Spherical polar system

This equation may be transformed into other systems of orthogonal coordinates, the most useful being the spherical polar system. (Carslaw and Jaeger, Conduction of Heat in Solids, gives details of the transformation.) When the operation is performed ... 

Use a spherical polar system on centre C to re-express the Cartesians on C in a form adapted to the ciction of the angular projectors ... 

Some coordinate transformations are non-linear, such as converting from a Cartesian to a spherical polar system. Here the r, 6, (p coordinates are related to the x, y, z coordinates by square root and trigonometric functions, as shown in Figure 16.1. Other coordinate transformations are linear, with the new coordinates given as linear combinations of the old ones. A linear transformation can be described as a rotation of the coordinate system. 

However, in the spherical polar system, the orbitals can be expressed as a product of fwo separate factors a radial factor, R, that depends only on r, and an angular factor, Y, that depends on 6 (theta) and (f) (phi) That is,... 

The divergences are obtained from analysis of the flow into and out of the volume element at each face. The spherical polar system is naturally one... 

For the interaction between a nonlinear molecule and an atom, one can place the coordinate system at the centre of mass of the molecule so that the PES is a fiinction of tlie three spherical polar coordinates needed to specify the location of the atom. If the molecule is linear, V does not depend on <() and the PES is a fiinction of only two variables. In the general case of two nonlinear molecules, the interaction energy depends on the distance between the centres of mass, and five of the six Euler angles needed to specify the relative orientation of the molecular axes with respect to the global or space-fixed coordinate axes. 

Consider the case of two neutral, linear, dipolar molecules, such as HCN and KCl, in a coordinate system with its origin at the CM of molecule A and the z-axis aligned with the intemiolecular vector r pointing from the CM of A to the CM of B. The relative orientation of the two molecules is uniquely specified by their spherical polar angles 0, 03 and the difierence <]) = - <])3 between their azimuthal angles. The leading temi in the... 

If the scattering system is isotropic, equation (Bl.9.54) can be expressed in spherical polar coordinates (the derivation is similar to equation (B 1.9.32)) ... 

It is convenient to employ two sets of coordinate systems. Spherical polar coordinates r, Q, A) are defined with the origin at the vertex of the cone the axis is 0=0, the surface of the conical portion of the cyclone is the cone 0 = 0% and the azimuthal coordinate is A. Using the same origin, cylindrical polar coordinates (R, A, Z) are defined, where R = r sin 0 and the Z-axis coincides with the axis 0=0. 

The simulations to investigate electro-osmosis were carried out using the molecular dynamics method of Murad and Powles [22] described earher. For nonionic polar fluids the solvent molecule was modeled as a rigid homo-nuclear diatomic with charges q and —q on the two active LJ sites. The solute molecules were modeled as spherical LJ particles [26], as were the molecules that constituted the single molecular layer membrane. The effect of uniform external fields with directions either perpendicular to the membrane or along the diagonal direction (i.e. Ex = Ey = E ) was monitored. The simulation system is shown in Fig. 2. The density profiles, mean squared displacement, and movement of the solvent molecules across the membrane were examined, with and without an external held, to establish whether electro-osmosis can take place in polar systems. The results clearly estab-hshed that electro-osmosis can indeed take place in such solutions. 

The most important parameters of the ionic atmosphere are the charge density Qv r) and the electrostatic potential /(r) at the various points. Each of these parameters is understood as the time-average value. These values depend only on distance r from the central ion, not on a direction in space. For such a system it is convenient to use a polar (spherical) coordinate system having its origin at the point where the central ion is located then each point can be described by a single and unique coordinate, r. 

The components of the direction vector are related in the usual way to the azimuthal () and polar (9) angles of a spherical polar coordinate system,... 

Figure 8.3. Spherical polar coordinate systems used to describe the fluorescence excitation problem.

The classical potential energy term is just a sum of the Coulomb interaction terms (Equation 2.1) that depend on the various inter-particle distances. The potential energy term in the quantum mechanical operator is exactly the same as in classical mechanics. The operator Hop has now been obtained in terms of second derivatives with respect to Cartesian coordinates and inter-particle distances. If one desires to use other coordinates (e.g., spherical polar coordinates, elliptical coordinates, etc.), a transformation presents no difficulties in principle. The solution of a differential equation, known as the Schrodinger equation, gives the energy levels Emoi of the molecular system... 

While it is possible to solve the vibrational problem using any one of a number of coordinate systems (Cartesian coordinates, spherical polar coordinates, valence coordinates, etc.), it is often most convenient to employ some type of valence... 

Spherical polar coordinates are used for conformational representation of pyranose rings in the C-P system. Unlike the free pseudorotation of cyclopentane, the stable conformations of cyclohexane conformers are in deeper energy wells. Even simong the (less stable) equatorial (6 = 90 ) forms, pseudorotation is somewhat hindered. Substitutions of heteroatoms in the ring and additions of hydroxylic or other exocyclic substituents further stabilize or destabilize other conformers compared to cyclohexane. A conformational analysis of an iduronate ring has been reported based on variation of < ) and 0 (28), and a study of the glucopyranose ring... 

Symbol for quantum yield. 2. Symbol for one of the space coordinates in the three-dimensional, spherical polar coordination system. 3. Symbol for electric potential. 4. Symbol for volume fraction. 5. With a subscript designation, symbol for a Dalziel coefficient. 6. Symbol for fugacity coefficient. 7. Symbol for osmotic coefficient. 8. Symbol for heat flow rate. 

Symbol (0) for characteristic temperature. 2. Symbol (0) for degree of saturation of binding sites as defined in the Langmuir isotherm treatment for adsorption of a ligand onto a surface. See Langmuir Isotherm. 3. Symbol (0) for plane angle. 4. Symbol (0) for one of the space coordinates in the three-dimensional, spherical polar coordinate system. 5. Symbol (0) for Celsius temperature. 

The Fourier transform of the spherical atomic density is particularly simple. One can select S to lie along the z axis of the spherical polar coordinate system (Fig. 1.4), in which case S-r = Sr cos. If pj(r) is the radial density function of the spherically symmetric atom,... 

For linear molecules, convention dictates that the high-symmetry axis be the z axis and then the Hessian of II(p) is diagonal in this coordinate system. Moreover, the expansion of Eq. (5.114) can be reduced to a two-dimensional one by using spherical polar coordinates to exploit the cylindrical symmetry. The expansion can be written as [355]... 

Fig. 16. Spherical polar coordinate system used to define metal ion (M) interactions with the side chain of methionine. In proteins the average polar angle d is 38°, and the average longitudinal angle is 169°. [Reprinted with permission from Chakrabarti, P. (1989) Biochemistry 28, 6081-6085. Copyright 1989 American Chemical Society.]...

Figure 4.1 Spherical polar coordinate system centered on a spherical particle of radius a.

First suppose that a spherical surface of unit radius is drawn and the center of this sphere is taken as the origin of a spherical polar coordinate system. Suppose further that p(0), the initial orientation of a diatomic molecule, is represented by the unit vector, k, along the positive Z axis of... 

Equation (11) is written in the form of Newton s second law and states that the mass times acceleration of a fluid particle is equal to the sum of the forces causing that acceleration. In flow problems that are accelerationless (Dx/Dt = 0) it is sometimes possible to solve Eq. (11) for the stress distribution independently of any knowledge of the velocity field in the system. One special case where this useful feature of these equations occurs is the case of rectilinear pipe flow. In this special case the solution of complex fluid flow problems is greatly simplified because the stress distribution can be discovered before the constitutive relation must be introduced. This means that only a first-order differential equation must be solved rather than a second-order (and often nonlinear) one. The following are the components of Eq. (11) in rectangular Cartesian, cylindrical polar, and spherical polar coordinates ... 

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