From Diffraction Data to Electron Density

Chapter 5. From Diffraction Data to Electron Density... 

These corrected ionic radii are a much better approximation to the experimental values derived from X-ray diffraction data and electron density contour maps, such as the one shown in Figure 12.6 for TiC. The experimentally determined values, defined by where the electron density of one ion ends in the contour map. 

Lastly, it should be pointed out that there are often significant differences between H positions determined by X-ray analysis and those determined from neutron data. Neutron diffraction provides true nuclear positions, whereas X-ray diffraction measures the electron density distribution. Thus, X-ray Fourier maps often give H peaks that, because of the perturbing influence of the M-H bonding electrons, appear closer to the M atoms than they really are A thorough analysis of this effect has... 

How is it possible to derive phase information when only structure amplitudes have been measured An answer can be found in what are called direct methods of structure determination. By these methods the crys-tallographer estimates the relative phase angles directly from the values of F hkl) (the experimental data). An electron-density map is calculated with the phases so derived, and the atomic arrangement is searched for in the map that results. This is why the method is titled direct. Other methods of relative phase determination rely on the computation of phase angles after the atoms in a trial structure have been found, and therefore they may be considered indirect methods. Thus, the argument that phase information is lost in the diffraction process is not totally correct. The phase problem therefore lies in finding methods for extracting the correct phase information from the experimental data. 

Accurate experimental electronic properties can now be obtained in only one day with synchrotron radiation and a charge-coupled device area detection technique. Recently spectacular electron densities were acquired on DL-proline monohydrate at 100 K.130 The accuracy of the data is comparable or even superior to the accuracy obtained from a 6-week experiment on DL-aspartic acid with conventional X-ray diffraction methods. A data acquisition time of one day is comparable to the time needed for an ab initio calculation on isolated molecules. This technique renders larger molecular systems of biological importance accessible to electron density experiments. The impact of the rapid collection of accurate data should not be underestimated. Indeed, if properly allocated a dedicated synchrotron source could now routinely produce accurate experimental densities at a dramatically increased rate. 

The discussion of this technique has been kept short because the diffraction spectra are very similar to Debye - Scherrer patterns the method is becoming very important for molecular structure analysis. An electron beam in a high vacuum (0.1-10 Pa) collides with a molecular beam, and the electrons are diffracted by the molecules. The film reveals washed-out Debye-Scherrer rings, from which a radial electron density distribution function for the molecule can be derived. Together with spectroscopic data, the distribution makes it possible to infer the molecular structure 129]. 

Some of the powerful biophysical techniques of structure determination can be and have been applied to fluid phase bilayers. Wiener and White published a series of papers in the early 1990s on a method to combine neutron and X-ray diffraction data to arrive at one of the most detailed pictures of a fluid phase bilayer determined by experiments thus far. Unfortunately, the degree of hydration of this system was much lower than what is biologically interesting and what is used in most simulations. Recently the structure of a fully hydrated DPPC bilayer was determined from X-ray methods. This study will provide a stringent test of simulations of DPPC. It provides accurate values for the area per lipid (62.9 1.3 A), bilayer form factors which can be obtained from electron density profiles, volume per lipid, and water stochiometry. 

The comparison with experiment can be made at several levels. The first, and most common, is in the comparison of derived quantities that are not directly measurable, for example, a set of average crystal coordinates or a diffusion constant. A comparison at this level is convenient in that the quantities involved describe directly the structure and dynamics of the system. However, the obtainment of these quantities, from experiment and/or simulation, may require approximation and model-dependent data analysis. For example, to obtain experimentally a set of average crystallographic coordinates, a physical model to interpret an electron density map must be imposed. To avoid these problems the comparison can be made at the level of the measured quantities themselves, such as diffraction intensities or dynamic structure factors. A comparison at this level still involves some approximation. For example, background corrections have to made in the experimental data reduction. However, fewer approximations are necessary for the structure and dynamics of the sample itself, and comparison with experiment is normally more direct. This approach requires a little more work on the part of the computer simulation team, because methods for calculating experimental intensities from simulation configurations must be developed. The comparisons made here are of experimentally measurable quantities. 

The amplitudes and the phases of the diffraction data from the protein crystals are used to calculate an electron-density map of the repeating unit of the crystal. This map then has to be interpreted as a polypeptide chain with a particular amino acid sequence. The interpretation of the electron-density map is complicated by several limitations of the data. First of all, the map itself contains errors, mainly due to errors in the phase angles. In addition, the quality of the map depends on the resolution of the diffraction data, which in turn depends on how well-ordered the crystals are. This directly influences the image that can be produced. The resolution is measured in A... 

The CK" ion can act either as a monodentate or bidentate ligand. Because of the similarity of electron density at C and N it is not usually possible to decide from X-ray data whether C or N is the donor atom in monodentate complexes, but in those cases where the matter has been established by neutron diffraction C is always found to be the donor atom (as with CO). Very frequently CK acts as a bridging ligand - CN- as in AgCN, and AuCN (both of which are infinite linear chain polymers), and in Prussian-blue type compounds (p. 1094). The same tendency for a coordinated M CN group to form a further donor-aceeptor bond using the lone-pair of electrons on the N atom is illustrated by the mononuclear BF3 complexes... 

The shortest cation-anion distance in an ionic compound corresponds to the sum of the ionic radii. This distance can be determined experimentally. However, there is no straightforward way to obtain values for the radii themselves. Data taken from carefully performed X-ray diffraction experiments allow the calculation of the electron density in the crystal the point having the minimum electron density along the connection line between a cation and an adjacent anion can be taken as the contact point of the ions. As shown in the example of sodium fluoride in Fig. 6.1, the ions in the crystal show certain deviations from spherical shape, i.e. the electron shell is polarized. This indicates the presence of some degree of covalent bonding, which can be interpreted as a partial backflow of electron density from the anion to the cation. The electron density minimum therefore does not necessarily represent the ideal place for the limit between cation and anion. 

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