X-PLOR provides several possibilities for the effective energy . The selection of the target is specified by the TARGet keyword. There are seven possible choices: RESIdual, AB, F1F1, F2F2, E1E1, E2E2, and PACKing.

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,
**k** is a scale factor, is an overall weight,
is the individual weights of the reflections,
**E**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 **R** value

for the first choice in Eq. 12.1,
the unweighted vector **R** value

for the second choice, or
the various correlation coefficients for the third to sixth
choices. The **R** 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 15), 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

where the angle brackets denote a weighted () averaging over all selected Miller indices . is defined as

where is ``partial" structure factors that can be used to represent a ``frozen" part of the molecule or bulk solvent contributions, and represents the structure factors that are computed from the current atomic model. provides individual weights for each reflection . The overall weight relates to the other energy terms (see Section 4.6).

The normalized structure factors
(**E**s) are computed from
the structure factors (**F**s) by averaging the **F**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. 12.1)
is to make the weight approximately independent of the resolution
range during SA-refinement. has been set to
.
The scale
factor **k** in Eq. 12.1 is set to

unless it is set manually by the FFK statement. Eq. 12.6 is a necessary condition to minimize the residual.

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

is a normalization factor set equal to the number of phase specifications occurring in the sum, is the phase centroid obtained from mir or other methods (PHASe specifications; see Section 12.4), is the phase of the calculated structure factors , is the individual figure of merit (FOM specifications; see Section 12.4), and is a well function with harmonic ``wells" given by

This form of the effective energy ensures that the calculated phases are restrained to .

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 16).
The third sum extends over all unique atoms **i** of the system.
The quantity denotes the orthogonal coordinates of atom **i**
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 **i**.
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 ** SCATter**
statement
and can be obtained from the * International
Tables for Crystallography* (Hahn ed. 1987).
The term
denotes an imaginary constant that can be
used to model anomalous
scattering.
Eq. 12.9
represents the space-group general form of the ``direct summation" formula,
which is used to
compute the structure factors. The fast Fourier transformation
(FFT) method consists of computing
by
numerical evaluation of the atomic electron density on a finite grid
followed by an FFT. The FFT method provides a way to
speed up the calculation. The METHod statement can be used to
switch between the FFT method and the direct summation method.

An approximation is used to reduce the computational requirements when multiple evaluations of Eq. 12.1 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. 12.1 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).

Sat Mar 11 09:37:37 PST 1995