Figure 1.9.4.1 

A selection of graphical representations of density modulations due to higher-order terms in the Gram–Charlier series expansion of a Gaussian atomic probability density function. All figures are drawn on a common scale and have a common orientation. All terms within any given order of expansion are numerically identical and refer to the same underlying isotropic second-order term; the higher-order terms of different order of expansion differ by one order of magnitude, but refer again to the same underlying isotropic second-order term. The orthonormal crystallographic axes are oriented as follows: x oblique out of the plane of the paper towards the observer, y in the plane of the paper and to the right, and z in the plane of the paper and upwards. All surfaces are scaled to 1% of the absolute value of the maximum modulation within each density distribution. Positive modulations (i.e. an increase of density) are shown in red, negative modulations are shown in blue. The source of illumination is located approximately on the [111] axis. The following graphs are shown (with typical point groups for specific cases given in parentheses). Third-order terms: (a) b²²²; (b) b²²³; (c) b¹¹³ = −b²²³ (point group ); (d) b¹²³ (point group ). Fourth-order terms: (e) b²²²²; (f) b¹¹¹¹ = b²²²²; (g) b¹¹¹¹ = b²²²² = b³³³³ (point group ); (h) b¹²²²; (i) b¹¹¹² = b¹²²²; (j) b¹¹²²; (k) b¹¹³³ = b²²³³; (l) b¹¹²² = b¹¹³³ = b²²³³ (point group ). Fifth-order terms: (m) b²²²²²; (n) b¹²²²³; (o) b¹¹¹²³ = b¹²²²³; (p) b¹¹¹²³ = b¹²²²³ = b¹²³³³ (point group ). Sixth-order terms: (q) b²²²²²²; (r) b¹¹¹¹¹¹ = b²²²²²²; (s) b¹¹¹¹¹¹ = b²²²²²² = b³³³³³³ (point group ); (t) b¹¹²²²²; (u) b¹¹¹¹³³ = b²²²²³³; (v) b¹¹³³³³ = b²²³³³³; (w) b¹¹¹¹²² = b¹¹²²²² = b¹¹¹¹³³ = b¹¹³³³³ = b²²²²³³ = b²²³³³³ (point group ); (x) b¹¹²²³³ (point group ).