Friedel Opposites Research Articles

Ethyl (3S)-3-[(3aR,5R,6S,6aR)-6-hy-droxy-2,2-di-methyl-tetra-hydro-furo[4,5-d][1,3]dioxol-5-yl]-3-{(3S)-3-[(3aR,5R,6S,6aR)-6-hy-droxy-2,2-di-methyl-tetra-hydro-furo[4,5-d][1,3]dioxol-5-yl]-5-oxoisoxazolidin-2-yl}propano-ate chloro-form monosolvate.

The title compound, C22H33NO12·CHCl3, was obtained as a product of a double aza-Michael addition of hydroxyl-amine on a Chiron with a known absolute configuration. The enanti-opure compound crystallized as a chloro-form solvate, in space group P1, and diffraction data were collected at room temperature with Ag Kα radiation. The Flack parameter refined to x = -0.01 (16); however, the Flack and Watkin 2AD plot clearly shows that differences between Friedel opposites (the D component of the plot) do not carry any reliable information about resonant scattering of Cl atoms, and are rather dominated by random and systematic errors. The RD factor calculated using 1941 acentric Friedel pairs is RD = 0.995. On the other hand, the 2A component of the plot, related to average intensities of Friedel pairs, shows that data are of good quality (RA = 0.069). This example illustrates that while using Ag Kα radiation (λ = 0.56083 Å), scatterers heavier than Cl should be present in a chiral crystal in order to determine confidently the absolute structure of the crystal.

An introduction to experimental phasing of macromolecules illustrated by SHELX; new autotracing features.

For the purpose of this article, experimental phasing is understood to mean the determination of macromolecular structures by exploiting small intensity differences of Friedel opposites and possibly of reflections measured at different wavelengths or for heavy-atom derivatives, without the use of specific structural models. The SHELX programs provide a robust and efficient route for routine structure solution by the SAD, MAD and related methods, but involve a number of simplifying assumptions that may limit their applicability in borderline cases. The substructure atoms (i.e. those with significant anomalous scattering) are first located by direct methods, and the experimental data are then used to estimate phase shifts that are added to the substructure phases to obtain starting phases for the native reflections. These are then improved by density modification and, if the resolution of the data and the type of structure permit, polyalanine tracing. A number of extensions to the tracing algorithm are discussed; these are designed to improve its performance at low resolution. Given native data to 2.5 Å resolution or better, a correlation coefficient greater than 25% between the structure factors calculated from such a trace and the native data is usually a good indication that the structure has been solved.

TiGePt – a study of Friedel differences

The X-ray single-crystal diffraction intensities of the intermetallic compound TiGePt were analysed. These showed beyond doubt that the crystal structure is non-centrosymmetric. The analysis revolves around the resonant-scattering contribution to differences in intensity between Friedel opposites hkl and \bar h\bar k\bar l. The following techniques were used: R(merge) factors on the average (A) and difference (D) of Friedel opposites; statistical estimates of the resonant-scattering contribution to Friedel opposites; plots of 2A(obs) against 2A(model) and of D(obs) against D(model); the antisymmetric D-Patterson function. Moreover it was possible to show that a non-standard atomic model was unnecessary to describe TiGePt. Two data sets are compared. That measured with Ag Kα radiation at 295 K to a resolution of 1.25 Å(-1) is less conclusive than the one measured with Mo Kα radiation at 100 K to the lower resolution of 0.93 Å(-1). This result is probably due to the fact that the resonant scattering of Pt is larger for Mo Kα than for AgKα radiation.

Absolute-structure reports

All the 139 noncentrosymmetric crystal structures published in Acta Crystallographica Section C between January 2011 and November 2012 inclusive have been used as the basis of a detailed study of the reporting of absolute structure. These structure determinations cover a wide range of space groups, chemical composition and resonant-scattering contribution. Defining A and D as the average and difference of the intensities of Friedel opposites, their level of fit has been examined using 2AD and selected-D plots. It was found, regardless of the expected resonant-scattering contribution to Friedel opposites, that the Friedel-difference intensities are often dominated by random uncertainty and systematic error. An analysis of data collection strategy is provided. It is found that crystal-structure determinations resulting in a Flack parameter close to 0.5 may not necessarily be from crystals twinned by inversion. Friedifstat is shown to be a robust estimator of the resonant-scattering contribution to Friedel opposites, very little affected by the particular space group of a structure nor by the occupation of special positions. There is considerable confusion in the text of papers presenting achiral noncentrosymmetric crystal structures. Recommendations are provided for the optimal way of treating noncentrosymmetric crystal structures for which the experimenter has no interest in determining the absolute structure.

Analysing Friedel averages and differences

Various practical applications of the average (A) and difference (D) of Friedel opposites are described. Techniques based on the resonant-scattering contribution to Friedel differences are applied to see whether a crystal is centrosymmetric or not, and to determine the point group of the crystal. For the validation of a structural study, plots of A(obs) against A(model), and D(obs) against D(model) are used extensively. Moreover, it is useful to display both plots on the same graph. Intensity measurements on a crystal of NaClO(3) were made at three different speeds, with two different radiations and two different diffractometers, and treated with two different software packages and four different absorption corrections. The evaluation of these numerous data sets reveals underlying deficiencies. For comparison, plots of A(obs) against A(model), and D(obs) against D(model) are presented for two centrosymmetric crystals.

Crystal structure of 3,4-bis-(acetyloxy)-N-phenyl-succinimide, C14H13NO6

C14H13NO6 ,h exagonal, P62 (no. 171), a =1 0.085(1) A, c =1 1.677(2) A,V =1 028.5 A 3 , Z =3 , R gt (F) =0 .0504, wRref(F 2 ) =0 .1386, T =2 98 K. Source of material Aniline (0.9 3g ,0 .0 1m ol) was added dropwise to as olution of dihydro-3,4-bis-(acetyloxy)-2,5-furandione (2.16 g, 0.01 mol) in THF. After the mixture was stirred at room temperature for 30 min, N-hydroxysuccinimide (1.73 g, 15 mmol) was added to the mixture and then the solution of N,N'-dicyclohexylcarbodiimide (DCC) (2.06 g, 0.01 mol) in THF was added slowly to the mixture. The reaction mixture was stirred for 19 ha tr oom temperature. The insoluble material was filtered off. Th es olvent was removed under reduced pressure to get ac olourless oil. The target compound was washed with NaHCO3 an dc rystallize df ro me thano l( 2.36 g, yield 81%, m.p. 111-112°C). Colourless blockshaped single crystals of the title compound suitable for X-ray diffractio na nalysis precipitated after several days. Experimental details Ha toms were positioned geometrically and refined with ar iding model using SHELXL-97 default values (Uiso(H) =1 .2 Ueq(C) for CH groups and Uiso(H) =1 .5 Ueq(C) for CH3). Refinement with all data (Friedel opposites not merged) led to an unsuitably large error of the Flack parameter. Th ef inal refinement was therefore performe dw ith ad ata set with merged Friedel pairs, hence the calculated Flack parameter is meaningless.

Response to Olczak's comment onProbability density functions of the average and difference intensities of Friedel opposites

A response is given to to Olczak's comment about the discussion of probability density functions (p.d.f.s) and simulations of Friedel intensity differences in Shmueli & Flack [Acta Cryst. (2010), A66, 669–675].

Comment onProbability density functions of the average and difference intensities of Friedel oppositesby Shmueli & Flack (2010)

A comment is made on the discussion of probability density functions (p.d.f.s) and simulations of Friedel intensity differences in Shmueli & Flack [Acta Cryst. (2010), A66, 669–675].

Practical applications of averages and differences of Friedel opposites

The practical use of the average and difference intensities of Friedel opposites at different stages of structure analysis has been investigated. It is shown how these values may be properly and practically used at the stage of space-group determination. At the stage of least-squares refinement, it is shown that increasing the weight of the difference intensities does not improve their fit to the model. The correct form of the coefficients for a difference electron-density calculation is given. In the process of structure validation, it is further shown that plots of the observed and model difference intensities provide an objective method to evaluate the fit of the data to the model and to reveal insufficiencies in the intensity measurements. As a further tool for the validation of structure determinations, the use of the Patterson functions of the average and difference intensities has been investigated and their clear advantage demonstrated.

Probability density functions of the average and difference intensities of Friedel opposites

Trigonometric series for the average (A) and difference (D) intensities of Friedel opposites were carefully rederived and were normalized to minimize their dependence on sin(theta)/lambda. Probability density functions (hereafter p.d.f.s) of these series were then derived by the Fourier method [Shmueli, Weiss, Kiefer & Wilson (1984). Acta Cryst. A40, 651-660] and their expressions, which admit any chemical composition of the unit-cell contents, were obtained for the space group P1. Histograms of A and D were then calculated for an assumed random-structure model and for 3135 Friedel pairs of a published solved crystal structure, and were compared with the p.d.f.s after the latter were scaled up to the histograms. Good agreement was obtained for the random-structure model and a qualitative one for the published solved structure. The results indicate that the residual discrepancy is mainly due to the presumed statistical independence of the p.d.f.'s characteristic function on the contributions of the interatomic vectors.

Intensity statistics of Friedel opposites and classification of reflections

Intensity statistics of Friedel opposites and classification of reflections

Concise intensity statistics of Friedel opposites and classification of the reflections

A previous extensive analysis of the mean-square intensity difference of Friedel opposites [Shmueli et al. (2008). Acta Cryst. A64, 476-483] is here concisely re-examined and confirmed by purely statistical methods. The analysis applies to noncentrosymmetric crystals only. For special reflections and centered lattices both mean-square intensity difference and average intensity of Friedel opposites depend on the centering factor of the crystal lattice and/or on the isotropy subgroup of the reflection. A complete classification of the reflections, based on the above intensity statistics, is presented. It is also shown that the experimentally important Bijvoet ratio is found to depend only on the chemical composition of the unit-cell content and the wavelength of the radiation.

Intensity statistics of Friedel opposites

A previous analysis of the average intensity and mean-square intensity difference of Friedel opposites, confined to the space group P1 [Flack & Shmueli (2007). Acta Cryst. A63, 257-265], is here extended to all the non-centrosymmetric space groups. The present analysis presumes purely non-centrosymmetric content of the unit cell. An important result of this study is that the average intensity and mean-square intensity difference of Friedel opposites have the same values for all the non-centrosymmetric space groups as those previously obtained for the triclinic space group P1. The ratios of average intensity and root-mean-square intensity difference to their triclinic equivalents were derived and exemplified for general as well as for special reflections. For the latter, enhancements were obtained which are shown to be due to those of average intensity and not to a mechanism related to Friedel opposites being explicitly considered.

The mean-square Friedel intensity difference inP1 with a centrosymmetric substructure

For non-centrosymmetric structures in space group P1 containing a centrosymmetric substructure, analytical expressions have been obtained for various functions of the diffraction intensity of Friedel opposites. These functions are the average intensity of Friedel opposites, the mean difference in intensity of Friedel opposites and the mean-square difference in intensity of Friedel opposites. A Bijvoet intensity ratio is defined for the evaluation of resonant-scattering effects in non-centrosymmetric and pseudo-centrosymmetric structures. Analysis of these expressions confirms that both resonant and non-resonant atoms are necessary to produce differences in intensity between Friedel opposites and also shows that in some circumstances atoms may lie on a centrosymmetric substructure without diminishing the Bijvoet intensity ratio. The effects of the real component of resonant scattering, of the variation of the scattering factors with sin theta/lambda, of isotropic atomic displacement parameters, of a crystal twinned by inversion, of atoms in special positions and of weak reflections are considered. Software is available for the evaluation of the Bijvoet intensity ratio.

Structure of isopropylidene sedoheptulosan

The title compound, 2,7-anhydro-4,5-O-isopropylidene-β-D-altro-2-heptulopyranose, C 10 H 16 O 6 , has a pyranoid conformation which is a distorted E 0 conformation with Q=0.609 (1) A, θ=147.4 (1) o and Φ=188.1 (3) o [Cremer & Pople (1975). J. Am. Chem. Soc. 97, 1354- 1358]. The structure is stabilized by hydrogen bonds. The absolute structure, determined through the Flack parameter, x=-0.08 (14) [Flack (1983). Acta Cryst. A39, 876-881], corresponds to the known chirality of the title compound (Friedel opposites collected and not merged)

Symmetry and the generalized maximum determinant rule

In the general case of a Karle-Hauptman matrix containing no symmetry equivalent reflections, maximizing the determinant as a function of the phases does not necessarily lead to an unambiguous solution of the phase problem. Individual phases may be shifted from their correct values in a seemingly completely arbitrary way. This problem is discussed in simple mathematical terms and a method is proposed allowing the identification of those elements in a Karle-Hauptman matrix possibly suffering from the effects discussed, given the space-group symmetry and the composition of the matrix. The conclusion reached in this paper is that only the presence of a sufficiently large number of symmetry equivalent reflections and/or Friedel opposites in a Karle-Hauptman matrix causes the Generalized Maximum Determinant Rule to be an effective tool in ab initio phase determination

Symmetry and the generalized maximum determinant rule

Abstract In the general case of a Karle-Hauptman matrix containing no symmetry equivalent reflections, maximizing the determinant as a function of the phases does not necessarily lead to an unambiguous solution of the phase problem. Individual phases may be shifted from their correct values in a seemingly completely arbitrary way. This problem is discussed in simple mathematical terms and a method is proposed allowing the identification of those elements in a Karle-Hauptman matrix possibly suffering from the effects discussed, given the space-group symmetry and the composition of the matrix. The conclusion reached in this paper is that only the presence of a sufficiently large number of symmetry equivalent reflections and/or Friedel opposites in a Karle-Hauptman matrix causes the Generalized Maximum Determinant Rule to be an effective tool in ab initio phase determination.

The determination of absolute structure. II. Absolute configurations from the Cambridge Crystallographic Data Centre files for 1982

A survey of 102 structures flagged with 'absolute configuration' in the Cambridge Crystallographic Data Centre files for 1982 reveals many unsatisfactory features in the original publications. These are associated with data collection (lack of Friedel opposites and of absorption corrections), misinterpretation of the term 'absolute configuration' (claimed for achiral molecules and space groups), methods used to determine absolute configuration (over-optimistic interpretation of discriminatory tests) and presentation of results (little experimental detail, no prominent statement that an absolute configuration was assigned, no clear description of the methods used). Some confusion arises because of unintentional flagging of absolute configurations determined by solely chemical means.