International
Tables for Crystallography Volume A Spacegroup symmetry Edited by M. I. Aroyo © International Union of Crystallography 2016 
International Tables for Crystallography (2016). Vol. A, ch. 1.6, p. 110

It is also possible to obtain information on the symmetry of the crystal after structure solution. The latter is obtained either in space group P1 (i.e. no symmetry assumed) or in some other candidate space group. The analysis may take place either on the electrondensity map, or on its interpretation in terms of atomic coordinates and atomic types (i.e. chemical elements). The analysis of the electrondensity map has become increasingly popular with the advent of dualspace methods, first proposed in the chargeflipping algorithm by Oszlányi & Sütő (2004), which solve structures in P1 by default. The analysis of the atomic coordinates and atomic types obtained from leastsquares refinement in a candidate space group is used extensively in structure validation. Symmetry operations present in the structure solution but not in the candidate space group are sought.
An exhaustive search for symmetry operations is undertaken. However, those to be investigated may be very efficently limited by making use of knowledge of the highest pointgroup symmetry of the lattice compatible with the known cell dimensions of the crystal. It is well established that the pointgroup symmetry of any lattice is one of the following seven centrosymmetric point groups: , 2/m, mmm, 4/mmm, , 6/mmm, . This point group is known as the holohedry of the lattice. The relationship between the symmetry operations of the space group and its holohedry is rather simple. A rotation or screw axis of symmetry in the crystal has as its counterpart a corresponding rotation axis of symmetry of the lattice and a mirror or glide plane in the crystal has as its counterpart a corresponding mirror plane in the lattice. The holohedry may be equal to or higher than the point group of the crystal. Hence, at least the rotational part of any spacegroup operation should have its counterpart in the symmetry of the lattice. If and when this rotational part is found by a systematic comparison either of the electron density or of the positions of the independent atoms of the solved structure, the location and intrinsic parts of the translation parts of the spacegroup operation can be easily completed.
Palatinus and van der Lee (2008) describe their procedure in detail with useful examples. It uses the structure solution both in the form of an electrondensity map and a set of phased structure factors obtained by Fourier transformation. No interpretation of the electrondensity map in the form of atomic coordinates and chemicalelement type is required. The algorithm of the procedure proceeds in the following steps:
Palatinus & van der Lee (2008) report a very high success rate in the use of this algorithm. It is also a powerful technique to apply in structure validation.
Le Page's (1987) pioneering software MISSYM for the detection of `missed' symmetry operations uses refined atomic coordinates, unitcell dimensions and space group assigned from the crystalstructure solution. The algorithm follows all the principles described above in this section. In MISSYM, the metric symmetry is established as described in the first stage of Section 1.6.2.1. The `missed' symmetry operations are those that are present in the arrangement of the atoms but are not part of the space group used for the structure refinement. Indeed, this procedure has its main applications in structure validation. The algorithm used in Le Page's software is also implemented in ADDSYM (Spek, 2003). There are numerous reports of successful applications of this software in the literature.
References
Le Page, Y. (1987). Computer derivation of the symmetry elements implied in a structure description. J. Appl. Cryst. 20, 264–269.Oszlányi, G. & Sütő, A. (2004). Ab initio structure solution by charge flipping. Acta Cryst. A60, 134–141.
Palatinus, L. & van der Lee, A. (2008). Symmetry determination following structure solution in P1. J. Appl. Cryst. 41, 975–984.
Spek, A. L. (2003). Singlecrystal structure validation with the program PLATON. J. Appl. Cryst. 36, 7–13.