Crystallographic Target Functions

is the Miller indices of the selected reflections, is the observed structure factors, is the computed structure factors, and are the real components, and are the imaginary components of the structure factors, is a normalization factor, is a scale factor, is an overall weight, is the individual weights of the reflections, s are normalized structure factors, and “Corr" is the standard linear correlation coefficient. The computation of the effective energy is accompanied by printing the unweighted value

for the first choice in Eq. 13.1, the unweighted vector value

for the second choice, or the various correlation coefficients for the third to sixth choices. The values are stored in the symbol $R, and the correlation coefficients are stored in the symbol $CORR. If the data are partitioned into a test and a working set (see Chapter 17), the corresponding values for the test set are stored in the symbols $TEST R and $TEST CORR.

The selection of reflections
is accomplished by the RESOlution and FWINdow statements (see below).
“Corr" is defined through

(13.4) |

(13.5) |

The normalized structure factors (s) are computed from the structure factors (s) by averaging the s in equal reciprocal volume shells within the specified resolution limits. The number of shells is specified by MBINs.

The purpose of the normalization factor
(first and second choice in Eq. 13.1)
is to make the weight approximately independent of the resolution
range during SA-refinement. has been set to
.
The scale
factor in Eq. 13.1 is set to

The term represents phase restraints
if is set
to a nonzero number.

(13.8) |

The structure factors (
) of the atomic model
are given by

The first sum extends over all symmetry operators composed of the matrix representing a rotation and a vector representing a translation. The second sum extends over all non-crystallographic symmetry operators if they are present; otherwise only the identity transformation is used (see Chapter 18). The third sum extends over all unique atoms of the system. The quantity denotes the orthogonal coordinates of atom in Å. is the 33 matrix that converts orthogonal coordinates into fractional coordinates; denotes the transpose of it. The columns of are equal to the reciprocal unit cell vectors . is the occupancy for each atom. is the individual atomic temperature factor for atom . Both quantities correspond to the Q and B atom properties (Section 2.16), which can be read along with the atomic coordinates (see Section 6.1). The atomic scattering factors are approximated by an expression consisting of four Gaussians and a constant

The constants and are specified in the

An approximation is used to reduce the computational requirements when multiple evaluations of Eq. 13.1 are required. The approximation involves not computing and its first derivatives at every dynamics or minimization step. The first derivatives are kept constant until any atom has moved by more than (TOLErance in xrefin statement) relative to the position at which the derivatives were last computed. At that point, all derivatives are updated. Typically, is set to 0.2 Å for dynamics and to 0-0.05 Å for minimization.

The PACKing target is defined for evaluating the likelihood of packing arrangements of the search model and its symmetry mates in the crystal (Hendrickson and Ward, 1976). A finite grid that covers the unit cell of the crystal is generated. The grid size is specified through the GRID parameter in the xrefin FFT statement. All grid points are marked that are within the van der Waals radii around any atom of the search model and its symmetry mates. The number of marked grid points represents the union of the molecular spaces of the search model and its symmetry mates. Maximization of the union of molecular spaces is equivalent to minimization of the overlap. Thus, an optimally packed structure has a maximum of the packing function. “Pack" in Eq. 13.1 contains the ratio of the number of marked grid points to the total number of grid points in the unit cell. For instance, a value of 0.6 means 40% solvent contents. is then set to 0.4 if .

For further reading on the crystallographic target functions in X-PLOR, see Brünger (1988,1989,1990).