Difference Fourier electron density maps

X-Ray diffraction from single crystals is the most direct and powerful experimental tool available to determine molecular structures and intermolecular interactions at atomic resolution. Monochromatic CuKa radiation of wavelength (X) 1.5418 A is commonly used to collect the X-ray intensities diffracted by the electrons in the crystal. The structure amplitudes, whose squares are the intensities of the reflections, coupled with their appropriate phases, are the basic ingredients to locate atomic positions. Because phases cannot be experimentally recorded, the phase problem has to be resolved by one of the well-known techniques the heavy-atom method, the direct method, anomalous dispersion, and isomorphous replacement.1 Once approximate phases of some strong reflections are obtained, the electron-density maps computed by Fourier summation, which requires both amplitudes and phases, lead to a partial solution of the crystal structure. Phases based on this initial structure can be used to include previously omitted reflections so that in a couple of trials, the entire structure is traced at a high resolution. Difference Fourier maps at this stage are helpful to locate ions and solvent molecules. Subsequent refinement of the crystal structure by well-known least-squares methods ensures reliable atomic coordinates and thermal parameters. 

For trypanosomal TIM we experimented with three different cocktails of 32 compounds (Table 4). Molecules were chosen in such a way that they would be compatible, soluble, cheap, and as varied as possible. Each compound was present at a concentration of 1 m M The final cocktail solutions were clear and devoid of precipitate. Since this was a pilot experiment both subcocktails were checked at each stage of the dichotomic strategy. Only the soak with cocktail 1 revealed electron density that could not be accounted for by water molecules, hereafter called peak X. The soaks with cocktails 2 and 3 led to featureless difference Fourier maps. The quality of the data and refinement can be inspected from Table 5, while Figure 9 illustrates the dichotomic search to identify peak X. An oxidized molecule of DTT, identified in the high-resolution structure of the native TIM crystals [24], served as an internal reference to judge the quality of the data and the noise level in the final difference Fourier maps. 

As a result of the recognized role of transition metal hydrides as l reactive intermediates or catalysts in a broad spectrum of chemical reactions such as hydroformylation, olefin isomerization, and hydrogenation, transition metal hydride chemistry has developed rapidly in the past decade (J). Despite the increased interest in this area, detailed structural information about the nature of hydrogen bonding to transition metals has been rather limited. This paucity of information primarily arises since, until recently, x-ray diffraction has been used mainly to determine hydrogen positions either indirectly from stereochemical considerations of the ligand disposition about the metals or directly from weak peaks of electron density in difference Fourier maps. The inherent limi-... 

P(3) bond distance. The largest residual electron-density peak in the final difference Fourier map is at that position and a hydrogen atom H(l) can be placed there with some confidence. 

The redistribution of the valence electron density due to chemical bonding may be obtained from summing the multipole populations or Fourier transforming appropriately calculated structure factors, having removed the contribution from neutral spherical atoms, to produce a so-called deformation density map [2], This function was introduced by Roux et al. [23] and has been widely used since then. The deformation electron density represents the difference between the electron density of the system, p(r), and the electron... 

I was characterized by powder X-ray diffraction (PXRD), energy dispersive analysis of X-rays (EDAX), chemical analysis, thermogravimetric analysis (TGA) and IR spectroscopy. EDAX analysis indicated the ratio of Mn S to be 3 2. The presence of fluorine was confirmed by analysis and the percentage of fluorine estimated by EDAX in a field emission scanning electron microscope was also satisfactory. Thermogravimetric analysis also confirms the stoichiometry of the compound. Bond valence sum calculations6 and the absence of electron density near fluorine in the difference Fourier map also provide evidence for the presence of fluorine. The sulfate content was found to be 30.8% compared to the expected 32% on the basis of the formula. 

Figure 4. Difference-Fourier maps of electron density associated with non-framework atoms in the precursor to fluoride-silicalite (a) section in mirror plane at y 0.75 (b)...

In the X-ray analysis of a protein crystal structure, solvent molecules appear as spheres of electron density in difference Fourier maps calculated at the end of a refinement. In a strict sense, the electron density map exhibits preferred.s/tes of hydration which are occupied by freely interchanging solvent molecules. This electron density is well defined for the tightly bound solvent molecules and can be as spurious as just above background for ill-defined molecules which exhibit large temperature factors and/or only partly occupied atomic positions. Since these two parameters are correlated in least-squares refinement, this gives rise to methodological problems. 

The determination of the atomic structure of a reconstruction requires the quantitative measurement of as many allowed reflections as possible. Given the structure factors, standard Fourier methods of crystallography, such as Patterson function or electron-density difference function, are used. The experimental Patterson function is the Fourier transform of the experimental intensities, which is directly the electron density-density autocorrelation function within the unit cell. Practically, a peak in the Patterson map means that the vector joining the origin to this peak is an interatomic vector of the atomic structure. Different techniques may be combined to analyse the Patterson map. On the basis of a set of interatomic vectors obtained from the Patterson map, a trial structure can be derived and model stracture factor amplitudes calculated and compared with experiment. This is in general followed by a least-squares minimisation of the difference between the calculated and measured structure factors. Of help in the process of structure determination may be the difference Fourier map, which is... 

The average remaining electron density in ZnAPO-34 sample was close to zero, indicating that zinc atoms were incorporated into the framework. The difference Fourier map of as-synthesised MnZnAPO-34 showed some positive diffuse electron density remaining in the cavity, that prevented the determination of the position of a manganese atom, which is probably situated at several positions in the cavity 7 and is most likely responsible for the observed electron density. Also in the case of calcined sample, where the cavities were free of template, manganese could not be located from the positive electron density in the cavities due to low crystallinity of the sample which caused low resolution of the Fourier maps. Low crystallinity was also the reason for non-stable refinement of calcined samples and that is only unit cell parameters are given for comparison (Table 1). Unit cells of calcined samples are smaller compared to as-synthesised ones, as expected. 

A difference Fourier map, calculated at this point, reveals an additional small electron density maximum in the tetrahedral cavity next to the partially occupied V2. Thus, it is reasonable to assume that the V2 site splits into two independent partially occupied positions with the coordinates, which distribute V atoms in a random fashion in two adjacent tetrahedral positions rather than being simply vanadium-deficient. We label these two sites as V2a (corresponding to the former V2) and V2b (corresponding to the Fourier peak). Refinement of this model slightly improves the fit. Subsequently, additional profile parameters (F, F , and sample displacement) were included in the refinement, followed by a typical procedure of refining the porosity in the Suortti approximation with fixed atomic coordinates and Ui o, and then fixing the porosity parameters for the remainder of the refinement. 

Simulated annealing refinement is usually unable to correct very large errors in the atomic model or to correct for missing parts of the structure. The atomic model needs to be corrected by inspection of a difference Fourier map. In order to improve the quality and resolution of the difference map, the observed phases are often replaced or combined with calculated phases, as soon as an initial atomic model has been built. These combined electron density maps are then used to improve and to refine the atomic model. The inclusion of calculated phase information brings with it the danger of biasing the refinement process towards the current atomic model. This model bias can obscure the detection of errors in atomic models if sufficient experimental phase information is unavailable. In fact during the past decade several cases of incorrect or partly incorrect atomic models have been reported where model bias may have played a role [67]. 

ELECTRON DENSITY, REFINEMENT, AND DIFFERENCE FOURIER MAPS... 

The difference Fourier map calculated from this equation should be near zero in value and featureless where the model corresponds to the correct structure, as the two electron densities would essentially subtract to zero (assuming F0bs and Fcaic are scaled properly to one another). Where they differ, however, is where we would expect features to appear. At those places where the model contains atoms, and therefore electron density, but the true structure does not, then p(x, y, z)expt —p(x, y, z)modei will be negative. Negative density will appear at that location in the difference Fourier map. If the model lacks atoms, hence electron density, at places occupied by atoms in the true structure, the p(x, y, z)eXpt -p(x, y, z,)model will produce positive density. 

FIGURE 10.11 Here is a two-dimensional representation of what one would likely see in an Fo — Fc difference Fourier map, where the Fc were calculated from a model that includes an arginine side chain, shown here, which was misplaced. A region of negative difference electron density (dashed contour lines) would superimpose upon the position incorrectly occupied by the side chain, while the positive density (solid contour lines) would indicate the location to which it should be shifted. 

Figure 3.1.4 Proton migration upon increasing temperature in the 0-H---0 hydrogen bonds between carboxyl groups in 2,4,6-trimethyl benzoic acid. X-ray difference Fourier maps showing the electron density associated with the carboxyl protons at 100 (a), 1 70 (b), 240 (c) and 290 K (d). Figure courtesy of Prof. Chick C. Wilson and Dr. Andrew Parkin, University of Glasgow.

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