The Spherical Harmonics Approach

1 Description of the Texture by Spherical Harmonics. The description of the texture by spherical harmonics was first reported by Roe and Bunge and later developed by Bunge.According to Btmge the orientation distribution 

The functions P/ are real for m + n even and imaginary for m + n odd. They have the following properties  

The last equation says that the functions PT of different harmonics indices 1 are orthogonal. By using the Equation (25) and taking account that the ODF is a real function one obtains the following condition for the complex coefficients cf  

On the other hand, the normalization condition (4) for the ODF together with Equation (28) impose cq = 1. 

According to equation (14.160) of Bunge, the pole distribution function defined by Equation (5) becomes  

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Only even orders are taken into account in Eqs. 2.83 and 2.84 due to the presence of the inversion center in the diffraction pattern. The number of harmonic coefficients C and terms k(h) varies depending on lattice symmetry and desired harmonic order L. The low symmetry results in multiple terms (triclinic has 5 terms for L = 2) and therefore, low orders 2 or 4 are usually sufficient. High symmetry requires fewer terms (e.g. cubic has only 1 term for L = 4), so higher orders may be required to adequately describe preferred orientation. The spherical harmonics approach is realized in GSAS. ... 

In eonneetion with the implementation in the Rietveld codes, the Dollase March model and the spherical harmonics approach, for pole distributions determination, is developed in the next two parts. The problem of pole figure inversion is outside the scope of this chapter. 

Arguments for recent developments of the spherical harmonics approach for the analysis of the macroscopic strain and stress by diffraction were presented in Section 12.2.3. Resuming, the classical models describing the intergranular strains and stresses are too rough and in many cases cannot explain the strongly nonlinear dependence of the diffraction peak shift on sin even if the texture is accounted for. A possible solution to this problem is to renounce to any physical model to describe the crystallite interactions and to find the strain/ stress orientation distribution functions SODF by inverting the measured strain pole distributions ( h(y)). The SODF fully describe the strain and stress state of the sample. 

Probably the most notable work on the structure in liquid water based upon experimental data has been that of Soper and co-workers [6,8,10,30,46,55]. He has considered water under both ambient and high temperature and pressure conditions. He has employed both the spherical harmonic reconstruction technique [8,46] and empirical potential structure refinement [6,10] to extract estimates for the pair distribution function for water from site-site radial distribution functions. Both approaches must deal with the fact that the three g p(r) available from neutron scattering experiments provide an incomplete set of information for determining the six-dimensional pair distribution function. Noise in the experimental data introduces further complications, particularly in the former technique. Nonetheless, Soper has been able to extract the principal features in the pair (spatial) distribution function. Of most significance here is the fact that his findings are in qualitative agreement with those discussed above. 

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 the literature three different approaches were reported based on the spherical harmonics representation of the SODFs by Wang et by... 

We first outline the approach of the following sections. Starting with Brown s equation we expand the distribution W(r, t) of orientations of M in spherical harmonics as in the previous chapter whence we obtain an infinite set of differential-difference equations. We then select the spherical harmonic of interest P for the longitudinal relaxation and P e" for the transverse relaxation. The relaxation times can then be expressed as in the previous section in terms of a continued fraction. Since we consider only the response with respect to a small applied field H, such that 1, it is only necessary to evaluate quantities linear in Furthermore we assume that equilibrium has been attained for the ratio R s) in the continued fraction and that this ratio is expressible in terms of the equilibrium values of the relevant spherical harmonics. This expression is achieved as follows. The average value of any spherical harmonic is... 

The most commonly used approach to the problem is to expand the correlation functions and their Fourier transforms in a series of orthogonal functions, usually the spherical harmonics. This approach was pioneered by Chen and Steele in the case of the Percus-Yevick approximation for hard diatomic fluids. More recently, the approach has been generalized to arbitrary... 

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... 

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. 

We have extended the linear combination of Gaussian-type orbitals local-density functional approach to calculate the total energies and electronic structures of helical chain polymers[35]. This method was originally developed for molecular systems[36-40], and extended to two-dimensionally periodic sys-tems[41,42] and chain polymers[34j. The one-electron wavefunctions here are constructed from a linear combination of Bloch functions c>>, which are in turn constructed from a linear combination of nuclear-centered Gaussian-type orbitals Xylr) (in ihis case, products of Gaussians and the real solid spherical harmonics). The one-electron density matrix is given by... 

For practical purposes two different approaches have been used. If the nuclear framework has a center with high degree of symmetry, it may be convenient to expand the Hartree-Fock functions fk(r) in terms of spherical harmonics Ylm(6, q>) around this center ... 

There are two approaches to map crystal charge density from the measured structure factors by inverse Fourier transform or by the multipole method [32]. Direct Fourier transform of experimental structure factors was not useful due to the missing reflections in the collected data set, so a multipole refinement is a better approach to map charge density from the measured structure factors. In the multipole method, the crystal charge density is expanded as a sum of non-spherical pseudo-atomic densities. These consist of a spherical-atom (or ion) charge density obtained from multi-configuration Dirac-Fock (MCDF) calculations [33] with variable orbital occupation factors to allow for charge transfer, and a small non-spherical part in which local symmetry-adapted spherical harmonic functions were used. 

Westlund developed also a theory for PRE in the ZFS-dominated limit for S = 1, which included a stringent Redfield-limit approach to the electron spin relaxation in this regime (118). Equations (35) and (38) were used as the starting point also in this case. Again, the correlation function in the integrand of Eq. (38) was expressed as a product of a rotational part and the spin part. However, since it is in this case appropriate to work in the principal frame of the static ZFS, the rotational part becomes proportional to exp(—t/3tb) (if Tfl is the correlation time for reorientation of rank two spherical harmonics, then 3t is the correlation time for rank one spherical... 

Variational one-center restoration. In the variational technique of one-center restoration (VOCR) [79, 80], the proper behavior of the four-component molecular spinors in the core regions of heavy atoms can be restored as an expansion in spherical harmonics inside the sphere with a restoration radius, Rvoa, that should not be smaller than the matching radius, Rc, used at the RECP generation. The outer parts of spinors are treated as frozen after the RECP calculation of a considered molecule. This method enables one to combine the advantages of two well-developed approaches, molecular RECP calculation in a gaussian basis set and atomic-type one-center calculation in numerical basis functions, in the most optimal way. This technique is considered theoretically in [80] and some results concerning the efficiency of the one-center reexpansion of orbitals on another atom can be found in [75]. 

In the MSE approach, the size and field homogeneity of the FOV are proportional to number of zeroed inner spherical harmonics, and number of vanished outer harmonics defines the size of the system footprint. We recall that the number of inner and outer spherical harmonics made to vanish refers to the order and degree of the design. Once the order and degree are specified based on the requirements of the FOV and the stray field, and the magnet domain has been established, then the MSE current density map is calculated. 

Let us return then to the problem of low symmetry in transition metal complexes. The most direct and unassuming approach would be to write a symmetry-based expansion of the ligand field potential in terms of spherical harmonics. For a completely unsymmetrical molecule (C,) this would be written as... 

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