4.25. GENEV: Normalized structure factors (E values)
Authors: Syd Hall and V. Subramanian
Contact: Syd Hall, Crystallography Centre, University of Western Australia, Nedlands 6907, Australia
GENEV calculates normalized structure factors (E values) from the measured structure factors, and calculates the scale and overall U. Unique aspects of GENEV include the estimation of |E| errors, the application of simple Bayesian statistics to weak data, the compensation for missing weak data, a facility for user-specified index rescale groups, and the calculation of group structure factor phases for use in subsequent phasing processes.
4.25.1. Introduction
Structure factor normalization converts measured |F| values into "point atoms at rest" coefficients known as |E| values. The normalization process used in GENEV is based on a Wilson plot procedure (Wilson, 1942). This approach also provides a reasonable estimate of the overall thermal displacement parameter and overall structure factor scale.
A general introduction to normalization procedures based on the Wilson plot can be found in the crystallographic texts such as Stout and Jensen (1968), or Luger (1980). For more detailed information on structure factor normalization the user should refer to the three papers by Subramanian and Hall (1982) and Hall and Subramanian (1982a,b). These studies form the basis for GENEV and provide detailed definitions and descriptions of the scaling functions and expectation expressions discussed here.
Central to the normalization approach of GENEV is the calculation of two separate normalized structure factors |E1| and |E2|. |E1| is calculated using the linear scale K exp(Bs2), the random-atom expectation value, and an overall rescale term. This combination of parameters has been shown to provide a consistently good estimate of |E| (S & H, 1982). Inflection point least-squares (H & S, 1982a) is used to obtain an estimate of the overall B which is largely independent of the Debye scattering effects. The basic process for calculating |E1| is fixed and cannot be varied by the user.
|E2| is calculated using scaling options selected by the user. These include the choice of linear or profile scale; random atom, random fragment, oriented fragment or positioned fragment |F2| expectation values; overall or index rescaling terms (see S & H for detailed description of these options). The control lines may be used to determine what form the calculation of |E2| should take. If no control lines are entered, |E2| will be calculated identically to |E1| except that index rescaling is the default rather than overall rescaling. Note that if GENEV is run just for calculating the scale or the overall U, the enot option should be used to prevent |E|'s from being output to the bdf. The default is eout .
Some of the options detailed below apply to both |E1| and |E2|. The fill and baye control line for example provides for the adjustment of weak data. It should also be emphasized here that group structure factor phases calculated from fragment information entered via frag and atom lines may be applied in subsequent phasing processes to both |E1| and |E2|. Details of this are given below.
4.25.2. Treatment Of Weak Data
One of the factors determining the reliability of |E| values is the precision
of the measured structure factors. This is true not only for the large
structure factors but also for the weak reflections that often make up the
majority of the data. In particular the precision of the weak data affects the
reliability of B and K estimated by the Wilson plot least-squares process.
GENEV provides two methods of treating weak data; one applies Bayesian
statistics to |F| and 
|F| values if |F| is less than 6|F|, and
the other compensates the Wilson plot for weak data omitted from the input
bdf.
4.25.2.1. Application of Limited Bayesian Statistics
The expected Bayesian
distribution of net intensities, and their associated standard deviations, have
been reported by French and Wilson (1978). A limited Bayesian treatment of
structure factor data is applied by entering baye on the GENEV
line. This assumes that the mean intensity for all shells of reciprocal space
is 20I. The procedure used in GENEV to do this requires the presence of
both the net intensity and the Lorentz-polarization factor on the input bdf (S
& H, 1982).
4.25.2.2. Compensation for Missing Weak Data
The Wilson plot process relies on a
relatively uniform distribution of data with s3 (s=
sin
/
). The omission of weak data from this calculation tends to
give rise to erroneously low values of B and to inaccurate estimates of |E|. In
GENEV the fill option on the GENEV line forces each shell of
reciprocal space to have the correct population. The values of |F| used for the
added reflections are based on half the minimum |F| of each range.
4.25.3. Application Of Scaling Options
GENEV provides for two basic scaling approaches: The linear scale Kexp(8p2Us2) (this is the default) and the profile scale ( prof ). In addition there are two rescaling procedures; overall (default) and index (using indexk). The definitions and the properties of these scales is detailed by S & H (1982).
4.25.3.1. Linear Scale (applied to E1; default for E2)
The application of the linear scale first requires the evaluation of the overall scale, K, and the overall thermal parameter U from the Wilson plot. The particular form of the linear scale used to calculate |E1| is based on U and K values estimated using a inflection-point least-squares procedure (see below). Use of the linear scale is optional for |E2| and, if applied in the default mode, will be identical to that used for |E1|. Entering the frag line will cause an independent linear scale to be evaluated for |E2|.
4.25.3.2. Profile Scale (prof option for E2)
If a prof is entered on the GENEV line the profile scale is used to evaluate |E2|. The profile scale is an interpolated curve based on 41 overlapped Wilson plot averages. Some degree of caution should be exercised in using this option because of the tendency of the radially-dependent structural contributions to |E2| to be reduced. It may, however, be useful for reducing the dominant features such as occur in "chicken-wire" structures.
4.25.3.3. Overall Rescaling (applied to E1; default for E2)
The rescaling options in GENEV are used to insure that the overall mean |E2| is precisely one. The simplest and most effective way of achieving this is by summing the |E| values determined using linear or profile scales, and then applying the inverse of the average |E2|. This is referred to as overall rescaling. It is mandatory for |E1| and optional for |E2|.
4.25.3.4. Index Rescaling (indexk option for E2)
With index rescaling, different groups of reflections defined by a particular combination of hkl indices are rescaled so that the mean value of |E2| is one. This option may be applied only to |E2|. The conditions for each index group may be specified on the indexk line with the 15 parameters p1, m1, n1, p2, . . . n3. A reflection belongs to a particular index group provided its indices jointly satisfy the following three equations:
[ h p1 + k q1 + l r1]mod(m1 ) = n1
[ h p2 + k q2 + l r2]mod(m2) = n2
[ h p3 + k q3 + l r3]mod(m3) = n3
Each group may be specified by a separate indexk line or, in the case of the even-odd parity groups, with a single blank indexk line. Only the specified index groups will be scaled separately, the remainder will be scaled together. Judicious use of the index parameters will permit single reflections to be scaled in this way. Specific scale values may also be entered on the indexk line for this purpose. No attempt will be made to make the mean |E2| = 1. in this case.
The index rescaling option must also be used carefully. As with the profile scale it can have the overall effect of reducing the structural content of the |E| values. The study of S & H (1982) showed that, in general, it provided less reliable |E|s than the overall rescaling option. Index rescaling can, however, be useful in the study of superstructure or hypersymmetry, since it ensures that groupings of reflections are given similar weight in the phasing process.
4.25.4. Choice Of Expectation Values
A squared normalized structure factor is the ratio of its scaled intensity to its expectation value. The expectation value for an intensity (or rather |F2|) depends on what is known about the structure. If only the atomic contents of the unit cell are known, then the best estimate of <F2> is the random-atom approximation (see S & H, 1982, for definition). Using the random-atom <F2> in the normalization process provides |E| values that will reflect how well the |F| values conform to those expected for a random structure. Significant departures of individual |E2| values from 1.0 (the overall mean) indicate whether a reflection is sensitive to the non-random aspects of the structure. The larger the departure from 1.0, the more important that reflection will be to a phasing process designed to investigate the non-random aspects of a structure. This is the basis for most structure invariant procedures.
If the coordinates of a structure are known (i.e. refined) then the value of <F2> is simply calculated |F2|, assuming atoms-at-rest. Application of this expectation value in the normalization procedure will result in all |E2| values being close to 1.0 (assuming of course, good data and a well-refined structure). Obviously |E| values determined in this way are of very limited use in direct methods since all reflections have equal weight. Those that are most sensitive to the non-random aspects of the structure cannot be identified.
Contrasting the application of random-atom and refined-atom expectation values illustrates a very important aspect of the normalization process. Structure information used in the expectation value will reduce that particular contribution in the resulting |E| values. In other words, the departures of |E2| values from unity reflect the differences due to structural information not used in evaluating the expectation value. In general, therefore, the higher-order expectation values, as provided with fragment information of type 2, 3, and 4 (see below), often have deleterious effects on the calculation of |E| values. There will, however, be situations when selective attenuation of structural information from |E| values, via the application of high-order expectation values, is extremely useful. The reduction of the dominant effects of a heavy atom or planarity are two obvious examples. In general, however, it is strongly recommended (S & H, 1982) that the random-atom expectation value be used in the initial stages of a structure solution, even when additional structure information is known (note well the comments in the next section).
4.25.4.1. Application of Known Structure Information
The general problem of applying known structural information to the structure invariant process is described by Main (1976). The definitions of the different categories of structural information as used by GENEV have been detailed by S & H (1982). These are treated in GENEV as the following categories,
| type 1 is for random atoms | ( wilson on frag line) |
| type 2 is for random fragments | ( random on frag line) |
| type 3 is for oriented fragments | ( orient on frag line) |
| type 4 is for positioned fragments | ( positn on frag line) |
While fragment information of type 2, 3, and 4 may not provide |E2| values that are more reliable than |E1| values based on random-atom expectation values, it should always be included in the GENEV calculation when available. This is because the group structure factors which are calculated as part of the evaluation of the expectation value of F squared, <F2>, provide phase information that is extremely useful in subsequent stages of the phasing process. This phase information can be applied to |E1|, as well as |E2|, in later calculations.
4.25.5. Inflection-point least-squares
For a random-atom structure, the Wilson plot is a straight line defined by the overall thermal displacement parameter of the constituent atoms, and the overall scale of the measure structure factors. For a real structure, a Wilson plot will often show significant systematic deviations from this line due principally to the short-range interatomic distances in the structure. The scattering effects of translational symmetry on the radial distribution of intensities is known as Debye scattering. For the majority of light-to-medium atom structures, the gross effects of Debye scattering are very similar. For instance the nodes, antinodes and inflection-points of a Debye scattering curves calculated for interatomic distances ranging from 1.30 to 1.55A in a 6-membered ring molecule are quite similar (H & S, 1982a). This means that for many structures inflection-points (the points where the Debye curve crosses the linear mean line) provide a means of finding a reliable linear fit to the Wilson plot, independent of the extent of the Debye scattering effects and the s2 truncation of the data.
GENEV uses the Wilson plot ratios for the 5 ranges clustered about the two cardinal inflection-points. The s squared default values for these inflection-points are set at 0.15 and 0.26 A-2, but these may be changed for non-typical structures with the din1 and din2 input on the GENEV line. In addition to the clusters of 5 points fixing the two inflection-points, the five largest Wilson plot ratios are used to fix the low angle part of the least-squares line. The user should always check that the assumptions embodied in the inflection-point least-squares process are valid for each structure. The Wilson plot points used for this purpose are shown in the printed graph as at signs (@). It is recommended that GENEV be rerun with specified values of U and K (using the fixu and fixk options) or different inflection-points (using din1 and din2 values), if the least-squares fit is unsatisfactory.
4.25.6. Estimation Of |E(hkl)| Errors
GENEV provides an estimate of the |E(hkl)| errors using a procedure described
by H & S (1982b). The principal source of error in |E| values arises from
inaccuracies in the measured structure factors. It follows that the legitimacy
of the errors estimated in GENEV will depend on the precision of the F
values entered on the bdf. The second most important contributor to the
|E(hkl)| errors arises from fitting the linear or profile scaling functions to
the Wilson plot (see below). The effect of Debye scattering on the Wilson plot
has already been discussed, and this is taken into account when estimating the
errors. The errors estimated for |E1| and |E2| are placed in the bdf for use in
subsequent calculations. The error distribution for a typical structure is
listed below.
4.25.7. BDF Output Of Genev Items
The user may decide which GENEV items are output to lrrefl: of the bdf. If an archiv line is not entered, the following items are automatically output to the bdf.
| Scales K | (item 100-) | in record lrexpl: |
| Overall U | (item 2) | in record lrdset: |
| Fragment count | (item 10) | in record lrdset: |
| |E1| | (item 1600) | in record lrrefl: |
| |E2| | (item 1601) | . . . . . . |
| |E1| | (item 1602) | . . . . . . |
| |E2| | (item 1603) | . . . . . . |
| Group structure factor 1 | (item 1606) | . . . . . . |
| Group structure factor 2 | (item 1607) | . . . . . . |
| . . . . . . . . | . . . . | . . . . . . |
| Group structure factor N | (item 1605+N) | . . . . . . |
If any archiv lines are entered, the items 1600 to 1630 must be named explicitly to be output to the bdf. Particular care must be taken if fragment information is used in GENEV calculation. The number of group structure factors (items 1606 on) is equal to the number of fragments, except for type 3 fragments where there is one group structure factor for each point group. The user must also check if any extra type 1 fragments have been added by GENEV to balance the cell content. Subsequent calculations that use the group structure factor phases require that the correct number be present. It is important to note that if GENEV items 1600 to 1630 are present on the input bdf, they will not be transferred to the output bdf. These are purged from lrrefl: before new GENEV items are appended. Additional items, other than 1600 - 1630, may also be deleted from lrrefl: using the archiv lines. A maximum of 30 items may be added or deleted in this way.
4.25.8. Some Normalization Tips
Tip 1 Check that sin/ maximum on the
GENEV line is as accurate as possible. The default value comes
from the bdf, otherwise it is set to 1.0. If ADDREF has been used with the two
pass option, the accurate value will be stored.
Tip 2 Assess the precision of the data. If some weak data are missing
from the input bdf, use the fill option to compensate for this.
If the weak data have not been processed with Bayesian statistics (e.g.negative
|F2| have been set to 0), then the baye option can
be used to apply a limited Bayesian treatment to |F| and F values.
Tip 3 Check what scaling and expectation options should be applied in the calculation of |E2|. These are fixed for |E1|.
Tip 4 If the values of either U or K need to be fixed, use the fixu and/or fixk options. These will apply only to |E2|.
Tip 5 The default rescaling mode for |E2| is index rescaling applied to the eight parity groups, provided that frag, prof , or fixu/k options are not used. If any of these lines are entered, the default rescaling mode for |E2| becomes 'overall'. Index rescaling may be specified explicitly with the indexk line(s) but care should be taken in selecting index groups appropriate to the problem.
Tip 6 Known structure information is entered using the frag,site, sitea, and siteg lines. Fragment information is entered for the asymmetric unit, as opposed to the celcon information which is entered for the whole cell. sitea lines containing coordinates in orthogonal Angstroms must be used for type 2 fragments. A grid line must precede the first siteg line entered. The frag line may be used to move the origin of atom coordinates that follow. This is sometimes useful for converting from type 3 to type 4 input.
Tip 7 The user may output the |F2| expectation values used in the GENEV calculation as items 1604 and 1605 in lrrefl:. This can be efficient for very large structures when GENEV is repeated for different normalization parameters but the same fragment information. This is specified by putting bexp in the GENEV line and leaving out the frag and site lines.
Tip 8 Cell content information is extracted from the input bdf (if entered through the program STARTX). This may be replaced by entering celcon lines.
Tip 9 Always check that the items to be used in subsequent calculations (e.g. GENSIN, GENTAN, and FOURR) are to be output. In most cases the default items will be sufficient, but in special cases the archiv line(s) can be used to add the items required. It is good archival practice to regularly check the contents of lrrefl: and remove any items that are no longer needed.
4.25.9. File Assignments
Reads structure factors from the input archive bdf
Writes normalized structure factors (E's) to the output archive bdf
4.25.10. Examples
|E1| will be calculated with linear scale, random-atom expectation value, and
overall rescale. |E2| will be the same except for index rescaling using hkl
parity groups. No |E| values will be listed and |E1| and |E1| will be
output on the bdf.
GENEV list 1.5 frag oriented :specify type 3 fragment site br1 .5 .5 0 *7 .5 :bromine in special position site c1 .73 .57 .333 site n3 -.15 .44 .62 archiv 1600 1602 1606 1607 :add |E1|, s|E1|, gsf1, gsf2 |
|E2| will be calculated with a linear scale, overall rescale and an expectation value derived from the type 3 fragment of atoms Br1, C1, and N3 and the remaining atoms (i.e. balance of cell contents) as a type 1 fragment.
GENEV smax .33 fixu .04 baye fill indexk :use index rescale (parity hkl) for |E2| archiv 1601 1603 -1800 :add |E2|, sigma|E2|, delete |Fc| from bdf |
|E2| will be calculated with linear scale (with u=0.04), random-atom
expectation value, and index rescaling with parity groups. All input |F|s and
|F|s are treated with limited Bayesian statistics and the Wilson plot is
adjusted for missing data. Only |E2| and |E2| are added to the bdf; |Fc|
is removed.
GENEV dset 3 bexp prof indexk 1 1 1 4 1 :set index scale group 1 indexk 1 1 1 3 *16.5 :set index scale group 2 and set scale |
|E2| will be calculated with a profile scale, random-atom expectation value, and index rescaling based on the groups (h+k+l)mod4=1, (h+k+l)mod4=3, and the remainder. The scale of the second index groups will be fixed at 0.5. In this example the two |E| estimates will be output without their error values.
4.25.11. References
French, S. and Wilson,K. 1978. On the Treatment of Negative Intensity Observations. Acta Cryst. A34, 517-525.
Hall, S.R. and Subramanian V. 1982a. Normalized Structure Factors. II. Estimating a Reliable Value of B . Acta Cryst., A38,590-598.
Hall, S.R. and Subramanian,V. 1982b. Normalized Structure Factors. III. Estimation of Errors . Acta Cryst. A38, 598-608.
Luger, P. 1980. Modern X-ray Analysis on Single Crystals. New York: de Gruter.
Main, P. 1976. Recent Developments in the MULTAN System - The Use of Molecular Structure.. Crystallographic Computing Techniques, F.R. Ahmed, K. Huml, B. Sedlacek, eds., Munksgaard. Copenhagen: 97-105.
Stout, G.H. and Jensen,L.H. 1968. X-ray Structure Determination: A Practical Guide: New York: MacMillan.
Subramanian, V. and Hall, S.R. 1982. Normalized Structure Factors. I. Choice of Scaling Function. Acta Cryst. A38, 577-590.
Wilson, A.J.C. 1942. Determination of Absolute from Relative X-ray Intensity. Nature 150, 151-152.