Showing papers in "Journal of Chemical Physics in 1960"

Ensemble Method in the Theory of Irreversibility

Abstract: We describe a new formulation of methods introduced in the theory of irreversibility by Van Hove and Prigogine, with the purpose of making their ideas easier to understand and to apply. The main tool in this reformulation is the use of projection operators in the Hilbert space of Gibbsian ensemble densities. Projection operators are used to separate an ensemble density into a ``relevant'' part, needed for the calculation of mean values of specified observables, and the remaining ``irrelevant'' part. The relevant part is shown to satisfy a kinetic equation which is a generalization of Van Hove's ``master equation to general order.'' Diagram summation methods are not used. The formalism is illustrated by a new derivation of the Prigogine‐Brout master equation for a classical weakly interacting system.

Correlation of Rates of Intramolecular Tunneling Processes, with Application to Some Group V Compounds

Abstract: A method is presented whereby the WKB expression for tunneling frequencies permits correlation of these frequencies for members of homologous series of molecules The necessary data required are the tunneling rate for one member of the series, and structural parameters and vibrational frequencies for all members of interest The method is applied to ND3, NF3, PH3, and AsH3, using NH3 as the molecule whose frequency is known The method also is used to support the hypothesis that pseudorotation occurs in the trigonal bipyramids PF5 and PCl5

Theory of ESR Linewidths of Free Radicals

Abstract: A theory of ESR linewidths for substances in which the magnetic anisotropy is small and for which the orbital magnetism has been essentially quenched is developed. Nuclear quadrupole moments, zero field splittings, anisotropic Zeeman terms, and intramolecular electron‐nuclear dipolar interactions, as well as motional and exchange effects are considered in the strong field case. The theory is developed in a manner that is particularly adaptable to the study of liquids but it is also applied to crystals since the resulting equations, though only applicable for small anisotropies, are particularly simple. The theory provides an extension of previous theories on ESR spectra in liquids. Several applications of the theory are discussed with particular emphasis on V4+ chelates. The theory of exchange in liquids, a phenomenon which is complicated by the noncommutivity of the motional and exchange Hamiltonians is considered in special detail. It is shown that this theory can be used to explain Hausser's results—th...

Crystal Structure of β‐Ga2O3

Abstract: The crystal structure of β‐Ga2O3 has been determined from single‐crystal three‐dimensional x‐ray diffraction data. The monoclinic crystal has cell dimensions a=12.23±0.02, b=3.04±0.01, c=5.80±0.01 A and β=103.7±0.3° as originally reported by Kohn, Katz, and Broder [Am. Mineral. 42, 398 (1957)]. There are 4 Ga2O3 in the unit cell. The most probable space group to which the crystal belongs is C2h3—C2/m; the atoms are in five sets of special positions 4i: (000, ½½0)±(x0z). There are two kinds of coordination for Ga3+ ions in this structure, namely tetrahedral and octahedral. Average interionic distances are: tetrahedral Ga–O, 1.83 A; octahedral Ga–O, 2.00 A; tetrahedron edge O–O, 3.02 A; and octahedron edge O–O, 2.84 A. Because of the reduced coordination of half of the metal ions, the density of β‐Ga2O3 is lower than that of α‐Ga2O3 which has the α‐corundum structure. Also the closest approach of two Ga3+ ions in β‐Ga2O3 is 3.04 A which is considerably larger than the closest approach of metal ions in the s...

Radiation Damage in Organic Crystals. II. Electron Spin Resonance of (CO2H)CH2CH(CO2H) in β-Succinic Acid

Abstract: An analysis of the electron spin resonance of x‐irradiated single crystals of β‐succinic acid shows that: (a) the principal long‐lived paramagnetic species produced by the radiation damage is (CO2H)CH2–ĊH(CO2H); (b) the radical is oriented in the crystal lattice in nearly the same way that the parent succinic acid molecule is oriented in the undamaged lattice; (c) the strongly anisotropic hyperfine interaction due to the σ proton is very nearly the same as that previously found for the σ proton in the malonic acid radical, (CO2H)ĊH(CO2H). In these molecules the σ proton is directly bonded to the carbon atom on which the odd electron is largely localized. The two methylene protons in the radical are not equivalent, and their hyperfine interactions are nearly isotropic, and in the range 80–100 Mc.

Studies in Molecular Dynamics. II. Behavior of a Small Number of Elastic Spheres

Abstract: The equation of state and the collision rate for systems ranging in size from four to 500 particles are described. The dependence of the results on the number of particles is qualitatively discussed and in this way insight is gained as to what is required of more accurate analytical theories. By comparing the results to various analytical theories now available their region of validity is established. The number of particles necessary at various densities to obtain a quantitative description of the equilibrium properties is delineated. Whether a first‐order phase transition exists for hard spheres remains open until larger systems are investigated.

Rotational Diffusion Constant of a Cylindrical Particle

Abstract: The torque constant of a closed cylinder rotating in a viscous medium has been calculated for length (2a) over width (2b) ratios larger than 3.5 to within a first order in b/a. The analysis demonstrates how the contributions to the viscous dissipation tend to be underestimated in hydrodynamic considerations so that the geometrical values deduced from them come out too high. Experimental results for the torque on cylindrical rods and ellipsoids for a/b values from 3.5 to 30 are close to the theoretical results. For a/b>10 the difference is about 10%; for shorter molecules 20%. With the rotational diffusion constant given by 3kT (σ—γ)/8πηa3ω, where σ=log2a/b we obtain best fit with γ(σ>2)=1.57–7 (1/σ—0.28)2±0.25. Experimental data for the rotational diffusion constant of a cylindrical virus (a/b=20) in water, obtained by O'Konski and Haltner agree with this result within 10%. The length of the protein fits within 3%.

Note on the Paramagnetic Resonance of Iron in Glass

Abstract: Fe+ + + and Fe+ + ions in glass have been studied by means of paramagnetic resonance. The correlation of the Fe+ + + concentration with an intense resonance at g=4.27 shows the resonance is due to Fe+ + +. A theory is proposed explaining the observed g value and the implications for the atomic surroundings of Fe+ + + are discussed. It is also proposed that the g=6 resonance observed in glass by Sands is due to Fe+ + +.

Self-diffusion of oxygen in single crystal and polycrystalline aluminum oxide

Abstract: The self‐diffusion coefficient of oxygen has been determined as a function of temperature in single crystal and polycrystalline aluminum oxide at temperatures up to 1780°C. The rate of exchange between a gas phase and solid particles was measured, utilizing the stable isotope, O18. In single crystals of aluminum oxide intrinsic diffusion occurs in a high temperature region, depending on temperature as D=1.9×103 exp (—152 000/RT). At temperatures below about 1600°C variable results were obtained depending on impurity content and previous heat treatment. For one set of samples experimental results could be represented as D=6.3×10—8 exp (—57 600/RT). The diffusion coefficient of oxygen in polycrystalline samples is about two orders of magnitude larger than that found for the single crystals, and has a somewhat smaller activation energy. With the polycrystalline oxide, variable results were also observed at lower temperatures.

Line Shapes of Electron Paramagnetic Resonance Signals Produced by Powders, Glasses, and Viscous Liquids

Abstract: The line shapes arising from randomly oriented particles in a fixed position have been calculated. It has been found that it is possible to determine the three principal g factors with relatively high accuracy and without investigating single crystals. Measurements on powders of CuCl2·2H2O and CuSO4·5H2O are used as examples. In addition, the influence of Brownian rotation in viscous liquids on line shapes has been studied. Equations have been derived for the case of high viscosities, supplementing the well‐known ones given by McConnell [J. Chem. Phys. 25, 709 (1956)]. To a first approximation, the effect of decreasing viscosity can be described by an over‐all diminishing line width and an unchanged line shape function. The hyperfine splittings are neglected, but can be taken into account.

Quantum Statistical Mechanical Theory of the Rate of Exchange Chemical Reactions in the Gas Phase

Abstract: A theory of the reaction rate of simple exchange reactions in the gas phase is developed on the basis of the quantum‐statistical mechanical theory of linear irreversible processes due to Kubo et al. A formal expression for the rate coefficient is found near the equilibrium point. A number of relations are derived concerning the scattering amplitudes for the collisions involved in the above reactions. By making use of these relations, the rate constant is expressed in terms of the reaction cross sections in a way which coincides with that known from a more intuitive collisional approach. Since no ad hoc assumptions are made, the present theory can be said to provide a statistical mechanical foundation for the collision theory in the particular case discussed.

Solvent Effects in Nuclear Magnetic Resonance Spectra

Abstract: Contributions to nuclear screening (chemical shifts) arising from molecular interactions with solvent molecules (excluding hydrogen bonding) are discussed in terms of appropriate theoretical models. These include contributions from van der Waals interactions σw, from the magnetic anisotropy of the solvent molecule σa, and from polar effects σE. By a suitable choice of solute‐solvent systems it has been possible to demonstrate each of these effects experimentally for proton resonances. For CH4 as a solute, σw was in all cases negative, its magnitude varying with the nature of the solvent and amounting to as much as 0.6 ppm for high molecular weight solvents. In agreement with the theoretical models, σa was found to be positive for disk‐shaped solvent molecules and negative for cylindrically symmetrical rod‐shaped molecules, its magnitude in extreme cases reaching 0.75 ppm. For CH3CN as a solute, σE was negative and showed the expected dependence on the dielectric constant of the solvent.

Perturbation Treatment of the Characteristic Vibrations of Polypeptide Chains in Various Configurations

Abstract: A perturbation treatment has been made of localized group vibrations of helical polymer chains. The amide I and II frequencies were derived in terms of adjacent group interactions as well as interchain and intrachain hydrogen bonding interactions. Frequency shifts caused by these interactions depend upon the configuration of polypeptide chains and also upon the chain packing in the crystalline region. Each characteristic vibration of the amide group gives rise to parallel and perpendicular bands. The parallel bands of the α helix of poly‐γ‐benzyl‐L‐glutamate were observed at 1650 cm—1 (amide I) and 1516 cm—1 (amide II) whereas the perpendicular bands were observed at 1652 cm—1 (amide I) and 1546 cm—1 (amide II). Both the parallel‐chain and antiparallel‐chain pleated sheets exhibit the perpendicular amide I band at ca 1630 cm—1. A characteristic band of β configurations at ca 1690 cm—1 was assigned to the parallel amide I band of the antiparallel‐chain pleated sheet.

Aspects of the Statistical Thermodynamics of Real Fluids

Abstract: By extending the ideas previously applied to the statistical mechanical theory of hard sphere fluids of Reiss, Frisch, and Lebowitz, an approximate expression has been determined for the work of creating a spherical cavity in a real fluid. In turn the knowledge of this entity permits an evaluation of properties such as the surface tension and the normal heats of vaporization of fluids and the Henry's law constants of fluid mixtures. The agreement between the calculated and experimental properties is satisfactory.

Quantum‐Mechanical Modified Boltzmann Equation for Degenerate Internal States

Abstract: A modified quantum‐mechanical Boltzmann equation has been derived for the general case in which the molecules have degenerate internal states. This is an equation of the Boltzmann type for a quantity which is simultaneously a Wigner distribution function in molecular phase space, and a density matrix in internal state space. In particular, the nondiagonal terms of this density matrix have been included in the formalism, resulting in the collision term being modified from the usual Boltzmann expression. Thus the collisions are described in terms of combinations of the Lippmann‐Schwinger scattering matrix rather than the collision cross section. For nondegenerate states the usual collision term is obtained again.

Dielectric Relaxation of Isoamyl Bromide

Abstract: The complex dielectric constant of isoamyl bromide has been measured at 1, 3, and 9 kMc between —75° and 25°C. Complex plane plots indicate an asymmetric, skewed‐arc distribution of relaxation times, with the shape of the distribution function not being appreciably temperature dependent. A defect diffusion model is proposed to explain the dielectric behavior of this system. This model implies that the relaxation of a molecule is more probable immediately after one of its neighbors has relaxed than at an arbitrary time. A distribution of relaxation times is derived which, under the appropriate conditions, closely resembles that of the empirical skewed‐arc function.

Microwave Spectrum and Molecular Structure of Formamide

Abstract: The microwave spectra of ten isotopic species of formamide (H2N–CHO) have been investigated. Type a and type b transitions have been identified and measured for all ten species. The inertial defect is found to decrease whenever a heavier isotope is substituted for any atom in the NH2 group, and in fact is negative for four of the species studied. It is concluded that the molecule is nonplanar with the H2N–C group forming a shallow pyramid. The structural parameters deduced from the rotational constants are: r(N–H′, where H′ is trans to the aldehyde hydrogen) = 1.014±0.005 A, r(N–H″, where H″ is cis to the aldehyde hydrogen) = 1.002±0.005 A, r(N–C) = 1.376±0.010 A, r(C–H) = 1.102±0.010 A, r(C=O) = 1.193±0.020 A, ∠H′NH″ = 118°53′ ±40′, ∠H″NC = 120°37′±40′, ∠H′NC = 117°9′±40′, ∠NCO = 123°48′±40′, ∠NCH = 113°14′±40′ and ∠OCH = 122°58′±40′. The dihedral angles between the H′NC plane and the NCO plane, and between the H″NC plane and the NCH plane, are 7°±5°, and 12°±5°, respectively.In all the spectra investiga...

Becquerel Photovoltaic Effect in Binary Compounds

Abstract: An experimental study has been made of photovoltaic effects which occur at semiconductor‐electrolyte interfaces. Single crystal specimens of CdS and several other compounds were used. It was found that in a number of cases the photovoltaic effect results from a chemical reaction of the electrode materials. In such cases the observations may be explained by a simple mechanism which relates the sign of the photo‐emf to the conductivity type of the semiconductor and to the chemical reaction which is occurring. The reaction may be predicted using readily available thermodynamic data. A different process occurs when the electrode material is GaAs. It acts as an inert electrode which exchanges electrons with an oxidation‐reduction couple in the solution.

Classical and Quantum Mechanical Hypervirial Theorems

Abstract: If W is a function of the coordinates and momenta, then in classical mechanics the time average of the Poisson bracket (H, W) is zero. In quantum mechanics it follows from the Heisenberg equation of motion that the expectation value of (WH — HW) for any wave function, corresponding to a stationary energy state of the system, is zero. For each selection of W there is a dynamical relationship, in a time‐average or space‐average sense, which the system must obey. Two classes of W's are considered in detail: W as a function only of the coordinates, and W as a function of the coordinates times the first power of a momentum. For the first class, neither the classical mechanical nor the quantum mechanical results provide useful information. The usual virial theorem of Clausius is a special case of the second class of relations. The classical treatment should be useful for the determination of the equation of state of liquids. The use of the quantum‐mechanical relations for determining the constants in an approximate wave function is discussed.

Microwave Spectrum, Structure, and Dipole Moment of Propane

Abstract: The microwave spectrum of ordinary propane and five isotopic species have been measured and analyzed. The rotational constants of C3H8 are 29 207.36, 8446.07, and 7458.98 Mc, and the dipole moment is 0.083±0.001D. In its equilibrium configuration the molecule has C2v symmetry, with both CH3 groups staggered with respect to the CH2 group. The complete structure has been determined by the substitution method. Important parameters are: r(CC)=1.526±0.002 A, ≰CCC=112.4°±0.2°; in the CH2 group, r(CH)=1.096±0.002 A, ≰HCH=106.1°±0.2°; in the CH3 groups, r(CH)=1.091±0.010 A, ≰HCH=107.7°±1.0°. The influence of zero‐point effects on the structure determination is discussed; these are found to be particularly bad for the CH3 group, probably because of its large vibrational amplitude.

``Forbidden'' Character in Allowed Electronic Transitions

Abstract: A theory for calculating the intensity of ``forbidden'' character in allowed electronic transitions is given based upon the early work of Herzberg and Teller. It is seen how both symmetric and nontotally symmetric vibrations may be active in the perturbation, and the temperature dependence of ``forbidden'' intensity is explicitly obtained for harmonic oscillator wave functions. As an example the case of certain p‐disubstituted benzenes is discussed in detail. A model derivative normal coordinate analysis is used to calculate perturbation energies in terms of electronic wave function parameters. It is seen how the magnitude of ``forbidden'' character is very sensitive to the detailed nature of the mixing electronic wave functions. Several trial functions are introduced to calculate the ratio of ``forbidden'' to allowed character in the near ultraviolet band. Comparison is made with recent observations of mixed polarization in two benzene derivatives. Finally, the possible significance of ``forbidden'' char...

Viscous Force Constant for a Closed Cylinder

Abstract: The viscous force constant of a moving cylinder closed at the ends has been calculated including first‐order effects in width (2b)/length (2a). Experimental results, obtained for macroscopic models, are in essential agreement with this. For a/b>20 the discrepancy is about 8%, for shorter cylinders near 20%. With the force on a cylinder moving sideways given by F=8πηav/(σ—γ), where σ=log2a/b, it is found that γ(σ>2)=0.35–4(1/σ—0.43)2±0.25. For a translation lengthwise, with F=4πηav/(σ—γ), the relation γ(σ>2)=1.30–8(1/σ—0.30)2±0.25 holds.

Dielectric Relaxation of Polar Liquids

Abstract: The statistical theory of the dielectric relaxation of polar liquids is developed using the fluctuation‐dissipation approach to linear dissipative phenomena, and an expression is derived relating the complex dielectric constant to a time‐dependent microscopic correlation function. It is found that a finite number of microscopic relaxation times leads to an equal number of macroscopic decay times, and, in the case of a single relaxation time τ0, the decay time is given by T0=[3e0/(2e0+e∞)]τ0, e0 being the static dielectric constant, and e∞ being the high frequency dielectric constant. Relaxation times are also determined for systems having two decay times, and for systems characterized by the circular‐arc and skewed‐arc distribution functions.

Theory of ``Melting'' of the Helical Form in Double Chains of the DNA Type

Abstract: Previous theoretical treatments of the transition between the helical and random forms of the desoxyribose nucleic acid (DNA) molecule are extended to include formally the explicit consideration of the dissociation into two separate chains and the consideration of the effects of the ends of the chains. An approximate form for the fraction of base pairs that are bonded is obtained in terms of two parameters, a stability constant for base pairing and a constant representing the interaction of adjacent base pairs. The matrix method of statistical mechanics proves to be adaptable to this problem. Some numerical examples are worked out for very long molecules, for which case it is found that the effect of concentration is small.

On the Effects of Intramolecular van der Waals Forces

Abstract: Evidence for the existence of strong nonbonded repulsions between atoms bonded to a given atom is cited. It is observed that bond angles in ethylene derivatives and related molecules can be correlated with a simple ``hard sphere'' model in which atomic positions are presumed to be governed by the sizes of spherical atoms packed around a given atom. Estimations of the magnitudes of the nonbonded interactions are presented. These indicate that the shortening of carbon‐carbon single bonds in unsaturated hydrocarbons may be principally attributable to the relaxation of nonbonded repulsions across digonal or trigonal bonds as compared with tetrahedral bonds. It is suggested that intramolecular van der Waals forces may be even more important than effects of hybridization and conjugation or hyperconjugation in governing bond angles, bond distances, energies of isomerization and hydrogenation, and bending vibrations of molecules.

Proton Spin Coupling by Pi Electrons

Abstract: A theoretical formulation is developed for the pi‐electron contribution, AHH′(π), to the spin coupling between pairs of protons in hydrocarbon molecules. By means of the correspondence between the σ—π interaction in unsaturated molecules and related free radicals, the proton spin coupling is expressed in terms of hyperfine constants and triplet state energies. With known values for these quantities, AHH′(π) is found to be in agreement with experimental measurements available for molecules in which the pi electrons are expected to dominate the coupling. Of particular interest are the large couplings (1.3–8 cps) calculated for certain systems with protons separated by three or four carbon atoms. Also absolute signs are predicted by the theory, with AHH′(π) equal to —6.7 cps for allene and +7.8 cps for butatriene.

Variational Theory of Chemical Reaction Rates Applied to Three-Body Recombinations

Abstract: A ``variational'' theory, which gives a least upper bound to the rate of a chemical reaction, is presented. The reaction is represented by the motion of a point in phase space across a trial surface dividing the ``initial'' and ``final'' chemical states. The trial surface is well defined in regions of phase space where interactions causing reaction are negligible, but is subject to arbitrary variations otherwise. It is shown that a least upper bound to the reaction rate can be obtained by calculating the rate at which representative points cross the ``trial'' surface and then minimizing this rate with respect to allowed variations of the surface. Explicit calculations of the recombination rate of attracting atoms in the presence of repulsive third bodies are made for a simple trial surface having one adjustable parameter. At low temperatures, the experimental rate constants are quite close to the theoretical bounds; at high temperatures, the experimental data fall away from the bounds in a manner which can be understood in terms of various approximations contained in the theory. Promising methods of improving the agreement between theory and experiment are discussed.

Quantum Mechanical (Phase Shift) Analysis of Differential Elastic Scattering of Molecular Beams

Abstract: For a spherically symmetrical intermolecular potential V(r)=ef(r/σ) the quantum calculation of the elastic scattering cross section dσ(Θ)/dΩ in the c.m. system is carried out as follows. For a given relative velocity (or deBroglie wavelength) and an assumed V(r), the radial wave equation is integrated for successive values of the angular momentum quantum number l, yielding the phase shifts ηι. Then dσ(Θ)dΩ is computed in terms of the series of ηι's in the standard way. A general computational program (following that of K. Smith) is outlined for the evaluation of the radial wave function and the phase shifts, utilizing an IBM 704 computer. Calculations are presented for the L‐J (12, 6) potential function. The results may be concisely represented using the framework provided by the semiclassical treatment of Ford and Wheeler, i.e., in terms of a set of reduced phase constants vs reduced angular momenta at various reduced relative kinetic energies K. Tables and graphs are presented from which the phases may ...

Electronic Spectra of Dimers: Derivation of the Fundamental Vibronic Equation

Abstract: The Hamiltonian describing the vibronic states of a dimer formed by two identical molecules is derived. The monomeric units are assumed to couple by resonance forces only and to have different equilibrium positions in the ground and excited states. The competition of these two effects is formulated mathematically. The strong coupling case occurs when the resonance forces dominate; the weak coupling case arises when the effect of the change of equilibrium positions dominates.

Infrared Spectra of Metal‐Chelate Compounds. I. A Normal Coordinate Treatment on Bis‐(Acetylacetonato)‐Cu(II)

Abstract: A normal coordinate treatment on the chelate ring of bis‐(acetylacetonato)‐Cu(II) gives calculated frequencies in good agreement with the observed ones in the 1700 and 350 cm—1 range obtained with NaCl, KBr, and CsBr optics. In order to assign the observed bands and to see the coupling between various vibrational modes, the L matrices and the potential energy distribution, Lii2Fi were calculated for all the in‐plane vibrations. The results reveal that (1) the band at 1580 cm—1 previously assigned to asym. CO stretching is asym. CC stretching, (2) the bands at 684 and 654 cm—1 formerly suggested to be metal‐oxygen stretching modes are metal‐oxygen stretching vibrations coupled with ring deformation and CCH3 bending modes, respectively, and (3) a new band found at 455 cm—1 is the sym. Cu–O stretching mode. By comparing the force constants of the Cu–O stretching vibration with those of metal‐nitrogen and metal‐carbon bonds of other compounds, it is suggested that the Cu–O bond in this compound has double bon...