Structure determination Fourier transform: electron density

A major application of QED is the accurate determination of crystal charge density. The scientific question here is how atoms bond to form crystals, which can be addressed by accurate measurement of crystal structure factors (Fourier transform of charge density) and from that to map electron distributions in crystals. 

Although it is possible to determine the complete electron density distribution using the Fourier transform of the observed structure factors, Eq. (1), the errors inherent in the structure factor amplitudes and, in the case of non-centrosymmetric structures, the errors in their phases introduce significant noise and bias into the result. Because of this, it has become normal practice to model the electron density by a series of pseudo-atoms consisting of a frozen, spherical core and an atom centered multipole expansion to represent the valence electron density [2,17]. 

Figure 4 Summary of the method for obtaining the structure of a bilayer from an oriented multilamellar sample. The diffraction pattern (shaded circles) is that expected from a multibilayer sample oriented perfectly parallel to the glass slide that is rotated in the x-ray beam. The intensities of the spots are proportional to square of the structure factors F h), which causes the phases (+1 or-1 in this case see text) to be lost. If the phases can be determined, then the electron density profile p z) can be determined by Fourier transformation, as shown. The correlations of the peaks and valleys of p(z) with the features of the bilayer are illustrated. This figure is based upon Fig. 1 of an article by White et al. (1986).

An electron density map can be obtained by a Fourier transform of the observed intensities and approximated phases, showing the electron density distribution around the atoms of a unit cell of the protein structure. An initial structure can be determined from this electron density map however, the maps are typically noisy and blurred, and evaluating such a map is a tedious process. 

Unlike the wave function, the electron density can be experimentally determined via X-ray diffraction because X-rays are scattered by electrons. A diffraction experiment yields an angular pattern of scattered X-ray beam intensities from which structure factors can be obtained after careful data processing. The structure factors F(H), where H are indices denoting a particular scattering direction, are the Fourier transform of the unit cell electron density. Therefore we can obtain p(r) experimentally via ... 

Affected by multiple scattering are, in particular, porous materials with high electron density (e.g., graphite, carbon fibers). The multiple scattering of isotropic two-phase materials is treated by Luzatti [81] based on the Fourier transform theory. Perret and Ruland [31,82] generalize his theory and describe how to quantify the effect. For the simple structural model of Debye and Bueche [17], Ruland and Tompa [83] compute the effect of the inevitable multiple scattering on determined structural parameters of the studied material. 

Fourier Transform NMR is very important for 13C NMR where the signals are very weak owing to the low natural abundance of 13C isotopes. Here computer accumulation of the responses, obtained after each RF pulse, gives spectra of sufficient quality in a relatively short time. The resolution of the NMR measurements is such that even very small changes in the position of the resonance line caused by the environment of the nuclei (electron density) can be determined with great accuracy. So a wealth of information on chemical and physical structure can be obtained from NMR. 

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 phenomenon of diffraction and its description as a Fourier transform (FT) is explained. The measured intensity of the diffracted X-rays related to the FT of the electron density, and the electron density - seen as an electron density map - is related to the (inverse) Fourier sum of the intensity of the diffracted X-rays. As we can only measure their intensity, we do not know the phases of the diffracted X-rays we have to determine them to solve the structure. Therefore, three principal methods are used, two experimental approaches (isomorphous replacement and anomalous scattering) and one based on known structures (molecular replacement). 

Crystal structure determination requires the development of a suitable model for periodic electron density distribution in the crystal that is related by Fourier transform to the structure factors that can be derived from the experimentally measured diffraction intensities. The model, in practice, is atomistic and consists of coordinates and atom types for all symmetry independent atoms as well as parameters that are used to describe the (mean square amplitude of) displacement of the atoms about their mean positions, which arises due to molecular motions and small variations in the mean position across the collection of unit cells that comprise the crystal. 

If there is one protein molecule per asymmetric unit, but it has some local symmetry such as a twofold axis of rotation relating two identical subunits, then use can be made of this information in order to solve the structure. For example, the orientation of the twofold rotation axis can probably be determined from the rotation function. In a similar way, the translation function can lead to information on the positions of the two related subunits. If the twofold axis can be located, then the electron density for the two symmetry related portions of the molecule can be averaged. An envelope is drawn in the electron-density map that essentially defines the edge of each molecule. The electron density in the two independent molecules is then averaged and the solvent region is flattened. By Fourier transformation a better set of phases is obtained for a new electron-density map, which may be more readily interpreted (63). 

The most usual way is effectively the same as is used for completion of crystal structures during structure solution (Section 10.7.6 andEq. 10.12). When a reasonable part of the structure has been described and some atoms positions have been defined, along with correct scattering factors and thermal parameters, a difference map can be calculated between this (incomplete) model and the Fourier transform of the experimental structure amplitudes. This map is a representation of the difference between the electron density of the model and that determined by the experiment. The biggest accumulations of electron density are assigned as new atoms until the structural model is complete. But even then there is residual electron density, which describes the difference between the true electron density and the model of a sum of independent spherical atom contributions, taking some simplified model of displacement into account There is also some random noise. 

Another way to represent the structure factor is shown in Eq. 18, where p(r) is the electron density of the atoms in the unit cell (r = the coordinates of each point in vector notation). As you may recall, this is in the form of a Fourier transform that is, the structure factor and electrOTi density are related to each other by Fourier verse Fourier transforms (Eq. 19). Accordingly, this relation is paramount for the determination of crystal stractures using X-ray diffraction analysis. That is, this equation enables one to prepare a 3-D electron density map for the entire unit cell, in which maxima represent the positions of individual atoms. ... 

Taking into account the mathematical properties of the Fourier transformation, the electron density p r), defining the high-resolution material structure (i.e., the atomic configuration), can be determined only if the complex function Aq) (modulus and phase) is known over a large volume in q space. On the other hand, if the amplitude Aq) is only determined within a rather small volume in q space around = 0, equation (8-2) exclusively yields the low-resolution features ofthe structure. 

Since the phase ofthe scattering amplitude A(q) cannot be determined, it seems useful to establish which is the function related to the structure and defined in the direct space that is obtained by Fourier transformation ofthe measured scattering intensity I T-The electron density p( ) can be written as the integral of an average density p plus its local deviations defined by Ap f), so as p f) = Pa + Ap r). Substituting this form for p r) in equation (8-1), the scattering amplitude becomes... 

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