If a structure, with space group H, is such that all its atomic positions r_i can be described as r_i^o+u_i, with u_i being small displacements, while the virtual atomic positions r_i^o have a higher symmetry described by a supergroup G⊃H, we say that the structure has pseudosymmetry G, or is pseudosymmetric for the space group G.

The detection of pseudosymmetry can be very useful for several purposes:

• Prediction of symmetry and structure of some other phase of the material.
• Prediction of phase transitions.
• Identification of ferroic materials: ferroelectrics, ferroelastics.
• Determination of an optimized virtual parent structure.
• Detection of false symmetry assignments (overlooked symmetry).
• Identification of the space group symmetry of a theoretical structure calculated without symmetry restrictions (ab-initio calculations).

The Bilbao Crystallographic server provides three tools for pseudosymmetry search in a given structure: PSEUDO, DOPE and BPLOT.

PSEUDO is a new version, still under construction, of a previous program [J. Appl. Cryst. (2001). 34, 783], which detects pseudosymmetry for a given supergroup G by applying on the structure the coset representatives of its left coset decomposition with respect to the observed symmetry H. The program then checks the approximate coincidence of the transformed structures with the original one through an identification of the atomic displacements relating the two configurations. If these latter are smaller than a tolerance, the structure is flagged as pseudosymmetric.

DOPE works similarly as PSEUDO, but does the quantitative comparison of the original and transformed structures by checking the degree of superposition of their virtual electronic densities, obtained from the standard atomic diffusion factors assigned to each atom. Results can be more difficult to interpret than those of PSEUDO, since superposition of atoms of different type may contribute to the calculated degree of pseudosymmetry.

BPLOT is a modified version (adapted for the Bilbao server with the collaboration of its author) of the program KPLOT [Hundt, R. (1979). KPLOT. A Program for Plotting and Investigating Crystal Structures]. BPLOT is intended in fact for searching and identifying symmetry rather than pseudosymmetry. It transforms the input structure to P1 and does not assume a priori any rotational symmetry. Generally BPLOT only works properly if the pseudosymmetry is very high, with displacements of the order of 0.1Å or less.

The Practical will concentrate on demonstrations and exercises using PSEUDO, which is the most powerful of the three programs.

Supplementary Material