Electron density from structure factors

Because the electron density we seek is a complicated periodic function, it can be described as a Fourier series. Do the many structure-factor equations, each a sum of wave equations describing one reflection in the diffraction pattern, have any connection with the Fourier series that describes the electron density As mentioned earlier, each structure-factor equation can be written as a sum in which each term describes diffraction from one atom in the unit cell. But this is only one of many ways to write a structure-factor equation. Another way is to imagine dividing the electron density in the unit cell into many small volume elements by inserting planes parallel to the cell edges (Fig. 2.16). 

So each reflection is described by an equation like this, giving us a large number of equations describing reflections in terms of the electron density. Is there any way to solve these equations for the function p(x,y,z) in terms of the measured reflections After all, structure factors like Eq. (2.4) describe the reflections in terms of p(x,y,z), which is precisely the function the crystallographer is trying to learn. I will show in Chapter 5 that a mathematical operation called the Fourier transform solves the structure-factor equations for the desired function p(x,y,z), just as if they were a set of simultaneous equations describing p(x,y,z) in terms of the amplitudes, frequencies, and phases of the reflections. 

The Fourier transform describes precisely the mathematical relationship between an object and its diffraction pattern. In Figs. 2.7-2.10, the diffraction patterns are the Fourier transforms of the corresponding objects or arrays of objects. To put it another way, the Fourier transform is the lens-simulating operation that a computer performs to produce an image of molecules (or more precisely, of electron clouds) in the crystal. This view of p(x,y,z) as the Fourier transform of the structure factors implies that if we can measure three parameters— amplitude, frequency, and phase — of each reflection, then we can obtain the function p(x,y,z), graph the function, and see a fuzzy image of the molecules in the unit cell. 

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Putz, M. V. (2003). Electronic Density from Structure Factor Determination in Small Deformed Crystals./nf.J. QuantumChem. 94(4),222-231 (DOI 10.1002/qua.l0475). Birau, O., Putz, M. V. (2000). The Standing Waves Concept in Dynamical Theory of X-Ray Diffraction (in Romanian), Mirton Publishing House, Timisoara, Chapter 5. 

As anticipated, the multipolar model is not the only technique available to refine electron density from a set of measured X-ray diffracted intensities. Alternative methods are possible, for example the direct refinement of reduced density matrix elements [73, 74] or even a wave function constrained to X-ray structure factor (XRCW) [75, 76]. Of course, in all these models an increasing amount of physical information is used from theoretical chemistry methods and of course one should carefully consider how experimental is the information obtained. 

I will discuss the iterative improvement of phases and electron-density maps in Chapter 7. For now just take note that obtaining the final structure entails both calculating p(x,y,z) from structure factors and calculating structure factors from some preliminary form of p(x,y,z). Note further that when we compute structure factors from a known or assumed model, the results include the phases. In other words, the computed results give all the information needed for a "full-color" diffraction pattern, like that shown in Plate 3d, whereas experimentally obtained diffraction patterns lack the phases and are merely black and white, like Plate 3e. 

As I described earlier, this entails extracting the relatively simple diffraction signature of the heavy atom from the far more complicated diffraction pattern of the heavy-atom derivative, and then solving a simpler "structure," that of one heavy atom (or a few) in the unit cell of the protein. The most powerful tool in determining the heavy-atom coordinates is a Fourier series called the Pattersonfunction P(u,v,w), a variation on the Fourier series used to compute p(x,y,z) from structure factors. The coordinates (u,v,w) locate a point in a Patterson map, in the same way that coordinates (x,y,z) locate a point in an electron-density map. The Patterson function or Patterson synthesis is a Fourier series without phases. The amplitude of each term is the square of one structure factor, which is proportional to the measured reflection intensity. Thus we can construct this series from intensity measurements, even though we have no phase information. Here is the Patterson function in general form... 

Displaying an electron-density map and adjusting the models to improve its fit to the map (see Plate 21 and the cover of this book). SPV can display maps of several types (CCP4, X-PLOR, DN6). I am aware of no programs currently available for computation of maps from structure factors on personal computers, but I am sure this will soon change. 

Refinement also may be accomplished in real space by calculating difference Fourier syntheses using as coefficients A F = F i,s — Fcaic and phases caic derived from the trial structure. In general, if an atom has been incorrectly placed near its true location, a negative peak will appear at its assigned position and a positive peak will appear at its proper location. That is, in the difference Fourier we have subtracted electron density from where the atom isn t, and not subtracted density from where it is. The atom is then shifted by altering x, y, z, improved atomic parameters are included in a new round of structure factor calculations, and another difference Fourier computed. The process is repeated until the map is devoid of significant features. 

First, we need to know what is meant by a periodic function. The crystal contains a periodic arrangement - a regular array - of atoms but, as mentioned above, X-rays scatter electrons. Therefore it is more convenient to think about the crystal and thus the unit cell in terms of its electron density not fixy,z) where /describes the scattering factor of the atoms, but p(xy,z), where p(xy,z) is the electron density at point xy,z. As the atoms are periodically arranged, so also is their electron density p(xy,z) is a periodic function. We can therefore approximate it with a Fourier series just as above. If we know the electron density function, we can use a FT to calculate the individual coefficients Fbyy However this is completely useless the shape of the electron density, that is, the arrangement of the atoms in the crystal its structure - is precisely what we want to find out. In order to achieve this we need to do quite the opposite - calculate the electron density from the diffraction pattern. Before we consider how, we will try to build up a physical picture of what the FT of the electron density means. 

These maps are obtained after subtracting the calculated structure factors (FJ from the observed structure factors (Fo), an operation which is, in a first approximation, equivalent to subtracting the calculated electron density from the observed electron density. Features which are present in the observed density, but not in the calculated density will give peaks, while atoms present in the model (in the Fc), but not in the observed electron density will result in holes (Figure 30.12). These maps are frequently used to detect errors in the model and can also be used to obtain an unbiased electron density of a bound inhibitor, for example. 

Now, the problem of determining the crystal structure can be reahzed. As the structure amphtude Fhki can be derived from the measurement of intensity of X-ray reflection, the phase angle hki cannot be directly determined and if these phases of the structure factor are known, then the crystal structure is known as one can compute the electron density from (8.4) and hence the positions of the atoms giving rise to the measured electron densities. Therefore, the lack of knowledge of the phases of the structure factors prevents from directly computing the electron density map and hence determines the positions of the atoms. Patterson suggested as an aid the use of the following equation instead of (8.4) [1,6,7] ... 

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. 

Deviations from this generalization may have several sources, including charge repulsion, steric effects, statistical factors, intramolecular hydrogen bonding, and other structural effects that alter electron density at the reaction site. Hague - ° P has discussed these effects. 

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