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Volume 72, Issue 7, 01 April 1980

Momentum space diffusion equations for chain molecules
View Description Hide DescriptionA Fokker–Planck equation is derived for the reorientational and torsional motion of a flexible butane molecule. The analysis starts with the Liouville operator for a chain molecule in a fluid and by projection operator techniques yields an equation of motion for the reduced distribution function of the Euler angles, torsion angles, and the molecule‐fixed angular and torsional momenta. A diffusionequation for butane is also derived by projection operator procedures. A fictitious metric potential arises in the diffusionequation and must be deleted by an offsetting potential. The time dependence of the diffusion coefficient, to lowest order in the streaming terms, can be obtained and yields a criteria for the applicability of Brownian motion models. The second moment of the torsion angle correlation function is also obtained and demonstrates a dependence on the dihedral potential, in contrast to the second moment of rigid body orientational correlation function which depends only on k _{ B } T and the appropriate components of the inertia tensor.

Valence bond model potential energy surface for H_{4}
View Description Hide DescriptionPotential energy surfaces for the H_{4} system are derived using the valence bond procedure. An a b i n i t i o evaluation of the valence bond energy expression is described and some of its numerical properties are given. Next, four semiempirical evaluations of the valence bond energy are defined and parametrized to yield reasonable agreement with various a b i n i t i o calculations of H_{4} energies. Characteristics of these four H_{4} surfaces are described by means of tabulated energy minima and equipotential contour maps for selected geometrical arrangements of the four nuclei.

Comparison of reactive and inelastic scattering of H_{2}+D_{2} using four semiempirical potential energy surfaces
View Description Hide DescriptionCollisions between hydrogen and deuterium molecules are examined using quasiclassical dynamical trajectory calculations with the intermolecular field specified by four semiempiricalpotential energy surfaces. Three of the surfaces are calculated within the valence bondmodel with semiempirical evaluation of the integrals, and the fourth is the London type. Various degrees of agreement are observed between these four surfaces and a b i n i t i o results. The trajectory calculations are performed at high system energies to permit the possibility of reactions. In addition to nonreactive collisions, four reaction paths are found on each surface with the product species 2H+D_{2}, H_{2}+2D, HD+H+D, and 2HD. The results are analyzed to determine the effect of surfaceproperties on reaction probabilities, average final state properties of the molecules and average final state energy distributions. Dynamical results are found to be strongly dependent on surface characteristics.

Electric deflection studies of metastable acetylene
View Description Hide DescriptionMolecular beams of a metastable electronic state of acetylene are prepared by electron impact excitation. The threshold for appearance of the state is near 4.5 eV. Electric deflection measurements of C_{2}H_{2}, C_{2}D_{2}, and HC_{2}D show the polarity of all three isotopes to be below the level of detectability. Comparison of the behavior of the metastable state with that of the ground electronic state of HC_{2}D, where polarity is readily established by electric deflection, is made. The polarity results are shown to establish a t r a n s geometry for the observed state.

The electronic structure of β‐alumina surface with an adsorbed sodium atom
View Description Hide DescriptionThe electronic states of a sodium atom adsorbed at a site on the a csurface of β‐alumina is studied by the SCF–Xα method. The system is represented by a stabilized ion cluster NaAlO_{3}. Electrons of all atoms except 1selectrons are included in the self‐consistency calculation. A singly occupied surface state is found to appear at 3.8 eV above the highest doubly occupied state. This surface state accepts the sodium 3selectron and represents an electronic state loosely bound to the sodium ion with the binding energy of 2.8 eV.

Generalized Langevin theory for many‐body problems in chemical dynamics: Modelling of solid and liquid state response functions
View Description Hide DescriptionThe equivalent harmonic chain heatbath modelling method [S. A. Adelman, J. Chem. Phys. 71, 4471 (1979)] for reducing problems in condensed phase chemical dynamics to effective few‐body trajectory problems is tested via the calculation of typical solid and liquid state velocity response (correlation) functions χ̇ (t). Finite nearest neighbor harmonic chain models for condensed phase atomic dynamics which may be used in trajectory simulations, are constructed from moments of the spectral density of χ̇ (t) and from additional information about the small z behavior of e.g., diffusion coefficients. Model response functions are then generated numerically as specified trajectories of the chain equations of motion. The convergence of chain model response functions is investigated for the bulk elasticmodel (isotropic continuum elastic treatment) of a solid and for the Gaussian approximation to the Levesque–Verlet molecular dynamics results for a Lennard‐Jones liquid. For both cases, the chain model heatbaths are found to give response functions which: (i) converge rapidly and systematically; (ii) exactly reproduce the chemically crucial short‐time response and thus account for caging (in liquids) and initial chemical system/heatbath energy transfer; (iii) reproduce the chemically less important long time response in a qualitatively correct manner. It is thus concluded that the chain modelling techniques yield a chemically and calculationally useful reduction of condensed phase dynamics.

O_{2}(b ^{1}Σ_{ g } ^{+}) production and deactivation following quenching of O(^{1} D _{2}) in O_{3}/O_{2} mixtures
View Description Hide DescriptionThe production of oxygen atoms, O(^{3} P _{ J }), following deactivation of photolytically generated O(2 ^{1} D _{2}) via was monitored using time‐resolved atomic absorption analysis in the vacuum ultraviolet. The measured overall rate constant for O_{2}(b ^{1}Σ_{ g } ^{+}) deactivation by O_{3}, 1.8±0.2×10^{−11} cm^{3} molecule^{−1} s^{−1}, and total yield of ground state O(^{3} P _{ J }) following quenching of O_{2}(b ^{1}Σ_{ g } ^{+}) by O_{3}, 0.61±0.06, are in good agreement with a previous study. These results suggest that the reaction of O_{2}(b ^{1}Σ_{ g } ^{+}) with O_{3} may play a role in fixing the overall density of oxygen atoms in the stratosphere.

The reactivity dependence of the recombination probability
View Description Hide DescriptionAn exact relation for the reactivity dependence of the recombination probability in the presence of a scavenger is derived. For a spherical symmetric system with a uniformly distributed scavenger this relation expresses the p a r t i a l l ydiffusion controlled recombination probability in terms of the t o t a l l ydiffusion controlled recombination probability. The relation is valid for a space‐dependent diffusion coefficient and for an arbitrary potential energy between the particles. The relation is proved only for a three‐dimensional system, but it appears to be valid also for one‐ and two‐dimensional systems.

Proton–H_{2} scattering on an a b i n i t i o CI potential energy surface. I. Vibrational excitation at 10 eV
View Description Hide DescriptionA complete configuration interaction (CI) ground statesurface for the H_{3} ^{+} system has been calculated using 5S and 3(P _{ x },P _{ y },P _{ x }) basis functions at each center. A total of 650 nuclear geometries has been considered which makes the new surface appropriate not only for scattering calculations, but also for the evaluation of the vibrational–rotational spectrum of the H_{3} ^{+} molecule. Significant deviations are found from the analytic Giese and Gentry potential used in many previous theoretical studies, especially for large and small nonequilibrium H–H separations which are important for vibrational excitation of the H_{2} molecule. Vibrational–rotational excitation cross sections have been calculated in the rotational sudden approximation where the vibrational degree of freedom is treated exactly by solving seven vibrationally coupled radial equations. The use of the new surface leads to increased vibrational excitation compared to previous calculations utilizing the same scattering approximation and to excellent agreement at 10 eV with the angle‐dependent measurements of Hermann, Schmidt, and Linder.

Proton–H_{2} scattering on an a b i n i t i o CI potential energy surface. II. Combined vibrational–rotational excitation at 4.67 and 6 eV
View Description Hide DescriptionInfinite‐order‐sudden calculations have been performed at 4.67 and 6 eV on the a b i n i t i o CI potential energy surface determined recently by Schinke, Dupuis, and Lester. The vibrational degree of freedom has been treated exactly by solving vibrationally coupled radial equations. The rotationally summed differential cross sections for vibrational excitation are in good agreement with the measurements of Schmidt, Hermann, and Linder. It is shown that the rotational excitation cross sections in the vibrational ground state near the rainbow angle are almost exclusively determined by the potential between 2.5a _{0} and 5a _{0} proton–H_{2} separations. In this region only the V _{2} term of an expansion into Legendre polynomials is nonvanishing and is a factor of ∼3 smaller for the new surface than for the Giese and Gentry analytic potential. These differences result in a dramatic decrease of the rotational excitation cross sections in the rainbow region so that the present theoretical transition probabilities are in much better agreement with the experiments than our previous sudden vib–rotor calculations utilizing Giese and Gentry’s surface.

Effect of fluctuations in internal coordinates on dimensions of macromolecules
View Description Hide DescriptionThe effect of fluctuations in the internal coordinates (bond lengths, bond angles, and bond rotational angles) on the end‐to‐end length of a macromolecule in a fixed rotational isomeric state is considered. Equations are derived for the first and second moments of the deviations in the end‐to‐end distance and the end‐to‐end vector. Approximations are then made for long chains that express these moments in terms of the relative position of the center of mass and chain ends, the radius of gyration, the end‐to‐end length, and the moment of inertia about the end‐to‐end axis. The distribution functions of both the end‐to‐end length and end‐to‐end vector are examined. The uncertainty in the position of the end of the chain can be a substantial portion of the end‐to‐end distance. Implications for chain dynamics are also discussed.

A quasiclassical trajectory study of collisional excitation in Li^{+}+CO_{2}
View Description Hide DescriptionWe present quasiclassical trajectory calculations of the state‐to‐state differential cross sections for vibrational excitation in Li^{+}+CO_{2} collisions and compare our results with analogous results of molecular beam experiments. In the trajectory calculations, the initial and final semiclassical eigenstates of CO_{2} are numerically determined before and after each collision by using a classical perturbation theory calculation of the good action‐angle variables associated with molecular vibrational motion. Two approximations are used to simplify this action‐angle analysis. First, an angular motion sudden approximation is introduced into the dynamics to separate angular from vibrational motion in solving the molecular Hamilton–Jacobi equation. Second, the off‐diagonal parts of the intramolecular potential are neglected to eliminate Fermi resonant coupling between the bending and symmetric stretch modes. This latter approximation precludes the accurate determination of state‐to‐state cross sections to certain nearly degenerate states such as (020) and (100), but should still enable the accurate determination of the sums of the cross sections to those states (which is all that is available from experiment). The intramolecular potential is approximated in two different ways, both using pairwise additive potentials. In Surface I, the usual ion‐induced dipole long range interaction is added to a sum of He–Ne pair potentials which simulate the short range Li^{+}–C and Li^{+}–O potentials. In Surface II, the sizes of the radius parameters in the short range part of Surface I are changed to correctly reproduce the anisotropy present in the experimentally derived He–CO_{2} interaction potential. The resulting ratios of inelastic to elastic differential cross sections (for the states (010), (020)+(100) and (030)+(110)) are in reasonable quantitative agreement with the experimental measurements, with errors typically smaller than a factor of two using Surface II at 4.72 eV translational energy and a factor of three at 6.87 eV. Some qualitative features of the angular distributions are actually quite accurately described, including the crossing of the (010) and (020)+(100) ratios near 24° at 4.72 eV and 18° at 6.87 eV, and the similar angular dependence of the (020)+(100) and (030)+(110) cross sections. In addition, a detailed interpretation of many features of the distributions of final vibrational states is developed, including relative propensities for certain types of overtone and combination mode excitation, and the variation in angular distributions as a function of final vibrational state.

Brownian dynamics study of a polymer chain of linked rigid bodies. II. Results for longer chains
View Description Hide DescriptionAs an extension of a previous paper, computer simulation results for a polymer chain of linked rigid bodies are presented for chains with between four and fifteen bonds. Previous theoretical results for a three‐bond chain are found by computer simulation to apply to longer chains. In particular, a Fixman potential for an N‐bond chain is developed as the sum of three‐bond Fixman potentials, and transition state rates measured in computer simulation for a four‐bond chain compare reasonably well with the theory for a three‐bond chain. A further study of the motion of the free end of a chain with one end fixed indicates correlation effects diminish the possibility of a ’’whipping motion’’ of the chain end.

The Raman excitation spectra and absorption spectrum of a metalloporphyrin in an environment of low symmetry
View Description Hide DescriptionThe effects of a symmetry lowering environment on the Raman excitation spectra of a metalloporphyrin have been studied using the generalized crude Born–Oppenheimer vibronic theory. Symmetry considerations reduce the number of matrix elements introduced by the interaction of the porphyrin with its environment. In practice, matrix elements of the model are determined by fitting the Raman excitation spectra. These matrix elements contain information about the interaction of the metalloporphyrin with its environment. This procedure is facilitated if the Raman excitation spectra of the metalloporphyrin in a more symmetrical environment have been measured and analyzed, since only the additional matrix elements describing the differences in the interaction with the environment must be determined. Some of the effects of environmental interactions on the excited states of metalloporphyrins are discussed. Calculations demonstrate that Raman excitation spectra and a new tool—Raman excitation difference spectra, provide a valuable means of examining these environmental effects.

Theory of hydrogen bonded chains in bioenergetics
View Description Hide DescriptionThree thermodynamic cycles in which hydrogen bonded chains play a central role are presented as models for biological energy transduction. These include (i) a primitive, highly irreversible proton pump, (ii) a simple reversible chemical to electrical energytransducer and (iii) a reversible molecular engine which transduces pH differences to mechanical energy. A careful analysis is given to the free energy diagrams which reveal that the rate limiting step of the protonconduction is similar in the three cases. The limiting rate is essentially the inverse of the equilibration time for a localized proton to delocalize over a free energy barrier. Using basic hopping rates obtained from studies in ice, the kinetic equations are set up and solved by three methods for the simplest case of first passage of the proton. The agreement of the exact solution with a computerized difference equation approximation lends confidence to calculations using the latter method for more complicated cases where the exact method is intractable. For free energy barriers of 250–450 mV (6–10 kcal), equilibration times are rapid (μsec–msec) on a biological time scale. The kinetics of the complete cycles are also investigated and shown to be in agreement with the preceding calculation. For steady state linear transport and the special case when the transport of a single defect type is rate limiting, our kinetic results reduce to those obtained for single‐file transport.

Non‐Markovian effects on vibrational energy relaxation in liquids
View Description Hide DescriptionWe consider the non‐Markovian effects on vibrational relaxation which can occur when the bath relaxation rate is comparable to or slower than the relaxation of the vibrational degrees of freedom. We carry out quantum mechanical simulations of a two level system strongly coupled to a bath of known statistical properties (either a Poisson or a Gaussian process) and show that the relaxation is surprisingly insensitive to the nature of the bath. Small oscillations in the relaxation are found for the Poisson simulation which maybe observable in high resolution picosecond experiments. We investigate both cumulant expansion and memory function descriptions of relaxation in the non‐Markovian limit and show that neither approach provides quantitatively accurate results.

A theoretical study on the water structure for nucleic acids bases and base pairs in solution at T=300 K
View Description Hide DescriptionMonte Carlo simulations are presented for the bases A, C, G, T, and U enclosed in a cluster of forty water molecules and for the base pairs A–T, G–C, and A–U enclosed in a cluster of fifty water molecules. These computations are carried out at a simulated temperature of 300 K. The importance of the temperature effect is shown by carrying the MC simulation for adenine at T=100 K. The intermolecular interactions are computed with atom–atom pair potentials obtained from quantum‐mechanical calculations on the water–water and water–base complexes. The identification of the positions and orientations for the water molecules in the first hydration shell is obtained from hydrogen and oxygen atoms probability distribution maps. Relations between the water structure around the bases and base pairs and the water structure in the major and minor groove of B‐DNA double helix are preliminarily presented.

Theory of polymer conformation based on the correlated walk model
View Description Hide DescriptionA model polymer is allowed to grow in the same direction as that of the previous unit with probability α or turn at right angles with probability β but not to reverse. The mean square end‐to‐end distance <R^{2}≳ of a polymer of N units, is calculated to be 〈R ^{2}〉/a _{0} ^{2}=[(1+α)/(1−α)]N−2α (1−α^{N})/(1−α)], where a _{0} is the unit length. An exact expression for the probability that the Nth unit points in the same direction as that of the zeroth unit is also obtained. A general way of interpreting the conformation of simple polymers in terms of the present model and the connection with other theories of the polymer conformation are discussed.

Polarized spectroscopy of partially oriented molecules. A general formulation for two‐photon processes in uniaxial samples
View Description Hide DescriptionGeneral expressions are given for linearly polarized intensity in a two‐photon experiment on a uniaxial assembly of partially aligned molecules of arbitrary shape and symmetry, without any assumptions concerning the detailed nature of the uniaxial orientation distribution. The formulas are valid for electric dipole interactions and apply to measurements such as ordinary and resonant Raman scattering, two‐photon absorption, photoluminescence, and transient or permanent photoinduced dichroismmeasured in stretched polymers,nematic liquid crystals, or lipid bilayers. Two examples are worked out in some detail: nonresonant Raman scattering and photoluminescence of symmetrical molecules. A discussion of the physical significance of the orientation factors and of their interrelation is given.

Rigorous and approximate relations between expectation values of atoms
View Description Hide DescriptionThe Z dependence of some expectation values of local operators, such as r^{α} and ρ^{α−1}, is derived within the Thomas–Fermi model for atoms. It is shown to have the general form F (α) =A (α) Z^{β} ^{(α)}, where β (α) =aα+b. This equation leads to further relations among the expectation values and the total energy. The parameters A (α) and β (α) are also treated variationally to get the best agreement with the Hartree–Fock expectation values and it is shown that for a certain range of α (depending on the operator) the agreement is quite good. Further, some inequalities relating the aforementioned expectation values to ρ (0), the charge density at the nucleus, are developed. Finally, the N dependence of the Z ^{−1} expansion coefficients is considered and a conjecture by March and White [N.H. March and R.J. White, J. Phys. B 5, 466 (1972)] concerning their asymptotic behavior is proved for ε_{1}(N).