Gram-Charlier expansion

With data averaged in point group m, the first refinements were carried out to estimate the atomic coordinates and anisotropic thermal motion parameters IP s. We have started with the atomic coordinates and equivalent isotropic thermal parameters of Joswig et al. [14] determined by neutron diffraction at room temperature. The high order X-ray data (0.9 < s < 1.28A-1) were used in this case in order not to alter these parameters by the valence electron density contributing to low order structure factors. Hydrogen atoms of the water molecules were refined isotropically with all data and the distance O-H were kept fixed at 0.95 A until the end of the multipolar refinement. The inspection of the residual Fourier maps has revealed anharmonic thermal motion features around the Ca2+ cation. Therefore, the coefficients up to order 6 of the Gram-Charlier expansion [15] were refined for the calcium cation in the scolecite. 

The two expansions discussed so far appear to be quite different. In the multistate Gaussian model, different functions are centered at different values of AU. In the Gram-Charlier expansion, all terms are centered at (AU)0. The difference, however, is smaller that it appears. In fact, one can express a combination of Gaussian functions in the form of (2.56) taking advantage of the addition theorem for Hermite polynomials [44], Similarly, another, previously proposed representation of Pq(AU) as a r function [45] can also be transformed into the more general form of (2.56). 

The three-dimensional Gram-Charlier expansion, first applied to thermal motion analysis by Johnson and Levy (1974), is a statistical expansion in terms of the zero and higher derivatives of a normal distribution (Kendal and Stuart 1958). If Dj is the operator d/du], the expansion is defined by... 

However, these refinements are made at a heavy cost in the observation-to-parameter ratio. The third-order terms, Cyk, of the Gram Charlier expansion add ten more parameters per atom to the nine Uy terms. These expressions are therefore only used when the experimental data are of exceptionally high quality, as in the neutron diffraction analysis of ice, Ih, discussed in Part IV, Chapter 21. They may also be necessary in experimental deformation density analysis, where a very precise description of the atomic thermal motion is required. 

A different expansion relies on using Gram-Charlier polynomials, which are the products of Hermite polynomials and a Gaussian function [41] These polynomials are particularly suitable for describing near-Gaussian functions. Even and odd terms of the expansion describe symmetric and asymmetric deformations of the Gaussian, respectively. To ensure that P0(AU) remains positive for all values of AU, we take... 

As Eq. (2.31) shows, the Gram-Charlier temperature factor is a power-series expansion about the harmonic temperature factor, with real even terms, and imaginary odd terms. This is an expected result, as the even-order Hermite polynomials in the probability distribution of Eq. (2.30) are symmetric, and the odd-order polynomials are antisymmetric with respect to the center of the distribution. 

Schulz 1982), corresponds to a probability distribution which is the Taylor expansion of Eq. (2.32), and similar to the Gram-Charlier distribution of Eq. 

Mallinson PR, Koritsanszky T, Elkaim E, Li N, Coppens P (1988) The Gram-Charlier and multipole expansions in accurate X-ray diffraction studies can they be distinguished Acta Crystallogr A 44 336-343... 

More complex mathematical treatment is necessary when the thermal motion is very large, as for hydrogen atoms in a room-temperature neutron structure analysis, or when it is curvilinear as in a hindered-rotor. The Uy second-rank tensor does not adequately describe the nuclear or electron-scattering density when the motion is far from harmonic or when it deviates from the familiar ellipsoidal probability form. Tb deal with such examples, more complex mathematic expressions using Gram Charlier or Edgeworth expansions are available [210]. 

Shapes of molecular electronic bands are studied using the methods of the statistical theory of spectra. It is demonstrated that while the Gram-Charlier and Edgeworth type expansions give a correct description of the molecular bands in the case of harmonic-oscillator-like potentials, they are inappropriate if departure from harmonicity is considerable. The cases considered include a set of analytically-solvable model potentials and the numerically exact potential of the hydrogen molecule. 

The application of this method to systems described by one-dimensional potentials is particularly simple (15, 18). Therefore, in this paper, the feasibility and the accuaracy of the approach has been illustrated by considering transitions between states described by several exactly solvable onedimensional models (20) and between X and B1 states of Henergy curves (21). It results, that with a proper choice of the functional form of the envelope, already three-moment curves give a very accurate description of the band shape. For harmonic oscillators (22), the Gram-Charlier-type expansions (23) are very accurate. They axe also rather good for the cases reasonably well approximated by harmonic-oscillator-type potentials (15,18). However, if the departure from harmonicity is considerable, these kinds of expansions are inappropriate. 

This expansion describes an arbitary distribution, P , for n>0 as a perturbation about a T-distribution, equivalent to the Gram-Charlier series resulting from a perturbation about a Gaussian distribution using Hermite polynomials [46]. 

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