Centrosymmetric projections

The true phases of the structure factors will, in general, be different from the phases calculated with the independent-atom model. In centrosymmetric structures, with phases restricted to 0 or n, only very few weak reflections are affected. In acentric structures, only the reflections of centrosymmetric projections, such as the hkO, hOl, and Okl reflections in the space group P212,21, are invariant. 

The effect of the neglect of Eq. (5.30) can be quite large. For the peptide N-acetyl-a,/ -dehydrophenylalanine methylamide (space group Cc), for example, the underestimate of the density is reported to be as large as 0.19 eA. In /V-acetyl-i.-tryptophan methylamide, which crystallizes in the space group F2,2121 with centrosymmetric projections, the features are somewhat smaller (Fig. 5.14), but still important. 

Similarly, a twofold axis parallel to 6, since it fixes pairs of atoms with the same y coordinate, should cause the OkO reflections to be, on the average, twice as strong as the general hkl reflections. The number of reflections along a central i qw of the reciprocal lattice may not, however, be large enough to make the statistical method applicable. A twofold axis also makes a pro jection along this axis a centrosymmetric projection. 

If two isomorphous derivatives in which the replaceable atoms have considerably different diffracting powers are available, an improved form of this method can be used. Strychnine sulphate pentahydrate and the corresponding selenate form isomorphous monoclinic crystals of space group C2 Bokhoven, Schoone, and Bijvoet (1951) solved the centrosymmetric b projection by the straightforward method first used... 

A more complex case of molecular overcrowding of the benzophenanthrene type is found in tetrabenzonaphthalene (86). The crystal structure of this molecule has been studied in its most favourable projection by Herbstein and Schmidt (1954b). The results of this partial analysis rule out a planar centrosymmetric structure and favour a molecule having 222 molecular symmetry with distortions of the benzene rings, as shown in (86), comparable to those found in 3,4-benzophenanthrene no molecular symmetry is required by the crystal. 

Figure 11 Diagrammatic representation of the ellipsoidal clathrate compound (10)4 (chloroform) projected in the ab plane. The black and white coding indicates the host enantiomers. Diol molecules are represented as two oxygen atoms (solid spheres) joined by their carbocyclic framework (solid rod). Dashed lines represent the hydroxy hydrogen bonds. The three 4 screw axes involve only one enantiomer of 10 (surrounded by white rods), and the three 43 screw axes the other (black rods). Each ellipsoidal clathrate cage site is, however, centrosymmetric.

Note that even though the crystal structure is centrosymmetric in projection view down a), in this space group the origin is, by convention, chosen so that the relative phase angle a 0kl) equals 0, 90, 180 or 270°. 

Figure 2.56. The structure amplitude, F(h), shown as a vector representing a complex number with its real, A(h), and imaginary, B(h), components as projections on the real and imaginary axes, respectively, in the non-centrosymmetric (left) and centrosymmetric (right) structures. The imaginary component on the left is shifted from the origin of coordinates for clarity.

Such observations immediately raise the question how reliable are projections of crystal-field effects onto multipoles The analyses of wavefunction-simulated X-ray data of small model compounds have revealed that the interaction density (8p = p(crystal) — p(isolated molecule)) manifests itself in low-order structure factors, and only to an extent that is comparable with the experimental noise [80]. Nevertheless, the multipole refinement was shown to retrieve this low signal (about 1% in F) successfully. A related study on urea, however, demonstrated that this is not the case if random errors of magnitude comparable with the effect of interaction density are added to the theoretical data [81]. The result also implies that indeterminacies associated with the interpretation of non-centrosymmetric structures can severely limit the pseudoatom model in distinguishing between noise and physical effects [82, 83]. 

Figure 1 The centrosymmetric ite is shown in projection on (001) in the (theob-romine)2H2l8 structure. The best plane has been calculated through the central Ij—13—13—12 unit and the deviations of all the iodine atoms from this plane are given in units of 10 A.

The structure of (theobromine)2H2l8 has revealed that this is a polyiodide salt containing protonated theobromine.The anion It6 is the largest discrete polyiodide ion that has been characterized until now. A projection of this centrosymmetric anion is shown in Figure 1 it is evident that the ion is nearly planar and is best described as being I3——1 3 —13—12—13. 

As has already been emphasized, only x-ray analysis could unequivocally establish the structure of the plro-heterocycle (58) (Table 9) <80CJC1645> as (2i ,3a 7 ,57 ,6a i )3,3a, 4,4 -tetraacetyl-5-hydroxy-5,5, 6 -trimethylspiro[furan-2(5F/),2 (3 /7)-furo(2,3-ft)furan]. Automatic centrosymmetric direct methods were applied to solve the structure. The study clearly delineates the spiran system (58) between a furan ring and a c -fused furo[2,3-ft]furan system. The study further reveals that in (58) the rings A and C are essentially planar, whereas obvious puckering in ring B is present with the 3 -carbon atom being projected (0.50 A) out of the plane of the four other atoms. 

It is actually possible to answer this question by the inspection of the projection p z) of the electron density along the normal to the layers which we call electron density profile in the following [28]. Indeed, the number of smectic reflections is related to p(z). p z) can be expanded in a Fourier series and, since it is a centrosymmetrical function, only cosine terms are relevant ... 

Fig. 2. Electron density of the centrosymmetrical isomer of 1,2,3,4-tetraphenylcyclobutane projected down the 5.77 k b axis (top) and 17.02 A a axis (bottom), showing the asymmetric unit in both cases (from [14]). In both maps, contours are drawn at intervals of approximately one electron per A. The interpretation of the bottom map is indicated.

Figure 8.49 Projection along the b axis showing extensive hydrogen bonding interactions around the centrosymmetric rhodizonate dianion in the host lattice of [(n-C4Hg)4N ]2C606 2(m-OHC6H4NHCONH2)-2H2O (24). Symmetry transformation a a - X, i - y, 1 - z)...

Figure 1. Pyrophyllite, reference model of dioctahedral smectites centrosymmetrical layer, (a) Projection on the biperiodic plane of the lattice (ab). (b) Projection on a plane perpendicular to the b axis. Only those atoms situated between the planes x and x of the projection (a) are represented. Arrows A indicate the centers of the hexagonal cavities of the surface of the layer. Arrows M and B indicate eventual localization of negative charges created by isomorphous replacements. Af—octahedral charges (montmorillonite) tetrahedral charges (beidellite).

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