feature articles
A nonmathematical introduction to the
description of modulated structures^{a}Novartis Institutes for BioMedical Research, 4002 Basel, Switzerland, and ^{b}Laboratory of Crystallography, University of Bayreuth, 95440 Bayreuth, Germany
^{*}Correspondence email: trixie.wagner@novartis.com
The Xray analysis of (6R,7aS)6(tertbutyldimethylsilanyloxy)1hydroxy2phenyl5,6,7,7atetrahydropyrrolizin3one, C_{19}H_{27}NO_{3}Si, revealed a diffraction pattern which is typical for modulated structures: strong Bragg peaks surrounded by weaker reflections which cannot be indexed with the same three vectors that are used to describe the strong peaks. For this class of crystal structures the concept of has been developed which, however, for many crystallographers still constitutes a Gordian Knot. As a possible tool to cut this knot the of the abovementioned tetrahydropyrrolizinone derivative is presented as an illustrative example for handling and describing the modulated structure of a typical pharmaceutical (i.e. molecular) compound. Having established a working knowledge of the concepts and terminology of the approach a concise and detailed description of the complete process of peak indexing, data processing, structure solution and structure interpretation is presented for the incommensurately of the abovementioned compound. The symmetry applied is P2_{1}(α0γ)0; the (incommensurate) q vector components at 100 K are α = 0.1422 (2) and γ = 0.3839 (8).
1. Introduction
Together with quasicrystals and composite crystals incommensurately modulated structures constitute the category of aperiodic crystals. Since modulated structures will be discussed in detail over the course of this paper the interested reader is referred to `An elementary introduction to ) for a comprehensible description of quasicrystals and composite crystals. The diffraction patterns of modulated structures are characterized by the existence of additional Bragg reflections so that an integer indexing with three indices hkl is not possible; instead, 3 + d indices are required (d = 1, 2 or 3). The necessity for using four or more indices must be understood as a loss of periodicity in three dimensions. To restore the periodicity, the concept of higherdimensional was developed (de Wolff, 1974, 1977). Atoms are no longer points in space, but are envisioned as ddimensional atomic domains (Janner & Janssen, 1977).
crystallography' by van Smaalen (2004The theoretical foundation for the ). Furthermore, considerable effort has been put into making JANA2006, a crystallographic computing system that can conveniently handle the data and structures of modulated compounds (Petricek et al., 2006), available and userfriendly.
approach is now well established (van Smaalen, 2007However, for the majority of the crystallographic community many of the concepts and terms used in this context are still not easily accessible because etc. will be introduced and explained. After a brief excursion into the historical and recent developments in the field of aperiodic structures this working knowledge will be illustrated in the third section where the complete process of peak indexing, data processing, structure solution and structure interpretation for an incommensurately modulated structure is presented for the of (6R,7aS)6(tertbutyldimethylsilanyloxy)1hydroxy2phenyl5,6,7,7atetrahydropyrrolizin3one, C_{19}H_{27}NO_{3}Si.
descriptions tend to be rather mathematical and based on inorganic examples. In this manuscript a more practical approach to this topic is presented, based on an illustrative example from a pharmaceutical crystallographic service lab. The manuscript is divided into three main sections: in the first section a working knowledge of modulated structures will be established and key terms such as modulation vector, atomic domain1.1. Modulated structures – a first definition
Most crystal structures are periodic in three dimensions and show a diffraction pattern that can be indexed with three integer numbers (Fig. 1). Modulated structures can be derived from those structures: here atoms, groups of atoms or even whole molecules are shifted or rotated with respect to their neighbours such that the threedimensional translational symmetry, often considered as characteristic feature of a is destroyed. These shifts and rotations in modulated structures, however, are not arbitrary; they follow distinct rules and within these distortions (or better: modulations) there is additional periodicity which can mathematically be described by socalled atomic modulation functions (AMFs). AMFs can be harmonic (continuous) and therefore be expressed by a sine/cosine combination or they may be discontinuous, in which case crenel or sawtooth functions are needed for their description (Fig. 2; Petricek et al., 1995). Based on the periodicity of the modulation wave a distinction can be made between commensurately modulated structures (periodicity matches an integral number of lattice translations of the basic cell) and incommensurately modulated structures (periodicity does not match an integral number of lattice translations of the basic cell and is therefore incommensurate with the periodic basic structure).
Due to the periodic character of the modulation, additional sharp peaks appear in the diffraction pattern, just as the Bragg reflections produced by a threedimensional crystal are a result of the periodic terms in the satellite reflections and usually they are weaker than the main reflections. The satellite reflections can lie parallel to one reciprocal axis, but they do not have to. It should be emphasized that modulated structures are not disordered but have longrange order which is reflected in those discrete additional peaks rather than in diffuse streaks. For displacive modulations as described above (i.e. the atomic positions are affected), the number of satellite reflections that can be observed depends on the amplitude of the AMFs or, in other words, on the degree of distortion/modulation present in the threedimensional The stronger the modulation, the stronger the satellite reflections and also the higher the order of the satellite reflections that have measurable intensity (see §1.2.3).
equation. These additional peaks are referred to asEven though the distribution of satellite reflections in h and/or k and/or l (Fig. 3). For an incommensurately modulated structure there is not any set of three vectors that allows an integer indexing of all (main and satellite) reflections.
is not arbitrary, they cannot be described with the same three vectors that allow an integer indexing of the main reflections. Any attempt to do so results in noninteger values for1.2. Handling the diffraction pattern of modulated structures
The diffraction pattern of a modulated structure (strong main reflections surrounded by weaker satellite reflections) can be approached in three different ways, two of which are based on a description in the traditional threedimensional space, and one that is not. Since the structure, however, is not periodic in three dimensions (except for commensurate cases) any threedimensional approach can only be regarded as an approximation. The threedimensional approaches shall be mentioned nonetheless since the results may contain valuable information that can be used at a later stage (spacegroup determination in etc.).
identification of groups with strongest modulation, type of AMF to describe the modulation1.2.1. Basic cell and average structure
The first option is to ignore the weak satellite reflections altogether and concentrate on the main reflections. The indexing procedure then delivers a socalled basic cell (sometimes also referred to as a subcell) with a volume that can accommodate a realistic number of the molecule under study (1, 2, 3 etc. ). In the threedimensional using the main reflections only corresponds to averaging the contents of several unit cells and looking at a structure which is commonly referred to as the average structure. It is usually characterized by large anisotropic atomic displacement parameters (ADPs) as well as unrealistic bond lengths and bond angles. In the hypothetical structure of Fig. 2, for example, the averaging would include eight unit cells (Fig. 4). The inherent disadvantage of this approach is that a certain amount of information provided by the diffraction pattern, i.e. satellite intensity, is neglected.
1.2.2. and superstructure
The second way of handling the diffraction pattern is to drop the distinction between main and satellite reflections and to use all reflections equivalently for indexing (Fig. 5). This results in a smaller reciprocal and a where, depending on the maximum order of the satellite reflections that are observed, many points may be empty, i.e. have no observable intensity. In the is accordingly larger and is commonly referred to as a supercell. The resulting is called a superstructure and is characterized by a large number of independent molecules in the (Z′; Z′ = 8 in the example of Fig. 5). In incommensurate cases where the positions of the satellite reflections do not perfectly fit the grid of the lattice a deformation of has to be accepted which usually manifests itself in poor agreement factors, large standard deviations of refined parameters, split atoms, large ADPs etc. In such cases the term approximation would be more appropriate (see also §1.4).
For commensurate cases the description as a (threedimensional) e.g. Hao et al., 2005; Siegler et al., 2008) and can even be used in combination with a (higherdimensional) treatment (Schmid & Wagner, 2005).
constitutes a valid approach (1.2.3. Moving into superspace
In the third approach, too, all reflections are used, but the distinction between main and satellite reflections is maintained. In a first step the reciprocal cf §1.2.1). The second step makes use of the fact that the distribution of satellite reflections in is not arbitrary; they can be divided into groups in which they are equidistant not only from each other but also from the main reflection to which they belong. This systematic distribution allows the definition of a fourth vector, the socalled modulation wavevector q, which describes the satellite reflections with respect to their main reflection. Now every satellite peak can be uniquely identified as being n·q (n = ±1, ±2, …) away from its main reflection, and one speaks of the nthorder satellite reflection of this main reflection. This manuscript will be restricted to fourdimensional cases, i.e. one modulation wavevector q. The theory for five or sixdimensional cases, where a second or even a third set of satellite reflections requires the use of one or two additional vectors, can be derived accordingly. Since five and sixdimensional cases, however, occur mainly with higher symmetry (hexagonal, cubic) and a standardized treatment is still in the process of being developed, they are beyond the scope of this introductory guide.
is established using the main reflections (Within the framework of the three a*, b* and c*, which describe the basic cell, the modulation wavevector q can be expressed only with the help of fractional components: q = αa* + βb* + γc*. In other words, the components of q are given with respect to the basis vectors of the of the basic cell or average structure.^{1} Please note that the number of nonzero components of q is not related to the dimensionality of the modulation, which is defined by the number of q vectors necessary for the description of Depending on the rationality of α, β and γ there is now a second way (cf. §1.1) to classify modulated structures as commensurate (all components rational) or incommensurate (at least one of the components irrational). In commensurate cases are characterized by a in which all lattice parameters are integer multiples (1, 2, …, n) of the lattice parameters of the basic cell. In practice, however, it is often not easy to distinguish between commensurate and incommensurate cases, especially when the multipliers involve larger integers and the differences between a commensurate and an incommensurate approach disappear.
vectorsWorking with one modulation wavevector q implies a transition into fourdimensional space (cf. §1.2.4) and every reflection of the diffraction pattern can now be described in a unique way by four integer indices h, k, l and m. This higherdimensional concept affects all subsequent steps of structure analysis and therefore will be discussed in detail in connection with the sample structure in §3. A schematic illustration of the relation between and threedimensional space is given in Fig. S1 of the supplementary material.^{2}
1.2.4. The definition of reciprocal superspace
It is difficult to envision the fourth dimension introduced in the preceeding section by looking at the q vector alone because q itself is only part of the definition of reciprocal To illustrate the relationship between threedimensional and reciprocal it is more convenient to focus on one line of the diffraction pattern shown in Fig. 6(a). In this figure q is chosen to be parallel to a* (i.e. only the α component of q is nonzero). The additional fourth dimension is defined by the unity vector e_{4}^{*} (Fig. 6b) which is perpendicular to a* in the plane of the paper (e_{4}^{*} is also perpendicular to b* and c*, which are not shown in Fig. 6). The linear combination of e_{4}^{*} and q defines the fourth vector, a_{s4}^{*} (cf. §1.3). Extending this vector a_{s4}^{*} illustrates that every satellite reflection along the (threedimensional) vector a* can now be understood as a projection of a reflection along the reciprocal vector a_{s4}^{*} (dotted lines), or, in other words: q is the result of the projection of the additional dimension a_{s4}^{*} into threedimensional R*.
1.3. and – where are the atoms?
Up to this point the terminology of d)dimensional description' (Janner & Janssen, 1977)?
has been used in conjunction with only. Now, how can the structure of an in be imagined? Where are the atoms? And what does it mean that `… the aperiodic structure in real space is interpreted as a cut through the (3 +Before answering these questions, the following convention shall be introduced: the vectors a, b and c and their respective coordinates x, y and z used in classical threedimensional crystallography will now be denoted a_{1}, a_{2} and a_{3}, and the coordinates as x_{1}, x_{2} and x_{3}. That way the notation can easily be extended to an arbitrary number of dimensions. Axes and coordinates in an ndimensional will then be referred to as a_{s1}, a_{s2}, a_{s3}, …, a_{sn} and x_{1}, x_{2}, x_{3}, …, x_{n} (just as the reciprocal direction in §1.2.4 was denoted a_{s4}^{*}).
Since in the a_{s1}, a_{s2} or a_{s3}) and the axis that runs parallel to the fourth dimension.
approach the additional dimensions are defined to be perpendicular to all three dimensions of physical space, it is impossible to illustrate all four dimensions on paper. One has to use projections (in reciprocal space) or sections (in direct space) of the higherdimensional space, just as one would draw a twodimensional projection of a threedimensional on graph paper. This way the complexity of fourdimensional space can be reduced by drawing twodimensional sections consisting of one of the three principal axes (Using this technique, a hypothetical fourdimensional (or better: (3 + 1)dimensional) modulated structure with one atom in the ). In the (D + d) nomenclature the first number D refers to the dimensions in physical space and the second number d to the dimensions in additional space. As mentioned above, the illustration is simplified by drawing four unit cells in a (1 + 1)dimensional section.
shall now be considered (Fig. 7To facilitate recognizing the threedimensional (c) a periodic one atom structure is shown first (Fig. 7a). In the aperiodic variant below (Fig. 7b) the atoms have been shifted from their original positions (small black circles) and the translational symmetry from Fig. 7(a) is lost: the distance between P and P′, for example, is smaller than the distance between P′ and P′′.
in the section that will be described in Fig. 7The periodic c) is now defined by two basis vectors a_{s1} (or a_{s2} or a_{s3}) and a_{s4}. This vector a_{s4} is perpendicular to the corresponding threedimensional axis a_{1} (if a_{s1} is chosen) and the angle between a_{s1} (superspace) and a_{1} (threedimensional space) is defined by the qvector component α: . In (3 + 1)dimensional atoms can no longer be envisioned as points in space. Instead they have to be understood as onedimensional objects, the socalled atomic domains, along the fourth dimension and they are represented by a curve (mathematically: the AMF) which is periodic along a_{s4}. The shape of the AMF has to be determined experimentally during structure solution and The same holds for the AMFs in the sections (a_{s2}, a_{s4}) and (a_{s3}, a_{s4}). Note that in the two dimensions shown in Fig. 7(c) the AMF does exhibit translational symmetry, or, in more general words: within the approach the additional dimensions are introduced to recover translational symmetry and the structure that is aperiodic in three dimensions becomes periodic in higherdimensional space.
structure (Fig. 7The aperiodic atomic structure in threedimensional space (cf. Fig. 7b) can now be deduced by a onedimensional (horizontal) cut R (parallel to a_{1}) through this twodimensional representation of The intersections of R with the AMFs of the atomic domains give the positions of the atoms in threedimensional space. Note again that along the threedimensional space line R there is no translational symmetry (periodicity).
If all intersections (P′, P′′, P′′′ etc.) are shifted by translational symmetry into the left (i.e. parallel to a_{s1}, dotted lines) the resulting curve is the for the depicted atom. Or, in more general words, the shape of the AMF for any atom can be reconstructed via the positions of this atom in threedimensional space and vice versa.
The type of representation depicted in Fig. 7(c) occurs very frequently in the description of modulated structures and will reappear in §3, for example in conjunction with twodimensional sections out of fourdimensional electrondensity maps.
The atomic modulation functions are wavefunctions with a period of 1, therefore a phase t of the modulation can be defined which runs parallel to a_{s4} and ranges from 0 (origin of cell) to 1 (origin of next cell). The coordinate of the fourth dimension, x_{4} (which also runs parallel to a_{s4}) has to be distinguished from this new parameter t: in Fig. 7(c) the atoms P, P′, P′′ and P′′′ have the same phase (t = 0), but different coordinates x_{4}. The value of t is found from a projection onto a_{s4} along R while the value for x_{4} is found by a projection onto a_{s4} along a_{s1}. A different value of t will lead to a different set of points P …P′′′ in threedimensional space which, however, is related to the one obtained for t = 0 by a simple origin shift. For a visual demonstration of the impact of t onto the description of the the interested reader is referred to the `superspace playground', a website maintained by the Laboratory of Crystallography at the École Polytechnique Fédérale de Lausanne (Chapuis & Orlov, 2005).
As the whole R of the (3 + 1)dimensional all atoms in the crystal (i.e. in the aperiodic threedimensional structure) have the same phase t, but, as mentioned before, not the same coordinate x_{4}. Therefore, for the crystalchemical analysis only atoms with the same phase of the modulation, i.e. with the same value of t, can be considered. Shifting these atoms from the threedimensional space line R into one (via the restored translational symmetry of the construction) provides a very elegant way for this analysis, realised, for example, in the socalled tplots shown in §3: varying t for a certain geometric parameter from t = 0 to t = 1 provides all values for this parameter occurring anywhere in the threedimensional along R. This is visualized in Fig. 7(d): the interatomic distance at the phase of the modulation t = t_{0} is equivalent to the distance between atoms P and P′ in threedimensional space, the distance for t = t_{1} represents the threedimensional space distance between P′ and P′′, the one for t = t_{2} the threedimensional space distance between P′′ and P′′′, and so on (note that the two atoms defining one distance still have the same t value). Of course, other structural features such as coordinates, bond lengths, torsion angles, etc. also change with t, and it is exactly this variation in geometry with t that is the core of the discussion of molecular geometry for modulated structures (cf. §3).
in threedimensional space is defined as a threedimensional cut1.4. Commensurate approximation or description?
Based on this Ultimate Question of Life, the Universe, and Everything from Douglas Adams' famous book The Hitchhiker's Guide to the Galaxy there is no simple answer to this question. In general, one can say that a commensurate/superstructure approach is valid as long as it properly reflects the threedimensional structure and does not lead to too many problems such as unrealistic geometric features, poor datatoparameter ratio, or strange displacement ellipsoids which have to be tamed with the help of (usually undesired) restraints or constraints.
description one of the fundamental questions related to modulated structures can be addressed: under what circumstances is a approximation of an incommensurately modulated structure valid enough and when is it necessary, or at least more appropriate, to use a higherdimensional approach? Unfortunately, unlike in the case of theIn incommensurate cases such a commensurate approach – as already mentioned – always has to be regarded as an approximation because the atomic positions used in the . To display the atomic positions of an aperiodic structure by a approximation, the lattice has to be deformed in such a way that one of the lattice points coincides with R. In Fig. 8(a) this is the lattice point of the fourth (grey circle). Therefore, the corresponding is fourfold, and the respective coordinates x_{1} of the commensurate approximation can be calculated from the fractional coordinates x_{1}, , and (Fig. 7) of the incommensurate structure.
are not the positions of the atoms in threedimensional space. This is illustrated in Fig. 8Owing to the lattice deformation which is necessary to approximate a q vector of (e.g.) 0.238 by the rational number 0.25 = ¼ (Fig. 8b) the resulting positions in the (green circles) are visibly different from those which are derived from the representation (Fig. 8a) and which are the real positions of the atoms in physical space (black circles). This effect becomes the more pronounced the further one proceeds along the threedimensional space line R and usually manifests itself in problems during the refinement.
In cases where the satellite reflections occur along one of the reciprocal axes and the multiplier n for the nfold is high, a commensurate threedimensional can still behave properly (and thus be pursued) but more often the mentioned warning signs (well known from classical threedimensional crystallography) appear already at an early stage of the and should be taken as an indication that a transition into a description might be more promising.
At first sight it may seem cumbersome and puzzling to introduce an additional dimension with no physical meaning. However, there is one inherent advantage: the e.g. in the case of phase transitions, one structural model for all different phases where the threedimensional of the nonmodulated high temperature structure may be derived as a of the of the modulated lowtemperature structure (Chapuis, 1996; Schönleber et al., 2003). In the case of an nfold the description might improve the datatoparameter ratio, which is especially important for superstructures with large n.
approach is not limited to aperiodic structures but can also be applied to commensurate superstructures and nonmodulated structures. Therefore, it can provide,2. Development and present situation
A glance into the crystallographic literature suggests that aperiodic structures have received much more attention in the past one or two decades than previously. However, the recognition of aperiodic structures is by no means a recent phenomenon in the long history of crystallography. The following section will summarize the developments in this field from early findings to the current status of research.
2.1. Historical perspective
Shortly after the law of constancy of angles by Nicolaus Steno heralded the birth of modern crystallography, JeanBaptiste RomédeIsle proposed in 1772 that crystals are made of microscopic building blocks (the unit cell). RenéJust Haüy renewed this thought by proposing that these building blocks are small parallelepipeds and he enunciated the law of rational indices, according to which each crystal face is uniquely characterized by three small integers. This law suggested that crystals are built up of a periodic array of those basic structural units in all three dimensions. This threedimensional periodicity was then believed for almost the next 200 years to be one of the most fundamental principles underlying crystalline architecture.
This concept of periodicity associated with spacegroup symmetry represents a powerful tool for structural studies and is the basis for successful investigations of crystals by optics and diffraction. Since the discovery of Xray diffraction by von Laue and coworkers in 1912 (Friedrich et al., 1912, 1913), it was believed that the discrete distribution of diffracted intensities was a direct consequence of the periodic arrangement of the atoms in the three dimensions of ordinary space. However, indications questioning the paradigm of threedimensional periodicity of crystals appeared all during the history of modern crystallography. Many crystals have been found which do not fulfil the periodicity criterion, but nevertheless give diffraction patterns with perfectly discrete peaks. The discovery of these new types of structures is mainly a result of the problem of trying to index the diffraction peaks with three small integers, but also of applying the to the crystal faces during studies of crystal morphology.
The law of Haüy on rational indices was questioned during optical studies on the mineral calaverite AuTe_{2} (Smith, 1903). Later Dehlinger explained the broadening of Debye diffraction lines in metals with a periodic deformation of the lattice (Dehlinger, 1927). As it was already known that periodic perturbations in optical gratings generate additional diffraction spots, Dehlinger thus concluded that in analogy a deformation of the should generate secondary lines next to the main lines in the Xray diffraction pattern, which he called Gittergeister (lattice ghosts). While reinvestigating calaverite, Au_{1 − p}Ag_{p}Te_{2} with p < 0.15, Goldschmidt and coworkers concluded that the is not a general law and cannot be applied to this mineral (Goldschmidt et al., 1931). While studying the diffraction pattern of the aluminium–copper alloy Duralumin, Preston observed additional reflections next to the diffraction peaks of the lattice (Preston, 1938). He called these additional diffraction spots satellites. Daniel and Lipson also found additional reflections while studying the alloy Cu_{4}FeNi_{3} (Daniel & Lipson, 1943). They identified a regular deformation of the original cubic lattice as the origin of those. Tanisaki explained additional sharp satellite reflections in the diffraction images of NaNO_{2} by a microdomain structure (Tanisaki, 1961, 1963).
de Wolff and his coworkers investigated γNa_{2}CO_{3} and found that the additional reflections could be indexed with small integers only if a fourth index was included (Brouns et al., 1964) and that the fourth direction (giving the fourth index) does not coincide with any set of lattice planes of the main lattice (defined by the first three directions). The structure of γNa_{2}CO_{3} was then refined as a modulated structure in an harmonic approximation (de Wolff, 1974; van Aalst et al., 1976), i.e. the was described with a continuous function by one harmonic wave. In a reinvestigation (Dusek et al., 2003) satellite reflections up to fifth order and additional harmonic waves were used to better model the anharmonic features of the structure.
At the same time Nowotny and coworkers identified the socalled chimney ladder compounds Cr_{11}Ge_{19}, Mo_{13}Ge_{23} and V_{17}Ge_{31} as incommensurate intergrowth compounds (Völlenkle et al., 1967). Some years later Johnson and Watson determined the atomic displacements, i.e. the atomic modulation functions of the molecular composite crystal hepta(tetrathiafulvalene) pentaiodide, which they called a dual sublattice system, via applying a approximation (Johnson & Watson, 1976).
The final step to recognizing that translational symmetry is not a conditio sine qua non for the crystalline state was the discovery of a (noncrystallographic) icosahedral in the diffraction pattern of aluminium–manganese alloy (Al with 14 at.% Mn; Shechtman et al., 1984). This alloy was then called a quasicrystal. The sharp diffraction spots are explained by a longrange orientational order. This discovery of quasicrystals forced the crystallographic community ultimately `… to reconsider the concept of crystals' (Yamamoto, 1996).
Following this new idea, calaverite was reinvestigated as an incommensurately modulated crystal and all faces could be indexed by four small integers. Now it was concluded, that `(…) the classical law of rational indices still holds for incommensurate crystals, provided the correct number of indices is used (four in the case of calaverite)' (Dam et al., 1985).
The two common features of all these examples are that the crystal exhibits flat faces and that they are characterized by sharp and well separated peaks, as is the case for classical threedimensional periodic crystals. But in contrast to `classical' crystals the examples discussed above lack threedimensional translational symmetry and consequently are classified in the new category of socalled aperiodic crystals. More detailed explanations can be found in reviews like those of Janssen & Janner (1987), Janssen (1988), Bertaut (1990), van Smaalen (1995), Chapuis (1996) and Yamamoto (1996) or in the International Tables for Crystallography, Vol. C in ch. 9.8 about Incommensurate and Commensurate Modulated Structures by Janssen et al. (2006).
2.2. Current status
The increasing number of publications in the past decades on modulated structures described within the Aperiodic Crystals by Janssen et al. (2007) and Incommensurate Crystallography by van Smaalen (2007) within the IUCr Monographs on Crystallography series. The interested reader is referred to these two volumes.
approach demonstrates that this concept is well established now and is becoming more and more known, accepted and familiar to the scientific community. This development is also reflected in the two textbooksThe recent technical developments (twodimensional detectors, powerful computers with large memory and fast processors etc.) allow the treatment of large numbers of reflections and complicated structures on an acceptable timescale. The implementation of higher dimensions, i.e. more than three indices, to allow handling of satellite reflections is available in almost all commercial and noncommercial diffractometer software packages. The release of JANA2006 (Petricek et al., 2006) enables not only `specialists' but `everybody' to apply the approach in their research.
Scanning through the reviews and databases on modulated structures, the dominance of inorganic compounds is striking. A search in the `Incommensurate Structures Database' at the `Bilbao Crystallographic Server' (Aroyo et al., 2006) for example reveals only 15% organic or metalorganic structures. However, despite this dominance of inorganic compounds, there are also some impressive examples of modulated organic materials.
For modulated molecular compounds there is a general agreement in the literature that one possible origin of modulation can be found in the interaction between crystal packing and i.e. in the interplay between intramolecular and intermolecular forces (Dzyabchenko & Scheraga, 2004; Herbstein, 2005). Another reason for the loss of threedimensional periodicity might be conflicting packing tendencies, even if the is fixed.
To understand the structural features and the chemical behaviour of modulated organic compounds an exact crystalchemical analysis is necessary, based on an accurate description of the
As discussed above, this can be performed in a very elegant and precise way by applying the approach. To illustrate the different implications of the modulation on molecular crystals the following structures are briefly presented:The classical example of the influence of the crystal packing on the _{12}H_{10} (Baudour & Sanquer, 1983; Dzyabchenko & Scheraga, 2004). In the gas phase the torsion angle φ between the two phenyl rings is 42 (2)° (Fig. 9), which can be presumed to be the optimal conformation of the biphenyl molecule. In the at room temperature the molecule is planar (φ = 0°), at least on average. This change of the is caused by intermolecular interactions in the crystal. During cooling to lower temperatures (T < 38 K) two other crystalline phases appear which are characterized by molecules with 〈φ〉 ≠ 0° (Baudour & Sanquer, 1983). In particular, for the incommensurately modulated lowtemperature structure at T = 20 K the average torsion angle varies between φ = −11° and φ = +11°. The modulation of the structure can be explained as the result of a competition between intra and intermolecular contributions to the lattice energy (Dzyabchenko & Scheraga, 2004).
is biphenyl, CA similar behaviour with respect to intramolecular torsion can be found in 4,4′dichlorobiphenylsulfone, (ClC_{6}H_{4})_{2}SO_{2} (Zúñiga & Criado, 1995) and in 2phenylbenzimidazole, C_{13}H_{10}N_{2} (Zúñiga et al., 2006). In these structures, too, the modulation is interpreted as a consequence of the interaction between intramolecular forces and the crystal packing. For 2phenylbenzimidazole, for example, a variation in the central torsion angle of ±5° is found.
In contrast, the incommensurately modulated structure of 3,4diphenyl2a,5a,6,7,8,8a,8bheptahydrofuro[4,3,2de]chromen2one, C_{22}H_{20}O_{3}, cannot be explained by a torsion of the (Guiblin et al., 2006). Here the molecule can be divided into three rigid bodies. Two of these are phenyl rings, but only one of them forms an intermolecular contact with a neighbouring molecule via a C—H⋯O interaction. Due to the modulation the complete molecule is shifted and the contact acts as a kind of mechanical spring. As a consequence, the shifts of the two phenyl rings with respect to the rest of the molecule are slightly different, and the origin of the modulation lies in the mutual frustration of and hydrogenbond formation (Guiblin et al., 2006).
The incommensurately modulated structure of the organic salt quininium (R)mandelate, C_{20}H_{25}N_{2}O_{2}^{+}·C_{8}H_{7}O_{3}^{−}, which exhibits a strongly pronounced (Schönleber, 2002; Schönleber & Chapuis, 2004), is slightly more complicated – different parts of the quininium cation follow different modulation functions: the quinolinium segment, for example, shows a rather undulating movement, while the vinyl group exhibits two preferred orientations, which are occupied as a function of the phase t of the modulation. As a consequence not only does the conformation of the molecules vary, but, due to the different atomic environments, the anisotropic ADPs also vary. Furthermore, the (intermolecular) hydrogenbond scheme is modulated, which means that the hydrogen bonds are formed and broken as a function of t (see §1.3).
Other recent examples of modulated organic structures comprise the channel _{3}(CH_{2})_{17}CH_{3}/(NH_{2})_{2}CO, which was refined as an incommensurate host–guest system at low temperature (Toudic et al., 2008) with an alternating intermodulation between host and guest substructures from channel to channel, and the modulated structure of 1phenyl4cyclohexylbenzene, C_{14}H_{20} (Evain et al., 2009), in which the cyclohexyl group is rotated with respect to the benzene ring.
of nonadecane/urea, CH3. Modulated structures – a case study
In the following section, the R,7aS)6(tertbutyldimethylsilanyloxy)1hydroxy2phenyl5,6,7,7atetrahydropyrrolizin3one (from now on referred to by its Novartis code VAR205), an organic compound (Fig. 10) with pharmaceutical impact, will be described and discussed. A detailed description of the complete procedure of structure analysis starting with the determination of the q vector from the diffraction pattern will be given. Intensity data were collected at T = 100 K on a Bruker AXS threecircle goniometer with a SMART6000 CCD detector, using Cu Kα radiation from a fine focus rotating anode generator equipped with Montel multilayer mirrors (Table 1). A standard data collection protocol was followed with 12 180° ω scans (scan width 0.6°) at different φ positions and two different detector positions (crystaltodetector distance 40 mm). The software utilized comprises the standard Bruker AXS suite SMART, RLATT, SAINT (Bruker AXS, 2007a,b,c) and SADABS (Bruker AXS, 2001),^{3} the SHELXTL program package (Sheldrick, 2008), and the crystallographic computing system JANA2006 (Petricek et al., 2006). Graphical representations of molecules were generated with PLATON (Spek, 2003) and DIAMOND (Brandenburg, 1999).
of (6

VAR205 is a key intermediate in the synthesis of potential lymphocyte functionassociated antigen 1 inhibitors (Zecchinon et al., 2006) bearing a pyrrolizinone scaffold (Baumann, 2007). Structural analysis was carried out to establish the relative stereochemistry at the fused ring core and the degree of pyramidalization at the central nitrogen atom.
3.1. The diffraction pattern
The diffraction pattern of VAR205 shows irregularities – visible either with a suitable program that allows a schematic representation of reflections in RLATT) or in a reconstruction of (Fig. 11): indexing using all observed spots (i.e. a approach) results in a huge monoclinic with a volume of 30 500 Å^{3}, which would correspond to approximately 70 molecules per Such large Z values usually indicate either a wrong a twinned crystal or a modulated structure. A closer look at various orientations of the diffraction pattern reveals a distinction between reflections based on their intensities: layers of very strong spots are surrounded in a nonarbitrary way by weaker reflections. This intensity pattern strongly suggests that the possibility of having a twinned crystal can be eliminated and that the of VAR205 is modulated.
(in the present studyAs discussed in §1 there are three ways of handling the diffraction pattern of a modulated structure: indexing the basic cell with main reflections only (ignoring the satellite reflections and therefore as a consequence neglecting a large fraction of diffracted intensities), indexing the using all reflections equivalently (which is in the case of incommensurateness only an approximation) and indexing the basic cell followed by the determination of the modulation wavevector q. These options along with their subsequent steps are summarized in the flowchart in Fig. 12. In many cases, especially when the modulation is not too strong, the fastest way to obtain structural information and an idea of the extent of the modulation is to start with the basic cell and try to determine the average structure.
3.1.1. Basic cell and average structure
The main reflections in the diffraction pattern of VAR205 yield a monoclinic basic cell with a volume of 950 Å^{3}, corresponding to two molecules per (Fig. 13). The satellite peaks were eliminated `by hand' with RLATT from the peak list, the remaining main reflections were reimported into the indexing routine. Data processing and scaling results in classical threedimensional data, which can then be inspected for to determine the spacegroup symmetry of the main reflections. The reflection condition 0k0, k = 2n and the enantiopurity of the compound^{4} led to the monoclinic P2_{1} with Z = 2. All attempts to solve the structure with classical direct (SHELXS) or dualspace recycling methods (SHELXM) failed, which was the first indication of a strong modulation.
If an average structure can be obtained this way, the coordinates from the structural (isotropic) P2_{1} is the of the average structure (even if the structure cannot be determined this way) P2_{1} will very likely also be the basis of the symbol (see §3.2.1).
can later be used as a starting model for the of the modulated structure (to find the proper shapes of the AMFs). Useful information can also be obtained from any disorder sites. Equally important for a later (3 + 1)dimensional treatment of the problem is the knowledge of the IfIf the modulation is weak, it is often not only possible to solve the average structure but also to refine it to very reasonable agreement factors, usually, however, at the cost of large displacement parameters or extensive disorder modelling.
3.1.2. and superstructure
For the qvector components (Fig. 13). In the case of VAR205 the three qvector components are α = 0.143 (2), β = 0 and γ = 0.384 (8) (cf. §3.1.3). To obtain an appropriate the nonzero components will be approximated by α = 1/7 (≃ 0.1429) and γ = 2/5 (= 0.4000), then the respective lattice parameters will be multiplied by the denominators, i.e. a by 7 and c by 5. As a result a is obtained with a volume which is 7 × 5 = 35 times that of the basic cell. Please note again that in this incommensurate case the represents only an approximation: the value for γ = 0.384 (8) was approximated with γ = 2/5; another possible approximation could be γ = 3/8 (= 0.375), resulting in a 7 × 8 = 56fold with a somewhat different description of the atomic arrangement for the same crystal structure!
processing the orientation matrix can either be obtained directly by indexing all reflections or, if this is not possible, by combining the information from the basic cell parameters andData processing and scaling can be pursued in the same way as for a classical threedimensional periodic structure. However, one has to be aware of the fact that even if satellite reflections to high order are observed, most of the i.e. no intensity will be detectable (Fig. 14). Accordingly, profile fitting will be poor, average intensities during data processing will be low and a large percentage of the reflections will be classified as weak or unobserved.
points are `empty',Nonetheless, there can be reasons to use a hkl file into higherdimensional data: conventional threedimensional software can be used and all data are available at any time so that e.g. the decision up to which order satellite reflections shall be included can be postponed.
integration and then later transform the resulting threedimensionalIn the present example of VAR205 structure solution was possible for the commensurate 7 × 5fold approximation in the P2_{1} using the dualspace recycling algorithm employed in SHELXM. Subsequent cycles of and differenceFourier map calculations allowed completion of the model (35 independent molecules = 840 nonH atoms). Owing to the large number of unobserved reflections the datatoparameter ratio was very poor and anisotropic (7560 parameters!) in SHELXL was only possible using blockdiagonal or conjugategradient leastsquares instead of fullmatrix leastsquares. Nonetheless, it was learned from the approximation that the molecules show a strong modulation along a (Fig. 15).
Even if the γ is equal to 0.384 (8), treating the data in a 7 × 8fold with γ = 3/8 is at least as valid as in the 7 × 5fold with γ = 2/5 discussed above. This problem occurs whenever the qvector components are between two rational numbers, e.g. 0.186, which is as close to 1/5 as it is to 1/6, so that neither a fivefold nor a sixfold will mirror the true structure (see, for example, the discussion in Schönleber & Chapuis, 2004). Usually this problem will reveal itself in strange geometric and/or displacement parameters in one or more of the molecules in the asymmetric unit.
of a could be carried out with a good datatoparameter ratio to acceptable agreement factors, it is important to emphasize once again that the solution in the present case can only be regarded as an approximation. SinceIn principle, one of the molecules in the ) by transforming the coordinates into the basic cell (in the present case by taking, for example, the molecule closest to the origin and multiplying the fractional coordinates x by 7 and z by 5). If the modulation is not too strong, i.e. if the amplitudes of the AMFs are not too large, the probability is high of finding appropriate starting values for the AMFs in the first cycles of In general, a starting model for (3 + 1)dimensional derived from the can be assumed to be better than a starting model from the threedimensional of the average structure in the basic cell. As in the present case the modulation is rather strong it was, however, not possible to establish a good structural model with proper AMFs for all atoms starting from the fractional coordinates of one molecule of the approximation.
could be used as a starting model for the of the structure in (in analogy to the average structure of §3.1.13.1.3. Indexing in superspace
The crucial step for all subsequent action in q vector from the original raw data. For this purpose it is important to separate main reflections from satellite reflections and obtain fractional indices (i.e. hkl values of the closest main reflection and the deviations from these integers) for the latter based on the orientation matrix of the basic cell defined by the main reflections. If this procedure is carried out with firstorder satellites only (the higherorder satellite peaks were eliminated from the reflection list with RLATT), a very clean and clear distribution of fractional indices (Fig. 16) was obtained (via the indexing routine in SMART) resulting in an initial q vector of (0.14, 0, 0.38).
is the determination of a suitableOnce the starting values for the qvector components have been determined this way, the data can be processed using the orientation matrix of the basic cell along with these qvector components and a maximum satellite order n_{max}, up to which satellite peaks should be processed.
Depending on the integration software, the output format of the resulting hkl file can vary. Using SAINT, the resulting reflection file is always (3 + 3)dimensional (indices h, k, l, m, n, p). In the present (3 + 1)dimensional case it contains main reflections (m = n = p = 0) and satellite reflections (m ≠ 0, n = p = 0). Overlapping satellite reflections (which could occur with one of the qvector components close to 0.5) are treated the same way as overlapping reflections from a twinned crystal, i.e. a `twin component' batch number is given at the very end of the lines in the reflection file.
The raw reflection files can then be scaled and corrected for absorption effects, e.g. with SADABS (unless the file contains overlapping reflections, this option is not yet implemented), and the resulting [again: (3 + 3)dimensional] hkl file can be directly imported into JANA2006. The guided input routine of JANA2006 (file > structure > new) leads through data reduction, analysis of and spacegroup determination. It closes with the option to attempt a structure solution in higherdimensional space via the external program SUPERFLIP (Palatinus & Chapuis, 2007).
3.2. Structure solution and completion in superspace
Just as for threedimensional periodic crystals the first step in the structure solution process for modulated structures in JANA2006 by checking the for both main and satellite reflections. The groups for (3 + 1) dimensions are tabulated along with their special in the International Tables for Crystallography, Vol. C in ch. 9.8, Incommensurate and Commensurate Modulated Structures, by Janssen et al. (2006).
is the determination of the correct This can be done manually or, as mentioned above, automatically during the data importing routine within3.2.1. Superspacegroup determination
A first glance at Table 9.8.3.5 `(3 + 1)dimensional International Tables for Crystallography, Vol. C, suggests that the superspacegroup symbol consists of the basic spacegroup symbol (here: P2_{1}) extended by an expression in parentheses which denotes the general form of the q vector followed by additional letters.
groups' in theHowever, this is only partially true. The Bravais symbol refers now to a (3 + 1)dimensional lattice. Looking at B2(00γ)s, for example, the letter B does not mean an additional centring of (½,0,½) as in threedimensional space groups, but of (½,0,½,0) since the centring can also occur along the fourth dimension. If this is the case, the Bravais symbol B′ (or X) is used to describe the additional centring of (½,0,½,½). The letters following the q vector indicate additional translational components of the symmetry operators along the fourth dimension (screw and glide components: if there is none, a `0' is written, an `s' stands for a translation by ½). For a more sophisticated discussion the interested reader is referred to ch. 3 of the IUCr textbook Incommensurate crystallography by van Smaalen (2007).
No. 53,In the case of VAR205 the choice of the k00 with k = 2n, indicating the twofold screw axis along b (which is already known from the basic cell). Therefore, the lattice in is primitive, the resulting is P2_{1}(α0γ)0.
is straightforward: the only special are 0Now imagine, for example, a monoclinic b unique lattice with a q vector (α0γ) and for the reflections 0k0m with m ≠ 2n. This indicates a screw component of the twofold axis along the fourth dimension, the resulting would be P2(α0γ)s. The might also be 0k00 with k ≠ 2n and h0lm with m = 2n. Here the first condition indicates as above a twofold screw axis along b, the second one a glide plane perpendicular to b with a glide component along the fourth dimension. The resulting would be P2_{1}/m(α0γ)0s.
3.2.2. Structure solution
Having established the SUPERFLIP can be generated and the program can be called directly from within JANA2006. SUPERFLIP extends the chargeflipping algorithm introduced by Oszlányi & Süto (2004, 2008) to an arbitrary number of dimensions and can therefore solve modulated structures ab initio in higherdimensional space. This capability is especially important if no threedimensional structure (average structure or superstructure) is available from which a starting model for the basic structure can be deduced. SUPERFLIP generates a (3 + 1)dimensional electrondensity map which is then searched for atoms by JANA2006. Since atoms in (3 + 1)dimensional have to be imagined as onedimensional waves, the interpretation of the electrondensity maps already requires the use of at least one modulation wave (automatically employed by JANA2006), i.e. the program puts out starting values for the fractional coordinates of the atoms and starting values for the parameters describing the AMFs.
the input file forFrom the SUPERFLIP solution of the present example a peak list was extracted from which the first six peaks were selected based on peak height and interatomic distances (see discussion below) as silicon (peak1), oxygen (peaks 2–3) and carbon (peaks 4–6) atoms. Starting from this set of six atoms the model was completed in subsequent cycles of
Fouriermap calculation, and peak search followed by manual identification of possible new atomic positions.Analyzing the surroundings of atoms in modulated structures differs considerably from doing so in nonmodulated structures: for every distance of interest there would be an infinitely long table of values, owing to the missing translational symmetry. Here the t (indicating the phase of the modulation, see §1.3 and Fig. 7) provides a very convenient way to visualize and analyse structural properties such as interatomic distances. A listing of a certain distance as a function of t, i.e. varying t from 0 to 1, provides all values for this distance occurring anywhere in the structure (van Smaalen, 2004). In this context it is important to keep in mind that the extent of variation which is found in the geometric parameters of modulated structures resembles the findings from packing analyses of nonmodulated structures: bond lengths are expected to show less variation than bond angles, which again vary less than torsion angles and intermolecular distances.
concept with the modulation parameterThere are two ways of checking the variation of a structural parameter with t in JANA2006: the subroutine Dist produces a listing of the structural parameter in userdefined intervals, the subroutine Grapht provides a plot of this relation. When looking for missing atoms in a structure based on a Fouriermap peak list it is usually faster to tabulate the distances at intervals of 0.1 or 0.2 in t than to make graphs. Fig. 17 shows the results of such a search in the vicinity of e.g. atom Si01. Atom C03 is one of the carbon atoms bonded to Si01 and at this early stage of shows a rather large variation in the bonding distance to the silicon atom (cf. the difference of 0.111 Å between minimum and maximum value). Peak max8 from the Fourier peak list to Si01 varies from 1.2 to 2.8 Å and therefore will probably not be a suitable candidate for one of the missing C atoms. Peak max7, on the other hand, shows a relatively even distribution of distances and therefore will be selected as carbon atom C04. Another cycle of Fouriermap calculation and peak search would then follow.
In threedimensional space refining a structure means finding values for the coordinates x, y, z and the six anisotropic atomic displacement parameters U^{ij} which result in the best agreement of F_{obs} and F_{calc}. In this concept has to be extended: the refined coordinates x, y, z (or better: x_{1}, x_{2}, x_{3}) are the positions of the atoms in the basic structure and, in addition, for each atom the deviation from this average position has to be determined and refined. This deviation can be expressed by a variety of AMFs (cf. Fig. 2), which, in the case of a superposition of harmonic waves up to second order for the x_{1} coordinate looks like
The coefficients A_{1}, B_{1} and A_{2}, B_{2} are the amplitudes of the harmonic waves of first and second order. These coefficients are determined by JANA2006 on the basis of the diffraction data and occur as parameters in the atomic parameter file m40. Therefore, for each of the positional parameters (fractional coordinates x_{1}, x_{2} and x_{3}) two more parameters per harmonic wave will occur to describe the AMF which characterizes the deviation from the basic position along the fourth dimension.
During the structure completion cycle it is important to verify for the atoms in the structural model that their position has been determined correctly and that their AMFs follow the observed electron density. This procedure resembles very much the 2F_{obs} − F_{calc} map calculation and inspection that structural biologists use to check if their protein model is placed correctly and fits the observed electron density. For this purpose an F_{obs} map is calculated in around the atom of interest as a function of x_{4} and an overlay of the map and the AMF is generated with the subroutine Contour (Fig. 18). Here again the twodimensional projection technique introduced in Fig. 7 is used: the vertical axis always represents the coordinate x_{4} (along a_{s4}), the second axis is one of the three remaining axes a_{s1}, a_{s2} or a_{s3} with their fractional coordinates x_{1}, x_{2} or x_{3} (as the angle between the two axes is defined via the respective qvector component, the images differ). The three sections in Fig. 18 show clearly that there is density around the position of Si01 and that the AMF follows the observed electron density. It also becomes obvious, however, that, at a later stage, higher harmonic funtions will have to be added to model the details of the electron density better. Please note again the onedimensional character of the atomic domain along a_{s4}, represented by its electron density and its AMF (red line).
When the structural model has been completed (i.e. all atoms have been found and assigned) according to the abovementioned procedure the is completed by adding harmonic functions of higher order, introducing anisotropic ADPs and calculating Hatom positions. For VAR205 R_{obs} for all (main and satellite) reflections at this stage had already dropped to 0.146, compared with 0.400 for the first model which included only six atoms.
3.2.3. Completion of refinement
In the case of a strong modulation, it is generally advantageous to use not only harmonic waves of first and second order, but also of the third order for the AMFs of the positional parameters before switching to anisotropic an example is shown where a hypothetical `electrondensity distribution' is fitted quite satisfactorily using a superposition of three harmonic waves of higher order, while the firstorder wave alone is a rather rough approximation. Introducing anisotropic ADPs too early can hinder the proper determination of the positional AMFs.
By summing up the higherorder waves, it is possible to better model the details of the electron density. In Fig. 19In the R_{obs} = 0.083) before switching to anisotropic ADPs (R_{obs} = 0.070). Just as in the of a threedimensional periodic structure it is important to make sure at all times that the introduction of additional harmonic waves (= more parameters) is justified by a statistically significant drop in the crystallographic agreement factors.
process for VAR205 three harmonic waves for the positional AMFs were added (Up to this point only AMFs for the positional parameters were used, i.e. only the socalled displacive modulation, which affects the positions of the atoms, was considered. However, just as the coordinates x_{1}, x_{2} and x_{3} can be modulated, other atomic parameters such as the occupation factors or the ADPs can also vary periodically throughout the structure, owing to the varying environments of the atoms. Again, these modulations can be described by harmonic, sawtooth or crenel functions (cf. Fig. 2). Occupational modulation is frequently encountered in alloys but can also occur when, for example, solvent positions in organic modulated structures are not always fully occupied and the amount to which they are occupied varies periodically. ADPs, on the other hand, tend to be modulated whenever a strong is present, because a change in interatomic distances and the variation of the atomic surroundings often go hand in hand with a change in the vibrational amplitudes. Accordingly AMFs can also be added for the atomic displacement parameters U^{ij} (thermal modulation). Since six values describe the shape of the anisotropic displacement ellipsoid, every additional harmonic wave for the AMFs will add 12 more parameters to the [one sine and one cosine component, see (1)]. Again, it is important to constantly monitor improvements in the fit of the electron density and the agreements factors to check whether the introduction of new higher harmonic waves is reasonable.
In the case of VAR205, in the final stages of R factor for all (main and satellite) reflections dropped to R_{obs} = 0.036.
four harmonic waves for the displacive and two harmonic waves for the ADP modulation were used and along with the addition of H atoms (riding model or free possible) the3.3. Interpretation and presentation of results
While presenting the ) and describe their movements relative to each other. During the discussion of individual geometric parameters of interest it is necessary to pay attention to the fact that all these values vary with the phase of the modulation t around a mean value which usually corresponds to the values expected for a nonmodulated structure. Sections of electron density can be used to show that the determined AMFs adequately describe the experimental electron density (resembling the 2F_{obs} − F_{calc} maps in structural biology papers) and to provide a graphical representation of the extent of modulation present in the structure. If there is interest in the crystal packing, the atomic coordinates of the modulated structure can be exported as a commensurate Here, however, the same approximations have to be accepted which have already been discussed in the context of the approach (cf. §3.1.2).
results for a modulated structure, it is important to keep in mind that there is not just `one bond length' or `one crystal environment' for a given group of atoms. Often, the first step following the summary of the details is to divide the molecule into socalled rigid units (if not already done during the process, see §3.3.23.3.1. parameters
The final structural model of VAR205 is based on continuous atomic modulation functions for displacive (x_{1}, x_{2}, x_{3}) and thermal (U^{ij}) modulation. Higher harmonics up to fourth order were applied for the of all atoms and higher harmonics up to second order for modulation of the ADPs of the 24 nonH atoms. In total 1396 parameters were refined. The atomic labels are introduced in Fig. 20. Except atom H17, all hydrogen atoms are coupled via a riding model to their respective carbon atoms. The position of atom H17 (of the OH group) was refined isotropically applying bondlength [O—H: 0.84 (10) Å] and bondangle [C—O—H: 109.47 (50)°] restraints.
The I > 3σ(I). The ratio between the number of refined parameters and the number of reflections used is > 20. In the last cycle no exceeded 0.7 (with three correlation coefficients larger than 0.6, all affecting the positional parameters of atom H17). The final residual parameters of the are given in Table 2.
was based on all main and satellite reflections up to fourth order, yielding a total of 28 265 reflections, of which 24 813 are observed with

3.3.2. Rigid units
Following the abovementioned concept of dividing the molecule into suitable rigid units the VAR205 molecule can be described as follows (Fig. 20): the silicon atom with its tertbutyl substituent and two methyl groups is connected via an oxygen atom to a nonplanar tetrahydropyrrolizinone moiety, to which a flat phenyl ring is attached. With respect to the modulation, these individual molecular units act like rigid bodies, the modulation mainly affects the torsion angles between the three segments, changing the conformation of the molecule as a whole, but not of the molecular units. This variation will be discussed in §3.3.3.
This concept of dividing the molecule into suitable rigid units can also be employed during the i.e. each with its own set of AMFs, while still retaining a very high datatoparameter ratio (see Table 2). If this is not the case a socalled rigidbody can help to reduce the number of parameters and/or avoid problems such as unreasonable geometries. For VAR205 this option was tested using the abovementioned rigid groups with Si01, C15 and C19 as reference atoms. For all nonH atoms in those rigid units the fractional coordinates and anisotropic ADPs were refined individually, while positional modulation waves up to fourth order and TLS (translation, libration, screw formalism) modulation waves up to second order were refined for each group as a whole (Schomaker & Trueblood, 1968). The H atoms were attached via riding models. The atoms O08, O17 with H17, and O18 were refined as individual atoms as described in §3.2.3.
procedure, especially in cases with poorer data quality or fewer observed satellite reflections. In the present study the high quality of the dataset with satellite reflections observable up to fourth order allowed all atoms to be treated individually,This strategy with common modulation parameters for all atoms in the respective rigid units reduced the number of refined parameters from 1396 to 775 and resulted in good agreement factors (R_{obs} = 0.040, R_{all} = 0.046 and wR_{all} = 0.047 for all reflections, Δρ_{max} = 0.31 e Å^{−1} and Δρ_{min} = −0.42 e Å^{−1}, compared with values in Table 2). However, the suffered heavily from correlations and did not converge properly. Therefore, this strategy of rigidbody was not pursued any further. However, as mentioned above, in cases with limited numbers of reflections, such a strategy is an option to consider.
3.3.3. Selected geometrical parameters
Mean bond lengths and angles in VAR205 are in very good agreement with the expected values for nonmodulated structures, as published in the International Tables for Crystallography, Vol. C in ch. 9.5 (Allen et al., 2006). The distances and angles remain constant within the standard uncertainties along the fourth dimension, i.e. along the phase of the modulation t (see Table 3 for some selected values).

Consider, e.g. the bond lengths and the bond angle within the Si01—O08—C09 group, connecting the tertbutyldimethylsilanyl substituent with the tetrahydropyrrolizinone. They are rather constant as a function of t (Table 3), the torsion angles around these bonds (e.g. C09—O08—Si01—C04), however, vary by ∼ 10°. The same holds for the C13—C19 bond linking the tetrahydropyrrolizinone moiety to the phenyl ring. Here the torsion angles vary by ∼ 5° (see Fig. 21). While the relative movement of the tertbutyldimethylsilanyloxy and the tetrahydropyrrolizinone moieties with respect to each other has a quite undulating character, the relative movement of the tetrahydropyrrolizinone moiety and the phenyl ring is less smooth (see animated gif attached to Fig. 22 in the electronic version of the paper).
The degree of pyramidalization of the central N atom is not affected by the modulation, N11 is on average 0.331 (1) Å out of the plane defined by the three carbon atoms C10, C12 and C15. This distance varies only slightly with t: the minimum and maximum distances are 0.320 and 0.343 Å. For related compounds with a fully sp^{3}hybridized nitrogen (four substituents) distances around 0.53 Å are found (Wagner, 2008).
The constancy with t also holds for the envelope structure C15—N11—C10—C09—C16, as can be seen from the torsion angles in Table 4. C16 is on average 0.625 (1) Å out of the plane defined by the four atoms C09, C10, N11 and C15 (minimum: −0.651 Å, maximum: 0.594 Å).

The relative stereochemistry of H15 and the tertbutyldimethylsilanyloxy substituent at the pyrrolidine fivering can be unambiguously defined as cis.
The two oxygen atoms O17 and O18 are clearly out of the best plane defined by the five atoms N11, C12, C13, C14 and C15. The average distance for O17 is 0.0934 (9) Å (minimum: 0.067 Å, maximum: 0.145 Å) and for O18 it is −0.2639 (9) Å (minimum: −0.253 Å, maximum: −0.283 Å).
As already mentioned above, the hydrogen atom H17 of the OH group was refined with restraints. As it is involved in the hydrogen bond O17—H17⋯O18^{i} along b [with symmetry code (i) x, y+1, z], the corresponding bond lengths and angles will be mentioned, but not further discussed. The average distance O17—O18^{i} is 2.564 (5) Å [minimum: 2.538 (6) Å, maximum: 2.588 (5) Å] and for O17—H17 it is 0.84 (5) Å [minimum: 0.78 (5) Å, maximum: 0.88 (5) Å], the angle O17—H17—O18 is almost linear with an average value of 176 (5)° [minimum: 172 (5)°, maximum: 179 (5)°].
3.3.4. Superstructure
The atomic coordinates from the modulated structure can be converted into a commensurate cf. §3.1.2). Looking at the of the alone may not provide any insights regarding the direction of the modulation, i.e. may not provide any answer to the question: how does the variation in geometry and/or environment proceed throughout the crystal? It can help to align the individual molecules of the according to the modulation. For the of VAR205 this arrangement is found by selecting molecules perpendicular to the direction of the q vector (for details, see supplementary material Fig. S3 ). The resulting line of molecules is shown in Fig. 23. While arranging the molecules in such a way reveals the lack of translational symmetry in the crystal very clearly, it is not the most convenient way to visualize the change of the intramolecular geometry with t. For this purpose it is much more illustrative to export the coordinates of the individual molecule in suitable steps of t and create an animated gif of these `snapshots along the fourth dimension' (cf. Fig. 22).
with, however, the same approximations that have to be accepted during a solution (4. Summary
The R,7aS)6(tertbutyldimethylsilanyloxy)1hydroxy2phenyl5,6,7,7atetrahydropyrrolizin3one, a typical organic compound, has been successfully refined using the approach. This approach was introduced in a nonmathematical way. All steps from data handling to structure solution, and presentation have been discussed in detail.
of (6The description of the diffraction pattern in (3 + 1)dimensional t. This variation is reflected in the torsion and dihedral angles between these rigid units and in the intermolecular distances. Nevertheless, all parameters associated with intra and intermolecular geometry correspond to values and distributions found for similar molecules in the Cambridge Structural Database (Allen, 2002).
delivered a unique indexing of the peaks, the final model during structure was based on 1396 parameters with a datatoparameter ratio above 20. Looking at the structure, the three individual molecular fragments act as rigid units, while the overall shape, or conformation, of the molecule varies significantly with the phase of the modulationIn contrast, treating the structure as a
(approximation) resulted in a series of problems: while indexing the diffraction pattern, it was not clear whether a 7 × 5fold or a 7 × 8fold should be used. In the structure because of the large number of molecules in the (35 and 56, respectively) and because of the large number of unobserved reflections, the datatoparameter ratio was very poor. For anisotropic 7560 parameters have to be considered for the 7 × 5fold and 12 096 for 7 × 8fold Reasonable geometric parameters and anisotropic displacement parameters can only be obtained by employing large numbers of restraints.Supporting information
10.1107/S0108768109015614/bk5084sup1.cif
contains datablocks global, I. DOI:Structure factors: contains datablock I. DOI: 10.1107/S0108768109015614/bk5084Isup2.hkl
Extra figures. DOI: 10.1107/S0108768109015614/bk5084sup3.pdf
Highresolution animated version of Fig. 22. DOI: 10.1107/S0108768109015614/bk5084sup4.gif
Lowresolution animated version of Fig. 22. DOI: 10.1107/S0108768109015614/bk5084sup5.gif
Data collection: SMART V5.630 (Bruker AXS, 2003); cell
Saint V7.36A (Bruker AXS, 2006); data reduction: Saint V7.36A (Bruker AXS, 2006); program(s) used to solve structure: SUPERFLIP (Palatinus & Chapuis, 2007); program(s) used to refine structure: Jana2006 (Petricek et al., 2006); molecular graphics: PLATON (Spek, 2003); software used to prepare material for publication: Jana2006 (Petricek et al., 2006).C_{19}H_{27}NO_{3}Si  F(000) = 372 
M_{r} = 345.5  D_{x} = 1.205 Mg m^{−}^{3} 
Monoclinic, P21(α0γ)0†  Cu Kα radiation, λ = 1.5418 Å 
q = 0.14216a* + 0.38390c*  Cell parameters from 9555 reflections 
a = 6.425 (2) Å  θ = 2.9–66.6° 
b = 6.522 (2) Å  µ = 1.22 mm^{−}^{1} 
c = 22.716 (6) Å  T = 100 K 
β = 90.739 (11)°  Platelet, colourless 
V = 951.8 (5) Å^{3}  0.30 × 0.07 × 0.02 mm 
Z = 2 
† Symmetry operations: (1) x_{1}, x_{2}, x_{3}, x_{4}; (2) −x_{1}, x_{2}+1/2, −x_{3}, −x_{4}. 
Bruker CCD diffractometer  28265 independent reflections 
Radiation source: finefocus rotating anode  24813 reflections with I > 3σ(I) 
Multilayer mirrors monochromator  R_{int} = 0.023 
ω–scans  θ_{max} = 66.8°, θ_{min} = 1.5° 
Absorption correction: multiscan SADABS (Sheldrick, 2007)  h = −8→8 
T_{min} = 0.712, T_{max} = 0.976  k = −6→7 
52466 measured reflections  l = −28→28 
Refinement on F  729 constraints 
R[F^{2} > 2σ(F^{2})] = 0.036  H atoms treated by a mixture of independent and constrained refinement 
wR(F^{2}) = 0.042  Weighting scheme based on measured s.u.'s w = 1/(σ^{2}(F) + 0.0001F^{2}) 
S = 2.38  (Δ/σ)_{max} = 0.008 
28265 reflections  Δρ_{max} = 0.27 e Å^{−}^{3} 
1396 parameters  Δρ_{min} = −0.39 e Å^{−}^{3} 
2 restraints 
C_{19}H_{27}NO_{3}Si  β = 90.739 (11)° 
M_{r} = 345.5  V = 951.8 (5) Å^{3} 
Monoclinic, P21(α0γ)0†  Z = 2 
q = 0.14216a* + 0.38390c*  Cu Kα radiation 
a = 6.425 (2) Å  µ = 1.22 mm^{−}^{1} 
b = 6.522 (2) Å  T = 100 K 
c = 22.716 (6) Å  0.30 × 0.07 × 0.02 mm 
† Symmetry operations: (1) x_{1}, x_{2}, x_{3}, x_{4}; (2) −x_{1}, x_{2}+1/2, −x_{3}, −x_{4}. 
Bruker CCD diffractometer  28265 independent reflections 
Absorption correction: multiscan SADABS (Sheldrick, 2007)  24813 reflections with I > 3σ(I) 
T_{min} = 0.712, T_{max} = 0.976  R_{int} = 0.023 
52466 measured reflections 
R[F^{2} > 2σ(F^{2})] = 0.036  2 restraints 
wR(F^{2}) = 0.042  H atoms treated by a mixture of independent and constrained refinement 
S = 2.38  Δρ_{max} = 0.27 e Å^{−}^{3} 
28265 reflections  Δρ_{min} = −0.39 e Å^{−}^{3} 
1396 parameters 
x  y  z  U_{iso}*/U_{eq}  
Si01  −0.37092 (2)  0.327443  0.659745 (8)  0.02007 (5)  
C02  −0.49419 (11)  0.55099 (13)  0.69617 (4)  0.0305 (2)  
C03  −0.56723 (12)  0.12354 (13)  0.64508 (4)  0.0303 (2)  
C04  −0.23589 (11)  0.40661 (13)  0.59038 (3)  0.0272 (2)  
C05  −0.14179 (13)  0.21792 (16)  0.56111 (4)  0.0383 (3)  
C06  −0.06346 (13)  0.56026 (15)  0.60545 (3)  0.0372 (3)  
C07  −0.39167 (14)  0.50777 (16)  0.54772 (4)  0.0402 (3)  
O08  −0.18123 (7)  0.23691 (9)  0.70179 (2)  0.02529 (15)  
C09  −0.20629 (10)  0.13164 (12)  0.75615 (3)  0.0231 (2)  
C10  −0.08436 (11)  −0.06992 (12)  0.75368 (3)  0.0239 (2)  
N11  0.12210 (8)  −0.01391 (9)  0.77662 (2)  0.01921 (16)  
C12  0.22778 (10)  −0.13298 (11)  0.81639 (3)  0.0182 (2)  
C13  0.34856 (10)  0.00057 (11)  0.85533 (3)  0.01811 (19)  
C14  0.28477 (10)  0.19596 (11)  0.84355 (3)  0.01862 (19)  
C15  0.13010 (10)  0.20130 (11)  0.79404 (3)  0.01854 (18)  
C16  −0.09831 (10)  0.25196 (12)  0.80553 (3)  0.0226 (2)  
O17  0.35849 (8)  0.36355 (9)  0.86877 (2)  0.02477 (15)  
O18  0.21029 (7)  −0.32253 (9)  0.81674 (2)  0.02422 (15)  
C19  0.51159 (10)  −0.06276 (11)  0.89747 (3)  0.0184 (2)  
C20  0.55142 (11)  0.05378 (12)  0.94779 (3)  0.0219 (2)  
C21  0.71368 (11)  0.00383 (13)  0.98554 (3)  0.0257 (2)  
C22  0.84021 (11)  −0.16203 (14)  0.97393 (3)  0.0288 (2)  
C23  0.80080 (11)  −0.28088 (13)  0.92462 (3)  0.0281 (2)  
C24  0.63787 (10)  −0.23176 (11)  0.88679 (3)  0.0221 (2)  
H02a  −0.606462  0.604054  0.67109  0.0457*  
H02b  −0.550668  0.507981  0.733914  0.0457*  
H02c  −0.390319  0.658571  0.702668  0.0457*  
H03a  −0.678814  0.178831  0.620147  0.0455*  
H03b  −0.501204  0.008181  0.625101  0.0455*  
H03c  −0.624909  0.076357  0.68228  0.0455*  
H05a  −0.07179  0.259368  0.525111  0.0575*  
H05b  −0.041335  0.153875  0.587994  0.0575*  
H05c  −0.251986  0.119662  0.551464  0.0575*  
H06a  0.006714  0.601449  0.569459  0.0558*  
H06b  −0.123714  0.68128  0.624118  0.0558*  
H06c  0.036851  0.496274  0.632394  0.0558*  
H07a  −0.320326  0.548353  0.511855  0.0602*  
H07b  −0.502282  0.410168  0.537903  0.0602*  
H07c  −0.451659  0.629233  0.566223  0.0602*  
H09  −0.357939  0.112783  0.762896  0.0278*  
H10a  −0.148219  −0.171421  0.780122  0.0287*  
H10b  −0.072913  −0.114622  0.712411  0.0287*  
H15  0.178039  0.312169  0.767162  0.0222*  
H16a  −0.138634  0.194466  0.843893  0.0272*  
H16b  −0.121985  0.400525  0.799589  0.0272*  
H17  0.3089 (12)  0.4673 (12)  0.8524 (3)  0.041 (3)*  
H20  0.466062  0.168538  0.956153  0.0263*  
H21  0.738239  0.083953  1.019619  0.0308*  
H22  0.953208  −0.194395  0.999489  0.0346*  
H23  0.885797  −0.396337  0.916759  0.0337*  
H24  0.612065  −0.314302  0.853195  0.0266* 
U^{11}  U^{22}  U^{33}  U^{12}  U^{13}  U^{23}  
Si01  0.01832 (7)  0.01957 (9)  0.02222 (9)  −0.00039 (7)  −0.00323 (6)  0.00023 (7) 
C02  0.0255 (3)  0.0258 (4)  0.0402 (5)  −0.0029 (3)  0.0019 (3)  −0.0056 (3) 
C03  0.0279 (4)  0.0253 (4)  0.0377 (5)  −0.0033 (3)  −0.0047 (3)  −0.0029 (3) 
C04  0.0281 (3)  0.0325 (4)  0.0207 (4)  −0.0010 (3)  −0.0041 (3)  0.0021 (3) 
C05  0.0352 (4)  0.0490 (6)  0.0309 (4)  0.0045 (4)  0.0038 (3)  −0.0073 (4) 
C06  0.0384 (4)  0.0464 (6)  0.0268 (4)  −0.0143 (4)  0.0025 (3)  0.0055 (3) 
C07  0.0466 (5)  0.0449 (6)  0.0287 (4)  0.0024 (4)  −0.0108 (3)  0.0098 (4) 
O08  0.0220 (2)  0.0312 (3)  0.0226 (3)  −0.00085 (19)  −0.00206 (19)  0.0079 (2) 
C09  0.0203 (3)  0.0276 (4)  0.0214 (4)  −0.0033 (3)  −0.0017 (3)  0.0058 (3) 
C10  0.0276 (4)  0.0208 (4)  0.0232 (4)  −0.0059 (3)  −0.0060 (3)  0.0007 (3) 
N11  0.0243 (3)  0.0136 (3)  0.0197 (3)  −0.0017 (2)  −0.0035 (2)  −0.0004 (2) 
C12  0.0217 (3)  0.0148 (4)  0.0180 (3)  0.0003 (3)  0.0030 (2)  0.0008 (2) 
C13  0.0211 (3)  0.0142 (4)  0.0190 (3)  −0.0004 (3)  0.0009 (2)  0.0007 (2) 
C14  0.0214 (3)  0.0151 (3)  0.0193 (3)  −0.0016 (3)  0.0002 (2)  −0.0002 (3) 
C15  0.0233 (3)  0.0131 (3)  0.0192 (3)  −0.0005 (2)  −0.0008 (2)  0.0011 (2) 
C16  0.0244 (3)  0.0209 (4)  0.0226 (3)  0.0020 (3)  0.0003 (3)  0.0020 (3) 
O17  0.0347 (3)  0.0115 (3)  0.0279 (2)  −0.0013 (2)  −0.0099 (2)  0.00043 (19) 
O18  0.0328 (3)  0.0122 (3)  0.0276 (2)  −0.00117 (19)  −0.0037 (2)  −0.00061 (18) 
C19  0.0172 (4)  0.0175 (4)  0.0204 (3)  −0.0028 (2)  0.0008 (2)  0.0049 (2) 
C20  0.0228 (4)  0.0192 (4)  0.0237 (3)  0.0007 (3)  0.0002 (3)  0.0018 (3) 
C21  0.0255 (5)  0.0270 (4)  0.0244 (3)  −0.0036 (3)  −0.0034 (3)  0.0019 (3) 
C22  0.0207 (4)  0.0340 (5)  0.0317 (4)  0.0014 (3)  −0.0050 (3)  0.0083 (3) 
C23  0.0238 (4)  0.0263 (4)  0.0341 (4)  0.0063 (3)  0.0018 (3)  0.0042 (3) 
C24  0.0241 (4)  0.0182 (4)  0.0242 (3)  0.0009 (3)  0.0017 (3)  0.0009 (2) 
Average  Minimum  Maximum  
O17—O18^{i}  2.564 (5)  2.538 (6)  2.588 (5) 
O17—H17  0.84 (5)  0.78 (5)  0.88 (5) 
O18—H17^{ii}  1.72 (5)  1.67 (5)  1.80 (5) 
O18^{i}—O17—H17  3(3)  1(3)  5(3) 
O17^{ii}—O18—H17^{ii}  1.2 (15)  0.3 (16)  2.7 (15) 
O17—H17—O18^{i}  176 (5)  172 (5)  179 (5) 
C16—C09—O08—Si01  −119.07 (5)  
C10—C09—O08—Si01  128.13 (5)  
C09—O08—Si01—C02  70.12 (6)  
C09—O08—Si01—C03  −51.62 (6)  
C09—O08—Si01—C04  −170.93 (6)  
C12—C13—C19—C20  −152.28 (7)  
C12—C13—C19—C24  31.60 (10)  
C14—C13—C19—C20  29.93 (10)  
C14—C13—C19—C24  −146.18 (7)  
C22—C23—N11—C15  25.00 (6)  
N11—C10—C09—C16  −26.37 (7)  
C10—N11—C15—C16  23.38 (6) 
Symmetry codes: (i) x_{1}, x_{2}+1, x_{3}, x_{4}; (ii) x_{1}, x_{2}−1, x_{3}, x_{4}. 
Experimental details
Crystal data  
Chemical formula  C_{19}H_{27}NO_{3}Si 
M_{r}  345.5 
Crystal system, space group  Monoclinic, P21(α0γ)0† 
Temperature (K)  100 
Wave vectors  q = 0.14216a* + 0.38390c* 
a, b, c (Å)  6.425 (2), 6.522 (2), 22.716 (6) 
β (°)  90.739 (11) 
V (Å^{3})  951.8 (5) 
Z  2 
Radiation type  Cu Kα 
µ (mm^{−}^{1})  1.22 
Crystal size (mm)  0.30 × 0.07 × 0.02 
Data collection  
Diffractometer  Bruker CCD diffractometer 
Absorption correction  Multiscan SADABS (Sheldrick, 2007) 
T_{min}, T_{max}  0.712, 0.976 
No. of measured, independent and observed [I > 3σ(I)] reflections  52466, 28265, 24813 
R_{int}  0.023 
(sin θ/λ)_{max} (Å^{−}^{1})  0.596 
Refinement  
R[F^{2} > 2σ(F^{2})], wR(F^{2}), S  0.036, 0.042, 2.38 
No. of reflections  28265 
No. of parameters  1396 
No. of restraints  2 
Hatom treatment  H atoms treated by a mixture of independent and constrained refinement 
Δρ_{max}, Δρ_{min} (e Å^{−}^{3})  0.27, −0.39 
† Symmetry operations: (1) x_{1}, x_{2}, x_{3}, x_{4}; (2) −x_{1}, x_{2}+1/2, −x_{3}, −x_{4}.
Computer programs: SMART V5.630 (Bruker AXS, 2003), Saint V7.36A (Bruker AXS, 2006), SUPERFLIP (Palatinus & Chapuis, 2007), Jana2006 (Petricek et al., 2006), PLATON (Spek, 2003).
Average  Minimum  Maximum  
O17—O18^{i}  2.564 (5)  2.538 (6)  2.588 (5) 
O17—H17  0.84 (5)  0.78 (5)  0.88 (5) 
O18—H17^{ii}  1.72 (5)  1.67 (5)  1.80 (5) 
O18^{i}—O17—H17  3(3)  1(3)  5(3) 
O17^{ii}—O18—H17^{ii}  1.2 (15)  0.3 (16)  2.7 (15) 
O17—H17—O18^{i}  176 (5)  172 (5)  179 (5) 
C16—C09—O08—Si01  −119.07 (5)  
C10—C09—O08—Si01  128.13 (5)  
C09—O08—Si01—C02  70.12 (6)  
C09—O08—Si01—C03  −51.62 (6)  
C09—O08—Si01—C04  −170.93 (6)  
C12—C13—C19—C20  −152.28 (7)  
C12—C13—C19—C24  31.60 (10)  
C14—C13—C19—C20  29.93 (10)  
C14—C13—C19—C24  −146.18 (7)  
C22—C23—N11—C15  25.00 (6)  
N11—C10—C09—C16  −26.37 (7)  
C10—N11—C15—C16  23.38 (6) 
Symmetry codes: (i) x_{1}, x_{2}+1, x_{3}, x_{4}; (ii) x_{1}, x_{2}−1, x_{3}, x_{4}. 
Footnotes
^{1}The q vector components α, β and γ have to be distinguished from the unitcell parameters α, β and γ. Alternative notations such as σ_{1}, σ_{2}, σ_{3} can be found in the literature (van Smaalen, 2007). In this manuscript, however, the nomenclature used in Vol. C of the International Tables of Crystallography (α, β and γ) will be followed (Janssen et al., 2006).
^{2}Supplementary data for this paper are available from the IUCr electronic archives (Reference: BK5084 ). Services for accessing these data are described at the back of the journal.
^{3}Please note that many other commercial and noncommercial software packages are also capable of handling diffraction patterns of modulated structures with satellite reflections.
^{4}The chemical synthesis of VAR205 started from naturally occurring hydroxyproline, the chiral carbon C09 remained unchanged in all synthetic steps. This was verified in one of the final cycles [Flack x = 0.04 (1)].
Acknowledgements
The authors thank Karl Baumann for providing the compound under study and Vaclav Petricek, Michal Dusek and Lukas Palatinus for very valuable discussions and assistance with the programs JANA2006 and SUPERFLIP. Ina Dix is acknowledged for the reconstruction of (Fig. 11) and Ekkehard Görlach for the calculations behind Figs. 14 and 16. Many thanks also to Carol Brock for her help and encouragement.
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