Gram-Charlier temperature factor

The Gram Charlier temperature factor is the Fourier transform of Eq. (2.30), which is given by... 

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. 

Compared with the Gram-Charlier temperature factor of Eq. (2.31), the entire series now occurs in the exponent, so, in the cumulant formalism, terms are added to the exponent of the harmonic temperature factor P0(H) = exp — fijkhjhk. ... 

Gram-Charlier temperature factors (chapter 2) were applied in this analysis, which made use of the XD programming package (Su and Coppens, to be published). In accordance with the results on H3P04, discussed in chapter 3, the values of n, were chosen as 6, 6, 7, and 7 for / = 1, 2, 3, and 4, respectively. However, the differences between the results from this choice and those of two alternative selections (4, 4, 4, and 4 and 6, 6, 6, and 6) are within one standard deviation. 

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. 

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