Showing papers in "Journal of Chemical Physics in 1963"

An Extended Hückel Theory. I. Hydrocarbons

Abstract: The Huckel theory, with an extended basis set consisting of 2s and 2p carbon and 1s hydrogen orbitals, with inclusion of overlap and all interactions, yields a good qualitative solution of most hydrocarbon conformational problems. Calculations have been performed within the same parametrization for nearly all simple saturated and unsaturated compounds, testing a variety of geometries for each. Barriers to internal rotation, ring conformations, and geometrical isomerism are among the topics treated. Consistent σ and π charge distributions and overlap populations are obtained for aromatics and their relative roles discussed. For alkanes and alkenes charge distributions are also presented. Failures include overemphasis on steric factors, which leads to some incorrect isomerization energies; also the failure to predict strain energies. It is stressed that the geometry of a molecule appears to be its most predictable quality.

Some Topics in the Theory of Fluids

Abstract: It is shown how certain thermodynamic functions, and also the radial distribution function, can be expressed in terms of the potential energy distribution in a fluid. A miscellany of results is derived from this unified point of view. (i) With g(r) the radial distribution function and Φ(r) the pair potential, it is shown that g exp (Φ/kT) may be written as a Fourier integral, or as a power series in r2 the terms of which alternate in sign. (ii) A potential‐energy distribution which is independent of the temperature implies an equation of state which is a generalization of a number of well‐known approximations. (iii) The grand partition function of the one‐dimensional lattice gas is obtained from thermodynamic arguments without evaluating a sum over states. (iv) If in a two‐dimensional honeycomb (three‐coordinates) lattice gas fr(r=0, 1, 2, 3) is the fraction of all the empty sites which at equilibrium are neighbored by exactly r filled sites, then at the critical density the values of all four of the f's ...

Atomic Screening Constants from SCF Functions

Abstract: The self‐consistent‐field function for atoms with 2 to 36 electrons are computed with a minimal basis set of Slater‐type orbitals. The orbital exponent of the atomic orbitals are optimized as to ensure the energy minimum. The analysis of the optimized orbital exponents allows us to obtain simple and accurate rules for the 1s, 2s, 3s, 4s, 2p, 3p, 4p, and 3d electronic screening constants. These rules are compared with those proposed by Slater and reveal the need of accounting for the screening due to the outside electrons. The analysis of the screening constants (and orbital exponents) is extended to the excited states of the groundstate configuration and to the positive ions.

Systematics of Vibrational Relaxation

Abstract: A large number of data points for the vibrational relaxation time (pτv in atm sec) of simple systems have been logarithmically plotted vs (T°K)—⅓. It appears that each system is well represented by a straight line, and that most of these straight lines when extended to higher temperatures intersect near the point [pτv=10—8 atm sec, (T°K)—⅓=0.03]. Systems with a small reduced mass μ are exceptions to such a simple convergence, and in an improved scheme, the location of the convergence point is dependent on the reduced mass. Such a presentation has lead to an empirical equation correlating available measurements of vibrational relaxation times: log10(pτv)=(5.0×10−4)μ12θ43[T−13−0.015μ14]−8.00, where θ is the characteristic temperature of the oscillator in K deg. This equation reproduces the measured times within 50% for systems as diverse as N2, I2, and O2–H2. In the worst case thus far, O2–Ar near 1000°K, it is off by a factor of 5.

Study of Moderately Rapid Chemical Exchange Reactions by Means of Nuclear Magnetic Double Resonance

Abstract: A nuclear magnetic double‐resonance method for the determination of chemical exchange rates has been developed. The method is applicable to systems in which a nuclear spin is reversibly transferred between two nonequivalent sites, A and B. The lifetime (τA) and spin—lattice relaxation time (T1A) in Site A are obtained through the study of the decay to a new equilibrium value of Signal A upon the sudden saturation of Signal B. The converse experiment permits the determination of τB and T1B. A number of data for cross checks are furthermore obtained through the study of the recovery of the signals upon the release of various combinations of saturating rf fields.A simple theory based on the Bloch equations as modified by McConnell to incorporate the effects of chemical exchange is given. Experimental results on the hydroxyl proton exchange in the system salicylaldehyde and 2‐hydroxyacetophenone are well described by this simple theory.The present method, which can readily be extended to systems with several ...

Electron spin resonance studies of transient alkyl radicals

Abstract: Electron spin resonance methods have been used to observe alkyl radicals in liquid hydrocarbon systems during irradiation with 2.8‐MeV electrons. These investigations provide detailed structural, radiation chemical, and kinetic information about a large number of radicals.In general, in these studies the ESR lines are found to be narrow; considerable fine structure is observable, permitting positive assignment of the radical species. Accurate hyperfine constants are reported for 21 alkyl and cycloalkyl radicals (including several deuterated species), vinyl, 1‐methylvinyl, 3‐butenyl, allyl, and cyclohexadienyl radicals, and hydrogen and deuterium atoms. Except for cyclopropyl radical, all the alkyl and cycloalkyl radicals have α coupling constants in the range 21–23 G. The β coupling constants in cases where they have been rotationally averaged isotropically are found to decrease with increasing substitution of alkyl groups on the α carbon atom. In general the values for primary, secondary, and tertiary ra...

Spin—Orbit Coupling and the Radiationless Processes in Nitrogen Heterocyclics

Abstract: The matrix elements of the spin—orbit operator between the zero‐order (spin‐free) (n, π*) states of nitrogen heterocyclics are examined. It is found that generally, to the first order, there is no spin—orbit coupling between singlet and triplet states of the same configuration. The coupling between states of different configurations due to differences in the occupancy of the π* orbitals, but having the same nonbonding orbital, is found to have matrix elements similar to those resulting from the coupling between (π, π*) states, which McClure has shown to be unimportant. For the diazines (or polyazines), the coupling between different configurations due to a different occupancy of the nonbonding orbitals is found to have matrix elements which cancel one another. The general conclusion is thus reached that spin—orbit (s.o.) coupling between singlet and triplet (n, π*) states is unimportant to the first order. Using the observed polarization and lifetime data, it is estimated that the ratio |〈S(n,π*)|Hs.o.|T(...

Equation of State for Hard Spheres

Abstract: Simple and exact expressions have been found for the compressibility and pressure equations of state predicted by the Percus—Yevick equation for hard spheres. The equations of state are in good agreement with the machine calculations of Wainwright and Alder, and Wood, Parker, and Jacobson.

Electronic Excitation Transfer and Relaxation

Abstract: One purpose of this paper is to point out that so‐called intermolecular resonance transfer, intermolecular nonresonance transfer, and intramolecular electronic relaxation in solid media are simply all special cases of radiationless transitions between nonstationary states of the entire system of molecules plus environment. Intermolecular resonance transfer is also a special case of the pure crystal exciton problem. The theoretical results from a previous paper are applied to radiationless transitions in π‐electron systems, and the importance of the Franck—Condon factors is emphasized. One role which the vibrational factors play in these processes is illustrated by the large isotope effects which can arise when the radiationless transition converts a large amount of electronic energy into vibrational energy of the system. Quantitative calculations are made of the radiationless transition probability for 1B2u→3E1u(S1→T2) and 1B2u→3B1u(S1→T1) ``intersystem crossing'' in C6H6. It is further shown that the 3B1...

Theory of Linewidths in Electron Spin Resonance Spectra

Abstract: A general theory of the linewidths in the electron spin resonance spectra of dilute solutions of free radicals has been developed in terms of the relaxation‐matrix theory of Bloch, Redfield, and Ayant. In contrast to previous theories, it is shown that a composite line arising from a set of degenerate nuclear‐spin states should, in general, consist of a sum of superimposed lines of Lorentzian shape with different widths rather than a single line with an over‐all Lorentzian shape. A single Lorentzian line is still obtained, however, as a limiting case when the variation of the widths of the different components of a composite line is small compared to the average width. Although the non‐Lorentzian shape of a composite line is often difficult to observe experimentally, a number of other observable properties are predicted by the present development that are outside the scope of the previous theories. For example, linewidth effects resulting from differences in the widths of the separate components of a composite line are predicted that explain the alternation in the linewidths from one hyperfine line to another recently observed in the ESR spectra in certain free radicals. The detailed form of the relaxation matrix is presented for intramolecular anisotropic and isotropic electron—nuclear dipolar interactions, quadrupole interactions, and g‐tensor relaxations. Modulations of the spin density and hyperfine splittings are included, as are internal motions, and a number of cross terms between the different relaxation mechanisms arise. In general the relaxation matrix of a composite line contains significant off‐diagonal elements, and the determination of the linewidths requires the evaluation of the eigenvalues of the matrix. Problems involving rapid chemical exchange, or modulation by jumps to a small number of sites, can be treated by the relaxation‐matrix theory and, under special restrictions, by either the modified Bloch equations or the Anderson theory of motional narrowing. When applicable, these latter procedures can be used over the entire range of exchange rates, while the relaxation‐matrix theory is limited to fast rates only.

Nuclear Magnetic Resonance Study of the Protolysis of Trimethylammonium Ion in Aqueous Solution—Order of the Reaction with Respect to Solvent

Abstract: It was found previously that the proton‐transfer reaction between trimethylammonium ion and trimethylamine in aqueous solution takes place via water. In the present investigation the number of water molecules involved in this reaction was determined. Proton exchange rates were determined from NMR measurements in trimethylamine—trimethylammonium chloride buffer solutions in O17‐enriched water. It is concluded that one water molecule is involved in the transfer reaction.In the appendix, theoretical equations are derived for the dependence of the observed spin‐echo decay rate on the 180° pulse rate for the case of fast exchange.

Theory of Carbon NMR Chemical Shifts in Conjugated Molecules

Abstract: A molecular orbital theory for carbon‐13 chemical shifts in conjugated molecules is formulated. Consideration of the various possible contributions to the shift shows that the local paramagnetic term σpAA is expected to dominate. By a detailed LCAO analysis of σpAA, its variation with the salient features of the electron environment is determined. It is demonstrated that there is a significant local‐charge dependence which agrees in sign and order of magnitude with an experimentally established correlation. In addition, the shielding is shown to be a function of the free valence of the atom under consideration and of the polarity of its sigma bonds. A linearized equation for the carbon‐13 chemical shift (with respect to benzene) is presented and compared with the available experimental data.

Direct Transformation of Hexagonal Boron Nitride to Denser Forms

Abstract: Hexagonal, graphitelike boron nitride may be changed directly into the zincblende cubic form reported earlier or into a new wurtzite form by the application of static high pressures. No catalyst appears to be necessary. At high temperatures, between about 2500° and 4000°K, the zincblende form is favored; at lower temperatures, down to 300°K, the wurtzite form is favored; frequently both forms appear together. The minimum pressure required for the transformations is about 115 kbar at 2000°K; somewhat higher pressures, of the order of 130 kbar, suffice at higher and lower temperatures. The crystallites of the dense phases are small but give good x‐ray diffraction patterns from which the crystal structures can be determined.

Rigorous Basis of the Frenkel-Band Theory of Association Equilibrium

Abstract: By means of the general theory of the classical grand ensemble, the complete formal relations for pressure and number of molecules in an open system, as a function of activity, are derived without approximation within the scheme of the Frenkel‐Band equilibrium cluster theory of association. The physical clusters, defined geometrically by overlap of spheres drawn about each molecule, are not regarded as statistically independent, but their mutual exclusion property gives rise to free energies of formation work for cavities necessary to contain those clusters. A connection is established between these free energies and a suitable generalization of the Mayer irreducible cluster sum. The appropriate modification of the theory to include external forces (such as the gravitational field) is displayed. A rigorous expansion of molecular distribution functions in terms of physical (rather than Ursell‐Mayer) cluster integrals is also presented.

Accurate and Facile Approximation for Vibrational Energy‐Level Sums

Abstract: Some exact sums of harmonic vibrational‐energy‐level densities for 45 molecules were obtained at various energies with a digital computer. An improved ``corresponding vibrational states'' type of approximation to these sums at low and moderate energies is described. The approximation has the great advantage or being very rapid, of requiring scarcely any calculational detail and of using the actual molecular frequencies. The agreement of the approximation with a smoothed true count is usually within a few percent at quite low energies, e≳0.1 ez. It has the further advantage of being good also for the difficult case of a complex polyatomic molecule with noncommensurable frequencies.The behavior of the sums for various vibrational patterns is considered. The importance of noncommensurability and number of frequency groups is displayed.

Perturbed Hartree—Fock Calculations. I. Magnetic Susceptibility and Shielding in the LiH Molecule

Abstract: The limited basis set Hartree—Fock problem is solved in the presence of a perturbation term in the Hamiltonian to obtain the first‐order perturbed wavefunction. The formulation is applied to the calculation of electric polarizability, magnetic susceptibility, and magnetic shielding. The effect of the limited basis set on the gauge invariance of the magnetic quantities is discussed. The magnetic susceptibility and shielding, the rotational magnetic moment, and the spin rotational constants are calculated for lithium hydride using two different wavefunctions. The agreement with experiment is reasonably good for the spin rotational constants and rotational magnetic moment. It is felt that further refinement of the calculation would not change the results by more than a few percent.

Domains in Liquid Crystals

Abstract: A regular domain pattern has been observed to form in nematic liquid crystals when an electric field is applied. The domains are visible in unpolarized light by either transmission or reflection. Both ac and dc fields produce similar domain patterns and these appear at a rather sharp threshold field of about 1000 V/cm. The general features of the phenomenon have been investigated. It is believed that the domains are due to ordering in the liquid of a kind which has not previously been recognized.

Do the ``Ligand Field'' Parameters in Lanthanides Represent Weak Covalent Bonding?

Abstract: Instead of explaining the seven different f‐orbital energies or five different d‐orbital energies by parameters Anm〈rn〉 of the electrostatic ligand field model, we propose to classify the energy levels according to the actual one‐electron energies and to interpret these quantities by the weak effects of σ antibonding on the partly filled shell Calculations of the relative angular dependence Ξ2 of such effects are made in a simple model and compared with experimental data for nine‐ and eight‐coordinated lanthanide compounds The agreement is judged to be much more satisfactory than when the electrostatic model is applied, and the number of freely chosen parameters is much smaller

Direct Conversion of Graphite to Diamond in Static Pressure Apparatus

Abstract: Static pressure apparatus has been developed which is capable of pressures up to 200 kbar and transient temperatures up to about 5000°K, using an electric flash‐heating technique. At pressures above about 125 kbar and temperatures in the 3000°K range it is found that graphite spontaneously collapses completely to polycrystalline diamond which may be retrieved quantitatively. The threshold temperature of the transformation is several hundred degrees lower than the melting temperature of graphite. The diamond/graphite/liquid triple point is found to be located at about 125 (+10, —0) kbar and 4100 (±200)°K. The shock compression results of DeCarli and Jamieson, and of Alder and Christian, are linked with the present results to construct a phase diagram of carbon extending to 5000°K and 800 kbar.

Structure of the f6 Configuration with Application to Rare‐Earth Ions

Abstract: Free‐ion energy‐level schemes for the 4f6 and 4f8 configurations of Eu3+, Sm2+, and Tb3+ are given, which have been calculated from energy matrices including interactions among the three highest multiplicities. The values obtained for the parameters are F2 = 401 cm—1, ζ = 1320 cm—1 for Eu3+; F2 = 330 cm—1, ζ = 1050 cm—1 for Sm2+; and F2 = 434 cm—1, ζ = 1705 cm—1 for Tb3+. The wavefunctions and g values for levels to 40 000 cm—1 are tabulated for Eu3+ and Tb3+. Application of the free‐ion wavefunctions to the Stark splitting of EuCl3 and TbCl3 is discussed.

Absorption Spectrum of Gaseous F- and Electron Affinities of the Halogen Atoms

Abstract: The absorption spectrum of gaseous fluoride ion has been observed in shock‐heated vapors of CsF, RbF, and KF. The spectrum is a continuum with two sharp thresholds at 3595 and 3542 A, similar to those of the other halide ions. The electron affinity determined from the low‐energy threshold is 3.448±0.005 eV. The cross‐section for photodetachment at 3565 A is about 2.5±2×10−18 cm2 and at 3525, about 3.3±2×10−18 cm2. The electron affinities of chlorine, bromine and iodine are revised, respectively, to 3.613, 3.363, and 3.063 eV, all with uncertainties of 0.003 eV.

Survey of the Spectra of the Divalent Rare‐Earth Ions in Cubic Crystals

Abstract: The rare‐earth ions may exist in the divalent state in suitable host crystals such as CaF2. All of the trivalent ions from Ce to Yb (and probably also La) are reduced in situ to the divalent state in CaF2 by gamma irradiation. The spectra of most of these ions show that the ground and first few excited states derive from fn configurations, but the weak absorption due to these is masked at higher energies by strong broad bands of the parity‐permitted fn→fn—1d transitions. The excitation energy of these spectra have been calculated in a first approximation as the energy difference between the ``Hund rule,'' single‐determinant states of the configurations fn—1d and fn. This procedure satisfactorily accounts for the remarkable variations in the excitation energy in passing from one ion to the next in the series with the exception of Gd++, Ce++, and Tb++. Gd++ probably has f7d for its ground configuration, while Ce++ and Tb++ are borderline cases. The spectral structure probably arises chiefly from the crystal...

Near‐Infrared Studies of the Structure of Water. I. Pure Water

Abstract: The absorption bands of water in the 1.1‐to‐1.3‐μ region have been measured. From the extinction coefficients, concentrations of water species involving 2,1, and 0 hydrogen bonds per molecule were calculated. The temperature dependence of these concentrations yielded a value of 3.7–4.5 kcal for the energy involved in breaking one mole of hydrogen bonds in water. On melting, 46% of the hydrogen bonds originally present in ice are broken; by 72°C the percentage broken has increased to 61. Further, the results were used in conjunction with the flickering cluster model of Frank and Wen to obtain an upper limit for the average size of the water polymers of 90 molecules of H2O at 20°C.

Theory of Dissociation Pressures of Some Gas Hydrates

Abstract: Dissociation pressures of some gas hydrates have been evaluated using the Lennard‐Jones 12–6, 28–7, and Kihara potentials in the Lennard‐Jones‐Devonshire cell model. The Lennard‐Jones 28–7 potential gives the least satisfactory results. The Lennard‐Jones 12–6 potential works satisfactorily for the monatomic gases and CH4 but poorly for the rodlike molecules C2H6, CO2, N2, O2, C2H4. This failure may be due to (i) distortions of the hydrate lattice, (ii) neglect of molecular shape and size in determining the cavity potential (iii) barrier to internal rotation of the molecule in its cavity. A crude model for the lattice shows that it is not distorted. The Kihara potential predicts better dissociation pressures for the hydrates of the rodlike molecules. Unlike the previously used Lennard‐Jones 12–6 potential, it depends on the size and shape of the interacting molecules. The absence of lattice distortions, improved dissociation pressures through the use of the Kihara potential and the restriction of the motio...

Microwave Spectra and Structure of Dimethyl Ether

Abstract: The microwave spectra of six isotopic species of dimethyl ether were investigated from 8–48 Gc. From the changes in moments of inertia with C13 and O18 substitutions the CO distance and

Padé Approximant Studies of the Lattice Gas and Ising Ferromagnet below the Critical Point

Abstract: The Pade approximant procedure is used to study the low‐temperature series for the Ising problem. The critical behavior of the spontaneous magnetization of an Ising ferromagnet and of the liquid‐vapor coexistence curve of the corresponding lattice gas is found to be I0(T)∼|ρliq−ρga5|∼|Tc−T|β where (in three dimensions) 0.303≤β≤0.318. It is conjectured that β=5/16=0.31250. The low‐temperature initial susceptibility and the compressibility of the lattice gas at condensation are found to diverge as χ0(T)∼κ0(T)∼|Tc−T|−γ′ where in two dimensions γ′=1.75±0.01 while in three dimensions γ′=1.25 (+0.07, —0.02). Consideration of a heuristic model partition function suggests the identity α′+2β+γ′=2 where α′ is the index of divergence of the specific heat below Tc. The amplitudes of the singularities are derived and extrapolation formulas are presented for the plane triangular, square, and honeycomb lattices and the three‐dimensional face‐centered, body‐centered, and simple cubic lattices. The results are compared br...

Theory of Field Desorption

Abstract: The theory of field desorption proposed previously has been extended by a more detailed quantum‐mechanical calculation of the rate constants of desorption for various cases, including an attempt to evaluate matrix elements for the electronic transition when the latter is slow. A simple, generalized model of chemisorption on metals is presented and used in these calculations. It is found that two‐electron transitions lead to somewhat smaller rate constants than one‐electron transitions, but that the effect is not very large. In general, a one‐dimensional theory leads to rate constants with essentially normal pre‐exponential terms, except under conditions where adsorbate tunneling becomes important. A tentative explanation for the compensation effect, i.e., decreases in pre‐exponential term with decreasing activation energy observed experimentally in a number of different cases, is given in terms of changes of reaction path with temperature.

Formal kinetic theory of transport phenomena in polyatomic gas mixtures

Abstract: A rigorous formal kinetic theory of multicomponent polyatomic gas mixtures is derived. The methods are essentially a combination of those used by Chapman and Enskog for monatomic gases (as extended to multicomponent mixtures by Curtiss and Hirschfelder), and those used by Wang Chang and Uhlenbeck for a single polyatomic gas for the case where the equilibration between internal and translational degrees of freedom is easy. The calculations correspond to the first approximation in the classical Chapman—Enskog theory. Expressions are derived for the coefficients of shear viscosity, volume viscosity (which is proportional to a relaxation time), ordinary diffusion, translational and internal thermal conductivity, and thermal diffusion. Although the results are rather formal, a number of useful conclusions about the effects of inelastic collisions can be drawn without the necessity of detailed calculations. For instance, it is comparatively simple to show that inelastic collisions have very little effect on shear viscosity and ordinary diffusion, but probably seriously affect thermal conductivity and thermal diffusion. The Hirschfelder—Eucken formula for the thermal conductivity of polyatomic gas mixtures appears as a limiting case of the present formulas. A number of other conclusions and implications of the present formulas are also discussed.