Spherical harmonics expansion

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. 

Table 5.10. Spherical harmonic expansion of the data listed in Table 5.9.

Morris RRJ, Najmanovich RRJ, Kahraman A et al (2005) Real spherical harmonic expansion coefficients as 3D shape descriptors for protein binding pocket and ligand comparisons. Bioinformatics (Oxford, England) 21 2347-2355... 

To obtain the interaction energy, Eq. (9.53) is substituted into Eq. (9.50). When the charge density B is expressed in terms of one or more spherical harmonic expansions, only terms of like symmetry will integrate to nonzero values, and we obtain the expression for the interaction between two distributions as... 

Payne, 1990 Sacks and Noguera, 1991). All those authors used a spherical-harmonic expansion to represent tip wavefunctions in the gap region, which is a natural choice. The spherical-harmonic expansion is used extensively in solid-state physics as well as in quantum chemistry for describing and classifying electronic states. In problems without a magnetic field, the real spherical harmonics are preferred, as described in Appendix A. 

In this Chapter, we present step-by-step derivations of the explicit expressions for matrix elements based on the spherical-harmonic expansion of the tip wavefunction in the gap region. The result — derivative rule is extremely simple and intuitively understandable. Two independent proofs are presented. The mathematical tool for the derivation is the spherical modified Bessel functions, which are probably the simplest of all Bessel functions. A concise summary about them is included in Appendix C. 

For magnet configurations in which coils are coaxial and symmetric about the illustrated xy-plane, such as the magnet configurations in Figure 2A and C, the spherical harmonic expansion results in the elimination of all even order terms within the expansion. To further reduce computational complexity, the strategy employed here considers only one quarter of the magnet domain, and thus, the constraints in Equation (5) simplify to ... 

Consider now the field scattered by an isotropic, optically active sphere of radius a, which is embedded in a nonactive medium with wave number k and illuminated by an x-polarized wave. Most of the groundwork for the solution to this problem has been laid in Chapter 4, where the expansions (4.37) and (4.38) of the incident electric and magnetic fields are given. Equation (8.11) requires that the expansion functions for Q be of the form M N therefore, the vector spherical harmonics expansions of the fields inside the sphere are... 

Then the nonstationary solution of the kinetic equation (4.27) with the energy function (4.55) is sought in the form of the spherical harmonics expansion... 

To evaluate the averages like those in Eq. (4.78), it is very convenient to pass from cosines ((en)k) to the set of corresponding Legendre polynomials for which a spherical harmonics expansion (addition theorem)... 

The general matrix equation of the problem that determines the amplitudes <2 /" is obtained by substitution of the spherical harmonic expansion (4.319) into the Fokker-Planck equation (4.313). After that the result is multiplied from... 

Molecular Integrals by Solid Spherical Harmonic Expansions. 

It has been found useful to represent the interaction potential for a dimer of homonuclear diatomic molecules [4,5,46,58] as a spherical harmonic expansion, separating radial and angular dependencies. The radial coefficients include different types of contributions to the interaction potential (electrostatic, dispersion, repulsion due to overlap, induction, spin-spin coupling). For the three dimers of atmospheric relevance, we provided compact expansions, where the angular dependence is represented by spherical harmonics and truncating the series to a small number of physically motivated terms. The number of terms in the series are six for the N2-O2 systems, corresponding to the number of configurations of the dimer (for N2-N2 and O2-O2 this number of terms is reduced to five and four, respectively). 

Figure 1 Structural (left column) and dynamical (right column) properties of the systems investigated. Upper left centre-of-mass radial pair distribution function gooo( ) lower left spherical harmonic expansion coefficient g2oo(r) upper right angular velocity correlation function lower right orientational correlation function. Dotted lines CO, 80 K, 1 bar thin lines CS2, 293 K, 1 bar thick lines CS2, 293 K, 10 kbar.

Figure 11.18 Predictions of the tumbling parameter A as a function of reduced concentration C/C2 from the Smoluchowski equation for hard rods with the Onsager potential. The exact result from the spherical-harmonic expansion is shown, compared to approximate results from an analytic formula and from the perturbation expansion of Kuzuu and Doi. The open circles (O) are estimates from the periods of shear stress oscillations in transient shearing flows for PEG solutions (see Walker et al. 1995), and the closed circle ( ) is from a direct conoscopic measurement of Muller et al. (1994).

We now discuss the analysis of the x-ray intensities. The atoms of the C6o molecule are placed at the vertices of a truncated icosahedron. - The x-ray structure factor is given by the Fourier transform of the electronic charge density this can be factored into an atomic carbon form factor times the Fourier transform of a thin shell of radius R modulated by the angular distribution of the atoms. For a molecule with icosahedral symmetry, the leading terms in a spherical-harmonic expansion of the charge density are Koo(fl) (the spherically symmetric contribution) and KfimCn), where ft denotes polar and azimuthal coordinates. The corresponding terms in the molecular form factor are proportional to SS ° (q)ac jo(qR)ss n(qR)/qR and... 

Yet another approach, which is based on the algorithm described by Bunge, uses spherical harmonics expansion to deal with preferred orientation in three dimensions as a complex radial distribution ... 

The subsequent refinement included profile parameters X, Y, X , Ya, peak asymmetry, sample displacement and transparency shift. Preferred orientation was switched from the March-Dollase to the 8 -order spherical harmonics expansion (6 variables total) and 12 coefficients of the shifted-Chebyshev polynomial background approximation were employed. A reasonably good fit, shown in Figure 7.36, was achieved as a result. 

The preferred orientation correction was accounted for in two ways during the refinement. First, the March-Dollase approach with one texture axis [001] resulted in x = 1.247(2) and correction coefficients ranging from 0.52 to 1.39, which gives the preferred orientation magnitude of 2.70. Second, the 8 -order spherical harmonics expansion, which corresponds in this crystal system to six adjustable parameters (200, 400, 600, 606, 800, and 806) was attempted with the March-Dollase preferred orientation correction (i) left as is but fixed (i.e. the spherical harmonics were in addition to the March-Dollase model), or (ii) eliminated. Both ways result in practically an identical result except for the magnitudes of the coefficients. In the second case, the correction coefficients ranged from 0.61 to 1.54, which corresponds to the preferred orientation magnitude of 2.52. 

In some respects, this approach is very attractive since, if the spherical harmonic expansions of the correlation functions are sufficiently rapidly convergent, the approximate solution of the Ornstein-Zernike equation for a molecular fluid can be placed upon essentially the same footing as that for a simple atomic fluid. The question of convergence of the spherical harmonic expansions turns out to be the key issue in determining the efficacy of the approach, so it is worthwhile to review briefly the available evidence on this question. Most of the work on this problem has concerned the spherical harmonic expansion of (1,2) for linear molecules. This work was pioneered by Streett and Tildesley, who showed how it was possible to write the spherical harmonic expansion coefficients as ensemble averages obtainable from a Monte Carlo or molecular dynamics simulation via... 

Thermodynamic property calculations based upon the solution of the molecular OZ equation using spherical harmonic expansions have not been extensive. This is probably due to the computational effort required in the implementation of these methods. However, quite promising results have been obtained for the equation of state of hard sphere diatomics (Chen and Steele, Freasier, Lado ) and for 12-6 diatomics (Lado ). Again, as we noted earlier in Section III.A, more extensive tests of this approach need to be made. 

Sokolowski and Steele have adapted the spherical harmonic expansion technique described in Section III.A to the calculation of the density profiles of nonspherical molecules in contact with solid surfaces. They have used the results to investigate molecular orientation effects in high temperature adsorption from the gas phase. The RAM theory described in Section III.E has been extended to the adsorption of fluids on solid surfaces by Smith et al. ° for hard-sphere interactions and by Sokolowski and Steele to the case of more realistic fluid-solid interactions. The principal deficiency of the approach is the accuracy of the predicted correlation functions for the bulk fluid which are required as input to the theory. 

Equation IV. 14 provides differential equations for the coefficients of the spherical harmonic expansion of when u —ua... 

For a molecule with atoms of unit mass on a unit sphere the moments of inertia may be expressed in terms of the following spherical harmonic expansions ... 

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