Crystals Manual

Chapter 9: Analysis Of Results

9.1: Scope of this section of the user guide

9.2: Analysis of residuals - \ANALYSE

9.3: Distance angles calculations - \DISTANCES

9.4: Distance-angles symmetry operations

9.5: Void Location - \VOIDS

9.6: TLS analysis, best planes and lines - \GEOMETRY

9.7: Torsion angles - \TORSION

9.8: Absolute Configuration - \TON

9.9: Publication listing of the atomic parameters - \PARAMETERS

9.10: Publication listing of reflection data - \REFLECTIONS

9.11: Summary of data lists - \SUMMARY

9.12: Estimation of Absolute configuration - \TONSPEK

9.13: CIF lists - \CIF

9.14: Graphics - CAMERON

 Analysis of residuals                          ANALYSE
 Distance and angles calculations               DISTANCES
 Void search                                    VOIDS
 Global Geometry (planes,lines & libration)     GEOMETRY
 Torsion angles                                 TORSION
 Absolute Configuration                         TON
 Publication listing of the atomic parameters   PARAMETERS
 Publication listing of the reflections         REFLECTIONS
 Summary of data lists                          SUMMARY
 CIF files                                      CIF
 Graphics                                       CAMERON

This analyses the residual, Fo-Fc, for systematic trends, which might either indiacate an incomplete model, or an unsatisfactory weighting scheme. It is described in the chapter Structure Factors and Least Squares.

 

 \DISTANCES INPUTLIST=
 OUTPUT MONITOR=  LIST= PUNCH= HESD=
 SELECT ALLDIST= COORD= SORTED= TYPE= RANGE= SYMM= TRANS=
 LIMITS DMINIMUM= DMAXIMUM= AMINIMUM= AMAXIMUM=
 E.S.D.S COMPUTE= CELL=
 INCLUDE atoms
 EXCLUDE atoms
 ONLY atoms
 PIVOT atoms
 BONDED atoms
 END

 \DIST
 E.S.D YES
 END

The distance angles routine is completely general with respect to crystal and lattice symmetry. For distances, the user may either use elemental radii specified in LIST 29 (see section 4.18 for input details), or specify minimum and maximum limits, and the program then calculates all possible contacts within these limits. All symmetry operations and unit cell translations are automatically generated. For the angles, LIST 29 or a separate set of distance limits may be used. At a given atom, angles are then calculated between all the atoms which bond to the central atom within the given limits.

The distance-angles routines can calculate the estimated standard deviations of the distances and angles that they produce. These e.s.d.'s are based upon the matrix stored in LIST 11 (see section 7.51), and as many variance and covariance terms as are present are used. (For a full matrix, therefore, the full variance-covariance matrix is used). For this reason, the calculation of e.s.d.'s takes at least ten times as long as a simple distance angles calculation.

When a set of e.s.d.'s are calculated, the variance-covariance matrix for the cell parameters (LIST 31, section 4.5) may also be used.
 

      5   -  Default value
      10

      OFF         -  no output
      DISTANCES   -  only monitors distances. (Default)
      ANGLES      -  only monitors angles.
      ALL         -  monitors distances and angles.

      OFF
      LOW   -  Default
      HIGH

No.H  No.H   U mult     dist
C-H
>4            1.5      .96 disorder
1      1      1.2      .93 C C-H (acetylene)
1      2      1.2      .93 C-C(H)-C
1      3      1.2      .98 (C)3-C-H
2      1      1.2      .93 C=C-H(2)
2      2      1.2      .97 (C)2-C-(H)2
3      1      1.5      .96 C-C-(H)3
N-H
>4            1.5      .89 NH4 or disorder
1      1      1.2      .86 N-N/H
1      2      1.2      .86 (C)2-N-H
1      3      1.2      .89 (C)3-N-H
2      1      1.2      .86 C-N-(H)2
2      2      1.2      .89 (C)2-N-(H)2
3      1      1.2      .89 C-H-(H)3
O-H
1      1      1.5      .82 O-H

Dist      esd = 0.02
Vib       esd = 0.002
Angle     esd = 2.0

      ALL      - (Default) Output all bond length and angle standard
                 uncertainties (if requested) to the CIF (if requested),
                 including those of bonds to fixed atoms (i.e. to atoms on
                 special positions, or to atoms that are not refined).
      NONFIXED - Exclude standard uncertainties of bond distances and angles
                 to Hydrogen atoms that have not been refined. (as
                 required by Acta's notes for authors).

      NO   -  Default value
      YES

      NO   -  Default value
      YES

      NO   -  Default value
      YES

      ALL   -  Default value
      INTRA
      INTER

      COVALENT      Use 'covalent' radii from LIST 29.
      VANDERWAALS.  Use 'VanderWaals' radii from LIST 29, but angles are
                    suppressed.
      IONIC.        Use 'ionic' radii from LIST 29.
      LIMITS.       Use specified or default ranges set by the LIMIT directive.

      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.

      NO   -  Default value
      YES

      NO   -  Default value
      YES

Accompanying each atom in a distance or angle calculation with LIST equal to HIGH are the symmetry operators that are necessary to bring the atom into the correct position in the cell to make a contact with the central atom. These symmetry operations are divided into six parts, which are indicated by five flags. These are explained in the section on Atomic and Structural Parameters.

 \
 \ distances from 0 to 2.5
 \ angles from 0 to 2.0
 \ the e.s.d.'s of the distances and angles are calculated
 \ distances from each atom to all other atoms are printed
 \ transformed coordinates are printed
 \ the distances are sorted in order of increasing magnitude
 \
 \DISTANCES
 SELECT ALL=YES,COORD=YES,SORT=YES,RANGE=LIMITS
 LIMITS DMAX=2.5, AMAX=2.0
 E.S.D. YES
 END

 \DIST
 EXCLUDE ALL
 ONLY C(1) C(3) C(4)
 END

 \VOIDS INPUTLIST=
 DISTANCE
 TOLERANCE
 CONTACTS
 RESOLUTION
 END

 \VOIDS
 DISTANCE 2.2
 END

This utility searches for the asymmetric unit for points which lie outside the known atoms. The 'radii' of the known atoms is independent of type, and in an input value. A pseudo atom in inserted at every point on a search grid outside the known atoms. The pseudo atoms are given a 'TYPE' dependant upon the number of neighbouring pseudo atoms. Atoms of type R are at the core of large voids, type L are intermediate, and M at the surface.
 

      5   -  Default value
      10

 \GEOMETRY INPUTLIST=
  ATOMS  W(1)  SPECIFICATION(1)  W(2)  SPECIFICATION(2) .
  PLANE
  LINE
  AXES
  TLS 
  EXECUTE
  EVALUATE  ATOM SPECIFICATIONS . . . .
  REPLACE ATOM SPECIFICATIONS . . .
  PUNCH
  SAVE
  DIHEDRAL  NP(1)  AND  NP(2)
  QUIT
  CENTRE   X=, Y=, Z=
  REJECT   NV=
  LIMITS   VALUE=   RATIO= 
  MODL L(11), L(22) L(33) L(23) L(13) L(12)
  MODT T(11), T(22) T(33) T(23) T(13) T(12)
  ZEROS
  DISTANCES  DL=   AL=
  ANGLES  AL=
  PLOT
  END

 \GEOMETRY
 ATOMS FIRST UNTIL LAST
 PLANE
 EXECUTE
 SAVE
 ATOMS FIRST UNTIL LAST
 TLS
 EXECUTE
 ANGLE 1 AND 2
 EXECUTE
 DISTANCES
 END

GEOMETRY is used for computing the following global derived parameters:

      Centroid (centre of gravity)
      Inertial Tensor
      Best Plane
      Best Line
      Shape Indices
      Principal Axes of adps
      Librational and Translational Thermal Tensors
      Dihedral Angles

It replaces the old \MOLAX, \AXES and \ANISO commands

PLANE & LINE are used for computing the principal axes of inertia through groups of atoms using the routines described in Computing Methods in Crystallography, edited by J. S. Rollett, Pergamon Press, 1965, p67-68.

The best plane for a series of N atoms whose positions have varying reliability, such that they can be assigned weights, w(1), w(2), . . . w(n), is defined as that for which the sum of the squares of the distances (in angstroms) of the atoms from the plane, multiplied by the weights, w(i), of the atomic positions, is a minimum. Note that the normal to the 'worst plane' is the 'best line', and if masses are used for weights, then the calculation gives the principal inertial axes.

The atomic positions are taken from LIST 5, possibly modified by symmetry information, to compute inertial axes & deviations of atoms from the planes or lines.

Each time a line or plane is computed, the direction cosines of the relevent axis are stored as AXIS number 'n'. The dihedral angles between these axes can be computed. Three geometry indices are also computed. The geometry is best described by the index closest to unity. (Mingos,D.P.M & Rohl,A.L., J.Chem.Soc. Dalton Trans (1991) pp 3419 - 3425)

If the three principal axes are "big, medium, small", then
Spherical = small/big Cylindrical = 1-((medium+small)/(2.big)) discoidal = 1-2.small/(big+medium)

TLS. This routine calculates the overall rigid-body motion tensors T, L, S (Shoemaker and Trueblood, Acta Cryst. B24, 63, 1968) by a least-squares fit to the individual anisotropic temperature factor components, together with librational corrections to bond lengths and angles.

Shoemaker and Trueblood's conventions and reductions are followed throughout; in particular, the trace of S, which is indeterminant, is set to zero. The program therefore determines 20 overall tensor components - the upper triangles of T and L together with the whole of S apart from S(33). (See also: Johnson in Crystallographic Computing, ed R.Ahmed, Munksgaard, 1970, pp 207-219)

Even when the trace-of-S singularity has been removed, however, the nature of the rigid body problem is such that ill-conditioned and singular normal matrices are much more common than in structure refinement and the program therefore proceeds via the eigenvalues and eigenvectors of the normal matrix. In most cases the largest and smallest eigenvalues are output for inspection, but if the ratio of these quantities is less than the LIMITing RATIO, a full eigenvalue/vector listing is produced. Further, if any eigenvalue is itself less than the LIMITing VALUE, the corresponding parameter combination is set to zero, thus removing the near- singularity. These actions can be modified by the use of the LIMIT and REJECT directives described below. If the TLS calcuation cannot be stabilised by means of these filters, the user can modify either T, L or S directly before applying the REPLACE or PUNCH commands. Though here is some danger in this, especially if the supposed rigid group is infact flexible, it may be preferable to using a model yielding negative vibrational or librational amplitudes.

The direction cosines of the principal axis of L are stored for use in inter-axis angle comutations.

Immediate execution of a directive can be forced by issuing an EXECUTE directive.
 

      5   -  Default value
      10

   Examples:
 
  These instructions compute a plane
  involving n(1),n(2),n(3) and c(1), and
  prints the co-ordinates of all the atoms with
  respect to this plane.  The positions of the
  nitrogen atoms have double weight
 
 \GEOMETRY
 ATOMS 2 N(1) UNTIL N(3)  1 C(1) C(2)
 PLANE
 EVALUATE ALL
 
  These instructions calculate another plane,
  printing only the co-ordinates of c(5) with respect to
  the second plane.  The angle between the two planes
  is then calculated
 
 ATOMS C(1) S(1) N(1)
 PLANE
 EVALUATE C(5)
 DIHEDRAL 1 AND 2
 END
 
 
  These instructions compute a TLS tensor for the specified atoms
  and then set up restraints to encourage the Uij to conform to a
  rigid body.  You might want to tighten the esd in TLSREST.dat to 0.005.
 
 \GEOM                                                                           
 ATOM ALL
 TLS
 PUNCH
 EVAL ALL
 EX
 END
 \LIST 16                                                                        
 \USE TLSREST.DAT                                                                
 END

 \TORSION INPUTLIST=
 ATOMS  SPECIFICATIONS
 PUBLICATION  PRINT=
 END

 \TORSION
 ATOM C(1) C(2) C(3) C(4)
 END

The routines described in this section calculate torsion angles which are defined as follows. The torsion angle about the bond j-k is the angle the bond k-l is rotated from the ijk plane. It is positive when, on looking from ij to kl, the rotation is clockwise.

The program uses atomic positions taken from LIST 5. These can be modified by the space group symmetry operators stored in LIST 2 (space group information, see section 4.8)
 

      5   -  Default value
      10

      NO   -  DEFAULT.  There is no publication listing
      YES    There is a publication listing sent to the PUNCH file
      CIF    The listing is in CIF format

 Example.
 \ the torsion angle about C(3)-C(4) is calculated
 \ two torsion angles about C(4)-C(5) are calculated
 \
 \TORSION
 ATOMS N(2) C(3) C(4) C(5)
 ATOMS C(3) C(4) C(5) C(6) O(1)
 END

 \TON 
 END

Compute the Hooft-Straver-Spek parameter for comparisonm with the Flack parameter
 

 \PARAMETERS
 LAYOUT INSET= ATOM= DOUBLE= CHOOSE= FLOAT= NCHAR= NLINE= LISTAXES= ESD=
 COORDINATES NCHAR= NDECIMAL= SELECT= TYPE= DISPLAY= PRINT= PUNCH=
 U'S NCHAR= NDEC= SELECT= TYPE= DISPLAY= PRINT= PUNCH=
 END

 \PARAMETERS
 LAYOUT ATOM-NAME=6,DOUBLE=YES
 END

This routine sends the atomic parameters to the PUNCH file in a suitable format for publication or binding into a thesis. As well as the current atomic parameters in LIST 5, the estimated standard deviations derived from the least squares normal matrix are also printed. THIS ROUTINE WILL NOT WORK if LIST 5 is modified in any way since the last round of refinement. If any changes, including renaming, are made, a further round of refinement must be done. If you wish to preserve parameter values, and create a valid matrix without changing the parameter values, compute a refinement cycle but set all the shifts to zero.

      \SFLS
      REFINE
      SHIFT GENERAL = 0.0
      END

The output is in two halves, the first containing the positional coordinates and any isotropic temperature factors, and the second containing all the anisotropic temperature parameters.

For the first part, a page is split into 6 separate fields. The first field is blank, and is an offset so that the information is centred on the page. The remaining fields contain the atom type and serial number, the three positional parameters, and a temperature factor. This will be the value of U(iso) with its e.s.d for isotropic atoms, otherwise U(equiv), without an e.s.d, for anisotropic atoms. U(equiv) is not simply related to the diagonal elements of U(aniso), and may be computed as either the arithmetic or geometric mean of the principal axes of the ellipsoid. See \SET UEQUIV in the chapter on IMMEDIATE commands. The width of each type of field may be altered by the user, using respectively the INSET , ATOM-NAME , and NCHARACTER parameters. The default length of a page of this type of output is that required for A4 paper.

The second part contains the anisotropic temperature factors, and each page is split into eight fields. As for the atomic coordinates, the first field is blank and represents an offset. The second field contains the atom type and serial number, and the remaining six fields contain the components of the anisotropic temperature factors. The width of each type of field may be adjusted by the user, using respectively the INSET , ATOM-NAME and NCHARACTER parameters. If a different value for INSET or ATOM-NAME is required in the first and second parts of the output, the job must be run twice. Depending upon the width across the page, the second part of the output occupies one sheet of A4 paper either across the page or down the page.

For both types of output, the user can select double spacing down the page with the DOUBLE parameter. Similarly for each of the numeric fields, the user can choose the number of decimal places to be printed (the NDECIMAL parameter), and whether the numbers are printed as integers or in floating point with a decimal point. (The FLOATING parameter). The e.s.d.'s are printed to the same accuracy as the atomic parameters, so that if the chosen field is too small and an e.s.d. appears to be zero, it will be omitted in exactly the same way as for a parameter that has not been refined. A parameter printed with 4 decimal places might thus appear as :

 0.0123(4)
 OR
    123(4)

Depending upon the format. In either case, the numbers are right justified in their field.

As an alternative to the user selecting the number of decimal places that should be printed, it is possible to get the program to choose the number of decimal places required for each parameter automatically. (The CHOOSE parameter). If the parameters are to be printed in floating point, the number of decimal places is chosen so that the e.s.d. Can be represented as a one digit number in the last decimal place. For numbers that are to be printed as integers, the field used is never less than that given by the NDECIMAL parameter. If the required field is larger than that defined by these s, a decimal point is inserted and the required number of extra digits is output. For example, if the number of decimal places required is four, but the e.s.d. is too small, it would appear as :

 0.12345(6)
 OR
  1234.5(6)

Depending upon whether floating point or integer output was required. For either type, if the parameter has not been refined, the number of decimal places is that given by the NDECIMAL instruction.

Since this routine prints the e.s.d.'s, it is vital that the least squares matrix (LIST 11, see section 7.51) belongs to the current LIST 5 (the model parameters). If LIST 5 has been modified in any way since the last Least Squares, this routine will abort.

When anisotropic atoms are present in LIST 5, U[EQUIV] is calculated according to the current setting of \SET UEQUIV.

 

 NO   No output is displayed on the terminal.
 YES  Output is displayed on the terminal.

 NO       No output is sent to the listing file
 YES      Output is sent to the listing file

 NO       No output is sent to the punch file
 YES      Output is sent to the punch file
 CIF      Output is in CIF format

 OFF   No output is displayed on the terminal.
 HIGH  Output is displayed on the terminal.

 NO       No output is sent to the listing file
 YES      Output is sent to the listing file

 NO       No output is sent to the punch file
 YES      Output is sent to the punch file
 CIF      Output is in CIF format

\REFLECTIONS INPUT=
LAYOUT NCOLUMNS= NLINES= INSET= NSPACE= SCALE= NCHARACTER=
OUTPUT PRINT= PUNCH= LIST28=
END

This routine prints the reflection data in LIST 6 (section 5.3) in a suitable format for publication or binding into a thesis. The information printed falls into one or more columns, each of which contains h, k, l, /Fo/, /Fc/, and the phase angle in degrees. Each column is 18 characters wide. Although the user has no control over the contents of each column, it is possible to vary the number of blank spaces at the start of each line, the number of columns across the page, the number of spaces between successive columns, and the number of lines per page. (The INSET , NCOLUMNS , NSPACE and NLINES parameters, respectively). /Fo/ and /Fc/ are both put on the same scale of /Fc/, using the scale factor in LIST 5, and both these two numbers may be modified by a scaling constant before they are printed. (The SCALE parameter). However, all the values of both /Fo/ and /Fc/ must be less than 10000 when they are printed.

LIST 28 is used for checking whether or not to print a reflection. Remember that if LIST 28 was used to reject some reflections when structure factors were last calculated, removing these restrictions before printing LIST 6 will mean that some reflections will have incorrect values of Fc and phase.

      6      Default
      7      Alternative reflection list

 NO       No output is sent to the listing file
 YES      Output is sent to the listing file

      NO       No output is sent to the punch file
      YES      Output is sent to the punch file

 \SUMMARY OF= TYPE= LEVEL=

 \SUMMARY LIST 5 HIGH
 END
 \SUMMARY EVERYTHING
 END

This command produces a summary on the terminal of the contents of a list. Use \PRINT if you need full details.
 

\TONSPEK CRITERION= INPUT= PLOT=
END

This routine contains code kindly donated by Ton Spek for the computation of the Hooft 'y' parameter. Rob W. W. Hooft, Leo H. Straver and Anthony L. Spek, J. Appl. Cryst. (2008). 41, 96–103. It required the current LIST 6 to be comverted to a LIST 7 in which Friedel pairs are adjacent and in which Fc has been computed with the Flack 'x' parameter set to zero.

The pre-preparation for using this routine has been packaged into a script available from the Analyse menu. The current value of the Flaxk paraemter should already be stored in LIST 30.

      6      Default
      7      Alternative reflection list

      NO      Default value
      FOFC    Plots the observed versus the calculated Friedel difference
      NPP     Plots the Normal Probability Plot of the Friedel differences

      n  a four-digit number giving the number of references to follow. 
         Other text on the line is ignored.
Next items repeated 'n' times:
      m  a three digit number preceded by a 'hash' symbol used as an 
         identifier for the reference. The numbers must be unique, not 
         necessarily in any order, with the largest one equal to 'n'
      
      The full reference. References are put in the file in alphabetic 
      order.

(1) if one adopts esd values to one digit precision (rule of 9) the values

      5.548(1)      1.453(2)  3.921(3)  1.2287(8)  are acceptable.

(2) if one permits two digits precision with a limit of 19 (rule of 19)...

      5.5483(9)     1.4532(16)  3.921(3)  1.2287(8)  are acceptable.

(3) if one permits two digits precision with a limit of 29 (rule of 29)...

      5.5483(9)     1.4532(16)  3.9214(28)  1.2287(8)  are acceptable.