. The last feature can be used to obtain variational bounds on relaxation times and sedimentation rates.

Limiting Laws and Counterion Condensation in Polyelectrolyte Solutions II. Self‐Diffusion of the Small Ions

Abstract: The concepts developed in Part I of this series are applied to the self‐diffusion of counterions and co‐ions. The condensed counterions are assumed to have no mobility, while the uncondensed counterions and all co‐ions execute Brownian motion subject to the locally inhomogeneous electric field created by the polyelectrolyte molecules. The polyelectrolyte field is assumed to have small amplitude, in analogy to the Debye–Huckel treatment of uncondensed mobile ions in Part I. The self‐diffusion coefficients of the counterion and the co‐ion are both less than the coresponding values in the absence of polyelectrolyte, but the coefficient for the counterion is much less than that for the co‐ion. The results for the self‐diffusion coefficient of the counterion in a salt‐free polyelectrolyte solution are in good agreement with experimental data.

New Method for Constructing Wavefunctions for Bound States and Scattering

Abstract: A new method is developed for integrating coupled differential equations arising in bound state and scattering problems in quantum mechanics. The wavefunctions are easily constructed in piecewise analytic form, to any prescribed accuracy.

Erratum: Conduction‐Induced Alignment of Nematic Liquid Crystals: Basic Model and Stability Considerations

Abstract: It is shown by way of model orientation patterns how conduction in nematic liquid crystals may cause a torque per unit volume that more than offsets the dielectric torque felt in the insulating state. This conduction‐induced torque arises from an anisotropy of the conductivity or of the dielectric constant. It is partly dielectric but its more important part is shear‐induced, the shear flow being a consequence of space charge which is generated by and interacts with the applied electric field. The stability of a uniform orientation pattern in electric fields parallel or perpendicular to the preferred axis is investigated, with respect to a complete set of sinusoidal perturbations and including distortional and (stabilizing) magnetic torques. The analysis shows that the shear‐induced torque cannot be treated like its dielectric counterpart. It depends in a peculiar way on the direction of the perturbational wave vector and may contain a nonconservative contribution. Formulas for the instability threshold, ...

Energy Transfer in Near‐Resonant Molecular Collisions due to Long‐Range Forces with Application to Transfer of Vibrational Energy from ν3 Mode of CO2 to N2

Abstract: The cross section for the near‐resonant transfer of vibrational energy from CO2(001) to N2(0), CO2(001) + N2(0)→CO2(000) + N2(1) + ΔE, is calculated for the isotopes 14N2 (ΔE = 18 cm−1) and 15N2 (ΔE = 97 cm−1). The impact parameter (semi‐classical) approximation is used, and it is assumed that the vibrational‐energy transfer is caused by the interaction of the instantaneous CO2 dipole moment with the N2 quadrupole moment. When proper account is taken of the rotational motions of the molecules it is found that in collisions of CO2 with 14N2 only the low rotational levels of the CO2 and N2 molecules contribute to Reaction (1). In collisions of CO2 with 15N2, only those rotational levels contribute which undergo transitions cancelling most of the relatively large (97 cm−1) vibrational‐resonance defect. For 14N2 below about 1000°K, where the cross section displays a negative temperature dependence, the results are in excellent qualitative and quantitative agreement with available experimental data, with no ad...

Semiempirical, Pseudopotential Calculation of Alkali–Noble-Gas Interatomic Potentials

Abstract: A relatively simple semiempirical method is suggested for calculating interatomic potentials of ground and low‐lying excited states of diatomic systems composed of an alkali atom and a ground‐state noble‐gas atom, and its applicability to other systems is discussed. Two types of interaction are included: an electrostatic interaction which yields the asymptotic R−6 behavior characteristic of van der Waals potentials, and a pseudointeraction which represents the effect of the Pauli exclusion principle on overlapping electron states. For the electrostatic interaction, the noble‐gas atom is treated as a polarizable dipole with two parameters, namely a polarizability α and a “radius” r0. The value of r0 is chosen by adjusting the depth of the ground‐state well to agree with recent scattering data. To determine the interatomic potentials, the total Hamiltonian including spin–orbit coupling is diagonalized in a limited basis set of atomic orbitals. Results are presented and compared with previous calculations an...

Band Shapes of the Electronic Spectra of Complex Molecules

Abstract: Structureless bands of ultraviolet absorption spectra of hydroxypyridine derivatives in solution have been examined. It is shown that the log–normal distribution provides a convenient four‐parameter empirical description of the bands and gives a much better fit than the two half‐width Gaussian. The log–normal yields a resolution of overlapping bands by a method of least squares that is very satisfactory. A simple configuration‐coordinate model is presented that lends some physical meaning to the parameters of the log–normal. It yields normalized third and fourth moments that are near those of the log–normal.

Compton scattering factors for aspherical free atoms.

Abstract: Incoherent scattering factors for all aspherical free atoms have been computed from numerical SCF Hartree–Fock wavefunctions. The complete Waller–Hartree theory with all exchange terms has been used.

Location of Energy Barriers. I. Effect on the Dynamics of Reactions A + BC

Abstract: The dynamics of exchange reactions A+BC→AB+C have been examined on two types of potential‐energy hypersurfaces that differed in the location of the energy barrier along the reaction coordinate. On “surface I” the barrier was in the entry valley of the energy surface, along the approach coordinate. On “surface II” the barrier was in the exit valley of the energy surface, along the retreat coordinate. The classical barrier height was Ec = 7.0 kcal mole−1 on both surfaces, and was displaced from the corner of the energy surface by the same amount; on surface I, r1‡ = 1.20 A, r2‡ = 0.80 A; on surface II, r1‡ = 0.80 A, r2‡ = 1.20 A (r1 ≡ rAB, r2 ≡ rBC, and the superscript ‡ refers to the location of the crest of the barrier). Three‐dimensional (3D) classical trajectory calculations were performed for the mass combination mA = mB = mC at several reagent energies. The reagent energy took the form of translation, vibration or an equilibrium distribution of the two. The main findings were that translation was markedly more effective than vibration in promoting reaction on surface I, and vibration markedly more effective than translation in promoting reaction on surface II. The total reactive cross section with the entire reagent energy vested in translation was symbolized ST, with the reagent energy (but for 1.5 kcal) in vibration, SV, and with an equilibrium distribution over reagent translation and vibration, Seq. On surface I ST ≫ SV: on surface II SV ≫ ST. Close to the threshold for ST on surface I, ST / Seq ∼ 10; close to the threshold for SV, on surface II, SV / Seq ∼ 10. At high reagent energies (2 × threshold) on surface I ST / Seq fell to 2, whereas on surface II SV / Seq increased to extremely large values. Product energy and angular distributions were recorded for two reagent energies. On surface I with low translational energy in the reagents a major part of the available energy appeared as vibration in the molecular product. At higher collision energy this fraction decreased. On surface II with low vibrational energy in the reagents only a small part of the available energy appeared as vibration in the product. At higher vibrational energy this fraction increased. The product angular distribution at low reagent translational energy on surfaces I and II corresponded to backward‐peaked scattering of the molecular product. At increased reagent energy on both surfaces the distribution shifted forward (this is a novel phenomenon in the case of increased reagent vibration; surface II).

Magnetoviscosity of Magnetic Colloids

Abstract: We have observed that the viscosity of a dilute (≈ 1015 particles cm−3) colloidal suspension of single domain ferromagnetic particles is a marked function of both the magnitude and direction of an externally applied magnetic field. For particles with dimensions of a few hundred angstroms, the viscosity saturates in a field of order 1 kG. The additional viscosity is greater by a factor of 2 for the field parallel to the flow than with it perpendicular. This magnetoviscous effect is qualitatively accounted for by the hindrance to particle rotation caused by the magnetic torque.

Absorptivity of Ice I in the Range 4000–30 cm−1

Abstract: The absorbance of several samples of ice Ih has been measured in the range 4000–30 cm−1, and scaled to that of a particular film of unknown thickness. The thickness of the film has been calculated by two methods, first from the known absorptivity at 4940 cm−1, and second by equating the appropriate Kramers–Kronig integral to the known infrared contribution to the microwave refractive index. The two thicknesses agreed well and allowed the absorptivity to be obtained in the range 4000–30 cm−1. The complex refractive index and permittivity and the normal incidence reflectivity have been calculated from the absorptivity. About three‐quarters of the infrared contribution to the microwave refractive index is caused by the translational lattice vibrations and about 15% by the rotational vibrations; the O–H stretching bands which absorb very strongly contribute relatively little. The maximum of the density of states in the transverse acoustic branch is at 65 cm−1 rather than below 50 cm−1 as reported earlier. Bel...

Structure and Free Energy of the Interface between Fluid Phases in Equilibrium near the Critical Point

Abstract: The theory of van der Waals and of Cahn and Hilliard, which yields the density or concentration profile through an interface and the associated interfacial tension, is generalized by replacing the equation of state assumed originally by one that reproduces more accurately the known thermodynamic singularities at the critical point. Though the correct equation of state is not known, its presumed homogeneity of form is alone sufficient to allow most features of the interface to be determined explicitly. In particular, the maximum density gradient in the interface, the asymptotic behavior of the interface profile at large distances from the position of its maximum gradient, and the surface tension, are each obtained up to a dimensionless, and presumably universal, proportionality constant that reflects only the functional form of the equation of state. Numerical evaluation of the surface tension requires a knowledge of the limits approached by the correlation length and compressibility in the homogeneous flu...

Structural Influences on Adsorbate Binding Energy. I. Carbon Monoxide on (100) Palladium

Abstract: The structural arrangement and binding energy of CO adsorbed on a (100) surface of palladium have been determined and related to each other over the submonolayer coverage range. There are four distinct equilibrium structural arrangements of CO: (1) At low coverages (< 0.4 monolayers) a random lattice gas configuration is observed in which each CO molecule is localized on a specific adsorption site. (2) Just below 0.5 monolayers a liquidlike short‐range order arrangement is found. (3) At exactly 0.5 monolayers a c(4 × 2) − 45° overlayer structure forms which is in complete registry with the substrate. (4) For coverages greater than 0.5 monolayers, the c(4 × 2) − 45° structure is uniaxially compressed to form an overlayer which is out of registry with the palladium substrate. The observed linear decrease in binding energy with coverage up to 0.5 monolayers suggests that the only important CO–CO interaction occurs between nearest neighbors. Near 0.5 monolayers an almost discontinuous decrease in the binding energy of 0.35 eV is attributed to loss of registry between the overlayer and substrate. As the coverage increased above 0.6 monolayers the binding energy again decreased sharply as a consequence of neighboring CO orbital overlap.

Configuration Interaction Studies of Ground and Excited States of Polyatomic Molecules. I. The CI Formulation and Studies of Formaldehyde

Abstract: A configuration interaction (CI) procedure which is designed to produce energies and wavefunctions from relatively large scale computations on ground and excited molecular states is presented based on an orthonormal set of molecular orbitals. The method of generating configurations is referenced to the particular state of interest and involves a two‐pass procedure: First, the generation of configurations by single and double excitations from selected important configurations (parents), subject to a threshold criterion, followed by diagonalization of the energy to obtain an initial approximation to the wavefunctions of interest; second, the use of these wavefunctions as parents and a repeat of the generation and diagonalization steps to produce the final CI wavefunction. CI studies of the 1,3A2 (n→π*), 1,3A1 (π→π*, n→3p), 1B1 (σ→π*, π→3s), and 1B2 (n→3s) states of formaldehyde are reported for several large Gaussian basis sets of near atomic Hartree–Fock quality based on molecular orbitals determined from ...

De‐excitation of Electronically Excited Sodium by Nitrogen

Abstract: A semiquantitative calculation is made of the cross section for the quenching of Na(32P) by molecular nitrogen, as a function of initial kinetic energy and of final vibrational quantum number, υf, of the nitrogen molecule. The large observed cross section, which is of gas‐kinetic order, can be explained in terms of an intermediate ionic state, involving Na+ and N2− (υ = υ−). This state is unstable at infinite separation of Na and N2, but because of the Coulomb attraction it becomes stable at collision distances below about 3 A. As a result of the vibrational structure of both the intermediate and final states, we treat the reaction in terms of a diffusion of the probability flux through a two‐dimensional network of potential‐energy curves parametrized by both the electronic state and also the vibrational quantum numbers υ− and υf. At each potential‐energy curve crossing we compute the transition matrix element for insertion into a Landau–Zener type of transition probability. The transition matrix element ...

Photoelectron Spectra of Molecules. I. Ionization Potentials of Some Organic Molecules and Their Interpretation

Abstract: The photoelectron spectra of 67 organic compounds, including paraffins, cycloparaffins, olefines, dienes, aromatic hydrocarbons, alcohols, ethers, aldehydes, ketones, and diene iron tricarbonyl complexes, have been measured, using a simple retarding‐potential grid‐type spectrometer, and the results interpreted in terms of a recently developed semiempirical SCF MO treatment (MINDO approximation). The observed ionization potentials show a remarkably close relationship to the calculated Hartree–Fock MO energies over the whole range of ionization potentials (<20 eV) studied. An estimate is given for the first ionization potential of cyclobutadiene.

Hydroxyl Radical Kinetics by Kinetic Spectroscopy. VI. Reactions with Alkanes in the Range 300–500°K

Abstract: Flash photolysis and kinetic spectroscopy were used to study the reactivity of OH radicals toward a representative set of alkanes. Previous studies have shown that OH radicals abstract H atoms from alkanes to produce H2O and an alkyl radical. The following formula was found to reproduce accurately all of the observed abstraction rate constants for the 10 model alkanes: ktot = 6.15 × 1011 N1exp(− 1635 / RT) + 14.1 × 1011 N2exp(− 850 / RT) + 12.6 × 1011 N3exp(+ 190 / RT) cc mol−1·sec−1, where N1, N2, and N3 are the respective numbers of primary, secondary, and tertiary H atoms in the alkane, and ktot is the total (in these experiments the observed) rate constant for the abstraction of H atoms from the alkane. Thus, there is a generally applicable frequency factor and activation energy for each of the three types of H atoms found in alkanes. Methane and ethane were somewhat expected exceptions to this rule. Their rate constants are 3.31 × 1012exp(− 3772 / RT) and 1.12 × 1013exp(− 2447 / RT) cc mol−1·sec−1, r...

Reflection Method for Obtaining the Infrared Spectrum of a Thin Layer on a Metal Surface

Abstract: Previous calculations have shown that a thin, moderately absorbing layer on a metal surface has an absorption factor in the infrared which varies greatly with the angle of incidence and the state of polarization of the incident radiation. The requirements on a spectroscopic system, which uses the insights of these calculations, are discussed under the realistic conditions of multiple reflections with a low f/number beam. A method is described for obtaining many reflections between closely spaced mirrors. Radiation emerges from a narrow aperture between the mirrors, the aperture serving as an effective source for the infrared monochromator. A system has been constructed and its use is illustrated by the spectrum of a cellulose acetate layer on a silver mirror.

Decay of Correlations in Linear Systems

Abstract: The conditions under which the decay of the pair correlation function at large distances is monotonic or oscillatory are investigated for one‐dimensional systems and discussed in detail for certain linear continuum and lattice models in which the molecules interact only with their nearest neighbors. In each case a locus is found in the pressure–temperature plane and in the density–temperature plane, such that in thermodynamic states of the one‐dimensional fluid that lie on one side of the locus, the decay of the correlation function is oscillatory, and in those that lie on the other side it is monotonic. At every temperature the decay is monotonic below a uniquely determined transition pressure or density. It is argued that such loci will likewise be found in real three‐dimensional systems, and that the critical point and a range of fluid states around the critical point, as well as states of the low‐pressure vapor, will lie in the region in which the correlation function at large distances is asymptotically positive and decays monotonically.

Perturbed Hartree–Fock Theory. I. Diagrammatic Double‐Perturbation Analysis

Abstract: The coupled Hartree–Fock scheme, for calculating atomic and molecular properties which are second order in some perturbing field, is studied with the help of a double‐perturbation expansion and diagrammatic techniques. The double‐perturbation expansion, with the external field as one perturbation (H(1)) and the difference between the true electron‐repulsion potential and the Hartree–Fock potential as the other (V), is compared with an iterative solution of the set of coupled Hartree–Fock equations for the first‐order perturbed orbitals. The term in the double‐perturbation expansion that is second order in H(1) and zeroth order in V is shown to be the Dalgarno “uncoupled” second‐order energy. The coupled Hartree–Fock second‐order (in H(1)) energy expression includes all terms first order in V and some of the terms to all orders in V, up to infinite order. It is shown how to separate the diagrams contributing to the CPHF energy from those that do not; in particular, the CPHF equations are composed of the so‐called “bubble diagrams” and their corresponding exchange diagrams. It is shown that each particle–hole pair, or “bubble,” is independent of all the other particle–hole pairs, in the sense that the total energy contribution from the diagram can be factored into contributions from the individual particle–hole pairs. It is shown that the first‐order perturbed wave‐function obtained from the coupled Hartree–Fock formalism includes effects due to all singly‐excited determinants and a certain class of linear combinations of doubly‐excited determinants, namely those singlet spin eigenfunctions which correspond to two electrons being excited without spin flips from Hartree–Fock unexcited orbitals to virtual orbitals; there exists a second, linearly independent, doubly excited singlet function which is not included in the CPHF wavefunction and does not contribute to the perturbation energy. The contribution to the second‐order energy from the doubly excited determinants amounts to about 10% of the total CPHF second‐order energy in the case of some electric and magnetic properties of H2 and 25% of the total second‐order energy corresponding to the electric dipole polarizability of the Be atom. The larger contribution of the terms involving doubly excited determinants for Be has its origin in the fact that a double excitation from the 2s to the 2p orbital gives a large fraction of the correlation energy between the 2s electrons and that the single excitation from the 2s to the 2p orbital is the largest single contribution to the polarizability. An alternate iterative scheme for the solution of the coupled Hartree–Fock equations is derived. This alternate iterative scheme, which includes certain terms summed to infinite order, is expected to be a considerable improvement over the simple scheme which starts with the “uncoupled” approximation as the zeroth iteration. Comparisons of the different schemes for the properties considered in this paper are made to confirm this conclusion. The CPHF energy (second order in H(1)) is partitioned approximately into a part due to correcting the unperturbed Hartree–Fock wavefunction for correlation and a part due to correcting the uncoupled Hartree–Fock perturbed wavefunction for correlation; the contribution from the former was shown to be about 10%–20% of that from the latter.

Transfer of Electronic Excitation Energy in Polyvinyl Carbazole

Abstract: The energy transfer in amorphous polyvinyl carbazole (PVCA) has been studied by fluorescence quenching experiments using perylene, trinitrofluorenone, and hexachloro‐p‐xylene as guest molecules. The PVCA shows excimer fluorescence. The results are discussed in terms of exciton diffusion and competition between excimer‐forming sites and guest molecules for the trapping of excitons. Single‐step transfer from the excimer state by dipole resonance is considered as an alternative mechanism of energy transfer; in all three cases, however, exciton migration is the more probable energy‐transfer mechanism. The concentration of excimer‐forming sites is estimated to be 10−3 mole/mole basic unit of the polymer. In the pure polymer the exciton covers about a thousand basic units during its lifetime.

Limiting Laws and Counterion Condensation in Polyelectrolyte Solutions. III. An Analysis Based on the Mayer Ionic Solution Theory

Abstract: The interaction with point ions of a line charge supplemented by a distance of closest approach is studied by means of the Mayer theory of ionic solutions. A number of points involved in a recent theory of polyelectrolyte solutions (Papers I and II of this series) are thereby clarified. It is demonstrated that the counterion condensation phenomenon occurs in the limit of zero concentration, that the Debye–Huckel approximation is applicable to the coions and uncondensed counterions, and that the relevant dielectric constant is that of the pure bulk solvent.

Electron Spin Resonance of High‐Spin d5 Systems

Abstract: The electron spin resonance spectra of polycrystalline samples of some sideramines (coprogen, ferrichrysin, ferricrocin, and ferrirubin), transferrin, and conalbumin are reported both at 9.3 and 36 kMc/sec and at various temperatures. These results have been interpreted in terms of a spin Hamiltonian of the form H = gβH·S + D[Sz2 − 13S(S + 1)] + E(Sx2 − Sy2). The application of this interpretation to other high‐spin d5 systems both in single‐crystal and polycrystalline form is discussed.

Ferroelectric Tungsten Bronze‐Type Crystal Structures. II. Barium Sodium Niobate Ba(4 + x)Na(2 − 2x)Nb10O30

Abstract: Ferroelectric Ba(4+x)Na(2 − 2x)Nb10O30, with a Curie temperature of 833°K, has a tungsten bronze‐type structure and crystallizes in the orthorhombic system, with subcell lattice constants a = 17.59182 ± 0.00001, b = 17.62560 ± 0.00005, and c = 3.994915 ± 0.000004 A at 298°K. The space group is Cmm2, and there are two formulas in the subcell. The c axis of the true cell is double that of the subcell. The integrated intensities of 6911 reflections within a reciprocal hemisphere of radius (sinθ)/λ = 1.02 A−1 were measured with PEXRAD, 1873 symmetry‐independent structure factors being significantly above background. The metal atom positions were determined from the three‐dimensional Patterson function and the oxygen atoms from metal‐phased Fourier series. The final agreement index between measured and calculated structure factors is 0.0579. The structure differs only in detail from previously determined tetragonal tungsten bronze structures. In the general formula (A1)2(A2)4C4(B1)2(B2)8O30, the B1 and B2 sites are fully occupied by Nb, the A2 sites by Ba and the A1 site by 87.0% Na and 6.5% Ba. Evidence from chemical analysis, x‐ray density calculations and the present determination suggests that the best approximation to the formula of the crystal studied is Ba4.13Na1.74Nb10O30. The Ba and O atoms at z ≃ 12 are disordered in a manner similar to the O atoms in the Ba layer in barium strontium niobate. The four crystallographically independent Nb atoms, each in octahedral coordination, are linked to O atoms by distances ranging from 1.765 ± 0.021 to 2.270 ± 0.021 A, with a mean value of 1.989 A. Ba is 10 coordinated, with Ba–O distances ranging upward from 2.671 ± 0.013 A. Na is 12 coordinated, with Na–O distances ranging from 2.660 ± 0.014 to 2.990 ± 0.015 A, with a mean of 2.788 A. The Nb‐atom displacements from the mean oxygen planes lie in the range 0.171–0.205 A; the parabolic relation with Curie temperature predicts a displacement of 0.204 A. The measured value of Ps at room temperature is 40 μC cm−2: the linear relation between displacement and polarization predicts a saturation value of 44–53 μC cm−2. All metal atoms are displaced from the oxygen planes in the sense given by the macroscopic positive polarity.

Ground States of σ‐Bonded Molecules. IV. The MINDO Method and Its Application to Hydrocarbons

Abstract: A simple semiempirical valence shell SCF MO treatment of hydrocarbons is described which enables their heats of formation to be calculated to within a few kilocalories per mole, and which can be applied to a wide variety of saturated, unsaturated, and aromatic hydrocarbons including strained microcyclic systems. Geometries have to be assumed since the method does not lead to correct predictions of bond lengths, and it also fails to predict correctly the relative energies of rotational isomers. Otherwise the method is both simpler and more accurate than the PNDO treatment described in Papers I–III.