Polar coordinates description

Here r,, and q> are polar coordinates used in the description of the orbitals of an atom, the nucleus being at the origin of the coordinate system. 

The initial description of the model is simple, as shown in Figure 3. The atomic coordinates of any suitable structure can serve as the input trial structure, even including a wrong monomer residue. The polar coordinates are calculated from the trial structure, adjusted and modified as necessary, and then subjected to refinement in accordance with the selected list of variables, limits and constraints. Any set of standard values and nonbonded potential function parameters can be used. Hydrogen bonds can be defined as desired, variables can be coupled, and the positions of solvent molecules can be individually refined. Single and multiple helices are equally easily handled, as are a variety of space groups. 

In addition to the Nyquist representation, the Bode plot is as well applied for the description of impedance spectrometry data. In this case, the impedance data is represented in polar coordinates... 

The motion of a free particle on the surface of a sphere will involve components of angular momentum in three-dimensional space. Spherical polar coordinates provide the most convenient description for this and related problems with spherical symmetry. The position of an arbitrary point r is described by three coordinates r, 0, 0, as shown in Fig. 6.2. 

In some way s p (r) does not provide the best description of the electron distribution, since the region around r = 0, where the wavefunction has its largest values, is a relatively small fraction of the volume accessible to the electron. Larger radii r have more impact since, in spherical polar coordinates, a value of r is associated with a shell of volume Anr dr. A more significant measure is therefore the radial distribution function (RDF)... 

The angular probability density function, P B, provides the basis for a pictorial description of the rotational states of dihydrogen. The function is a surface represented in polar coordinates. Any point on the surface can be joined to the origin by a line. The 0, (f) coordinates of the line correspond to the orientation of the dihydrogen molecular axis and the probability of any particular orientation occurring is given by the... 

For systems of chemical interest the amplitude function ip that occurs as a solution of (4.19) is postulated to give a complete description, provided the potential energy V, is correctly specified. In reality, the only chemically significant problem that has been solved is of an electron associated with an isolated stationary proton, with potential energy V = jr, in atomic units. The differential wave equation is separable in spherical polar coordinates. Separate solutions, as functions of radial (r) and angular 9, ip) coordinates, describe the quantized energy and angular momentum of the electron as ... 

A formal description of the filtered back-projection process is best served by expressing the object function /(jc, y) defined in Eq. (26.64), in an alternative coordinate system. Here, die rectangular coordinate system in the frequency domain (u, v) is exchanged for the polar coordinate system iyv, 0). so that... 

The above Cartesian coordinates do not provide the optimal description of the conical intersection. We will show below that a set hyperspherical coordinates provides the best representation. We begin by defining spherical polar coordinates QO, (f>C) for the z, x,y (g , h ) axes and... 

The three-dimensional velocity distribution can be expressed in a number of ways. We have, so far, considered a Cartesian description of the velocity vector. If, however, we switch to spherical polar coordinates in which... 

The lamellar reflections are not flat, but are curved i.e., there is a continuous shift in the z-position of the maxima (z ) in the lamellar peaks as a function of x (Figure 1) Because of this curvature, the two-dimensional (2-D) data could not be fitted in Cartesian coordinates. But they are not curved enough to be a circle, hence the polar coordinates ordinarily used in analyzing the wide-angle x-ray diffraction patterns cannot be used either. It appears that the description in elliptical coordinates provides the best fit to the data. This feature of the scattering curve will be analyzed in detail in this paper. 

It s very straightforward (using the polar coordinate system of Figure 27) to solve for the rotational velocity and the radial velocity as a function of radius, and so to model the velocity field with only two free parameters. And when you build that model you get Figure 28c, which, as you see, is a tolerable description of what we ve observed. 

The diammine [Hg(NH3)2Cl2], descriptively known as fusible white precipitate , can be isolated by maintaining a high concentration of NH4+, since reactions (2) and (3) are thereby inhibited, or better still by using non-polar solvents. It is made up of a cubic lattice of Cl ions with linear H3N-Hg-NH3 groups inserted so as to give the common, distorted octahedral coordination about Hg (Hg N = 203 pm, Hg-Cl = 287 pm) (Fig. 29.4a). 

A generalization of Harriman s orbitals has been proposed by Gosh and Parr [63] it is based on a description similar to that of Zumbach and Maschke [61] but uses polar spherical coordinates ... 

Fig. 2. The angular coordinates used to describe the orientation of the magnetic field, B0, and an internuclear vector, r, relative to an arbitrarily chosen coordinate frame which is fixed to the molecule. Description in terms of (A) direction cosines and (B) polar angles.

Some of the bonding schemes set out in Table 6.9 should not be taken too literally. In scheme (11) for Xe04, the promotion of four electrons to 5d orbitals must require a colossal amount of energy. The alternative semi-polar structure Xe4+(0 )4, with coordinate/dative bonds between Xe and O, is equally far-fetched. A complete VB description of the bonding in Xe04 would require resonance among a large number of structures such as ... 

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