Functions angle-dependent

The angle-dependent functions irn and t appear to pose no particular computational problems—at least no one has complained about their misbehavior in... 

Figure 4.3 Polar plots of the first five angle-dependent functions n, and t . Both functions are plotted to the same scale.

The angle-dependent functions mn and rn are computed by the upward recurrence relations (4.47). They need be computed only for scattering angles between 0 and 90° because of the relations (4.48). 

Other measures of properties in 3D, such as Molecular Lipophilicity Potential (MLPot) and Molecular Hydrogen Bond Potential (MHBP), have been used to characterize 3D properties. They are defined for points on a molecular surface created around the molecule and calculated from the summation of contributions from the substructural fragments making up the molecule weighted by the distance function. The hydrogen bond potentials include an angle-dependent function. 

The detected intensity is obtained by calculating Eout Eout tnd the angle dependent function can be specified ... 

It is useful to remember that integration over the and 6 parts of dv gives 4 7r as the result if no other angle-dependent functions occur in the integral.) Comparing (4-17) with our expression for indicates that r is 1.5 times greater than r p. 

In the band structure methods that use the concept of an atomic sphere (muffin-tin sphere), a wave function in an atomic sphere is represented by the product of a radial function and an angle-dependent function Yf (spherical harmonic). By weighting the total DOS by the square of the contribution of the partial functions with a specific / value to the total wave function of each state, a local (site projected) Hike partial DOS is obtained. The total DOS, g(E), is thus spatially divided according to... 

The single center term would be given as usual by an expansion in angle-dependent functions and may contain short- and long-range contributions. The multicenter term may in turn be written as... 

We assume that there exists a function which we represent by P(0)-in recognition of the fact that it is angle dependent-which can be multiplied by the scattered intensity as predicted by the Rayleigh theory to give the correct value for i, even in the presence of interference. That is. 

Fig. 17. The angle-dependent integrated opacity function dan(00 —> v = 0,1, f = 0 6, Eq, Jmax) versus Jmax computed for the experimental energy Eq = 1.200eV. This quantity is computed by restricting the partial wave sum in the DCS to the terms J < Jmax- The result is shown for forward and backward scattering to illustrate the J-contributions to scattering at different 0.

For a wave function with no angle dependence one therefore has a onedimensional Schrodinger equation... 

Thus () is an eigenvalue of Lz with eigenvalue The angle-dependent part of the wave equation is seen to contain wave functions which are eigenfunctions of both the total angular momentum as well as the component of angular momentum along the polar axis. 

The first step beyond the statistical model was due to Hartree who derived a wave function for each electron in the average field of the nucleus and all other electrons. This field is continually updated by replacing the initial one-electron wave functions by improved functions as they become available. At each pass the wave functions are optimized by the variation method, until self-consistency is achieved. The angle-dependence of the resulting wave functions are assumed to be the same as for hydrogenic functions and only the radial function (u) needs to be calculated. 

Figure 4 Schematic view of the polar-angle dependence of the gap functions at the Fermi surface, (a) for m = 0 and (b) for m / 0.

As we mentioned already, the problem of the rotation function is its score, leading to a difficult energy landscape to be searched we can now describe another way to tackle this problem. Since the Translation Function score is much more sensitive, one might try to run a translation for every possible rotation angle, therefore exploring the 6D space exhaustively. The space to be searched in eule-rian angles depends on the space group of the crystal and can be found in Rao et al. (1980). It turns out that it is doable in most cases within reasonable cpu time with a normal workstation. 

The mathematical basis of the Mie theory is the subject of this chapter. Expressions for absorption and scattering cross sections and angle-dependent scattering functions are derived reference is then made to the computer program in Appendix A, which provides for numerical calculations of these quantities. This is the point of departure for a host of applications in several fields of applied science, which are covered in more detail in Part 3. The mathematics, divorced from physical phenomena, can be somewhat boring. For this reason, a few illustrative examples are sprinkled throughout the chapter. These are just appetizers to help maintain the reader s interest a fuller meal will be served in Part 3. 

There are many other angle-dependent scattering functions in the scientific literature, which is a source of endless confusion. In the hope that it will help the confused—among whom we count ourselves—to reconcile the notation and terminology of various authors, some of the more commonly encountered functions are expressed in our notation. 

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