Crystals Manual

Chapter 8: Fourier Routines

8.1: Scope of the Fourier section of the user guide

8.2: Input of the Fourier section limits - LIST 14

8.3: Printing the contents of LIST 14

8.4: Fourier calculations - \FOURIER

8.5: Calculation of superposition minimum functions

8.6: Processing of the peaks list - LIST 10

8.7: Printing the contents of LIST 10

8.8: Elimination of duplicated entries in LISTS 5 and 10 - \PEAKS

8.9: Slant fourier calculations - \SLANT

In this section of the user guide, the lists and instructions relating to the Fourier routines are described.

      Input of the Fourier section limits                  -  \LIST 14
      Fourier calculations                                 -  \FOURIER
      Processing of the peaks list                         -   LIST 10
      Elimination of duplicated entries in LISTS 5 and 10  -  \PEAKS
      Slant fourier calculations                           -  \SLANT

 \LIST 14
 X-AXIS MINIMUM= STEP= MAXIMUM= DIVISION=
 Y-AXIS MINIMUM= STEP= MAXIMUM= DIVISION=
 Z-AXIS MINIMUM= STEP= MAXIMUM= DIVISION=
 X-PAT MINIMUM= STEP= MAXIMUM= DIVISION=
 Y-PAT MINIMUM= STEP= MAXIMUM= DIVISION=
 Z-PAT MINIMUM= STEP= MAXIMUM= DIVISION=
 ORIENTATION DOWN= ACROSS= THROUGH=
 SCALEFACTOR VALUE=

 \LIST 14
 X-AXIS 0.0 0.0 0.5 0.0
 Y-AXIS 0.0 0.0 0.9 0.0
 Z-AXIS -2 2 32 60
 ORIENTATION Z X Y
 SCALE VALUE = 10
 END

The Fourier routines will calculate a map with section edges parallel to any two of the cell axes (a, b or c). The starting and stopping points must be given for each direction (in crystal fractions). The user should choose the asymmetric unit to have one range as small as possible, and the other two approximately equal. Orientate the computation so that the sections are perpendicular to the short range direction. If the instruction \SPACEGROUP has been used to input the symmetry information, a LIST 14 will have been generated. This will be a valid choice, but may not be optimal.
 

 
\LIST 14

 

 
X-AXIS MINIMUM= STEP= MAXIMUM= DIVISION=

This directive specifies how the x-axis is to be divided.
MINIMUM= This parameter gives the initial value along the x-direction. If it is omitted, a default value of 0.0 is assumed for MINIMUM.
STEP= This parameter, which has a default value of 0.3, gives the step along the x-direction.
MAXIMUM= This parameter, which has a default value of 1.0, gives the final value along the x-direction.
DIVISION= If DIVISION is greater than zero, it defines the number of divisions into which the x-axis is to be divided. In this case, the three remaining parameters are expressed in terms of DIVISION and give the first point ( MIN ), the increment between successive points ( STEP ) and the final point to be calculated ( MAX ). If the divisions of the unit cell along the x-axis are given in this way, the user must ensure that sufficient map is calculated for the map scan, by adding one extra point beyond the asymmetric unit at both ends along the x-axis. If this is not done, peaks at the edge of the asymmetric unit may be missed by the peak search.

If DIVISION is equal to zero, which is its default value, the Fourier routines will calculate the number of divisions required along the x-axis. In this case, STEP is the interval between successive points along the axis in angstrom. If this parameter is less than 0.05, a default value of 0.3 angstrom is used. MINIMUM And MAXIMUM define the first and last points to be calculated and are given in fractional coordinates. When the values of MIN and MAX are converted into unit cell divisions, an extra point is added at each end to ensure that the peak search functions correctly.
 

 
Y-AXIS MINIMUM= STEP= MAXIMUM= DIVISION= Similar to X-AXIS above.
 

 
Z-AXIS MINIMUM= STEP= MAXIMUM= DIVISION= Similar to X-AXIS above.
 

 

 
X-PAT MINIMUM= STEP= MAXIMUM= DIVISION= This instruction is similar to the X-AXIS instruction, but refers to the Patterson asymmetric unit.
 

 
Y-PAT MINIMUM= STEP MAXIMUM= DIVISION= Similar to X-PAT above.
 

 
Z-PAT MINIMUM= STEP= MAXIMUM= DIVISION= Similar to X-PAT above.
 

 
ORIENTATION DOWN= ACROSS= THROUGH=

Controls the orientation parameters for the map calculation and printing.
DOWN=

      X  -  Default value
      Y
      Z

The contents of LIST 14 can be listed to the line printer by issuing the instruction :
 

 
\PRINT 14

There is no instruction available for punching LIST 14.
 

 

 \FOURIER
 MAP TYPE= NE= PRINT= SCAN= SCALE= ORIGIN= NMAP= MONITOR=
 REFLECTIONS WEIGHT= REJECT= F000= CALC=
 LAYOUT NLINE= NCHARACTER= MARGIN= NSPACE= MIN-RHO= MAX-RHO=
 PEAKS HEIGHT= NPEAK= REJECT=
 TAPES INPUT= OUTPUT=
 END


 \FOURIER
 MAP TYPE=DIFF
 PEAK HEIGHT = 3
 END

Before a Fourier is computed, a LIST 14 must have been created or input. The routine will compute a map in any space group, the relevant symmetry being found in LIST 2.

In the ouput listing, new peaks are labelled, with the following meanings

      GOOD PEAK - The peak centre was determined by Least-Squares.
      POOR PEAK - The peak centre was determined by interpolation.
      DUBIUOS PEAK - The peak centre is only a local maximum.
      MALFORMED PEAK - The peak centre is extrapolated to be out side
                 of the asymmetric unit - usually due to very poor phasing.

      F-OBS     -  Default value
      F-CALC
      DIFFERENCE
      2F0-FC
      OPTIMAL
      FO-PATTERSON
      FC-PATTERSON
      EXTERNAL

      NO   -  Default value
      YES

      NO
      YES  -  Default value

      NO
      AUTOMATIC  -  Default value
      YES

If SCALE is NO, the scale factor is taken from LIST 14 for all types of Fourier maps. If SCALE is AUTOMATIC, there is automatic scaling for an external or Patterson map, while other maps take their scale factors from LIST 14.
ORIGIN= The default value for this parameter is 999, and is used when the program calculates a scale factor (see SCALE above).
NMAP Controls negation of the density values, with default NO. Use YES, in which case the density values are negated, when looking for minima. This feature permits location of hydrogen in Neutron maps, and the location of minima (which become maxima) generally. Set the Peak Height positive even when searching for minina, since at the time of the search the minima are inverted. The out put density values have the correct sign. Use \COLLECT 10 5 rather than \PEAKS on negated maps, since PEAKS cannot handle minima.
MONITOR=

      LOW
      MEDIUM  -  Default value
      HIGH

      SIM
      NO     -  Default value
      LIST-6

If WEIGHT is NO , its default value, then the map is not weighted.

If WEIGHT is set equal to SIM , then SIM weights are computed. This option requires both LIST 29 and LIST 5. The occupation factors in LIST 5 are used to determine how many atoms of each type are present, and LIST 29 indicates how many should be present. See the notes under 'TYPE', above.

If WEIGHT is LIST-6 , then the map is weighted with the weight stored in LIST 6.
REJECT=

      NONE
      SMALL  -  Default value
      QUARTER
      HALF

Some users like to omit reflections if Fc is smaller then a fraction of Fo. The options QUARTER and HALF are available.
F000= The default value for this parameter is zero, and specifies the value of F(000) to be used.
CALC

      NO   -  Default value
      YES

This directive specifies how the map should be printed, if the value of the PRINT parameter on the MAP directive is YES.
NLINE= This parameter sets the number of lines per row of map, and has a default value of 2.
NCHARACTER= This parameter controls the number of characters for each grid point, and has a default value of 4.
MARGIN= This parameter, whose default value is 4, defines the number of characters per division number down each side of the map.
NSPACE= This parameter has a default value of 2, and defines the number of spaces between the division number and the grid number down each side of the map. The minimum value for NSPACE is 2.
MIN-RHO= This parameter has a default value of -1000000, and points less than MIN-RHO are left blank when the map is printed.
MAX-RHO= This parameter has a default value of 1000000, and points greater than MAX-RHO are left blank when the map is printed.
 

 
PEAKS HEIGHT= NPEAK= REJECT=

Controls the search for peaks when the map is searched, i.e. if the value of the SCAN parameter on the MAP directive is YES.
HEIGHT= This parameter sets the search of the map for all peaks with an electron density greater than HEIGHT. If this parameter is omitted, a default value of 50 is assumed for an external or Patterson map. For all other maps, the map is scanned for peaks greater than 1.5*SCALE, where SCALE is the map scale factor, either taken from LIST 14 or computed using SCALE = YES above.
NPEAK= This parameter, whose default value is 0, determines the number of peaks to be retained after they have been ranked by peak height. If NPEAK is zero or negative, the number of peaks saved is computed from

      NPEAK = (Cell volume) / (18 * Space Group multiplicity)

                               18 is an average atomic volume.

      \OPEN MT1 filename

The tape is unformatted.

 Record 1: 'INFO  DOWN ACROSS SECTION'
 Record 2: 'TRAN'       9 elements of a transformation matrix
 Record 3: 'CELL'       Cell parameters, angles in radians
 Record 4: 'L14 '       List 14 information
 Record 5: 'SIZE'       number of points down, across, and number of sections
 Record 6:  number of values,  values for a section
           Record 6 is repeated for every section.
 Record n:  number of atoms, number of items per atom
 Record n+1: Items for an atom, repeated for all atoms

Record 4 contains 6 integers, (No of points down and across the page, number of sections, and the index of these directions, 1 = x). Subsequent records contain a whole section line by line, prefixed by the total number of points in the section.
INPUT=

      NO   -  Default value
      YES

*** THIS FACILITY IS NOT CURRENTLY IMPLEMENTED ***
 

OUTPUT=

      NO   -  Default value
      YES

(Issue 7 - implementation incomplete, 1984)

(Issue 9 - implementation still incomplete, 1993 - no one seems to want it anyway!- use SHELXS if you need to).

(Issue 10 - still no change, 1996)

The Fourier routine provides a way of calculating superposition minimum functions. For each map that is produced, it is possible to specify that another map should be read in from magnetic tape at the same time (the TAPES directive). Each point of the resulting map is taken as the minimum of the newly computed map and that read off the magnetic tape. This output map may be written to a second magnetic tape, also by use of the TAPES directive.

When the input map and the calculated map are superposed, the first point calculated and the first point read off the tape are compared, the second point calculated and the second point input are compared, and so on. This implies that the first point on each map must represent the same point in real space for the output map, and that each map must contain the same number of points. The origin of each map that is to be calculated is altered by changing LIST 14. For example, if a 2x, 2y, 2z vector has been identified at 0.36, 0.14 and 0.28, and the 2x, 1/2-2y, 0 vector resulting from a two-fold axis has been found at 0.36, 0.36, 0, then the two LIST 14's for the superposition function might appear as :

 \LIST 14
 X-AXIS 14 4 122 400
 Y-AXIS 5 2 59 100
 Z-AXIS 12 2 66 100
 ORIENT X Y Z
 SCALE 10
 END

      and

 \LIST 14
 X-AXIS 14 4 122 400
 Y-AXIS 16 2 70 100
 Z-AXIS -2 2 52 100
 ORIENT X Y Z
 SCALE 10
 END

For the first map, the origin of real space is at 0.18, 0.07 and 0.14 in vector space. This point is moved so that it is one grid point in along each axial direction, to allow for the map scan. For the second peak, the origin in real space is at 0.18, 0.18 and 0.0. The second LIST 14 places this point one grid point in along each of the axial directions so that the real space origin of the two maps coincides. To convert the coordinates that result from the second map scan to real space coordinates, it is necessary to subtract 0.18 from x and 0.18 from y, since the coordinates are printed in Patterson space for all the maps calculated.
 

LIST 10 cannot be input bythe user. When the map scan has been completed, the resulting peaks are output to the disc as a LIST 10. Except for an external or Patterson map, the atoms already in LIST 5 are placed at the beginning of the LIST 10.

A LIST 10 is usually converted to a LIST 5 by one of the following instructions :

 \EDIT 10 5                \PEAKS 10 5
 \COLLECT 10 5             \REGROUP  10 5

The contents of LIST 10 can be listed with:
 

 
\PRINT 10

There is no instruction available for punching LIST 10 out on cards.
 

 \PEAKS INPUTLIST= OUTPUTLIST=
 SELECT REJECT= KEEP= MONI= SEQ= TYPE= REGROUP= MOVE= SYMM= TRANS=
 REFINE DISTANCE= MULTIPLIER=
 END

 \PEAKS
 SELECT REJECT=0.0001
 REFINE DISTANCE=.5
 END

This routine eliminates atoms or peaks which duplicate other entries in an atomic parameter list. When using this routine, a set of distances is calculated about each atom or peak in turn. Atoms or peaks further down the list than the current pivot are then eliminated if they have a contact distance less than a user specified maximum (the REJECT parameter). Thus, when peaks have been added to a LIST 5, the peaks corresponding to the atoms can be eliminated.
 

 
\PEAKS INPUTLIST= OUTPUTLIST=
INPUTLIST and OUTPUTLIST specify where the atoms are to be taken from, and where they will be put.
INPUTLIST=

      5
      10  -  Default value

      5   -  Default value
      10

      LOW
      HIGH  -  Default value

      NO  -  Default value
      YES

      PEAK  -  Default value
      ALL
      AVERAGE

      NO  -  Default value
      YES

      SPACEGROUP  -  Default value. The full spacegroup symmetry is used in
                                    all computations
      PATTERSON.     A centre of symmetry in introduced, and the translational
                     parts of the symmetry operators are dropped.
      NONE.          Only the identity operator is used.

Controls action of Fourier refinement.
DISTANCE= This parameter has a default value of zero, and is the distance below which atoms and peaks are considered to be coincident. The coordinates of an existing atom are replaced by those of a coincident peak. Refinement takes precedence over deletion of peaks.
MULTIPLIER= This parameter has a default value to give automatic refinement. It is set to 1 for a centric space group and is set to 2 for a non-centric space group. It can be set to 0.0 to preserve original coordinates but be given new peak heights.

      X(new) = x(atom) + mult(x(peak) - x(atom)).

 \ reject atoms or peaks with contact distances less than 0.7
 \ keep 30 entries in the output list
 \ list the atoms and peaks rejected because of both 'KEEP'
 \ and 'REJECT'
 \
 \PEAKS 10 5
 SELECT REJECT=0.7,KEEP=30,MONITOR=HIGH
 END

 \SLANT
 MAP TYPE= MIN-RHO= SCALE= WEIGHT=
 SAVED MATRIX=
 CENTROID XO= YO= ZO=
 MATRIX R(11)= R(12)= R(13)= R(21)=  .  .  . R(33)=
 DOWN MINIMUM= NUMBER= STEP=
 ACROSS MINIMUM= NUMBER= STEP=
 SECTION MINIMUM= NUMBER= STEP=
 END

A Slant Fourier is one that is calculated through any general plane of the unit cell. For such a Fourier, the normal Beevers-Lipson expansion of the summation cannot be used, so that it will take many orders of magnitude longer than a conventional one. The algorithm adopted here is as follows :

 X    A general vector expressed in fractions of the
     unit cell edges (i.e. x/a, y/b and z/c)
 XO   The centroid of the required general fourier section,
     also expressed in crystal fractions.
 XP   The coordinates of the point 'X' when expressed
     in the coordinate system used to define the
     plane of the general section.
 'X' and 'XP' are related by the expression :  XP = R.(X-XO)
 R    'R' is the matrix that describes the transformation
     of a set of coordinates in the crystal system to
     a set of coordinates in the required plane.
 therefore :  X = S.XP + XO

     'S' is the inverse matrix of 'R'.

 The required expression in the fourier is :

     H'.X = H'.S.XP + H'XO

 H    H is a vector containing the Miller indices of
     a reflection and H' is the transpose of H.
 This may be re-expressed as :

     H'.X = H'.S.DXP + H'.(S.XPS + XO)

 DXP  'DXP' represents the increment in going from the
     first point on the section to be calculated.
 XPS  'XPS' is the coordinate of the first point on the
     section to be calculated.
     obviously :  XP = XPS + DXP.

When the Fourier is calculated, the term H'.(S.XPS + XO) is constant for each section to be calculated. The term H'.S , which may be regarded as the transformed indices, is also constant for each reflection, so that a two dimensional recurrence relation may be used to change DXP and thus Cos(2*PI*H.X - ALPHA)' over the required section for each reflection. ( ALPHA is the phase angle for the current reflection).

The input for the slant Fourier thus must include the rotation matrix R, the centroid XO, and the steps and divisions in the required plane.
 

 
\SLANT

This is the instruction which initiates the slant fourier routines.
 

 
MAP TYPE= MIN-RHO= SCALE= WEIGHT=
TYPE=

      F-OBS
      F-CALC
      DIFFERENCE
      FO-PATTERSON
      FC-PATTERSON

      NO   -  Default value
      YES

If WEIGHT = YES, the observed and calculated structure factors are multiplied by the weights in LIST 6 (usually SQRT(w)). The user should be aware that this might have a major effect on the scale if the map density, and that SCALE may need adjusting.
 

 
SAVED MATRIX=

This directive, which excludes CENTRIOD and MATRIX, uses the matrix and centroid stored in LIST 20 by a previous MOLAX or ANISO command.
MATRIX=

      MOLAX
      TLS
      AXES

This specifies the slant Fourier map centroid, in crystal fractions, and excludes SAVED.
XO=
YO=
ZO= The defaults value for XO,YO,ZO, the coordinates of the centroid, are 0.0.
 

 
MATRIX R(11)= R(12)= R(13)= R(21)= . . . R(33)=

This gives the elements of the rotation matrix R, and excludes SAVED. The trnsformation generally used is from crystal fractions to orthogonal Angstroms.
R(11)= R(12)= R(13=) R(21)= . . . R(33)= There are no default values for any of these parameters.
 

 
DOWN MINIMUM= NUMBER= STEP=

This directive defines the printing of the map down the page.
MINIMUM= There is no default value for this parameter, the first point, in Angstrom, down the page of the plane to be calculated.
NUMBER= There is no default value for this parameter, the number of points of the plane to be printed down the page
STEP= There is no default value for this parameter, the interval in Angstrom between successive points down the page.
 

 
ACROSS MINIMUM= NUMBER= STEP=

This directive defines the printing of the map across the page. The parameters have similar meanings to those for 'DOWN'.
 

 
SECTION MINIMUM= NUMBER= STEP=

This directive defines the printing of the map sections. The parameters have similar meanings to those for 'DOWN'.

The units of MINIMUM and STEP are based on the coordinate system used to describe the plane, with the new 'x' axis going down the page and 'y' across. In general the most convenient axial system for the plane is one expressed in Angstrom, so that the initial points and the steps are all expressed in Angstrom. (The least squares best plane program prints out the centroid in crystal fractions and the rotation matrix from crystal fractions to best plane coordinates in Angstrom, which are the numbers required, and may be saved for use in SLANT by the directive 'SAVE').

 \ the map will be a difference map
 \ we wish to compute the section 0.3 anstrom above the plane
 \ numbers less than zero will be printed as zero
 \ the molecule lies at a centre of symmetry
 \ so that the centroid in crystal fractions is 0, 0, 0
 \ the plane coordinates are in angstrom
 \ for printing the plane both across and down the page,
 \ we will start 4 angstrom from the centroid,
 \ and go 4 angstrom the other side of the centroid,
 \ making a grid 8 angstrom by 8 angstrom
 \
 MAP DIFFERENCE 0.3 0
 CENTROID 0 0 0
 MATRIX 3.4076 10.0498 6.1794
 CONT   5.0606  8.287 -9.5483
 CONT  -6.9181 11.0121 1.546
 DOWN -4 33 0.25
 ACROSS -4 33 0.25
 END