Crystals User Guide

Chapter 4: Fourier And Patterson Functions

4.1: EXAMPLE 1

4.2: Patterson maps

4.3: Sharpened Patterson maps

4.4: Fourier refinement

4.5: Assembling a single molecule

Fourier summations are computed by the instruction \FOURIER with an appropriate keyword. Patterson functions can be computed directly from the observed structure amplitudes, Fourier summations require that phase angles have been computed previously. This may be done by issuing the instruction \SFLS with the directive SCALE or CALCULATE. Remember, if you have been doing Least-Squares refinement, that after a new parameter list has been generated the structure factors will need recomputing if they are to reflect these new parameters.

If the user has not used the SPACEGROUP command, he is required to choose for himself that fraction of the unit cell which must be computed, and enter this information as a LIST 14. Where there are alternative choices possible, computational efficiency is the main factor to be concidered if the map is not going to be printed. If the X-ray intensity data are to the same resolution along all axes, the user should, if the space-group permits, choose an asymmetric volume in the form of a thin 'slab', and use an ORIENT directive to compute sections parallel to the slab face to minimise computation time. For example, in an orthorhombic cell 16x11x24, a volume 1/2 x 1/4 x 1 computed in 3min 27secs, and a volume 1 x 1/2 x 1/4 computed in 1min 40secs.

When the map is to be printed, then convenience in handling the final map is the main consideration. For example, if the structure has an essentially planar fragment which needs looking at, the sections should be chosen approximately parallel to the plane. Also, though CRYSTALS will compute maps with any number of points in each direction, and will split the sections up onto several sheets of paper if they are too wide to fit directly, the output is clearly more manageable if the user can arrange the width to fit onto a single page. The fraction of the cell chosen should be such that it contains at least an asymmetric unit, with a small additional border if the user is explicitly defining the volume and grid intervals. Failure to include this border may lead to the peak-search routine finding (during the least-squares peak fitting) very large or small peaks several Angstroms beyond the edge of the volume computed. Such spurious peaks are easily found by inspection, but could be troublesome if some sort of automatic interpretation of the map was being attempted. When the system is allowed to chose its own sampling interval, a border is automatically included, and these spurious peaks seldom occur.

Parameters on the \FOURIER instruction enable the figure fields to be scaled and formatted, and the user should well consider what precision he requires in the output map. If he is looking for features that can be represented to two integer places, and there are other previosly characterised features requiring three places, a format permitting printing printing of signed two digit integers will be adequate if rho-max is set to 99.

The default map type is an F-OBS map. The peak search routines automatically reject all but one peak from sets of peaks related by a symmetry element. Additionally, if an atoms list, LIST 5, is present, the program evaluates the electron density at atomic sites. The message 'NOT FOUND IN THE MAP' inevitably means that the user has used an inappropriate LIST 14, and has not computed a complete asymmetric volume (though he may have computed some part twice!).

 \FOURIER
 MAP PRINT=YES
 LAYOUT NLINE=1 NCHAR=3 MARGIN=2 NSPACE=2 MIN-RHO=0 MAX-RHO=99
 END

Patterson functions are computed by the instruction FOURIER with the directive MAP TYPE = FO-PATTERSON. CRYSTALS automatically modifies the symmetry information taken from LIST 2 into that appropriate for a Patterson map, i.e. includes a centre of symmetry and drops the translational components. In some space groups this may lead to an origin shift, so that the user will have to input appropriate values in LIST 14.

The modified LIST 2 is used for the implied distance-angle calculation called by FOURIER to detect degenerate peaks, but it is not written back to the DISK, which retains the original LIST 2. If the user wishes to explicitly compute distances and angles on the Patterson peaks list, he must include the parameter SYMMETRY=PATTERSON in the distance calculation.

Any of the weighting schemes described for LIST 4 can be applied to the Patterson coefficients. Unit weights can be modified using Scheme 13 to sharpen the map. The formula of Dunitz (Acta Cryst(1973), B29, 589) should be used with P(1)=1 and P(2)=10 to avoid over-sharpening. The weight for a reflection at theta=30 will be 50 for Mo radiation. Do not forget you will probably wish to restore the original weighting scheme.

 \LIST 4
 \ Use the default scheme, which produces unit weights.
 END
 \WEIGHT
 \ To apply unit weights
 \LIST 4
 SCHEME NUMBER=13 NPARAM=2
 PARAM 1 10
 END
 \WEIGHT
 \ To apply weights for sharpening
 END

CRYSTALS provides a mechanism for performing elementary Fourier refinement. Such a procedure is very cost-effective during the initial development of poorly phased structures, such as those based on E-maps or heavy-atom phased maps. At that stage, non-linearities degrade the convergence of simple least-squares, though the use of constraints may substantially improve the situation (see the section on Least-squares). Fourier refinement has a good range of convergence and is relatively insensitive to errors in the model, though it does require the atomic sites to be resolved.

 \SFLS
 \ we will compute phases and put Fo on an 
 \ approximately absolute scale
 SCALE
 END
 \FOURIER
 \ set the reject limit very small so that peaks 
 \ which lie close to input atomic sites are not 
 \ rejected
 PEAKS HEIGHT=10 NPEAK=16 REJECT=.0005
 END
 \PEAKS
 \ hang on to all the peaks again
 SELECT REJECT=.0005
 \ this does the refinement.
 \ peaks within .15 of an input atomic site are 
 \ identified with that atom.
 \ The positional coordinates of the atom are computed 
 \ from the corresponding peak coordinates
 REFINE DISTANCE=.15
 END

In addition to maps computed with /Fo/, /Fc/ or /Fo-Fc/ as coefficients, maps can be based on /2*Fo-Fc/. Such maps are commonly used in protein crystals structure analysis, and are useful in small structure analyses when there is disorder. The reduced contribution from /Fc/ means that a diminished image of the 'known' structure is retained in the map, and serves as a guide through those features due to /Fo/.

It is rare that the user can use LIST 14 to choose a fraction of his cell so that the resulting map will contain a discrete and connected molecule. In general, the peaks taken from the map must have symmetry operators applied to them to generate a unified molecule. This can often be done automatically for organic molecules using the instruction \REGROUP. By using \REGROUP with different collection ranges, it may be possible to construct a very acceptable molecule.

 \REGROUP
 \ build up the main frame of non-hydrogens first
 SELECT MOVE=1.6
 END
 \REGROUP
 \ now re-move the hydrogens, important if the 
 \ molecule is crowded or distorted
 SELECT MOVE=1.1
 END

The REGROUP instruction reorders (and optionally renumbers) LIST 5. \COLLECT, which computes connectivity based on radii taken from LIST 29, leaves the order of LIST 5 unchanged, but applies any necessary symmetry to the coordinates. COLLECT with the parameter TYPE=PEAK is useful for moving new found peaks as close as possible to existing atoms.

While the user can often get some idea of the shape of his structure by looking at peaks lists and inter-atomic distances and angles, simple line diagrams may also be useful. In CRYSTALS, these are obtained via the molecular axes (q.v.) calculation \MOLAX. This instruction enables a plot to be made onto the best plane through a specified list of atoms or peaks. The list may include all the items in LIST 5, only selected items, and symmetry operators may be introduced if the user thinks that this may produce a more interpretable figure. The example below is based on trans-1,4 dimethyl cyclohexane, which lies on a centre of symmetry and has 4 unique carbon atoms and 8 hydrogen atoms.

 \MOLAX
 \ use the whole of LIST 5, together with its 
 \ image in the centre at 0,0,0
 ATOM FIRST UNTIL LAST FIRST(-1) UNTIL LAST
 PLOT
 EXEC
 \ now just look at the methyl group
 ATOM C(4) H(41) H(42) H(43)
 PLOT
 EXEC
 END

The output from these calculation is:

 ATOM FIRST UNTIL LAST FIRST(-1) UNTIL LAST
 
 
                   H32                     H22
                      .                    .
                       .                 ..
                       H+13..........CH*1
                        ..            ..
 H43                  ..                .                  H42
   ..                .                   ..                .
     .             ..                      .             ..
     H+14........+1*                       H+11.......C.4
      .           ..                       ...         H41
    ..              .                     .               .
   .                 ..                  .                 ..
 H42                   .               ..                   H43
                       H+12..........C.3
                       ...            H31
                      ..                .
                     .                   .
                   H22                   H32
 

 ATOM C(4) H(41) H(42) H(43)                
                                                
                           H43                  
                             ..                 
                               ..               
                                C.4.......H42   
                                .
                               .
                            H41

Note that atom types are abbreviated to their initial letter only, and serials are represented by only the last two digits. The symbols + and * mark character overlap.