Molecules nonspherical

Parker G A and Pack R T 1978 Rotationally and vibrationally inelastic scattering in the rotational lOS approximation. Ultra-simple calculation of total (differential, integral and transport) cross sections for nonspherical molecules J. Chem. Phys. 68 1585... 

By the last two assumptions the theory, strictly speaking, is only applicable to the monatomic gases A, Kr, Xe, to a somewhat lesser extent to the almost spherical molecules CH4, CF4, SFe, and perhaps to nonpolar diatomic molecules. The rotation of even slightly nonspherical molecules like Q2 and N2 will not be free in the entire cavity when such a molecule comes close to the wall of its cage it will have to orient itself parallel to this wall. Furthermore, some of the cavities are somewhat oblate (cf. Section I.B), and thus the rotation of relatively large, oblong molecules may be seriously... 

The quantity riV/RT is equal to six times the rotational period. The rotational relaxation time, p, should he shorter than the fluorescence lifetime, t, for these equations to apply. It is possible to perform calculations for nonspherical molecules such as prolate and oblate ellipsoids of revolution, but in such cases, there are different rotational rates about the different principal axes. 

Time Resolved Fluorescence Depolarization. In Equation 3, it is assumed that the polarization decays to zero as a single exponential function, which is equivalent to assuming that the molecular shape is spherical with isotropic rotational motion. Multiexponential decays arise from anisotropic rotational motion, which might indicate a nonspherical molecule, a molecule rotating in a nonuniform environment, a fluorophore bound to tbe molecule in a manner that binders its motion, or a mixture of fluorophores with different rotational rates. 

Ionization surfaces calculated for several of the molecules listed in Table 1 are shown in Figures 10 and 11. The volume averaged cross sections determined from the volumes enclosed by these surfaces for nonspherical molecules such as C02 are in much better accord with experiment, consistent with the idea that the poor performance of Cartesian averaging for molecules such as C02 is due to the large departure from a spherical shape. Improved agreement with experiment is also... 

The original work by van de Waals and Platteeuw (1959) used the Lennard-Jones 6-12 pair potential. McKoy and Sinanoglu (1963) suggested that the Kihara (1951) core potential was better for both larger and nonspherical molecules. The Kihara potential is the potential currently used, with parameters fitted to experimental hydrate dissociation data. However, it should be noted that the equations presented below are for a spherical core, and while nonspherical core work is possible, it has not been done for hydrates. 

On the basis of these approximations, ideal solubility can then be estimated on the basis of the structure of molecules. Usin n = 13.5 for rigid and nonspherical molecules as described in Equation 3.17, ideal solubility given by Equation 3.14 can be approximated by... 

Because the radius of a nonspherical molecule cannot be defined precisely, molecular friction coefficients and diffusion coefficients are often related to the Stokes radius (or Stokes diameter). This is defined as the radius (or diameter) of a sphere having / and D values identical to those of the molecule under consideration. 

Figure 11-2. Steric interference. (A) To avoid steric violations in placing die centers of mass of two spheres in space, their separation must be greater than one sphere diameter. (B) For nonspherical molecules, steric violations are determined by a complex coupling of center of mass positions and orientations (for rods) or chain conformations (for polymers), (a) and (b) show two identical center of mass positions for two molecules. In (a) the relative orientations lead to no steric conflict, while die configurations in (b) are impossible because of steric violations. The figure is taken from Chan and Dill [18] with permission...

Steele, W. A., and R. Pecora Scattering from fluids of nonspherical molecules, I. X-Rays and Neutrons. J. Chem. Phys. 42, 1863 (1965). 

S. Gupta, Comput. Phys. Commun. 48, 197 (1988). Vectorization of Molecular Dynamics Simulation for Fluids of Nonspherical Molecules. 

The time dependence of the anisotropy r(t) depends on the underlying dynamics of reorientational motion. For rotational diffusion (tumbling) of a spherical object, the expected anisotropy decay is exponential with a rotational diffusion time given in the hydrodynamic limit by the Stokes-Einstein-Debye equation. For nonspherical molecules, more complex time dependence may be detected. (For more on these topics, see the book by Cantor and Schimmel in Further Reading.)... 

Table 2 Contributions to the Energy of Interaction Between Nonspherical Molecules ...

Evans. G.T.. She. R.S.C. and Bernstein. R.B. (1985) A simple kinetic theory model of reactive collisions of rigid nonspherical molecules. J. Chem. Phys. 82, 2258-2266... 

Teja and Rice [134] have proposed another relationship to calculate the viscosity of mixtures of polar liquids. It is based on a corresponding states treatment for the mixture compressibility factors. This method is more accurate than the previous one for polar-polar mixtures, particularly for aqueous solutions. It uses two reference-solution models for nonspherical molecules and a single interaction parameter. For methanol-water systems, the accuracy was within 9% [5]. 

Polarizabilities for nonspherical molecules are averages over the three principal axes. 

For a system of rigid, nonspherical molecules, the derivation of the pressure equation is essentially the same as that for spherical molecules. The result is... 

A variety of approaches have been used in the development of models for the interactions between nonspherical molecules, reflecting both the complexity of such interactions and the paucity of truly accurate quantitative information about them that could be used to assess the suitability of different models. Perhaps the most widely studied approach, especially over the last decade or so, is the interaction site formalism, in which the intermolecular pair potential is written as a sum of potentials between interaction sites on each molecule. We have... 

In this section, we review some of the important formal results in the statistical mechanics of interaction site fluids. These results provide the basis for many of the approximate theories that will be described in Section III, and the calculation of correlation functions to describe the microscopic structure of fluids. We begin with a short review of the theory of the pair correlation function based upon cluster expansions. Although this material is featured in a number of other review articles, we have chosen to include a short account here so that the present article can be reasonably self-contained. Cluster expansion techniques have played an important part in the development of theories of interaction site fluids, and in order to fully grasp the significance of these developments, it is necessary to make contact with the results derived earlier for simple fluids. We will first describe the general cluster expansion theory for fluids, which is directly applicable to rigid nonspherical molecules by a simple addition of orientational coordinates. Next we will focus on the site-site correlation functions and describe the interaction site cluster expansion. After this, we review the calculation of thermodynamic properties from the correlation functions, and then we consider the calculation of the dielectric constant and the Kirkwood orientational correlation parameters. 

THE STATISTICAL MECHANICS OF INTERACTION SITE FLUIDS to fluids of rigid nonspherical molecules... 

The general formulation of density-functional theory for molecular fluids has been described by Smithline et al. They have applied a simplified version of the theory to the freezing of hard core diatomic fluids. A suitable starting point for such theories for rigid nonspherical molecules is the following expression for the grand potential... 

Sokolowski and Steele have adapted the spherical harmonic expansion technique described in Section III.A to the calculation of the density profiles of nonspherical molecules in contact with solid surfaces. They have used the results to investigate molecular orientation effects in high temperature adsorption from the gas phase. The RAM theory described in Section III.E has been extended to the adsorption of fluids on solid surfaces by Smith et al. ° for hard-sphere interactions and by Sokolowski and Steele to the case of more realistic fluid-solid interactions. The principal deficiency of the approach is the accuracy of the predicted correlation functions for the bulk fluid which are required as input to the theory. 

The COR eos is made up of three terms the CS hard-sphere eos, the rotational pressure of liquation (4.181), and Alder et al. s double-series attractive pressure modified for nonspherical molecules. 

Barker [92-94] has presented a general formulation of the cell theory and we give a brief review of his approach here. We will restrict our discussion to single-component atomic solids and discuss the application to mixtures and nonspherical molecules later. Suppose we have a system of N molecules in the canonical ensemble. The configurational partition function, Eq. (2.205), may be rewritten by breaking the volume into N identical subvolumes or cells so that... 

Many of the papers on DFT have focused primarily on the hard-sphere system, and it is for this system that most success has been achieved. However, DFT has also been applied to the Lennard-Jones 12-6 system, binary mixtures, nonspherical molecules, and coulombic systems. We will discuss some of these applications later in the chapter as we review what is known about the phase diagrams of various models systems. 

The application of this approach to the hard-sphere system was presented by Ree and Hoover in a footnote to their paper on the hard-sphere phase diagram. They made a calculation where they used Eq. (2.27) for the solid phase and an accurate equation of state for the fluid phase to obtain results that are in very close agreement with their results from MC simulations. The LJD theory in combination with perturbation theory for the liquid state free energy has been applied to the calculation of solid-fluid equilibrium for the Lennard-Jones 12-6 potential by Henderson and Barker [138] and by Mansoori and Canfield [139]. Ross has applied a similar approch to the exp-6 potential. A similar approach was used for square well potentials by Young [140]. More recent applications have been made to nonspherical molecules [100,141] and mixtures [101,108,109,142]. 

Glass formation, the other common fate of a compressed fluid, however, gives a much larger viscosity increase. For mixtures and fluids with nonspherical molecules, it is common to produce a metastable glass at some pressure in excess of the equilibrium crystallization point. Methanol is one example. It can be easily superpressed past its crystallization pressure of 3.5 GPa at T = 25°C towards its glass-transition pressure of 11.4 GPa and beyond. Being a... 

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