Chemistry 4000
Additional topic
1.
Determination of Absolute Structure
Lecture 28
Determination of Absolute Structure
Chiral, non-centrosymmetric and centrosymmetric space groups
References: Massa 10.2, pp. 132-139
Earlier in our discussion of space group determination, the three categories chiral, non-centrosymmetric (or polar) and
centrosymmetric (or non-polar) were used. Often the same set of systematic absences apply to several different space
groups which differ precisely by these additional properties. They have to do with the chemical property of optical activity.
1.1
Optical activity
If you need a good review of symmetry, including the concepts of optical activity, I recommend the web site
maintained by Dr. Stefan Immel at
http://csi.chemie.tu-darmstadt.de/ak/immel/tutorials
Optical activity is a symmetry property that applies to molecules and larger objects. In chemistry, we are familiar with
the concept of chiral molecules. It is often stated that when a molecule lacks a mirror plane it is optically active, but
this is not a sufficient test.
Most accurately, all molecules which have an n-fold alternating axis of symmetry (equal to an improper rotation
axis or a rotary-reflection axis, symmetry element Sn) are achiral (and thus superimposable with their mirror
images). A Sn axis is composed of two successive transformations, first a rotation through 360°/n, followed by a
reflection through a plane perpendicular to that axis; neither operation alone (rotation or reflection) is a valid
symmetry operation for these molecules, but only the combination of both. Note, that an S1 axis is identical to a
simple mirror plane m, and a S2 axis is equivalent to a center of inversion i.
In HM notation, these are the symmetry elements: m, , , ,  and  Thus a molecule that lacks all such symmetry
elements is chiral and therefore has the property of rotating plane polarized light as shown by the typical experiment
of polarimetry which must students encounter in an undergraduate laboratory at some point or another.
The most common source of optically active substances is those derived from natural sources. In fact, all the natural
amino acids are chiral, i.e. pure optical isomers. Substances produced in biologicaly processes are very often optically
active. A very common example is found among the sugars. Sugars are also good examples of the process of
racemization where as a consequence of chemical reactions at the chiral center the optical activity is lost through a
process of randomization.
This natural process of randomization is the reason that the normal products of chemical reactions in a synthetic
chemistry lab cannot produce optically active substances. Thus synthetic samples which contain chiral centers are
normally produced in equal ratios of the optical isomers and this is called a racemic mixture.
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Determination of Absolute Structure
Lecture 28
What is less well known to many chemists is that solid-state substances can also be optically active. A very good
example of this is crystals of quartz, a polymer of SiO2. Quartz has two crystal forms, so-called α quartz in the
trigonal class and β quartz which is hexagonal. Both are optically active. Thus, a crystal of quartz will rotate a beam
of plane polarized light just as will a solution of a natural sugar. Important useful properties including the pressuredependence of its electrical resistance, so called piezoelectricity depends on this crystal chirality.
The impact of optical activity on crystallography is two-fold. First of all, molecules which are pure enantiomers can
only crystallize in the chiral space groups, i.e. those space groups belonging which lack the symmetry elements listed
above that render things achiral. One of the features of the XPREP software in the unit on space group assignment is
the option of specifying that the molecule is chiral. This means, enantiomerically pure. Chemists working with
compounds that are derived from natural sources, or who have gone to the laborious work of rendering a completely
synthetic sample enantiomerically pure, will suspect that their product is also going to be optically active and
therefore can force the software to consider only the subset of suitable space groups. Of course, the whole field of
natural protein crystallography falls within this class, but also large components of the pharmaceutical industry are
concerned with making enantiomerically pure substances.
However, issues of optical activity also affect “normal” chemical synthetic research as follows: some molecules
spontaneously crystallize in chiral space groups. These include both achiral molecules and racemic mixtures. When
this happens the chirality of the specific crystal being investigated must be considered.
1.2
Determining crystal chirality or polarity
The most common chiral space groups are P212121, P21, P1 and C2221. In addition there are higher symmetry space
groups that involve 31, 41, 61 or 62 axes. The check for the chirality of the crystal is done by two separate refinements
(best separated into either two file names or two sub-directories on the computer hard disk), in one of which the
coordinates are inverted. In SHELX, inversion is accomplished by the MOVE 1 1 1−1 command. If nothing else is
changed between the two refinements (essential!) then the lower R-factor gives the correct “hand” of the crystal. In
the case of chiral space groups with 31, 41, 61 or 62 axes, the sense of screw direction must also be changed to give the
so-called entantiomeric space group. For example, the enantiomer of P31 is P32, P41 is P43, P61 is P65 and P62 is P64.
The remaining non-centrosymmetric space groups are known as the polar space groups. In these space groups, the
reversal of the direction of one of the three crystal axes leads to an arrangement distinguishable from the other when
anomalous scattering is taken into account (see below), even though neither group is chiral and any group of atoms
that is present is accompanied within the lattice by its enantiomer. Thus in general the "handedness" of any crystal
belonging to a non-centrosymmetric space group should have its crystal absolute structure determined. Failure to do
so could actually reduce the accuracy of the structure.
For the test between the two “hands” of the crystal, it is best to use the weighted R factor. Hamilton proposed a test
for the statistical validity of the R-factor difference. In earlier days many data sets were insufficiently accurate to get
meaningful differences with small changes in R factor. However, that situation has now changed and with typical
modern high-quality data sets any reduction in R factor is usually statistically meaningful. For organic structures even
using Cu radiation, the difference from chiral space groups may be as small as 0.001. The presence of e.g. one S atom
can cause a difference of ten times larger, and heavier atoms can make a difference as much as 0.03.
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1.3
Determination of Absolute Structure
Lecture 28
Racemic twinning
When a substance is a racemic mixture and still crystallizes in a chiral space group, it is possible to co-crystallize the
two “hands” within a single crystal. In this case, there is a smaller difference than expected in the weighted R factor
(and also often anomalous E-statistics) between the two enantiomeric unit cells. There are regions in the crystal –
usually distinct crystal domains – of one absolute structure and regions which have the other. They have crystallized
in such a way that the crystallographic axes align with one or more axes having the opposite absolute direction. This
often happens in the chiral space group P21 where the direction of the b axis is polar. For example, in the figure below
the unit cell of a metallocene is shown in the two different “hands” of the chiral lattice. If the two different mirror
images crystallize in different domains within the same crystal, a racemic twin will result. The diffraction pattern in
this case gives no obvious clue that inversion twinning has occurred. It is only detectable when anomalous scattering
is present in measureable amounts. The example structure in these notes for Dipp3As is an example of racemic
twinning.
The Flack parameter is an elegant way to detect the presence of racemic twinning. An additional parameter called x is
refined such that the calculated structure factor contains a fraction of 1 – x of the model being refined, and x of its
inverse. If the anomalous scattering contribution is very small, the standard error for the Flack parameter can be as
high as or higher than the parameter. This tells you that anomalous scattering has a negligible effect on the two
enantiomers. On the one hand, this means it will not be possible to tell which is the right “hand”; on the other, you
also know that it does not matter as it will not negatively affect the model you are creating. The original suggestion
was published as H.D. Flack, Acta Crystallogr. A39 (1983) 876.
If the Flack parameter is significant (larger than standard error) then it must be heeded. If x is a small number (much
less than one) then you have the correct “hand”, probably by pure chance. If on the other hand x approaches to 1, then
you should create and refine the inverted structure which will be the correct “hand” of the unit cell.
2.
Anomalous scattering
Up till now we have assumed the Friedel’s law holds exactly. Thus the reflections hkl and h k l are always supposed to be
identical in absence of measurement error. The origin of the behavior implied by this law is that scattering is assumed to be
both classical and elastic. So far we have allowed the atomic scattering factors fi to be affected by thermal vibrations, so that
we adjust the magnitude of fi by the Temperature Factor but we have ignored inelastic scattering. However, this simplifying
assumption is not completely true and it particularly breaks down when the energy of the incident X-ray photons is slightly
larger than an adsorption edge of one of the types of atoms present. We discussed absorption edges for the source back
in lecture 10. Here we saw that the condition of an adsorption edge being lower in energy than the source requires a lower
atomic number. For example, niobium has an edge just below the energy of the Mo Kα line and can therefore be used as a
filter by absorbing the Mo Kβ X-ray photons. But this rule applies equally to other absorption edges, i.e. the L or M edges.
Thus for a given X-ray source, several atom types in the crystal may have one of their adsorption edges a bit lower than the
energy of the incident radiation. If absorption occurs for a given photon, then that only affects the intensities as well as
causing background radiation from the subsequent reemission at a different wavelength. However, those photons that
interact with an absorption edge but are still diffracted are altered slightly in both magnitude and phase. This effect is
known as anomalous dispersion or anomalous scattering. The size of the anomalous scattering terms (in units of electrons
per atom as for all scattering factors) is shown for many elements in the Table 10.1. Data is provided for both Cu and Mo Xrays. Note that there is a strong parallelism between the size of the anomalous scattering term and the absorption coefficient
for given atoms. Strong absorbers will also be strong anomalous scatterers, as a rule.
2.1
The phase shift of anomalous scattering.
Because of the phase shift that is associated with the anomalous contribution to the scattering, it may be divided into a
real part ∆f' and an imaginary part ∆f". The real part can have either a positive or a negative sign, while the imaginary
part is always positive. Since the elastic scattering f and the real part always act in the same direction, they can be
combined together to give a composite: f' = f + ∆f'. The direction of the imaginary part is different, and will either
slightly advance or slightly retard the phase of that reflection.
It is important that anomalous effects be taken into consideration. For Cu radiation, the presence of atoms heavier
than carbon, and for Mo those heavier than sodium are capable of giving measureable anomalous effects. In contrast
to normal scattering, the magnitude of the anomalous scattering is independent of scattering angle. Thus, the effects
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Chemistry 4000
Determination of Absolute Structure
Lecture 28
are more pronounced at high scattering angle where the "normal" scattering is weak. This again speaks to the
importance of the high-angle weak diffraction data for establishing the nuances of structures by X-ray diffraction.
2.2
The effect in centrosymmetric space groups
Without anomalous scattering the imaginary parts of the scattering vector derived from a pair of atoms related by an
inversion center are equal and opposite and the phase angle Φ of the resultant must be either 0 or 180°, Fig. 10.5a.
But if anomalous scattering is present, then the ∆f" term contributes in the same direction for both atoms so that there
must be a resultant phase angle despite the presence of an inversion center. Nevertheless, the contribution that both
make to the structure factor Fhkl remains the same. Thus the Friedel law still holds:
Fc ( hkl ) = Fc ( h k l )
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Determination of Absolute Structure
Lecture 28
The result of anomalous dispersion is to add a small phase shift ∆Φ that would not b e present in its absence. Note that
these effects are small - the diagram in Fig. 10.5b greatly exaggerates the effect. However, for accurate
crystallography the anomalous terms must be considered and software such as SHELX includes anomalous scattering
terms in its atomic scattering factor tables that are used for all calculations of structure factors.
2.3
The effect in non-centrosymmetric space groups
In the case of non-centrosymmetric space groups the Friedel law no longer holds. Inverting the indices now changes
the sign of the "normal" imaginary contribution to the scattering but not the anomalous imaginary contribution. Thus
the vector addition of the scattering terms leads to both different amplitudes and different phases for Fhkl and the
inverse as shown in Fig 10.6.
These are known in the literature as the Bijvoet differences and all the reflections related by  are known as the set of
Bijvoet pairs. Here too, the figure greatly exaggerates the magnitude of these differences.
Bijvoet (pronounced "bay-foot") pairs:
Fhk l
and
Fh k l
By comparing (and if necessary, specifically re-measuring a large number of) reflections that are Bijvoet pairs, with
the presence of significant anomalous scatterers, it is often possible to determine the correct "hand" of the unit cell as
mentioned in section 1.2 above. In the specific case of enantiomerically pure chiral compounds crystallized in a
chiral space group (of necessity) the absolute structure of the chiral molecule is also established. This technique is
the primary way in which absolute structures of enantiomers have been determined. Note that it is not necessary to
establish each and every resolved chiral compound directly as it is often possible to use classical chiral chemistry
(formation of diastereomers, polarography) to establish the relative handedness of different molecules. But it is
necessary to have available a good number of unambiguously determined absolute structures as starting points for
such comparisons. These absolute structures have all been established through crystallography by this method of
comparison of the Bijvoet pairs when atoms in the structure have significant anomalous scattering.
3.
Comparison of resolved and racemic alanine
A lot can be learned from looking at examples, and you may therefore find the following comparison of the crystal
structures of the amino acid alanine, C3H7NO2, instructive. In the following pages of the notes, the structure of
enantiomerically pure natural alanine is compared with that of a synthetic sample which is a racemic mixture. The two
forms crystallize in different space groups, in this case one that is chiral, P212121 and one that is polar, Pna21. The
description of these two structures is taken from the book by Richard Tilley, Crystals and crystal structures.
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