Spherical symmetry polar coordinates

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. 

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]... 

From the definition of a particle used in this book, it follows that the motion of the surrounding continuous phase is inherently three-dimensional. An important class of particle flows possesses axial symmetry. For axisymmetric flows of incompressible fluids, we define a stream function, ij/, called Stokes s stream function. The value of Imj/ at any point is the volumetric flow rate of fluid crossing any continuous surface whose outer boundary is a circle centered on the axis of symmetry and passing through the point in question. Clearly ij/ = 0 on the axis of symmetry. Stream surfaces are surfaces of constant ij/ and are parallel to the velocity vector, u, at every point. The intersection of a stream surface with a plane containing the axis of symmetry may be referred to as a streamline. The velocity components, and Uq, are related to ij/ in spherical-polar coordinates by... 

The choice of generating functions is dictated by whatever symmetry may exist in the problem. In this chapter we are interested in scattering by a sphere therefore, we choose functions ip that satisfy the wave equation in spherical polar coordinates r, 6,

[Pg.84]

The CD signs of the ICD of P-CDx complexes with mono-, di-, and triazanaphthal-enes in aqueous solutions depend on the polarization directions of these azanaphthal-enes. The polarization assignments come from the CNDO/S-CI calculation 165) coupled with the MCD data of these azanaphthalenes 165). In the first place, the origin of the coordinates was choosen in the center of a ring of P-CDx that is assumed to have a 7-fold symmetry axis. Next, ep which is put at the origin of the p-CDx-fixed Cartesian coordinates, is expressed by the spherical polar coordinates (Fig. 23). 

The standard approach to solving the Schrodinger equation for hydrogenlike atoms involves transforming it from Cartesian (x, y, z) to polar coordinates (r, 6, (p), since these accord more naturally with the spherical symmetry of the system. This makes it possible to separate the equation into three simpler equations, fir) = 0, fid) = 0, and fiip) = 0. Solution of the fir) equation gives rise to the n quantum number, solution of the/(0) equation to the l quantum number, and solution of the fifi) equation to the mm (often simply called m) quantum number. For each specific n = n, 1 = 1 and mm = inm there is a mathematical function obtained by combining the appropriate fir), fi fJ) and /([Pg.101]

The spherical symmetry of the interaction implies, in particular, that the angular momentum of the relative motion is conserved. That is, since the angular momentum is a vector, both direction and magnitude are conserved quantities. The collision process will, accordingly take place in the plane defined by the initial values of the radius vector and the momentum vector. This implies that only two coordinates are required in order to describe the relative motion. These coordinates are chosen as the polar coordinates in the plane (r, 0). The scattering in the center-of-mass coordinate system is shown in Fig. 4.1.7. 

The steady state diffusion equation in spherical polar coordinates relates concentration c to only the radius r because of spherical symmetry ... 

Systems of spherical symmetry are more amenable to analysis in a spherical polar, rather than cartesian, coordinate system. In this case the Lapla-cian operator is a function of (r,9,(p) rather than (x,y,z). Separation of... 

Here we use polar coordinates and can be satisfied with r on account of the spherical symmetry of [Pg.124]

We choose a spherical polar coordinate system (r, 6, cp) in which the origin O is located at the center of sphere 1, the 6 = 0 hne coincides with the hne joining the centers of the two spheres, and (p is the azimuthal angle about the 6 = 0 line. By symmetry, the electric potential ij/ in the electrolyte solution does not depend on the angle (p. In the Debye-Hilckel approximation, il/ r,6) satisfies... 

Systems of molecules with linear symmetry can be viewed as a special application of spherical-polar coordinates. Placing the polar axis along the molecular axis of symmetry allows one to average over the angle 0. The spatial distribution function then becomes a two-dimensional function, g(r,6), which can be much more readily calculated and visualized (for an example, see Figure 1). 

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. 

Since it is possible to formulate the nuclear motion problem for a diatomic in terms of the spherical polar coordinates of the intemuclear vector, it is possible to describe the rotational motion of the nuclei without leaving the Cartesian space R3. It was thus possible for Combes and Seiler to consider how this rotational motion approximated the rotational motion as a whole. However, it is not generally possible to do so, and for rotational motion to be considered explicitly, it is necessary to decompose to the manifold form discussed above. However, since the required kind of fiber bundle can be constructed only upon a Cartesian space that means that there is no single form for the electronic Hamiltonian but one for each of the possible bundles. This corresponds to approximating only that subset of states which are accessible in the chosen formulation. The fiber bundle structure here is thus nontrivial and is only generalizable locally. The nontrivial nature of the separated fiber bundle form has so far prevented a mathematically sound account of the Bom-Oppenheimer approximation from being given with explicit consideration of rotational symmetry. 

The usual spherical polar coordinate system (r, 6, right-handed Cartesian coordinate system whose coordinates occur in the equations. The Z-axis is the unique axis around which r-functions (A = 0) have rotational symmetry. Functions with 0 < A g / are characterized 19) through A by their standard transformation properties when they undergo rotations around the Z-axis ... 

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