Rotation Search

EXTEnd=BOX XMIN=-45.0 XMAX=45.0 YMIN=-45.0 YMAX=45.0 ZMIN=0.0 ZMAX=30.0where one makes use of the centrosymmetry and 30.0 is the maximum vector length. Patterson vectors are selected according to the value of the Patterson map at the particular grid point (THREshold) and according to the distance from the grid point to the origin (RANGe). Of this selected subset of Patterson vectors, only the strongest NPEAKs Patterson vectors are chosen for the rotation search.

The selected Patterson vectors are rotated using
Eulerian angles (, , ) as defined
by Rossmann and Blow (1962), pseudo-orthogonal Eulerian angles
as defined by Lattman (1985),
or spherical polar angles (
)
(see Section 2.4 for the conventions).
The
Lattman angles are related to
the Eulerian angles by

(19.1) | |||

The angles are sampled at a constant interval . Following Lattman (1985), the variable interval for is given by , and the variable interval for is given by . For both Eulerian angles and spherical polar angles, the sampling interval is constant () for the three angular variables. The rotation search should be restricted to the asymmetric unit of the rotation function (Rao, Jih, and Hartsuck, 1980). Using the convention in the Rao et al. paper, the Patterson symmetry of the rotated map is “1", and that of is the Patterson symmetry of the space group.

The values of the Patterson map P2 at the positions of the rotated Patterson vectors of map P1 are computed by linear eight-point interpolation (Nordman, 1980).

For each
sampled orientation the product function

(19.2) |

Upon completion of the rotation function,
all sampled grid points are sorted with respect to their product
correlation value,
and a simple peak search is carried out using the matrix metric defined
in Brünger (1990). For two given rotation matrices
, the metric is defined as

i.e., the metric assumes the crystal symmetry for both Patterson maps P1 and P2. In the case of space group , the metric is related to a rotation angle , which defines a rotation around a certain axis :

(19.5) |

Two RF grid points are considered as being in the same cluster if the corresponding rotation matrices yield . For example, if is set to 0.25, matrices belong to the same cluster if they can be transformed by a rotation of 10 or less around a certain axis. The incorporation of crystallographic symmetry in Eq. 19.3 ensures that clusters of grid points at the boundaries of the asymmetric unit of the RF are treated properly.

This peak search removes grid points that are close to grid points with larger RF values. It is not a true peak search, but rather reduces the number of points to be checked by subsequent analysis. The reduced set of the highest grid points is written to a specified file. For example, if is set to 0.25, the file will contain grid points that are mutually different by at least 10. This file will be read by the -refinement procedure described below. The value for should be chosen to be less than the radius of convergence of the -refinement (around 10). It should be noted that the peak search procedure maps the grid points into an asymmetric unit of the rotation function.