Volume 59, Issue 6, 15 September 1973

Lattice dynamics of the pyrazine crystal studied by coherent inelastic neutron scattering
View Description Hide DescriptionThe lattice vibrational excitations at a number of points in the Brillouin zone for a deuterated pyrazine crystal at room temperature have been measured by means of coherent inelastic scattering of neutrons. These give values for all 9 adiabatic elastic constants, 25 excitations at symmetry points in the Brillouin zone, and dispersion curves in the (0 η0) direction. These have been interpreted in the quasiharmonic, rigid molecules, approximation using an atom‐atom 6‐exp potential plus electrostaticintermolecular forces. This gives 3 parameters to be fixed. Fitting to the observed excitations gives physically reasonable values for these parameters. The fit obtained is not perfect (χ = 3.4) but good enough [(ω_{obs} − ω_{calc})/ω_{calc} ∼ 7%] to say that this model describes the intermolecular forces in pyrazine well. The C–H ··· N hydrogen bond at long ranges is well described by a 6‐exp potential plus an important electrostatic interaction between the N lone‐pair electrons and the C–H bond dipole. The asymmetry in the Nitrogen atom electronic distribution does not appear to have any effect on the dynamics other than its electrostatic contribution. It is suggested, from the pattern of the deviations of the dynamics and crystal structure from the model, that the errors in the model derive from neglect of three body atom‐atom electronic correlations.

Vibrational relaxation of shock‐heated N_{2} by atomic oxygen using the ir tracer method
View Description Hide DescriptionWe have studied the effects of oxygen atoms on the vibrational relaxation of shock‐heated N_{2} over the temperature range from 1200 to 3000 °K, using the CO tracer technique. The measuredrelaxation timespτ_{N2–O} vary from 0.8 to 2 atm · μsec in this range with an uncertainty of ± 50%, in excellent agreement with existing high‐temperature shock tube and low‐temperature flow tube results. The weak temperature dependence of all these results is in marked contrast to theoretical predictions. Some limited measurements of the relaxation of CO by oxygen atoms yield values of pτ_{CO–O} of approximately 0.04 atm · μsec, in good agreement with other recent measurements. This rate is approximately 25 times as fast as the N_{2}–O rate and, coincidentally, is approximately the same as that for the relaxation of O_{2} by O atoms. The greater rapidity of the CO–O and O_{2}–O relaxation, compared with that of N_{2}–O, may be the result of atom‐exchange processes in the former cases.

Vibrational spectral study of molecular orientation in vitreous fibers
View Description Hide DescriptionPreferential orientation of polymeric chains with respect to the principle axis in completely noncrystalline, vitreous fibers is demonstrated by laser Raman and infrared reflectance study of sodium metaphosphate glass fibers. The polarization dependence of Raman intensities in 24 scattering arrangements have been obtained by varying the incident and scattered radiation propagation and polarization directions and are compared to calculated Raman intensity ratios for uniaxial orientation of the chains. Infrared dichroic ratio data were obtained from a Kramers‐Kronig transformation analysis of polarized reflection spectra on oriented fibers, and employed to compute the averages of the angles the metaphosphate chains make with the fiber axis.

Burnett theory of planar thermal force with surface accommodation
View Description Hide DescriptionThe force on a plane surface in a rarefied gas caused by a temperature gradient normal to the surface is studied. In the free‐molecule regime the force depends on the normal‐momentum accommodation coefficient. For the transition regime the second‐order Burnett constitutive equations for stress and heat flux provide the necessary coupling between force and temperature gradient. Force versus Knudsen number has a maximum which is related to both normal‐momentum and thermal accommodation coefficients. The results compare favorably with the dusty gas model predictions and with experimental data. Relationships to other rarefied gas transport phenomena, e.g., thermal transpiration, are demonstrated.

ESR of the quartet states of randomly oriented molecules: Calculation of the line shape and detection of the zero‐field splitting
View Description Hide DescriptionThe ESRspectrum of the quartet states of randomly oriented molecules with axial symmetry is discussed. The absorption curves of the Δm = 1, Δm = 2, and Δm = 3 transitions are calculated in first order perturbation approximation. The line broadening is explicitly taken into account. The experimental spectrum of an alternant hydrocarbon agrees sufficiently well with the theoretical predictions. The shape of the absorption curve permits the ready determination of the zero‐field parameters of quartet molecules.

Dynamics of the CS_{2}–O_{2} flame
View Description Hide DescriptionThe kinetics of the CS_{2}–O_{2}flame for laser applications is described by a computer program (utilizing 24 contributing reactions) allowing vibrationally excited CO to be created from CS + O. The validity of individual contributing rate coefficients has been established through a comparison with experimental evidence on the oxidation of CS_{2}. [K. H. Homann, G. Krome, and H. Wagner, Ber. Bunsenges. Phys. Chem. 74, 654, (1970); K. H. Homann, G. Krome, and H. Wagner, Ber. Bunsenges. Phys. Chem. 72, 998, (1968); K. H. Homann, G. Krome, and H. Wagner, Ber. Bunsenges. Phys. Chem. 73, 967 (1969)]. Induction time, reaction time, and vibrational energy produced in CO are presented as functions of the initial amount of atomic oxygen so that a comparison can be made between the effects of predissociation, as by an electric discharge, and the formation of atomic oxygen in the chemical chain. The inclusion of simple mixing and heat loss at the boundary in an aerodynamic/thermodynamic mass flow environment has provided a fit with published thermal data [C. Wittig, J. C. Hassler, and P. D. Coleman, J. Chem. Phys. 55, 5524 (1971)] for an operating CS_{2} – O_{2} laser. The computer program was used to verify simpler kinetic models for specific cases, for example, a four reactionmodel: (1) CS_{2} + O → CS + SO, (2) CS + O → CO + S, (3) S + O_{2} → SO + O, (4) SO + O_{2} → SO_{2} + O to describe the CS_{2}–O_{2}flame under conditions where a chain reaction is required. Also, the production of vibrationally excited CO at diluent pressures up to 50 torr was found to be adequately described by only Reactions (1)‐(3) for input ratios of CS_{2}/O ≃ 1.

Perturbation theories for the calculation of molecular interaction energies. I. General formalism
View Description Hide DescriptionIn the calculation of the interaction energy of two atoms, it is particularly convenient to take the zero order Hamiltonian H ^{(0)} to be the sum of the atomic Hamiltonians. Then H ^{(0)} does not reflect the full permutational symmetry of the exact Hamiltonian H. Many different perturbation expansions with such a ``nonsymmetric'' H ^{(0)} have been proposed. In this paper, these expansions are reviewed and classified within a very general perturbation framework. Also, a new ``optimized'' method is proposed which, by virtue of the Rayleigh‐Ritz variational principle, is guaranteed to give better results than any of the previously suggested methods. For simplicity the analysis is restricted to the case that the spin‐free zero order function φ^{(0)} and the spin‐free exact wavefunction ψ are both nondegenerate. As an example, the construction of the first order wavefunction in each method is discussed in detail for the simple case of a single symmetry operation.

Perturbation theories for the calculation of molecular interaction energies. II. Application to
View Description Hide DescriptionThe many perturbation expansions reviewed in Paper I [D. M. Chipman, J. D. Bowman, and J. O. Hirschfelder, J. Chem. Phys. 59, 0000 (1973)] are applied to the interaction of a ground state hydrogen atom with a proton to form the 1s σ_{ g } and 2p σ_{ u } electronic states of . The calculations were made with high precision for the range of separations R = 0.2–15a _{ o } using a large basis set of Slater‐type orbitals. For the polarization and symmetrized polarization methods the energies were obtained analytically, these results providing a check on the completeness of the basis set. Also, long range and short range asymptotic formulas are given for the first order wavefunctions and the perturbation energies. The HS and MSMA methods gave the best second order energies. The EL‐HAV second order energy is good at intermediate separations but becomes bad at large separations. The expectation value of the Hamiltonian using the wavefunction truncated after the first order (where the coefficient of the zero order function is energy optimized) gave very good results in the OPT, COR, CH, HAV, EL, HS, DEM‐I, AM, and SYM‐P methods in the order listed; the MSMA and P results were poor due to the use of unsymmetrized wavefunctions. The OPT treatment, which energy optimizes the coefficients of three separate first order functions, has an error of less than 0.04 kcal/mole for the 2p σ_{ u } state, over the full range of separations, and for the 1s σ_{ g } state has an error of only 1.4 kcal/mole at the equilibrium separation R = 2a _{ o }, becoming much better than this at larger separations. Reasonably good interaction energies can be calculated from the polarization function without using the auxiliary functions θ and ω. In the limit of large separations, we find that the EL‐HAV second order energy approaches 41/54 = 0.759 ··· times the exact Coulombic energy, rather than the previously estimated factor of one‐half. Also, at large R, the SYM‐P, HS, and DEM‐I exchange energy agrees with Herring's exact asymptotic expression to within 0.1%.

Molecular formulation of the internal viscosity in polymer dynamics, and stress symmetry
View Description Hide DescriptionInternal viscosity, the property of polymer ``submolecule'' resistance to its rate of extension, is derived from the mechanics of atomic bond rotations within the submolecule. The result for these deformational forces and velocities is F̂^{ d } = c C · V̂^{ d } in supervector notation for the whole set of submolecules, where C is the Zimm matrix and c is related to rotational energy barriers E(φ) and other molecular parameters. The controversy about stress symmetry predictions of all such models is resolved.

Theory of chemically induced dynamic electron polarization. II
View Description Hide DescriptionThe earlier theoreticalanalysis for chemically induced dynamic electron polarization (CIDEP), based on the stochastic‐Liouville equation, is generalized to explicitly include the spin‐dependent exchange forces in the diffusive trajectories, thus permitting a consistent analysis of the simultaneous effects of exchange on both the spin‐selective chemical reaction and CIDEP effects. The semiclassical treatment of diffusion under a ``classical'' force field due to the valence interactions requires the introduction of spin‐dependent diffusive and reactive trajectories, and this is discussed for the Brownian‐motion model utilized. Our results show that the polarization generated per fractional probability that singlets react , is not sensitive to the actual details of the spin‐selective reactive process (although the absolute polarizationP ^{∞} is sensitive to the reactive process), due presumably to the spatial distinction between interradical separations (r) for which the reaction may occur vs those for which CIDEP polarizations are developed. The former require ℏJ(r)/kT > 1 while for the latter ℏJ(r)/kT < 1, where J(r) is the exchange interaction. It is found that differences in the (nonreactive) diffusive trajectories for singlets and triplets give polarizations that are generally negligible compared to those which develop as a result of the spin‐selective reaction (for our overdamped diffusive model). However, our results for more long‐range Coulomb interactions between charged radicals show they can produce significant changes on that are quite sensitive to the magnitude of J. Thus ionic‐concentration effects on should be an important indicator of the CIDEP mechanism. Results are also given for the spin‐depolarization process, whereby the effects of spin exchange on a radical pair, which initially collide with residual nonthermal polarization, are to destroy this polarization. The effective range of the spin exchange is found to be weakly enhanced as the range of J(r) is increased. Also, it is shown that, for several variations of a simple exponential dependence of J(r) on r, is hardly affected, although nonexponential dependences can introduce marked changes.

Interpretation of the magnetic anomalies in Cu(II):Ni(II) bis(diselenocarbamate) crystals
View Description Hide DescriptionA possible interpretation of the relevant magnetic anomalies, recently observed in Cu(II):Ni(II) bis(diselenocarbamate) mixed crystals, is presented. The Cu–Se bond is considered responsible for the large ``circularization'' (to the value for the free electron) of g about a twofold axis in the molecular plane. Thus it is possible that a weak C _{2h } distortion of the D _{2h } symmetry in the complex [mixing of (xz) into the (3z ^{2}−r ^{2}) ground state, or (y z) into the (x y) ground state] may produce both the characteristic angle of approximately 45° between the low values of g and the molecular plane and the appearance of a principal value of g slightly lower than the free electron value. It is also clarified why A does not exhibit an analogous rotating effect. Relevant expressions of g and Atensors are deduced in terms of a simple LCAO‐MO formalism and implications on the covalent character of the magnetic bonds in the complex are finally discussed.

Deactivation of electronically excited sulfur atoms, S(3 ^{1} S _{0})
View Description Hide DescriptionDeactivation of S(3 ^{1} S _{0}) produced in the gas phase photolysis of carbonyl sulfide at 147.0 nm was studied with various molecules. The time dependence of the concentration of excited sulfur atoms is followed by their emission at 772.5 nm (^{1} S _{0} → ^{1} D _{2}) and 458.9 nm (^{1} S _{0} → ^{3} P _{1}). Absolute rate coefficients for removal (reaction plus collisional relaxation) of S(3 ^{1} S _{0}) were measured for O_{2}, CO, CO_{2}, OCS, H_{2}, NO, NO_{2}, N_{2}O, H_{2}S, CS_{2}, SO_{2}, CH_{4}, C_{2}H_{6}, C_{2}H_{4}, C_{2}H_{2}, and Ar. For many of these gases, removal rate coefficients for S(3 ^{1} S _{0}) and O(2 ^{1} S _{0}) are similar, but exceptions are noted as well.

Observation of an electron transfer process
View Description Hide DescriptionElectron transfer has been observed by ESRspectroscopy in x‐irradiated single crystals of partially deuterated glycine hydrochloride. The crystals were irradiated at 4.2°K and the transfer occurs on warming to 77°K. It consists of the movement of electrons between carboxyl groups with the electrons exhibiting greater affinity for protonated carboxyl groups than for deuterated carboxyl groups. A slight perturbation of β‐proton hyperfine couplings in carboxyl anion radicals due to substitution of a deuteron for a proton on the carboxyl group was also observed. The shift in coupling, which amounts to a fraction of a megahertz, was measured by ENDORspectroscopy.

Energy transfer between transition‐metal complexes in double complex salts
View Description Hide DescriptionIn a double complex salt consisting of cationic complex tris (2, 2′‐bipyridine) ruthenium (II) [Ru(bipy)_{3}]^{2+} and anionic chromium (III) complex [CrL_{6}]^{3−}, an excitation energy transfer from the excited ruthenium (II) complex to the chromium (III) complex is observed. When the ^{4} T _{2g } state of [CrL_{6}]^{3−} is lower than the ``metal to ligand'' charge‐transfer singlet excited state^{1} CT of [Ru(bipy)_{3}]^{2+}, the phosphorescence of [Ru(bipy)_{3}]^{2+} is completely quenched and the counterpart complex [CrL_{6}]^{3−} emits an enhanced phosphorescence even if it is irradiated at a wavenumber which excites exclusively [Ru(bipy)_{3}]^{2+} but not [CrL_{6}]^{3−}. An efficient path of the excitation energy transfer in the double complex salt is the one from the ^{1} CT state of [Ru(bipy)_{3}]^{2+} to the ^{4} T _{2g } state of [CrL_{6}]^{3−}. When the ^{4} T _{2g } state of [CrL_{6}]^{3−} is higher than the ^{1} C T state of [Ru(bipy)_{3}]^{2+}, the excitation transfer is competitive with emission and both [Ru(bipy)_{3}]^{2+} and [CrL_{6}]^{3−} in the double complex salt phosphoresce Even such a chromium (III) complex behaves as an efficient quencher to excited [Rh(bipy)_{3}]^{3+} in the double complex salt and emits a strong phosphorescence upon excitation of the [Rh(bipy)_{3}]^{3+}. Since the ^{4} T _{2g } state of such a chromium (III) complex is lower than the lowest singlet excited state of [Rh(bipy)_{3}]^{3+}, an excitation transfer takes place from the excited [Rh(bipy)_{3}]^{3+} to [CrL_{6}]^{3−}.

Laser‐induced rate processes in gases: Phase coherence in an N‐level model
View Description Hide DescriptionWe consider a gas of molecules absorbing light from a monochromatic laser source and also undergoing both phase randomizing and thermalizing collisions with a heat bath of inert molecules. The generalized master equation formalism, used previously to study a system of this type with a two‐level absorber, is extended to an absorber with N equally spaced levels. The possibility of unimolecular decay from any or all of the N levels is included in the formalism. The well‐known relaxation model of Rubin and Shuler and the chemical reactionmodel of Montroll and Shuler are built into the formalism as special cases when the light source is turned off. Specific calculations for an eight‐level model are presented and compared with results expected for an incoherent light source. A coherent effect which increases the upper level populations at steady state is found to exist. A corresponding increase in the macroscopic chemical rate constant is calculated for the coherent model when unimolecular decay from the top level is included. A particularly simple and general result is found for the power absorbed by a harmonic oscillator. It is also shown that, for the N‐level model, one does not necessarily find equal populations in the limit of infinite light field intensity.

Laser stimulation of chemical reactions and scattered field detection
View Description Hide DescriptionWe propose a model to account for some experimental observations on stimulating effects of light absorption on the rate of unimolecular chemical decomposition for gas phase molecules suffering collisions in a thermal bath. Our model calculations show a maximum in the rate constant when plotted against pressure at constant light intensity. A significant additional rate stimulation is predicted when molecules respond coherently to a laser field. This effect is most pronounced as the efficiency of energy transfer collisions increases. Numerical calculations applied to multilevel systems delineate the effects of temperature, pressure, light intensity, and efficiency of energy‐changing collisions on the chemical rate constant. The dynamic behavior of a two‐level system undergoing chemical decomposition is given in detail. We predict that the unimolecular rate constant appears as a linewidth for a broadened elastic component in the scattered electromagnetic field and the inelastically scattered band intensity is directly proportional to the energy‐changing relaxation time. These scattered field measurements may be useful for measuring rapid chemical reactions and vibrational relaxation times.

Generalization of Hamilton's principle to continuous dissipative systems
View Description Hide DescriptionSome variational principles describing the behavior of nonstationary dissipative processes are presented. These criteria appear as an extension of Hamilton's principle to dissipative systems. Firstly, processes with vanishing barycentric velocity, i.e., of purely dissipative nature, are considered. The principle proposed is a slight modification of a criterion derived previously by the authors: The central quantity in the expression of the principle is however no longer the Legendre transform of the entropy per unit mass but is here the Legendre transform of the internal energy per unit volume. As an illustrative example, thermodiffusion is considered. Afterwards, the principle is generalized to describe the flow of a compressible, Newtonian fluid in which temperature effects are not neglected. After having shown that the Euler‐Lagrange equations are equivalent to the conservation laws, some particular expressions of the principle, corresponding to special situations like the stationary case and fixed boundary conditions, are given.

Hund's basic potential from direct summation
View Description Hide DescriptionThe Coulomb potential at points inside NaCl and CsCl structures was calculated previously by a method of direct summation. This method consists, simply, of adding all the Coulomb potentials, at the point of consideration, due to the point charges of the ions in a piece of the crystal. The shape of the piece of crystal should be determined correctly. Although this method succeeded when applied to NaCl and CsCl structures, it failed when applied to ZnS, CaF_{2}, and Cu_{2}O structures; the reason is that the last three structures lack a certain type of symmetry. However, a hypothetical structure H was constructed such that this method succeeded when applied to it. Moreover, Hund's basic potential can be expressed in terms of the potentials of H and CsCl by a simple relation derived from Hund's identity. Thus, Hund's basic potential was computed at a grid of points of intervals a/16 and tabulated. The potential inside all structures having the cubic Bravais lattices as components can, therefore, be computed easily from this table.

New equation for the influence of pressure on melting
View Description Hide DescriptionA new equation for the variation of melting points T_{m} of metals as a function of pressure p has been developed on the basis of simplified atomic considerations. The equation, [ΔT_{m}/T_{m} (0)]/βp≈ 2(γ_{ G } − 1/3)(1 + C _{2}βp), is thought to be in a more convenient form for comparison with experimental data than previous relationships. In this equation, β is the compressibility, γ_{ G } the Grüneisen parameter, and C _{2} is a constant related to the anharmonicity. In checking the equation it was found that a value of C _{2} = −2.9 appears to be more accurate for the ratio than the value −2.5 reported by Slater (a _{1} and a _{2} are the coefficients in the expression −ΔV/V _{0} = a _{1} p + a _{2} p ^{2}, for the compression of a material). Obviously, the proposed equation for ΔT_{m} should not be applied when the compression is not well described by the relation involving a _{1} and a _{2}.

^{75}As Zeeman perturbed NQR in antiferroelectric NH_{4}H_{2}AsO_{4}
View Description Hide DescriptionThe ^{75}As EFG tensors corresponding to the four ``nonferroelectric'' H_{2}AsO_{4} Slater configurations have been determined and the correctness of the antiferroelectricproton ordering scheme proposed by Nagamiya is demonstrated. A comparison of the theoretical and experimental temperature dependences of the ^{75}As NQR frequencies for T > T_{c} seems to show that the Slater parameter ε has the same sign in ADA as in KDA. The antiferroelectric transition thus seems to be triggered by long range dipole‐dipole forces rather than by short range ones.