Spherical internal coordinates

The SHAPES force field" has been implemented in CHARMM and used to examine the structures of several square planar rhodium complexes. This force field is based on angular overlap considerations and treats angular distortions for a variety of geometries. Spherical internal coordinates and Fourier potential functions form the basis for the description of these molecular shapes. The parameters for this force field were derived from normal coordinate analysis, ab initio calculations, and structure-based optimizations. The average rms deviation for bond lengths was 0.026 A, and the average rms deviation for bond angles was 3.2°. 

For molecules that are non-linear and whose rotational wavefunctions are given in terms of the spherical or symmetric top functions D l,m,K, the dipole moment Pave can have components along any or all three of the molecule s internal coordinates (e.g., the three molecule-fixed coordinates that describe the orientation of the principal axes of the moment of inertia tensor). For a spherical top molecule, Pavel vanishes, so El transitions do not occur. 

The moment equations of the size distribution should be used to characterize bubble populations by evaluating such quantities as cumulative number density, cumulative interfacial area, cumulative volume, interrelationships among the various mean sizes of the population, and the effects of size distribution on the various transfer fluxes involved. If one now assumes that the particle-size distribution depends on only one internal coordinate a, the typical size of a population of spherical particles, the analytical solution is considerably simplified. One can define the th moment // of the particle-size distribution by... 

The classical kinetic energy of the system has now been separated into the effect of displacement of the center of mass of the system, with momentum P and that of the relative movement of the two particles, with momentum p. In the absence of external forces, the interaction of the two (spherical) particles is only a function of (heir separation, r. That is, the potential function appearing in Eq. (37) depends only on the internal coordinates x, y, z. 

Clodius, W. B., and Quade, C. R. (1985), Internal Coordinate Formulation for the Vibration-Rotation Energies of Polyatomic Molecules. III. Tetrahedral and Octahedral Spherical Top Molecules, /. Chem. Phys. 82, 2365. 

In our non-BO calculations performed so far, we have considered atomic systems with only -electrons and molecular systems with only a-electrons. The atomic non-BO calculations are much less complicated than the molecular calculations. After separation of the center-of-mass motion from the Hamiltonian and placing the atom nucleus in the center of the coordinate system, the internal Hamiltonian describes the motion of light pseudoelectrons in the central field on a positive charge (the charge of the nucleus) located in the origin of the internal coordinate system. Thus the basis functions in this case have to be able to accurately describe only the electronic correlation effect and the spherically symmetric distribution of the electrons around the central positive charge. 

Another problem comes in examining the polarizability. In the physical picture, the spherically symmetric molecule, just like an atom, has isotropic polarizability. In the chemical picture, for a diatomic molecule we have two unique polarizabilities (1) and in the internal coordinate system or (2) dzz = 5 (o xc + (isotropic polarizability) and Aa = — [polar-... 

Coordinates of molecules may be represented in a global or in an internal coordinate system. In a global coordinate system each atom is defined with a triplet of numbers. These might be the three distances x,, y,-, z, in a crystal coordinate system defined by the three vectors a, b, c and the three angles a, / , y or by a, b, c, a, P, y with dimensions of 1,1,1,90°, 90°, 90° in a cartesian, i. e. an orthonormalized coordinate system. Other common global coordinate systems are cylindrical coordinates (Fig. 3.1) with the coordinate triples r, 6, z and spherical coordinates (Fig. 3.2) with the triples p, 9, . 

Because the orbital angular momentum of the positron-hydrogen system is zero for s-wave scattering, the total wave function is spherically symmetric and depends only on the three internal coordinates which specify the shape of the three-body system. The kinetic energy operator... 

Distances and angles. Structures can be presented in an internal coordinate system (symmetry adapted coordinates used in spectroscopy or Z-matrices, that is interatomic distances, three center angles and four center angles) instead of a global coordinate system (coordinate triples, for example cartesian, crystal, cylindrical or spherical coordinates). 

When the elements of the disperse phase can be classified as equidimensional, namely they have nearly the same size or spread in multiple directions, and have constant material density, typically a single internal coordinate is used to identify the size of the elements. This could be particle mass (or volume), particle surface area or particle length. In fact, in the case of equidimensional particles these quantities are all related to each other. For example, in the trivial cases of spherical or cubic particles, particle volume and particle surface area can be easily written as Vp = k d and Ap = k d, or, in other words, as functions of a characteristic length, d (i.e. the diameter for the sphere and the edge for the cube), a volume shape factor, k, and a surface-area shape factor, k. For equidimensional objects the choice of the characteristic length is straightforward and the ratio between kp, and k is always equal to six. The approach can, however, be extended also to non-equidimensional objects. In this context, the extension turns out to be very useful only if... 

Ap is the difference in material density between the liquid and gas phases. This situation is typically handled by describing the bubbles with a single internal coordinate (i.e. the equivalent-sphere diameter) and by introducing an aspect ratio, defined as the ratio between the minor and the major axes of the bubble. This aspect ratio E can be calculated by using the empirical equation proposed by Moore (1965) as a function of the Morton number E = 1/(1 + 0.043RCp Mo ). An alternative to this is the use of the correlation proposed by Wellek et al. (1966) for liquid-liquid droplets E = 1/(1 + 0.1613Eo° ), which is valid for Eo < 40 and Mo < 10 , whereas for Eo > 40 and RCp > 1.2 fluid particles are typically of spherical shape. Once the characteristic E value is known, the ratio of the real area of the bubble Ap and the area Aeq of a sphere with an equivalent volume can be calculated as follows ... 

The spatial arrangement of the atoms constituting a material is specified completely by the topology and geometry. Topology is simply the pattern of interconnections between atoms. It is often expressed in the form of a connectivity table. Geometry also encompasses the coordinates of the atoms, usually in Cartesian (x, y and z) coordinates but sometimes in alternative coordinate systems such as spherical, cylindrical or internal coordinates. 

We start with a model of polar molecules in which the effects of polarizability are neglected. More precisely, we assume that in the absence of external fields, the potential energy associated with N particles is a sum of pair potentials < >( /), each of which depends on the positions r, and tj and orientations S2, and itj of particles / and j. Thus the particles are regarded as rigid, with no internal coordinates, and we assume for simplicity that they are all identical. Extensions of the results of Section II to mixtures are for the most part straightforward, as discussed by Hoye and StelP and in references they cite. Pertinent references to the mean spherical approximation generalized to mixtures are also given at an appropriate point in this chapter. 

Associated with these coordinates we define a system of internal q-space mathematical axes OXkYZk in which a general point P has spherical polar coordinates p, oox, and yk. The cartesian coordinates of P in this space are... 

Figure 5 Equipotential energy surface for a system of three atoms in symmetrized internal coordinates. The OXK = OYk, OY = OZ, and OZx = OXK define two related nonphysical internal nuclear coordinate frames and spaces. A point P in those spaces has spherical polar coordinates p, cox, yx in the OXkYZx frame and p, 0,

If we transform to the center of mass and restrict attention to spherically symmetric states, then the Laplacians can be rewritten in terms of the internal coordinates r.-... 

Consider the collision of two particles initially in internal states described by an index i. To simplify notation, it is convenient to use a single index to specify the states of both particles. The angle between the initial and final relative velocities v and 1/ is given by spherical polar coordinates and , where 0 is the deflection angle in the center of mass frame. We start with a well-defined beam of particles with a flux li (number of particles per unit area per unit time). After the collision, the flux Ij (number of particles per unit solid angle per unit time) is a function of the deflection angle 0 and is different for each possible set of final internal states j. We define the differential cross-section as... 

The particulate nature of solids is characterized by a number of properties, such as size, shape, liquid and gas content, porosity, composition, and age. These are denoted as internal coordinates, whereas Euclidian coordinates, such as rectangular coordinates (x, y, z), cylindrical coordinates (r, 4>, z), and spherical coordinates (r, 6, 4>) used to specify the locations of particles, are defined as external coordinates. 

One manner of using the presented results is to incorporate them in the traditional way of tackling fluid bed granulation theoretically, namely population balance modeling. This can be achieved by expanding the population balance to more internal coordinates than just particle size (see Volume 1 of this series. Chapter 6, Section 6.9.1). The additional property in the case of the present example would be wet a lomerate composition, defined either by the mass fraction of solids within one particle or the binder/solid ratio. The latter can be further spht up to account for the spatial distribution - and, thus, accessibUity - of the hquid binder and for the thickness of the binder layer on the outer surface of the agglomerate. Alternatively, discrete models of agglomeration (see Section 7.7) could be expanded to account for non-spherical primary particles. 

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