
Crystals ManualChapter 8: Fourier Routines
8.1: Scope of the Fourier section of the user guideIn 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 8.2: Input of the Fourier section limits  LIST 14\LIST 14 XAXIS MINIMUM= STEP= MAXIMUM= DIVISION= YAXIS MINIMUM= STEP= MAXIMUM= DIVISION= ZAXIS MINIMUM= STEP= MAXIMUM= DIVISION= XPAT MINIMUM= STEP= MAXIMUM= DIVISION= YPAT MINIMUM= STEP= MAXIMUM= DIVISION= ZPAT MINIMUM= STEP= MAXIMUM= DIVISION= ORIENTATION DOWN= ACROSS= THROUGH= SCALEFACTOR VALUE= \LIST 14 XAXIS 0.0 0.0 0.5 0.0 YAXIS 0.0 0.0 0.9 0.0 ZAXIS 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.
This directive specifies how the xaxis is to be divided.
If DIVISION is equal to zero, which is its default value,
the Fourier routines will calculate the number of divisions
required along the xaxis. 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.
Controls the orientation parameters for the map
calculation and printing.
X  Default value Y Z The default value X indicates that the x coordinate goes down the printed page. ACROSS= As DOWN above, but with the default value Y indicating that the y coordinate goes across the page. THROUGH= As DOWN above, but with the default value Z indicating that the z coordinate changes from section to section. SCALEFACTOR VALUE= VALUE= This parameter specifies the value by which the electron density, on the scale of /Fc/, is multiplied before it is printed. If this parameter is omitted, a default value of 10 is assumed. 8.3: Printing the contents of LIST 14The contents of LIST 14 can be listed to the line printer
by issuing the instruction :
There is no instruction available for punching LIST 14.
8.4: Fourier calculations  \FOURIER\FOURIER MAP TYPE= NE= PRINT= SCAN= SCALE= ORIGIN= NMAP= MONITOR= REFLECTIONS WEIGHT= REJECT= F000= CALC= LAYOUT NLINE= NCHARACTER= MARGIN= NSPACE= MINRHO= MAXRHO= 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 LeastSquares. 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. \FOURIER MAP TYPE= NE= PRINT= SCAN= SCALE= ORIGIN= NMAP= MONITOR= TYPE= FOBS  Default value FCALC DIFFERENCE 2F0FC OPTIMAL FOPATTERSON FCPATTERSON EXTERNAL The map type 'OPTIMAL' implements a suggestion of Peter Main. It is a form of weighted Fo map, with coefficients w*Fo if the reflection is in a centrosymmtric class, otherwise (2*w*Fo)Fc, where w is the Simm weight. NOTE this is not the same as w(2*FoFc), a Sim weighted 2FoFc map. It has the property that known and unknown atom peak heights are approximately the same, and should be usefull for Fourier refinement. NE= This parameter indicates which solution should be used to compute the externally phased map, and has a default value of 1. NE is only used in conjunction with TYPE = EXTERNAL. PRINT= Controls the printing pf the map. NO  Default value YES SCAN= Controls automatic scanningof the map for peaks. NO YES  Default value SCALE= Controls the scaling of the electron density in the map. NO AUTOMATIC  Default value YES If SCALE is YES, the program computes a scale factor rather than take one from LIST 14. The scale factor is computed by summing the modulus of all the contributors to the map, and dividing this total into ORIGIN (see the next parameter). For a Patterson, therefore, the origin is scaled to be ORIGIN, while for other maps a scale factor is computed which guarantees that every number is less than ORIGIN. 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.
LOW MEDIUM  Default value HIGH If MONITOR is MEDIUM the, the peak coordinates are printed as they are found. If HIGH, density at known sites is also printed. REFLECTIONS WEIGHT= REJECT= F000= CALC= WEIGHT= SIM NO  Default value LIST6 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 LIST6 , then the map is weighted with the weight
stored in LIST 6.
NONE SMALL  Default value QUARTER HALF If REJECT is NONE, all the reflections in LIST 6 which are allowed by LIST 28 are included. In this case, no check is made on the /Fc/ value. For an /Fo/, /Fc/ and difference Fourier, the program expects that there should be an /Fc/ value if the phase is to be defined. Accordingly, reflections where /Fc/ < 0.001 are normally rejected for such Fouriers, and this is the default option of SMALL. Some users like to omit reflections if Fc is smaller then a fraction of
Fo. The options QUARTER and HALF are available.
NO  Default value YES Value YES causes structure factors (i.e. Fc and phase) to be calculated immediately before the map is computed. This option can only be activated if some previous task with the current DSC file has computed phases via a \SFLS instruction and left a LIST 33 on the disk. LAYOUT NLINE= NCHARACTER= MARGIN= NSPACE= MINRHO= MAXRHO= This directive specifies how the map should be printed, if the value of the
PRINT parameter on the MAP directive is YES.
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.
NPEAK = (Cell volume) / (18 * Space Group multiplicity) 18 is an average atomic volume. REJECT= This parameter, with a default value of 0.01, specifies that peaks within a distance of REJECT angstrom of a peak already ranked on peak height, will be rejected from the list. TAPES INPUT= OUTPUT= This directive is used if a map is to be read off magnetic tape, or a computed map is to be written to a magnetic tape. Remember that CRYSTALS will use scratch files unless given named files. To assign a named output file, issue \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.
NO  Default value YES If INPUT is YES, a map will be read in from the 'input magnetic tape', and the resulting map will be the minimum of each point of the calculated and input maps. The input map sections must be on device 'MT2' *** THIS FACILITY IS NOT CURRENTLY IMPLEMENTED ***
NO  Default value YES If OUTPUT is YES, the map produced is written to the 'output magnetic tape'. You may need to OPEN a permanent file on device 'MT1'. 8.5: Calculation of superposition minimum functions(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, 1/22y, 0 vector resulting from a twofold 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 XAXIS 14 4 122 400 YAXIS 5 2 59 100 ZAXIS 12 2 66 100 ORIENT X Y Z SCALE 10 END and \LIST 14 XAXIS 14 4 122 400 YAXIS 16 2 70 100 ZAXIS 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.
8.6: Processing of the peaks list  LIST 10\LIST 10 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 \PEAKS is the normal choice, since duplicate peaks related by symmetry, or peaks corresponding to known atoms can be eliminated. It is described below; EDIT, COLLECT and REGROUP are in the section on Atomic and Structural Parameters. 8.7: Printing the contents of LIST 10The contents of LIST 10 can be listed with:
There is no instruction available for punching LIST 10 out on cards.
8.8: Elimination of duplicated entries in LISTS 5 and 10  \PEAKS\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.
5 10  Default value OUTPUTLIST= 5  Default value 10 SELECT REJECT= KEEP= MONI= SEQ= TYPE= REGROUP= MOVE= SYMM= TRANS= REJECT= REJECT is the distance above which connected atoms or peaks are assumed to be distinct. If a contact is found which is less than REJECT the second atom or peak of the pair in the list is eliminated, and defaults to 0.5. KEEP= This parameter indicates how many entries are to be kept in the output list. The default value of 1000000 is the maximum possible. MONITOR= LOW HIGH  Default value If MONITOR is given as LOW only the atoms or peaks that are deleted because of the REJECT limit are listed. If MONITOR is HIGH, all the atoms deleted because of both KEEP and REJECT are listed. SEQUENCE= NO  Default value YES If SEQUENCE is YES, then the program will give sequential serial numbers to the atoms and peaks in the final output list . TYPE= PEAK  Default value ALL AVERAGE If TYPE is PEAK, then the program will only delete PEAKS which are within REJECT of an existing atom. It TYPE is ALL, atoms are also deleted. If TYPE is AVERAGE, coincident atoms or peaks are averaged. The radius for coincidence is taken from the DISTANCE keyword on the REFINE directive. The default radius is .5 Angstrom. REGROUP= This parameter has two allowed values : NO  Default value YES If REGROUP is YES, then the program will reorganise LIST 5 so that bonded atoms and peaks are adjacent. MOVE= The value of this parameter is the maximum separation for 'bonded' atoms. The default is 2.0 A. SYMMETRY= This parameter controls the use of symmetry information in the calculation of contacts, and can take three values. 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. TRANSLATION= This parameter controls the application of cell translations in the calculation of contacts, and can take the values YES or NO REFINE DISTANCE= MULTIPLIER= Controls action of Fourier refinement.
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 8.9: Slant fourier calculations  \SLANT\SLANT MAP TYPE= MINRHO= 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 BeeversLipson 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.(XXO) 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 reexpressed 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.
This is the instruction which initiates the slant fourier routines.
FOBS FCALC DIFFERENCE FOPATTERSON FCPATTERSON There is no default value for this parameter MINRHO= This parameter has a default value of zero, and is the value below which all numbers on the map are replaced by MINRHO. SCALE= The terms used in the Fourier are put on the same scale as Fc, and then before the map is printed the numbers are multiplied by SCALE . (i.e. SCALE is the map scale factor). The default is 10. WEIGHT= 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.
This directive, which excludes CENTRIOD and MATRIX, uses the matrix and
centroid stored in LIST 20 by a previous MOLAX or ANISO command.
MOLAX TLS AXES CENTROID XO= YO= ZO= This specifies the slant Fourier map centroid, in crystal fractions,
and excludes SAVED.
This gives the elements of the rotation matrix R, and
excludes SAVED. The trnsformation generally used is from crystal
fractions to orthogonal Angstroms.
This directive defines the printing of the map down the page.
This directive defines the printing of the map across the page. The
parameters have similar meanings to those for 'DOWN'.
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 