Spherical harmonics approximation

In order to take advantage of moment methods, the radiation intensity is expressed as a series of products of angular and spatial functions. If the angular dependence is expressed using a simple power series, the moment method (MM) is obtained if spherical harmonics are employed to express the intensity, then the method is called the spherical harmonics (SH) approximation. In principle, the first-order moment and the first-order spherical harmonics approximations are identical to each other, as well as to the lowest order discrete ordinates (DO) approximation [34]. 

M. P. Mengu and R. K. Iyer, Modeling of Radiative Transfer Using Multiple Spherical Harmonics Approximations, Journal of Quantitative Spectroscopy and Radiative Transfer, 39, pp. 445-461,1988. 

R. K. Iyer and M. P. Mengii , Quadruple Spherical Harmonics Approximations for Radiative Transfer in Two-Dimensional Rectangular Enclosures, AIAA Journal of Thermophysics and Heat Transfer, 3, p. 266,1989. 

There are three broad categories of deterministic whole reactor calculation methods currently in use (a) diffusion theory (b) spherical harmonics approximations, or Pn methods, and (c) discrete ordinates or Sn methods. Diffusion theory is widely used for reactor core calculations although the faster Pn and Sn methods developed in recent years are now replacing them. 

Figure 14 Representation of the solvent-excluded molecular surface of crambin. Central molecule (white) shows the solvent-excluded molecular surface the surrounding surfaces show various resolution representations of the surface using the globally analytic spherical harmonic approximations (clockwise from the upper left orders 5, 10, 15, and 20. Color coding is by shapeindex (hue) and curvature (saturation). " (Image courtesy of Arthur J. Olson, The Scripps Research Institute, La Jolla, CA)...

By the argument in Section IIB, the presence of a locally quadratic cylindrically symmetric barrier leads one to expect a characteristic distortion to the quantum lattice, similar to that in Fig. 1, which is confirmed in Fig. 7. The heavy lower lines show the relative equilibria and the point (0,1) is the critical point. The small points indicate the eigenvalues. The lower part of the diagram differs from that in Fig. 1, because the small amplitude oscillations of a spherical pendulum approximate those of a degenerate harmonic oscillator, rather than the fl-axis rotations of a bent molecule. Hence the good quantum number is... 

Third, a further simplification of the Boltzmann equation is the use of the two-term spherical harmonic expansion [231 ] for the EEDF (also known as the Lorentz approximation), both in the calculations and in the analysis in the literature of experimental data. This two-term approximation has also been used by Kurachi and Nakamura [212] to determine the cross section for vibrational excitation of SiHj (see Table II). Due to the magnitude of the vibrational cross section at certain electron energies relative to the elastic cross sections and the steep dependence of the vibrational cross section, the use of this two-term approximation is of variable accuracy [240]. A Monte Carlo calculation is in principle more accurate, because in such a model the spatial and temporal behavior of the EEDF can be included. However, a Monte Carlo calculation has its own problems, such as the large computational effort needed to reduce statistical fluctuations. 

Function approximation comes naturally with the Fourier transition. Since tiny details of a function in real space relate to high-frequency components in Fourier space, restricting to low-order components when transforming back to real space (low-pass filtering) effectively smoothes the function to any desirable degree. There are special function decomposition schemes, like spherical harmonics, which especially build on this ability [128]. 

Most atomic transitions are due to one electron changing its orbital. Using the central-field approximation, we have the angular part of the orbital function being a spherical harmonic, for which the selection rule is A/= 1 [(3.76)]. Hence for a one-electron atomic transition, the / value of the electron making the jump changes by 1. 

In the Born-Oppenheimer approximation, the molecular wave function is the product of electronic and nuclear wave functions see (4.90). We now examine the behavior of if with respect to inversion. We must, however, exercise some care. In finding the nuclear wave functions fa we have used a set of axes fixed in space (except for translation with the molecule). However, in dealing with if el (Sections 1.19 and 1.20) we defined the electronic coordinates with respect to a set of axes fixed in the molecule, with the z axis being the internuclear axis. To find the effect on if of inversion of all nuclear and electronic coordinates, we must use the set of space-fixed axes for both fa and if el. We shall call the space-fixed axes X, Y, and Z, and the molecule-fixed axes x, y, and z. The nuclear wave function of a diatomic molecule has the (approximate) form (4.28) for 2 electronic states, where q=R-Re, and where the angles are defined with respect to space-fixed axes. When we replace each nuclear coordinate in fa by its negative, the internuclear distance R is unaffected, so that the vibrational wave function has even parity. The parity of the spherical harmonic Yj1 is even or odd according to whether J is even or odd (Section 1.17). Thus the parity eigenvalue of fa is (- Yf. 

In this section, we will examine the role of interelectronic repulsion in the perspective of the internal symmetries of the shell. The key observation is that in a d-only approximation — i.e. if the t2g-orbital functions can be written as products of a common radial part and a spherical harmonic angular function of rank two - the interelectronic repulsion operator and the pseudo-angular momentum operators commute [2]. This implies that the dominant part of the... 

The sudden approximation is easy to implement. One solves the onedimensional Schrodinger equation (3.43) for several fixed orientation angles 7, evaluates the 7-dependent amplitudes (3.47), and determines the partial photodissociation amplitudes (3.46) by integration over 7. Because of the spherical harmonic Yjo(x, 0) on the right-hand side of (3.46), the integrand oscillates rapidly as a function of 7 if the rotational... 

The crystal field interaction can be treated approximately as a point charge perturbation on the free-ion energy states, which have eigenfunctions constructed with the spherical harmonic functions, therefore, the effective operators of crystal field interaction may be defined with the tensor operators of the spherical harmonics Ck). Following Wyboume s formalism (Wyboume, 1965), the crystal field potential may be defined by ... 

Of a different nature is the PI approximation (Modest, 2003), also known as the diffusion approximation (Ishimaru, 1997). Here the number of propagation directions is not restricted, but instead it is assumed that energy distributes quite uniformly over all these directions, as will be described in the next section. This approximation is the lowest order of the spherical harmonics method (also known as the Pn approximation). It is more versatile than two-and four-flux models, because it lends itself more easily to different geometries. 

As can be seen, these orbitals are the product of a radial part R dependent on r, the distance from the electron to the nucleus, and Y, called a spherical harmonic, a function detailing the angular dependence of the atomic wavefimction. For all many-electron atoms, the radial term must be approximated because of the aforementioned problem of electron-electron repulsions. 

The free energy differences between the real and the reference clathrate hydrates are also given in Table 1. The anharmonic free energy change from the spherical guests with harmonic approximation to the nonspherical guests is -0.25... 

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