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Crystals User GuideChapter 8: Refinement
The refinement of crystal structures is complex and rarely a well behaved process, and only occasionally can it be performed reliably without some human supervision. In CRYSTALS we have tried to provide a structure that will reduce the amount of supervision necessary under normal circumstances, and still be very flexible for use in difficult or unusual cases. Before discussing the facilities available in CRYSTALS, some of the aims and factors relevent to refinement need considering. [Top] [Index] Manuals generated on Wednesday 27 April 2011 8.1: Aims of RefinementThe overall aim is to find an explanation of some observable phenomena
in terms of a model which is intellectually acceptable. Inevitably, the
thresholds for acceptability will change from observer to observer,
and from time to time. However, certain norms exist for the models,
and it is deviations from these norms that increase our understanding
of the physical process. The steps in arriving at a suitable model are:
1. Deciding what the observations are, and what errors are associated
with them. Postulating a model that will enable the observations to be simulated. Deciding if the differences between the actual and the simulated
observations are acceptably small. If they are, then we understand
the phenomena, otherwise we must try to modify the model.
Step 1 poses the problem of resolving the 'observation' from the 'model'.
In Xray crystallography, are the observations 'I' or 'F'? If an absorption
correction is applied, is the correction part of the observation, or of the
model? The answer to these questions really depends on what additional
questions the model must answer.
[Top] [Index] Manuals generated on Wednesday 27 April 2011 8.2: Factors Influencing RefinementThe refinement process cannot be made to yield more information than that present in the data (though the data may include items other than the diffraction observations). Deficiencies in the data may include: Systematic Errors
These really are shortcommings in the model. If the model being refined contains parameters highly correlated with the systematic errors, then the model will be prejudiced in a way depending upon the form of the systematic error. For example, failure to correct the diffraction data for the theta dependent component of the absorption effect will systematically reduce the atomic temperature factors. Random Errors
These are inevitable in any experimental observation. They should be minimised by careful design of the experiment (e.g. choosing appropriate crystal size, counting times and radiation). Often reducing random errors is something that must be purchased (by increasing the time spent in performing the experiment), or is in conflict with reducing systematic errors (large crystals giving good counting statistics may introduce nonrandom errors). Shortage of Data
If some of the data is unobservable in a way that is not highly correlated with the model, then the result is that the random errors in the data lead to random errors in the model. Increasing the amount of data, or its quality, will improve the model. Note that reducing the number of parameters in the model implies increasing the amount of data, since those parameters not refined will be given (either explicitly or implicitly) values that are themselves observations of high precision. Their accuracy will depend upon the approximation made in simplifying the model. The user should always apply Occams Razor (The Principal of Minimum Assumption) to any new model. Lack of Resolution
If some of the data is not observable in a systematic way, this may
mean that it will not adequately define some parameters in the model. The
most common example is the link between the angular range of the diffraction
data and the detail that can be resolved in the structure. Lack of resoluton
in the data usually shows itself as large e.s.d.s in the refined parameters,
though these can also arise from a very inadequate model. For example,
an anisotropic temperature factor may fail to give a satisfactory
representation for an atom that is disordered over two widely separated sites.
[Top] [Index] Manuals generated on Wednesday 27 April 2011 8.3: Restraints and ConstraintsGenerally restraints, also called 'soft constraints', are used to test the hypothesis that the diffraction data will yield parameters not incompatible with those taken from some other source. For example, the restraint may request the refinement to verify that the diffraction data is not incompatible with a certain bond having a length of 1.39(1)A. The user should realise that just because a restraint is satisfied, he has no guarantee that no other hypothesis would fit the data just as well. Restraints may be used to help speed up the convergence of a refinement.
If the restraints are imposed with large weights (i.e. small e.s.d.s)
in the initial stages, they will force the refinement rapidly towards the
preconceived structure, possibly helping to keep it from false minima.
As convergence approaches, the weights can be reduced, or the restraints
removed altogether.
[Top] [Index] Manuals generated on Wednesday 27 April 2011 8.4: Notes on WeightsThe reflection weights for least squares must reflect errors in BOTH Fo and the model since we are using (FoFc) as the argument in the LS procedure. The most serious approximations in the model are usually made in: 1. Absorption corrections. 2. The form of the temperature factors. 3. Atomic scattering factors. 4. Extinction corrections.
Errors in the diffraction data should ideally follow Poisson statistics. However, it has been our experience that except in the case of painstaking work on very high quality crystals, other errors are present. These generally seem to be approximately correlated with the reflection intensity, and can be represented by an expression like: sigma(F) = a*sigma(I) + b*I + c*I**2
Since values for the constants a, b and c are arrived at by inspired guesses, we now prefer to represent sigma(F) by a smooth function in Fo. The function chosen is a Chebychev polynomial with the minimum number of coefficients needed to make w(FoFc)**2 constant as a function of Fo. This procedure only yields weights appropriate for the current model, and so is inappropriate for intermediate stages of the analysis. Futher, introduction of a model dependent weighting scheme too early in a structure analysis may lead to important features being concealed. Until all the parameters that need to be refined have been identified, unit weights or the scheme due to Hughes are used. A valid weighting of the reflections makes w(FoFc)**2 constant for all rational samplings of the data. Changes of weighting scheme usually have the most dramatic effects on thermal and extinction parameters, which should be rerefined whenever the scheme is modified. The user should not be unduly surprised to find his normal R factor
increasing after the application of weights (he is not trying to minimise
'R'), and will generally find Rw higher than R (Rw uses delta squared).
If Rw is much higher than R, then the model may be very inadequate in
some aspects, or there may be a few 'rogue' reflectins in the data. These
should be found, considered, and possibly rejected.
[Top] [Index] Manuals generated on Wednesday 27 April 2011 8.5: A StrategyCurrently, the only effective way for completing the refinement of a crystal structure at atomic resolution is by the method of leastsquares in which we minimise w(FoFc)**2 summed over the diffraction data. In restrained refinement we also simultaneously minimise w(PtPc) ( where Pt and Pc are the theoretical and calculated values for some structural parameter) summed over the required parameters. In both cases 'w' is the weight appropriate for the term in the summation. Fc and Pc are usually nonlinear functions of the atomic parameters. The functions are thus expanded as a Taylor series, and the second and higher order terms neglected. The zero'th order terms (Fc and Pc) depend upon the current model, and the neglected terms only really become insignificant as the process approaches convergence. e.g. Fc(i) + [x(k).d(Fc(i))/d(p(k))] = Fo(i) Pc(i) + [x(k).d(Pc(i))/d(p(k))] = Po(i)
where the terms k in [] are summed over the parameters being varied and we solve for the shifts x(k) in parameter p(k). The terms Fc(i) and Pc(i) are effectively observations for the current model, and can be moved to the right hand side. Because of the approximations made, the process need not be convergent, but this becomes more likely the better the starting model. Fc is more sensitive to reasonable errors in some types of parameter (e.g. x,y,z) than in others (e.g. U's). At the outset of refinement therefore the least sensitive parameters can be given reasonable values (e.g. 0.05 for U[iso] of a carbon atom) which is not refined until the more volatile parameters have stabilised. In this example, the refinement is heavily constrained with the constraint that U[iso] = 0.05. Once the model for the geometry has begun to stabilise, the thermal model can be refined, though here again it is generally wise to constrain the temperature factors to be isotropic initially. Because all the parameters are correlated, it makes no sense to refine these intermediate models to convergence. An r.m.s.(shift/e.s.d.) of about 1 to 3 is adequate before the model can be relaxed (though any individual anomalies should be investigated). As convergence approaches, the gradient terms (d(F)/d(p)) change only slowly compared with Fc, and it is thus appropriate in large structures only to recompute FoFc (the Right Hand Side) every cycle, and reuse the matrix of derivatives for alternate cycles, thus saving considerable machine time. Throughout this section, the importance of the starting model has been emphasised. The user should now see why Fourier refinement and Regularisation were discussed in detail earlier. Nonlinear leastsquares is a powerful method for acheiving a satisfactory result only when it is given a good starting model. While it can give some indications as to redundant parameters (by refining them to absurd values or attributing them large e.s.d.s), it cannot introduce new parameters. The final test of a 'good' structure is not a low 'R' factor or minimisation function  it is more likely to be an intuitive assesment based on the relationship between the current model and similar structures, and a critical examination of difference Fourier syntheses. [Top] [Index] Manuals generated on Wednesday 27 April 2011 8.6: A Practical SchemeThe following list is a flow diagram for a typical crystal structure refinement assuming that there are no problems with disorder, pseudosymmetry, twinning etc. 1. Get a trial structure (Patterson or direct methods) 2. Calculate structure factors and a Scale Factor. 3. Do an Fo map and Fourier refinement. 4. Regularise if you can. Use REGULARIZE or MOLAX 5. If you have found new atoms, go to 2. 6. Refine the positional parameters. 7. Refine isotropic temperature factors. 8. Locate or compute hydrogen atoms. 9. Refine positions again. 10. Refine aniso temperature factors. 11. Locate or compute any missing hydrogen atoms. 12. Enter the final stages of refinement a. If a small structure, FULL matrix. b. If a medium structure, LARGE BLOCK approximation. If refinement is stable and correlation coefficients are small, then use DIAGONAL approximation c. If a large structure, DIAGONAL approximation or CASCADE refinement. 13. Look at the agreement (variance) analysis (\ANALYSE). If <w(FoFc)**2> not constant, change weights to SCHEME 1 with a parameter P(1) = F(min<w delsq>). 14. Do 1 cycle of refinement of temperature factors plus a CALCULATION. 15. Look at the agreement analysis again, and decide if you need a Chebychev weighting scheme. 16. Refine to convergence, reusing matrix on alternate cycles. 17. If there are unusual features (distances, angles, planarity, Uaniso) use REGULARIZE, MOLAX, ANISO to correct the feature, and use restrained refinement to test the hypothesis that the 'corrected' model is compatible with the Xray data.
[Top] [Index] Manuals generated on Wednesday 27 April 2011 8.7: Structure Factor Control List  LIST 23The overall conditions governing the calculation of structure factors and monitoring the refinement process are generally kept constant throughout a refinement, and so have been collected together into the Structure Factor Control List, LIST 23, which should be input with the other initial data. The directive MODIFY consists of a series of switches for causing the application (or not) of data contained in other lists, thus removing the need to reinput those lists as circumstances change. Note that, for example, though LAYER scale factors or an EXTINCTION parameter may be present in LIST 5, they will only be included in the calculation of Fc if the switches are set in LIST 23. The MINIMISE directive controls whether or not the restraints are to be added into the minimisation function (i.e. a LIST 16 may be present on the DISK but will not be actioned unless LIST 23 requests it), and whether F or F**2 is to be used for the diffraction data. Note that if you request F**2, you will probably need to concider changing the weighting scheme. The directives ALLCYCLES and INTERCYCLE control the convergencedivergence parameters. [Top] [Index] Manuals generated on Wednesday 27 April 2011 8.8: Defining the Matrix  LIST 12In PART 5 we introduced the definitions for parameter names and showed how these were associated with parameter values via LIST 5. For least  squares we need to set out a table showing the relationship between leastsquares parameter names, leastsquares parameter shifts, and the physical parameters. This table is held as LIST 22, which the interested user may care to print (for a small matrix!). For those cases where there is one leastsquares parameter per physical parameter and the refinement is full matrix, this table is relatively trivial. For refinements in which physical parameters are EQUIVALENCed or RIDING, or where the parameters are divided into several matrix blocks, it would be unreasonably complex for the user to have to construct this table for himself. The information which CRYSTALS needs to construct this table is presented in a symbolic form as a LIST 12. LIST 12 consists of directives that define matrix blocks (FULL, BLOCK,
DIAGONAL) and directives that modify the contents of the current block
(RIDE, EQUIVALENCE, FIX, WEIGHT, PLUS). The directive CONTINUE merely extends
the directive begun on the previous line.
[Top] [Index] Manuals generated on Wednesday 27 April 2011 8.9: Simple examplesFor many simple structures a simple full, largeblock or atomblock diagonal matrix refinement may be sufficient. Parameters implicitly included by a preceding FULL, BLOCK or DIAGONAL card may be removed from the refinement by the FIX directive. Rules for parameters
In the examples, 'parameters' may be either EXPLICIT (e.g. C(1,X) C(3,X,Y,Z) C(2,X'S) ) or IMPLICIT (e.g. X Y U'S ). Parameters defined by an 'UNTIL' sequence (e.g. C(1,U[11],U[22]) UNTIL C(6) ) are regarded as implicit for LIST 12 processing. No parameter may be given implicitly twice, nor given explicitly twice, in the same LIST 12. Rules for FULL, BLOCK, DIAGONAL, FIX and PLUS
LIST 12 must begin with FULL, BLOCK or DIAGONAL. LIST 12 must not contain any two of FULL, BLOCK or DIAGONAL. LIST 12 must not contain more than 1 FULL directive.
Parameters on a BLOCK card must not appear on another BLOCK or DIAGONAL card. Parameters on a DIAGONAL card must not appear on another DIAGONAL or BLOCK card. Parameters on a PLUS card must not appear on another PLUS, FULL, BLOCK or DIAGONAL card. \LIST 12 This defines a single matrix block FULL parameters containing all the specified parameters END PLUS the overall scale factor. \LIST 12 This defines two independant matrix blcks, BLOCK parameters(1) one containing parameters(1), the other BLOCK parameters(2) containing parameters(2) and (3), i.e. the PLUS parameters(3) PLUS directive is acting rather like CONTINUE. END The overall scale factor is not implied, and must be specified explicitly in the appropriate block if required. Note that the same parameter must not occur more than once.
[Top] [Index] Manuals generated on Wednesday 27 April 2011 8.10: EXAMPLES\LIST 12 Refine the positions and anisotropic FULL X'S U'S temperature factors of all atoms in a single END matrix block with the overall scale factor. \LIST 12 As above, only now don't refine the y FULL X'S U'S coordinate of the lead atom (polar axis? FIX PB(1,Y) see also restraints). END \LIST 12 Refine the positions for all the carbon BLOCK C(1,X,Y,Z) UNTIL LAST one block BLOCK C(1,U'S) UNTIL C(10) SCALE and the anisotropic PLUS H(11,U[ISO]) UNTIL LAST t.f.s for the carbons CONT UNTIL LAST together with an isotropic t.f.s for the END hydrogen atom and the scale factor in another. Note that PLUS could be replaced by CONTINUE.
[Top] [Index] Manuals generated on Wednesday 27 April 2011 8.11: Advanced examplesIn some situations a simple onetoone relationship between structural and least squares parameters is not suitable. For example, the x and y coordinates of an atom on the special position X,X,1/6 must move synchronously, that is, be represented by a single least squares parameter. Similarly, an atom on X,X,11/12 has the x and y coordinates linked, but here the shift to y is in the opposite sense of that for x. Sometimes the relationship between parameters is not due to space group symmetry, but to some other physical requirement. In the example given above (Restraints), where either Na or K can occupy the same site, we may have site occupancy disorder such that the total occupancy for the ions is fixed, but the ratio is to be determined. In this case, the shifts in the occupation factors for the two ions are equivalenced to a single least squares parameter, but made to move in opposite senses. Linking of physical parameters like this provides a powerfull way for dealing with certain types of instability in the refinement, and for reducing the cost by reducing the number of parameters refined. For example, the hydrogen atoms on a carbon can be shifted synchronously with it, thus preserving the local geometry (but watch second neighbour relationships). This procedure is sometimes known as a 'riding' refinement. Another use for riding parameters is to give a poorly defined residue a single overall anisotropic temperature factor (not as good as proper TLS of course, but perhaps more realistic than individual atomic temperature factors). For EQUIVALENCED LINKED or RIDING parameters, the partial derivative for each parameter is computed, multiplied by the WEIGHT if requested, and added into the normal equations. After solution of the equations, the resultant shifts are multiplied by the same weights and then applied to the corresponding structural parameters. Rules for FIX, EQUIVALENCE, LINK, COMBINE, GROUP, RIDE, WEIGHT
Parameters on a FIX, EQUIVALENCE, LINK, COMBINE, GROUP, RIDE or WEIGHT card are added to or modify the action of parameters on the last previous FULL, BLOCK or DIAGONAL card. Parameters given explicitly over ride those given implicitly. Parameters on an EQUIVALNCE, LINK, COMBINE, GROUP or RIDE card must not occur on another EQUIVALENCE, LINK, COMBINE, GROUP, RIDE or a PLUS card. Parameters on a FIX card may not appear on another FIX or a WEIGHT card. Parameters on a WEIGHT card may not appear on another WEIGHT or a FIX card. Parameters on an EQUIVALENCE, LINK, COMBINE, GROUP, or RIDE card need not have been given on a preceeding FULL, BLOCK or DIAGONAL card. Parameters on an EQUIVALENCE, LINK, COMBINE, GROUP, or RIDE card will modify the action of parameters given on a preceeding FULL, BLOCK or DIAGONAL card. Parameters on a WEIGHT card should also occur on an EQUIVALENCE, LINK, COMBINE, GROUP, or RIDE card. Parameters on a RIDE card must be either explicit definitions or UNTIL sequences. There must be the same number of parameters in each argument list. Parameters on a LINK card must be either explicit definitions or UNTIL sequences. There must be the same number of parameters in each argument list. Parameters on a COMBINE card must be either explicit definitions or UNTIL sequences. There must be the same number of parameters in each argument list. Only atom names can be given on a GROUP card
[Top] [Index] Manuals generated on Wednesday 27 April 2011 8.12: EXAMPLESThis example is also given in the MANUAL. The structure contains C,N and O in the space group P6122, with N at x,x,11/12, O at x,x,1/6 and C isotropic. The refinement will be by full matrix. The following 3 LIST 12s all lead to the same matrix. The example in the manual has only 9 directives. 10 \LIST 12 \LIST 12 \LIST 12 FULL C(1,X'S) FULL X'S FULL X'S U'S FIX C(1,U'S) PLUS C(1,U[ISO]) PLUS C(1,U[ISO]) PLUS C(1,U[ISO]) PLUS N(1,U[33],U[12]) PLUS N(1,U[33],U[12]) EQUIV N(1,X) N(1,Y) EQUIV N(1,X) N(1,Y) EQUIV N(1,X) N(1,Y) WEIGHT 1. N(1,Y) WEIGHT 1. N(1,Y) WEIGHT 1. N(1,Y) FIX N(1,Z) FIX N(1,Z) EQUIV N(1,U[11],U[22]) EQUIV N(1,U[11],U[22]) EQUIV N(1,U[11],U[22]) EQUIV N(1,U[23],U[13]) EQUIV N(1,U[23],U[13]) EQUIV N(1,U[23],U[13]) PLUS O(1,U[33],U[12]) PLUS O(1,U[33],U[12]) EQUIV O(1,X) O(1,Y) EQUIV O(1,X) O(1,Y) EQUIV O(1,X) O(1,Y) FIX O(1,Z) FIX O(1,Z) EQUIV O(1,U[11],U[22]) EQUIV O(1,U[11],U[22]) EQUIV O(1,U[11],U[22]) EQUIV O(1,U[23],U[13]) EQUIV O(1,U[23],U[13]) EQUIV O(1,U[23],U[13]) WEIGHT 1. O(1,U[13]) WEIGHT 1. O(1,U[13]) WEIGHT 1. O(1,U[13]) END END END
The next example is for H(11) H(12) and H(13) riding upon C(1). Note that the hydrogen atoms are all given the same temperature factor shifts. They would also probably start with the same temperature factor values, though for the isotropic refinement of a phenyl group (for example) the investigator may have some a priori reason for starting the atoms off with slightly differing values. We are not concerned here with the rest of the structure. \LIST 12 FULL (parameters) The x, y and z coordinates for RIDE C(1,X'S) H(11,X'S) UNTIL H(13) the four atoms are handled by EQUIV H(11,U[ISO]) UNTIL H(13) 3 ls parameters. The 3 U(iso)s END are handled by 1 parameter.
[Top] [Index] Manuals generated on Wednesday 27 April 2011 8.13: Matrix blocking schemesThere are no fixed rules for deciding how or even whether a given structure should be refined by full or block approximations to the matrix. The choice will change from institution to institution and from time to time. Broadly, full matrix methods are the most certain to converge. They are also likely to require less cycles of refinement than a block matrix method. However, the time per cycle will always be more than for a block matrix method, and there will be increased storage requirements. Thus, small structures should be refined using the full matrix, and large ones using a block approximation. On computing machinery where time is paid for whether the machine is idle or not (perhaps the users own machine) it makes sense to use the largest approximations to the full matrix compatible with other users of the machine and its efficiency of operation. At some stage, for large structures, a substantial number of the offdiagonal elements of the matrix will be negligable, and the cost of accumulating and paging these may not be justified. Such large structures need a special strategy. For medium large structures, it is probably convenient to break the matrix into a small number of largish blocks. Experience has shown what parameters can be expected to be correlated: these MUST be included in the same matrix block. Correlated Parameters
There is no definative way for predicting which parameters will be highly correlated, but experience has shown that the groups of parameters in the following table should normally be refined together.  bonded atoms  nonorthogonal coordinates  scale factors, extinction parameters and temperature factors.  molecules or fragments related by a pseudo symmetry operator.
\LIST 12 BLOCK ATOM(1,X'S) UNTIL ATOM(1n/2) BLOCK ATOM(1n/2+1,X'S) UNTIL ATOM(n) END \LIST 12 BLOCK ATOM(1,X'S) UNTIL ATOM(1n/4) PLUS ATOM(3n/4+1,X'S) UNTIL LAST BLOCK ATOM(1n/4+1,X'S) UNTIL ATOM(3n/4) END
Singularities
High correlation in itself does not necessarily lead to invalid
parameter estimates. For example, the effect of bond length restraints is
to deliberately increase the offdiagonal terms between the coordinates
of the atoms involved. If some parameters are so highly correlated that
the illconditioning of the matrix makes it (almost) singular, the user is
left with few solutions. Essentially his problem is that he has an
inapropriate model, or that his data are not resolving or defining his model.
The first approach is to ascertain whether more or different
data would cure the problem, for example collecting data to a higher theta
value, at a lower temperature, or by adding restraints. If no solution
of this type is possible, then the approximate relationship implied
between the highly correlated parameters should be formalised, that is
a new model is required. For example, if the temperature factor and
occupation factor for an atom fail to refine properly, the user may choose
to define one and refine the other, and perhaps later reverse the roles.
He should be aware however that he is not performing a free refinement of
both parameters, and that in the final cycle he is making the statement
that he 'knows' the correct value for the unrefined parameter. Separating
correlated parameters into different refinement cycles (or into different
matrix blocks in the same cycle  the effect is almost mathemtically
equivalent) is an extremely hazardous way of avoiding singularities, and
implies effectively uncontrolled assumptions on the part of the user.
Reparameterisation may help with pseudosymmetry problems, using the COMBINE
instruction to combine the approximately equivalent parameters.
[Top] [Index] Manuals generated on Wednesday 27 April 2011 8.14: Special positions and floating originsAtoms on special positions can be handled either through constraint or through restraints. The latter method can be applied automatically, and is to be prefered if LIST 12 is already complex because of GROUPed, LINKed RIDING or COMBINed parameters. Constraints
Constraining atomic parameters to remain in fixed positions or have certain relationships between them is done in LIST 12 with the directives FIX, EQUIVALENCE and WEIGHT. See VOL5 for examples. The special atomic coordinates must have correct values or relationships before starting the refinement Restraints
Restraining atomic parameters to remain in fixed positions or have
certain relationships between them is done in LIST 17 using the RESTRAIN
directive. LIST 12 must be set up to let the special parameters refine,
but their refinement is controlled by the restraints. The instruction
\SPECIAL APPLY automatically generates the correct restraints and warns of
nonunity site occupancies. The special atomic parameters need not be
exactly correct before refinement starts  the restraint will correct them.
Floating Origins
These can be fixed by not refining the appropriate coordinates of a heavy
atom, using FIX in LIST 12. However, a much better solution is to restrain
the centroid of the structure, using SUM in either LIST 16 or LIST 17
[Top] [Index] Manuals generated on Wednesday 27 April 2011 8.15: Tips for least squares1) Look at the output. 2) If refinement is going well, consider using the DIAGONAL approximation. 3) If refinement is slow to converge, or gives poor molecular parameters, or temperature factors are anomalous, use a large block approximation to identify correlated parameters. 4) Watch out for high correlations or physically implausible parameters. 5) Do NOT 'FIX' an unexpected singularity by putting the related parameters into different matrix blocks. Make a better guess at the parameter values, or use restraints. 6) If your data dont define a parameter, dont try to refine it. 7) Consider using EQUIVALENCE to reduce the number of refined parameters. 8) Look at mean shifts and 'reversals'. At convergence, shifts should be small, and reversals ca. 50%. If reversals are <50% and shifts are large, structure is still converging. If reversals are large and shifts are large, structure is not converging. If the model is the best you can think of, use geometric restraints and the LIMIT restraint to stabilise divergence or oscillations. 9) Think. Use LIST 22 and CHECK online before issuing batch jobs. Use AXES and DISTANCES to verify molecular reasonableness. 10) Remember that the mathematical 'best solution' depends on the maths you use, and may not be the intellectually best solution. 11) Get good data. If you know the data are bad, consider a partial refinement only. 
© Copyright Chemical Crystallography Laboratory, Oxford, 2011. Comments or queries to Richard Cooper  richard.cooper@chem.ox.ac.uk Telephone +44 1865 285019. This page last changed on Wednesday 27 April 2011.