Blake A.J.(ed.) Crystal Structure Analysis

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172 Reﬁnement of crystal structures

13.1.4 Structure factor from a parameterized model

hkl

≈



2πi(hx

+ky

+lz

)

. (13.4)

The continuous electron density in a unit cell can be replaced by a

parameterized model, that provides a convenient approximate repre-

sentation of the true electron density. It is this model which is normally

called the ‘crystal structure’. Remember that X-rays do not see atoms –

they see average electron density. We replace this by discrete atoms as

a convenience.

13.2 Reasons for performing reﬁnement

Reﬁnement usually means adjustment of the values in the param-

eterized model according to some criteria. Both Fourier and least-

squares methods are regularly applied in crystallography. Reﬁnement

is undertaken for a number of reasons, as follows.

13.2.1 To improve phasing so that computed electron

density maps more closely represent the actual

electron density

Commonly used variations of (13.3) are as follows.

F = F

, α = α

. The resultant map contains features from both the

observed intensities and the computed phases. It is easily demonstrated

that the phases have a profound inﬂuence on the features in the map,

so that their veracity is fundamental. Better phases lead to better maps,

which in their turn can lead to better phases, and so on.

F = F

− F

, α = α

. The ‘difference map’. If the structure amplitudes

generated from the model lead to F

values differing only from the

observed amplitudes by their random errors, then this map will be fea-

tureless. This criterion is normally assumed to indicate that the structure

has been properly parameterized, and it is assumed that, if F

is the

same as F

, then α

will be the same as the (unmeasured) α

. The prob-

lem is that there are unspeciﬁed (and possibly systematic) errors in F

The Fourier transform of the residual F

− F

and α

may contain fea-

tures that will enable the model to be improved. Isolated positive peaks

indicate atoms missing from the model. Positive peaks adjacent to neg-

ative peaks indicate misplaced atoms, and positive or negative peaks

surrounded by ripples indicate either inappropriate atomic scattering

factors or inappropriateisotropic displacement parameters. Pairs of pos-

itive and negative peaks forming a ‘clover leaf’ indicate inappropriate

anisotropic displacement parameters.

F = 2F

− F

or F = 3F

− 2F

, α = α

. Hybrid maps having proper-

ties in common with both the normal F

map and a difference map. They

reveal features from the current model, and as-yet unparameterized

13.2 Reasons for performing reﬁnement 173

features, and are commonly used in macromolecular crystallography. If

the data are subject to a θ cut-off before the majority of the reﬂections

are unobservably weak, the map will show series termination ripples

with a wavelength slightly less than the resolution limit.

F = xF

.(1 − x)E, α = α

. E is the normalized structure amplitude, as

used in direct methods, so that the peaks appearing in the map are

accentuated. The optimal value for x is resolution dependent.

The phases, α

, are generally obtained from the current parameter-

ized model, which may itself have been reﬁned by either Fourier or

least-squares methods. Rarely, the computed phase may contain a con-

tribution from the discrete Fourier transform of part of an existing map

(SQUEEZE in PLATON; Spek, 2003). Interpretation of maps is the most

common way of locating items missing from the parameterized model,

though other forms of model building may also be applicable (addition

of hydrogen atoms at their expected positions, geometric completion of

other regular shapes, e.g. PF

−

groups).

13.2.2 To try to verify that the structure is ‘correct’

Because there is no way of directly and unambiguously computing the

structure from the data, there is always the possibility that the proposed

structure is ‘wrong’. Thereare broadlytwo waysof assessingthe validity

of a structure.

1. From the X-ray data alone. This is the most fundamental method,

and is the only one applicable to totally novel materials. It is also

the most difﬁcult method to apply. Techniques for parameter opti-

mization that rely on ﬁtting F

to F

cannot be certain that there

is not another solution giving about the same goodness of ﬁt for

the amplitudes, but a better ﬁt for the (unmeasured) phases. In

addition, in reality we know little about the actual errors in the

observed data, and there is the risk that over-parameterization of

the model will simply result in a model that better ﬁts the uniden-

tiﬁed errors. In protein crystallography, where there is a very real

possibility that a structure will be novel (in that there are still no

reliable ways for predicting protein folding), R

free

is used to mon-

itor parameterization. A random subset of the data, often 10%, is

excluded from the parameterization stage of the reﬁnement, but

has its value calculated whenever there is a substantial change in

the modelling. If R

free

fails to fall by a signiﬁcant amount, it is prob-

able that the change in the modelling is following noise in the data

(the signal represents a common trend throughout all the data, the

noise is individual to each data point, though systematic errors

can be regarded as correlated noise). R

free

is rarely used in small-

molecule work since 10% of a typical data set does not contain

enough reﬂections to give a reliable estimator.

2. From comparisons with the ‘known’ properties of structures. This

is a Bayesian method, but requires that the analyst correctly

174 Reﬁnement of crystal structures

identify which properties are known, and what their values are.

It cannot reveal if a structure is ‘right’, but may reveal that it is

‘wrong’.

‘Wrongness’ comes in several forms.

(a) Correct local geometry, but the whole structure is misplaced

in the cell. This phenomenon was common with early direct

methods programs, and may also occur for structures solved

by Patterson methods when the ‘heavy’ atom is not all that

heavy. Symptoms include the following.

i. The structure looks generally OK, but gives a high R factor,

often as low as 20%, but failing to fall any lower.

ii. Unusual bond lengths and adps. These rarely fall into a

systematic pattern, as might be the case for reﬁnement in a

space group of too-low symmetry.

iii. Unreasonable intermolecular contacts. Translational mis-

placement of a structure within a cell will generally bring

it too close to a symmetry element, leading to very short

non-bonded contacts.

iv. Noisy difference maps. The difference map may be gener-

ally noisy, show some inexplicable peaks, or occasionally

show ghost structural fragments.

v. Strongly featured DIFABS map

or highly anisotropic R-

DIFABS ﬁts a Fourier series in polar co-

ordinates in reciprocal space to the ratio

of F

. The value of this function can be

plotted out to reveal if this ratio deviates

stronglyfrom unity in any partofreciprocal

space (Walker and Stuart, 1983).

tensors.

These can also be due to uncorrected systematic

The R-factor tensor is a tensor that rep-

resents the variation of local R-factor as

a function of direction in reciprocal space

(Parkin, 2000).

errorsin the data, e.g. absorption, crystal decay, crystal mis-

centring, icing, beam inhomogeneity.

(b) Incorrect but plausible local geometry. This is relatively rare in

small-molecule crystallography. The most common occurrence

is generally due to disorder. Symptoms are as follows.

i. R factor higher than anticipated, though often as decep-

tively low as 6–8%.

ii. Novel molecular features. These must sometimes occur,

otherwise there is little point in much structure analysis.

However, if they cannot be rationalized by accepted chem-

ical or physical reasoning,there remains the possibility that

the structure is false.

iii. Weird adps. Weird may mean unexpectedly large, small or

anisotropic. Usually, if something is simply ‘wrong’, there

is no evident relationship between the adps of adjacent

atoms.

iv. A few particularly large F

− F

discrepancies, though the

most common cause for this is some kind of failure in the

data collection or pre-processing.

v. Noisy difference maps. The maps are generally rather fea-

turelessexcept for a few substantial peaks, though cases are

known where the maps were quite featureless even though

subsequent events showed that the proposed structure was

in serious error.

13.3 Data quality and limitations 175

13.2.3 To obtain the ‘best’ values for the

parameters in the model

Once the analyst is convinced that the structure is essentially correct,

weighted least-squares reﬁnement is used to optimize the parameter

values.

13.3 Data quality and limitations

The quality of the data should be as high as is economically practical.

Spending a little time choosing a good crystal from a mixed batch is time

well spent, though it also harbours the risk that the chosen crystal is not

truly representative. Spending very long times on data collection is use-

ful only if the crystal is of excellent quality, since the signal-to-noise ratio

only doubles if the counting time is increased by a factor of 4. Factors

arising in data collection that can affect later reﬁnement are as follows.

13.3.1 Resolution

With copper radiation and organic molecules, useful data can usually

be measured to the instrument θ limit. With molybdenum radiation,

the operator needs to use experience. If there are a few heavy ele-

ments present, they will dominate the high-angle data, and they should

be measured to reduce series termination (diffraction ripple) effects.

If there are only light atoms, the mean (Wilson) temperature factor B

will show when reﬂections become indistinguishable from background.

Figure13.3showsthevariation of relativeintensity as a function ofBragg

angle for B values of 4 and 8.

0.8

Variation of mean intensity (I/I

)

as a function of theta

Theta

B = 8 B = 4

Mean intensity, I/I

0.6

0.4

0.2

5 101520253035404550

Fig. 13.3 The variation of mean intensity as a function of Bragg angle for two different B

values.

176 Reﬁnement of crystal structures

Lack of properly measured high-angle data will restrict the quality

of the analysis. However, if the Wilson temperature factor is unusu-

ally high, this indicates that the crystal itself is of poor quality, with

substantial static or dynamic disorder. Re-collection of the data at a

lower temperature should reduce dynamic disorder. Unless there is

a phase change with conservation of crystallinity, struggling to work

at extremely low temperatures rarely has much inﬂuence on this

type of problem. The best strategy is either to try to understand the

nature of the disorder, or to try growing better crystals by a different

method. Simply including high-angle refections that are indistinguish-

able from noise into a reﬁnement in order to achieve some required

observation-to-parameter ratio is neither productive nor desirable.

13.3.2 Completeness

All kinds of reﬁnement are to some extent sensitive to the systematic

omission of substantial sections of the reciprocal lattice. Fourier meth-

ods, depending upon a summation over the whole reciprocal lattice, are

most sensitive to missing reﬂections. Least-squares methods are more

robust. Loss of data in the ‘cusp’ region of some diffractometers is rarely

important. More serious is not collecting data to similar resolution in all

directions in the asymmetric unit of reciprocal space.

13.3.3 Leverage

While all the reﬂections should be used to compute the electron density

at any point in the unit cell, in least-squares reﬁnement some reﬂec-

tions will have a particular inﬂuence on certain parameters. This effect is

known as leverage. Once a structure has been approximately solved, it is

possible to determine which reﬂections are important for which param-

eters. These reﬂections can be carefully remeasured and introduced into

the reﬁnement with appropriate weights or used as discriminators to

test the effectiveness of new parameterizations.

13.3.4 Weak reﬂections and systematic absences

As noted above, high-angle weak reﬂections have very little information

content. Weak lower-angle reﬂections can be really important, for exam-

ple in deciding between a centrosymmetric or non-centrosymmetric

space group. The problem is to get systematic-error-free determinations

of these reﬂections. Programs that perform analysis of the systematic

absences generally reveal that, within a given θ range, these reﬂections

have a net positive value, i.e. are not strictly absent. Reasons for this

include the Renninger effect, thermal diffuse scattering, λ/2 contamina-

tion and deviations from strict space group symmetry. Thesenet positive

values for absences suggest that all weak low-angle reﬂections may be

systematically in error, a condition that cannot be properly handled by

least-squares reﬁnement.

13.4 Reﬁnement fundamentals 177

13.3.5 Standard uncertainties

Even when most data were collected on serial diffractometers there were

differences of opinion about the correct way to compute the standard

uncertainties, particularly of the very strong and very weak reﬂections.

With the advent of area detectors, with their very sophisticated numer-

ical data pre-processing, the uncertainty in the uncertainty has grown.

Even the massive redundancy possible with these machines creates a

false sense of security (remember the parable of the Emperor of China’s

new robes

). If the errors in the individual observations are large but

Discussedinanarticle‘How precise are mea-

surements of unit-cell dimensions from single

crystals?’ (Herbstein, 2000).

more or less randomly distributed about the ‘true’ value, then repeated

measurements will yield a result approaching the true result (central

limit theorem). In this case, even if R

int

is large, the structure can be well

reﬁned to a reasonably trustworthy solution. If the individual observa-

tions are highly reproducible, but systematically wrong, this will lead

to a low R

int

, but an incorrect ﬁnal structure.

13.3.6 Systematic trends

One might expect that, for a given instrument, all structures of a similar

constitution would reﬁne to about the same R value, and this is often

the case. In those cases where the R value is anomalously low, we might

suspect that the R factor is dominated by an anomalous feature in the

structure, such as the presence of heavy atoms. In those cases where

it is anomalously high, the analyst should seek an explanation. The

following are possible explanations.

i. Weak data, i.e. the intensities disappear into the background at a

relativelylow θ angle.This can be due to static or dynamic disorder,

poor crystallinity, or large solvent content.

ii. Features in the data that cannot be represented by the normal struc-

tural model. Either the experiment should be repeated, taking care

to avoid these unwanted trends in the data, or some kind of ‘cor-

rection’ should be applied to the data to remove them. Statistically,

this kind of tampering with the data is not recommended, and the

statistician would prefer the model to be extended to reﬂect these

perturbations in the data. In practice, it turns out that the tinker-

ing works reasonably well (e.g. empirical absorption corrections,

SADABS-, DIFABS- and SORTAV-like procedures).

13.4 Reﬁnement fundamentals

For the bulk of small-molecule structures, reﬁnement means some

process related to least squares. The process seeks to minimize some

function

M =



w(Y

− Y

)

. (13.5)

178 Reﬁnement of crystal structures

13.4.1 w, the weight

For a statistically well-understood problem, it can be shown that, for

uncorrelated errors in the observations, the optimal parameter values

and their standard uncertainties are obtained when the weight is equal

to the inverse of the variance of the observation. In crystallography,

we are by no means certain about the distribution of the errors in our

observations, nor can we be certain that our model is capable of fully

explaining the observations; for example, there may be systematic errors

in the data that should be accounted for by additional parameters in the

model. Weights are now usually chosen to satisfy the criterion that there

is no appreciable trend in M for any rational ranking of the data. This

enables the weights to reﬂect errors in both the data and the model, and

leads to optimal parameter values and s.u.s for that model. Computing

these model-dependent weights when the model is substantially incor-

rect may lead to the error being concealed. It is therefore recommended

that purely statistical or modiﬁed unit weights be used in the early

stages. Note that unit weights are robust for F

reﬁnement, and should

be replaced by (1/2F) or 1/σ

) for F

reﬁnement. Note also that the

assumption that the errors in observations are uncorrelated is generally

unfounded, though no widely available single-crystal programs use the

full data variance-covariance matrix.

Omission of data according to some s.u. threshold is a brutal way of

down-weighting (to zero!) the weak reﬂections, and in principle can-

not be justiﬁed. However, there is little justiﬁcation either in including

hundreds of high-angle data that are indistinguishable from noise. An

appropriate θ cut-off is more acceptable than an I/σ (I) cut-off. The small

numbers of weak reﬂections remaining in the reﬁnement, even if posi-

tively biased as explained above, probably do little harm, and may even

be required for some kinds of analysis.

Weighting schemes can be modiﬁed in various ways to accelerate

convergence, to reduce the inﬂuence of outliers, or to enhance some

feature of the structure.

13.4.2 Y

, the observations

The community is still divided as to whether Y

should be F

, F

or I. The

debate is ﬁnely balanced, and in practice perfectly acceptable results are

obtained whichever representation is used. Traditional nomenclature,

in which the structure factor is written |F

|, has added to the confusion.

The moduli signs were introduced to remind the reader that one can

measure only the magnitude of the diffracted beam but not the phase. It

has nothing todo with takingthe square root of F

. Negative F values are

permitted in least-squares reﬁnement. The application of a non-linear

transformation (taking the square root) to the data raises several issues.

Rollett and Prince have independently shown that, with appropri-

ate weighting, both F and F

reﬁnements yield the same parameters

and s.u.s.

13.4 Reﬁnement fundamentals 179

Non-linear transformations of the data lead to a skewing of the error

distribution. For medium and strong reﬂections, a 1/2F term success-

fully accommodates this, but there is an evident problem if F is ≤ 0.

In the absence of a proper knowledge of the error distribution in F

for weak reﬂections, an approximate value has to be estimated for the

s.u. of F

Why was reﬁnement against F

ever introduced? Transformation of

data or variables is a technique commonly used to improve the numer-

ical stability of a calculation, which would certainly have been an issue

before digital computers. In the case of transformation from F

to F,

except for very small reﬂections, unit weights are close to the optimal

weighting, again an advantage before computers. Neither of these con-

siderations is important now for routine work, though F reﬁnement

remains resistant to the effects of outliers (especially partially occluded

observations), particularly among the strong reﬂections.

Although the minima for both reﬁnements should be the same, the

pathsfromagiven starting model willbedifferent.Therearesuggestions

that F

reﬁnement is less inﬂuenced by false minima, but I have only

seen contrived artiﬁcial examples of this.

Contrary to common belief, for the software user F

reﬁnement has no

advantage over F

reﬁnement for either determining the Flack parame-

ter, or treating twinned data. The programmer has a little more work to

do in the case of F

reﬁnement.

Note that the F versus F

debate is quite independent of the debate

about the inclusion/exclusion of weak reﬂections.

13.4.3 Y

, the calculations

This is generally F

or F

to match Y

. In maximum-likelihood reﬁne-

ment, the following function is minimized:

M =



−





)

. (13.6)

F

 is the expected value of F

, and is a modiﬁed form of F

.Itis

intended to include a contribution to the minimization function due to

the uncertainty in the phase angle, and seems to be particularly useful

in protein crystallography, where the models rarely achieve the same

resolution as those for small molecules. As the model improves, F



approaches F

, and such is the power of modern direct methods pro-

grams that it is generally believed that initial structures are so good

that maximum-likelihood methods have little to offer small-molecule

crystallographers.1/σ

isa special weighting function. Maximumlike-

lihood may have somerolein cases of pseudo-centred cells, where a very

large percentage of the data is systematically weak. Edwards (1992) has

shown that maximum-likelihood least squares is unaffected by the F

to F transformation.

180 Reﬁnement of crystal structures

13.4.4 Issues

During least-squares reﬁnement, (13.7) is solved. There are some

important things to notice.

i. The observations of restraint are handled in the same way as the

X-ray observations.

ii. The matrix of constraint is applied to the whole matrix of deriva-

tives, so will override any conﬂicting restraints.

iii. The shift-limiting restraints are included as equations in the matrix

work – they inﬂuence the terms in the matrix, and hence its

numerical processing.

applied

= P.(M



.W.A.M)

−1



.W.Y

A =

⎛

⎝

∂F

/∂x

∂R

/∂x

⎞

⎠

A is the matrix of derivatives

is a restraint target value

Y =



−F

−R

0.0



Y is the vector of residuals



applied

= P.δ

leastsquares

P is the matrix of partial shifts

physical

= M.x

leastsquares

+ c

M is the matrix of constraint

c is a vector of constants.

(13.7)

iv. The matrix of partial shifts is applied after the matrix inversion,

and so cannot help in the control of singularities.

v. The terms in the design matrix do not involve the values of the

observations, only terms computed from the current model. If the

model is seriously wrong, these terms will also be seriously wrong.

vi. The vector of residuals involves both the actual observations and

values calculated from the model. If the values of F

are hopelessly

wrong, these residuals will drive the reﬁnement towards a false

minimum.

13.5 Reﬁnement strategies

If there were a known, single, reproducible reﬁnement strategy, it would

have been programmed long ago. In the 1970s it was hoped that such a

strategy was on the point of being discovered. Now, 30 years later, com-

puterized strategies do exist for the processing of good-quality data

from well-behaved structures, but there is an ever-growing number

of structures requiring some kind of human insight and intervention.

If a structure does not develop and reﬁne in the normal way, some

13.5 Reﬁnement strategies 181

possible actions are listed below. A reﬁnement ‘blows up’ when the

R factor begins to rise uncontrollably, or the displacement parameters

take on nonsensical values, or massive parameter shifts make the struc-

ture unrecognizable. At later stages, it can also mean that the extinction

parameter or some displacement parameters have gone substantially

negative.

The crucial thing to remember about reﬁnement is that both Fourier

and least-squares methods have limited ranges of convergence. Both

(13.3) and (13.7) involve the current model. The better the currentmodel,

the greater the chance of computing reliable estimates of changes to

make to the parameters. In difﬁcult cases, model development should

be undertaken cautiously, with non-crystallographic information used

to hold parameters at sensible values. Many cases are known where the

structure solution was difﬁcult, yet for which the ﬁnal fully parame-

terized model reﬁned quite stably. Exclusion of valid atoms is generally

less harmful than inclusion of false ones, and in any case the valid atoms

generally reappear in subsequent maps.

i. If the direct methods solution does not appear to reveal the struc-

ture, try changing some of the initial parameters, or try changing

program. If SIRxx initially shows a promising structure, which falls

to pieces during the automatic reﬁnement, turn off the reﬁnement

and process the initial solution manually. Note that very poor data

or data with systematic errors can yield an interpretable E-map that

may never reﬁne well.

ii. If the direct methods give a reasonable ﬁgure of merit but the

structure looks wrong, try alternative programs for assembling

molecules. Molecule assembly routines are affected in different

ways by missing or spurious peaks. If this fails, compute a pack-

ing diagram and set the bonding criteria to a little more than their

usual values. Packing diagrams are especially useful if the molecule

straddles symmetry elements. A chemist, knowing what to look

for, may spot molecular fragments amongst a jumble of spurious

peaks.

iii. If the structure is plausible overall, but ‘blows up’ on normal reﬁne-

ment, try Fourier reﬁnement. For Fourier reﬁnement it is best to

omit really dubious atoms from the model used to compute phases.

iv. If the main features are recognizable, use geometric model build-

ing to regularize the molecular parameter values (bonds, angles,

planarity).

v. Least squares can be used to identify potentially spurious atoms.

A few cycles of reﬁnement of U

iso

(with ﬁxed x, y, z) may reveal

spurious atoms as those with very high values.

vi. Least squares has no mechanism for introducing new atoms. This

can be done only by Fourier methods or model building. 2F

− F

or 3F

−2F

maps are often the easiest to interpret.

vii. If the model persists in blowing up, try increasing the effect of

shift-limiting restraints. Fix the isotropic displacement parameters