Self-rotation function

The self-rotation function can give independent information about the contents and organisation of the asymmetric unit. The radius of integration should be approximately the diameter of search model. It may also provide some knowledge of the oligomeric state in the case of an unknown structure and therefore suggest which model oligomer could be used as a search model in MR.  [If it is likely that the new structure has point group symmetry, the NCS operators from the self-rotation function can be used in the Locked Rotation Function.] However self-rotation results can be very confusing or misleading when there is high crystallographic symmetry as well as NCS.

The self-rotation function has always a huge value (high peak) for zero rotation. This means that the Patterson function agrees perfectly with a copy that has not been rotated (that is: rotated through a zero angle). So this peak should always be ignored, the so-called 'origin peak' (when the rotation function is plotted as a function of the rotation angles).

Any rotation that represents a crystallographic symmetry operation will have an equally huge peak. If a crystal has 2-fold rotational symmetry, this 2-fold operation will create a copy of the Patterson which is identical to the original. In the calculated rotation function, peaks equal to the origin peak will be found at every rotation representing a crystallographic symmetry operation.

Huge origin and crystallographic peaks must be ignored. We look for smaller peak/peaks, substantially larger than any other peaks which do not represent self-rotation operations, and several sd more than the 'noise' or meaningless peaks. Also, very large peaks found around the origin and symmetry peaks should be ignored.

Stereographic projection

The crystal is imagined to be at the center of a sphere (the stereographic sphere); the normals to the crystal faces are imagined to radiate out from the center and to intersect the sphere in an array of points. Each point on the sphere therefore represents a crystal face or plane (labeled with the appropriate Miller index). The (angular) distance between two points is equal to the angle between the corresponding planes.

How to construct a stereographic projection, from: Point groups and crystal systems
J. Drenth - International Tables for Crystallography (2006). Vol. F, ch. 2.1, pp. 47-52 
http://it.iucr.org/Fa/ch2o1v0001/sec2o1o3/

Imagine a sphere around the crystal with O as the centre. O is also the origin of the coordinate system of the crystal. Symmetry elements of the point groups pass through O. Line OP is normal to a crystal plane. It cuts through the sphere at point a. This point a is projected onto the horizontal plane through O in the following way: a vertical dashed line is drawn through O normal to the projection plane and connecting a north and a south pole. Point a is connected to the pole on the other side of the projection plane, the south pole, and is projected onto the horizontal plane at a'. For a normal OQ intersecting the lower part of the sphere, the point of intersection b is connected to the north pole and projected at b'. For the symmetry elements, their points of intersection with the sphere are projected onto the horizontal plane.

Another nice representation can be found here: 
http://www.ysbl.york.ac.uk/~alexei/molrep.html#self_rotation_function

And a useful link here: 
http://www.tele.ed.nom.br/est5i.html




Example 1:
in the presence of 222 molecular symmetry
inspection k=180° section of the self-rotation function this should reveal the position of the molecular two-fold axes. If for example there are two tetramers in the ASU, 6 peaks corresponding to the three 2-folds of each tetramer should be observed, and these peaks should fall into two mutually orthogonal sets.

Example 2:
Qamra et al. Acta Crystallogr Sect F (2005) vol. 61 (Pt 5) pp. 473-5 http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=2219981

Space group C2 2 molecules per ASU inspection k=180° section of the self-rotation function 
The two strongest peaks at 90° indicate the crystallographic two-fold symmetry (b*), the other two peaks marked with "x" correspond to the non-crystallographic two-fold axes.

Example3:
 
A self-rotation puzzle, Cao, Z. and Isaacs, NW 

Monoclinic space group C2; presence of decamers suspected; SRF to locate 2-fold and 5-fold axes:
absence of substantial peaks on chi=72° interpreted as lack of 5-fold symmetry. Three peaks on chi=60° section interpreted as 3 6-fold symmetry axes, each of which perpendicular to 6 2-fold axes shown as lines of 6 peaks in the chi=180° section. Peaks in each line located at ~30° to each other and perpendicular to a 6-fold peak also coincident with a 2-fold peak. These observations suggest a 12mer structure rather than 10mer. (And see the text in the link above for answer to the puzzling 3 different 6-fold axes).