Spherical polar

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... 

Figure Bl.9.5. Geometrical relations between the Cartesian coordmates in real space, the spherical polar coordinates and the cylindrical polar coordinates.

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)) ... 

Pio. 2.1 The relationship between spherical polar and Cartesian coordinates. di pends on 6 and 0 ... 

Contrary to the impression that one might have from a traditional course in introductory calculus, well-behaved functions that cannot be integrated in closed form are not rare mathematical curiosities. Examples are the Gaussian or standard error function and the related function that gives the distribution of molecular or atomic speeds in spherical polar coordinates. The famous blackbody radiation cuiwe, which inspired Planck s quantum hypothesis, is not integrable in closed form over an arbitiar y inteiwal. 

Plot this orbital with appropriate scale factors to deteiiiiine the behavior of tE in rectangular coordinates. Describe its behavior in spherical polar coordinates. 

The hydrogen atom is a three-dimensional problem in which the attractive force of the nucleus has spherical symmetr7. Therefore, it is advantageous to set up and solve the problem in spherical polar coordinates r, 0, and three parts, one a function of r only, one a function of 0 only, and one a function of [Pg.171]

The relationships among cartesian and spherical polar coordinates are given as... 

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. 

In spherical polar coordinates, the corresponding expression for grad / is... 

Atoms are spherical objects, and what we do is to write the electronic Schrddinger equation in spherical polar coordinates, to mirror the symmetry of the problem. 

The first assumption of the Debye-Hiickel theory is that is spherically symmetric. With the elimination of any angular dependence, the Poisson equation (expressed in spherical-polar coordinates) reduces to... 

To interpret the information in each atomic orbital, we need a way to identify the location of each point around a nucleus. It is most convenient to describe these locations in terms of spherical polar coordinates, in which each point is labeled with three coordinates ... 

FIGURE 129 The spherical polar coordinates r is the radius, which gives the distance from the center, tt is the colatitude, which gives the angle from the z-axis, and c >, the "longitude," is the azimuth, which gives the angle from the x-axis. 

Figure 4. The following data were used leak radius r0 (40 micron), leak conductance F = 1 cc. sec.-1, gas pressure (air) p (20 torr), diffusion coefficient (26) for ions Di — 2 sq. cm. sec.-"1, for electrons De = 2000 sq. cm. sec.-1 (at 20 torr air). The flow velocities were assumed independent of 6 and (spherical polar coordinates). The spherically symmetrical flow pattern, which obviously overestimates the flow velocity for large values of 6 was chosen because of its simplicity. The velocity of the radially directed flow at a distance r is v = F/2irr2. The time re-...

Here, Yx m( j) denotes a spherical harmonic, coj represents the spherical polar angles made by the symmetry axis of molecule i in a frame containing the intermolecular vector as the z axis. The choice of the x and y axes is arbitrary because the product of the functions being averaged depends on the difference of the azimuthal angles for the two molecules which are separated by distance r. At the second rank level the independent correlation coefficients are... 

Our next objective is to find the analytical forms for these simultaneous eigenfunctions. For that purpose, it is more convenient to express the operators Lx, Ly, Zz, and P in spherical polar coordinates r, 6, q> rather than in cartesian coordinates x, y, z. The relationships between r, 6, q> and x, y, z are shown in Figure 5.1. The transformation equations are... 

The volume element dr = dv dj dz becomes dr = sin OdrdOdtp in spherical polar coordinates. 

Equation (6.12) cannot be solved analytically when expressed in the cartesian coordinates x, y, z, but can be solved when expressed in spherical polar coordinates r, 6, cp, by means of the transformation equations (5.29). The laplacian operator in spherical polar coordinates is given by equation (A.61) and may be obtained by substituting equations (5.30) into (6.9b) to yield... 

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