Symmetry in the generation of trial structures

In the generation of trial structures by the systematic variation of the position and orientation of a molecule of known dimensions, the ranges to be scanned by the positional and orientational parameters depend on the symmetry of the molecule and on the space group. The set of transformations of the crystal axes that leave invariant the coordinates of equivalent positions for a given space group defines a corresponding derivative symmetry, conforming to one of thirty distinct `Cheshire' groups. The direct product of this group with the molecular point group specifies the symmetry of the six-dimensional space of the trial-structure parameters. The asymmetric unit in this space is the region to be scanned by the several parameters.