The electron density in a crystal

The link with the reciprocal lattice treatment of the same problem is given by the following expression, which is Bragg s law in vector form (again, recall Figs 5.10 and 5.13)  

Since Ohki = arc sin(A,/(2 rfhki)). in order to And the angle 0 for any Bragg reflection one needs to And the interplanar spacing for any triplet of Miller indices, either in direct or in reciprocal space. Working out the direction of reciprocal space vectors is easy in orthogonal coordinates but becomes more and more tedious for monoclinic and triclinic crystals [5]. 

It can be shown, at the price of a short course in Fourier analysis, that the structure factors are the coefficients of the Fourier synthesis of the electron density in the cell. We will not give the derivation here, and the reader should be satisfied that the 

The use of this equation obviously requires a knowledge of the phases of the structure factors. Once the phase problem is solved, the moduli of the observed stmcture factors can be used in a Fourier synthesis with calculated phases, to find the centroids of the electron density peaks, which indicate the positions of atomic nuclei. If the phases are nearly correct, the Fourier map will reveal the position of most if not all of the atoms. The Fourier synthesis is, in brief, a convenient way of transforming the phase information in terms of phase angles into the same information in terms of atomic positions (recall equations 5.28 to 5.30). If the first try does not reveal the position of all atoms, then a Fourier synthesis using as coefficients the differences between observed and calculated structure factors, plus the nearly correct phases, will reveal the position of the missing atoms. 

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The mathematical theory of topology is the basis of other approaches to understanding inorganic structure. As mentioned in Section 1.4 above, a topological analysis of the electron density in a crystal allows one to define both atoms and the paths that link them, and any description of structure that links pairs of atoms by bonds or bond paths gives rise to a network which can profitably be studied using graph theory. 

To be more precise about diffraction, when we direct an X-ray beam toward a crystal, the actual diffractors of the X rays are the clouds of electrons in the molecules of the crystal. Diffraction should therefore reveal the distribution of electrons, or the electron density, of the molecules. Electron density, of course, reflects the molecule s shape in fact, you can think of the molecule s boundary as a van der Waals surface, the surface of a cloud of electrons that surrounds the molecule. Because, as noted earlier, protein molecules are ordered, and because, in a crystal, the molecules are in an ordered array, the electron density in a crystal can be described mathematically by a periodic function. 

The electron density in a crystal is a three-dimensional periodic function. As the most general case consider the crystallization of a molecular compound. If, in the process, two identical molecules interact more efficiently for a specific mutual orientation, it is most likely that all molecules will adopt this... 

Fourier synthesis A method of summing waves (such as scattered X rays) to obtain a periodic function (such as a representation of the electron density in a crystal). (See Chapter 6 glossary for a more detailed definition.)... 

Any periodic function (such as the electron density in a crystal which repeats from unit cell to unit cell) can be represented as the sum of cosine (and sine) functions of appropriate amplitudes, phases, and periodicities (frequencies). This theorem was introduced in 1807 by Baron Jean Baptiste Joseph Fourier (1768-1830), a French mathematician and physicist who pioneered, as a result of his interest in a mathematical theory of heat conduction, the representation of periodic functions by trigonometric series. Fourier showed that a continuous periodic function can be described in terms of the simpler component cosine (or sine) functions (a Fourier series). A Fourier analysis is the mathematical process of dissecting a periodic function into its simpler component cosine waves, thus showing how the periodic function might have been been put together. A simple... 

The electron density in a crystal precisely fits the definition of a periodic function in which an exact repeat occurs at regularly fixed intervals in any direction (the crystal lattice translations). Therefore the electron density in a crystal with a periodicity d can be described by a Fourier synthesis in which each component cosine wave (which we will call an electron-density wave) has a periodicity (i.e., wavelength) d/n, and the amplitude of the rath-order Bragg reflection. 

The electron density in a crystal is periodic from unit cell to unit cell. Therefore it can be represented as a Fourier series (as discussed in Chapter 6). The coefficients of this Fourier series are the amplitudes of the Bragg reflections the periodicities h,k, and /) are the indices of each Bragg reflection. Only the relative phase angles a hkl) are still needed, and once these have been estimated (see Chapter 8), all of the information for calculating the electron density becomes available. 

The electron density in a crystal, p (xyz), is a continuous function, and it can be evaluated at any point x,y,z in the unit cell by use of the Fourier series in Equations 9.1 and 9.2. It is convenient (because of the amount of computing that would otherwise be required) to confine the calculation of electron density to points on a regularly spaced three-dimensional grid, as shown in Figure 9.3, rather than try to express the entire continuous three-dimensional electron-density function. The electron-density map resulting from such a calculation consists of numbers, one at each of a series of grid points. In order to reproduce the electron density properly, these grid points should sample the unit cell at intervals of approximately one third of the resolution of the diffraction data. They are therefore typically 0.3 A apart in three dimensions for the crystal structures of small molecules where the resolution is 0.8 A. 

Based on the Fourier inverse transformations properties, the inverse Fourier transformation of Eq. (5.17) involves only the inversion of the sign in the appeared exponential. Generally, the Fourier transformation of integral form has a inverse transformation still as an integral form, yet, in this case being about the electronic density in a crystal, i.e., presenting a periodicity. Thus, the inverse transformation of the Eq. (5.17) will be written as a sum, generating the electronic density equation ... 

The traditional eharaeterisation of an electron density in a crystal amounts to a statement that the density is invariant under all operations of the space group of the crystal. The standard notation for sueh an operation is (Rim), where R stands for the point group part (rotations, reflections, inversion and combinations of these) and the direct lattice vector m denotes the translational part. When such an operation works on a vector r we get... 

Figure 2.2 A contour plot of the electron density in a plane through the sodium chloride crystal. The contours are in units of 10 6 e pm-3. Pauling shows the radius of the Na+ ion from Table 2.3. Shannon shows the radius of the Na+ ion from Table 2.5. The radius of the Na+ ion given by the position of minimum density is 117 pm. The internuclear distance is 281 pm. (Modified with permission from G. Schoknecht, Z Naiurforsch 12A, 983, 1957 and J. E. Huheey, E. A. Keiter, and R. L. Keiter, Inorganic Chemistry, 4th ed., 1993, HarperCollins, New York.)...

It is impossible to directly measure phases of diffracted X-rays. Since phases determine how the measured diffraction intensities are to be recombined into a three-dimensional electron density, phase information is required to calculate an electron density map of a crystal structure. In this chapter we discuss how prior knowledge of the statistical distribution of the electron density within a crystal can be used to extract phase information. The information can take various forms, for example ... 

Several formulations were proposed [65, 66], but the intuitive notation introduced by Hansen and Coppens [67] afterwards became the most popular. Within this method, the electron density of a crystal is expanded in atomic contributions. The expansion is better understood in terms of rigid pseudoatoms, i.e., atoms that behave stmcturally according to their electron charge distribution and rigidly follow the nuclear motion. A pseudoatom density is expanded according to its electronic stiucture, for simplicity reduced to the core and the valence electron densities (but in principle each atomic shell could be independently refined). Thus,... 

A promising simplification has been proposed by Bader (1990) who has shown that the electron density in a molecule can be uniquely partitioned into atomic fragments that behave as open quantum systems. Using a topological analysis of the electron density, he has been able to trace the paths of chemical bonds. This approach has recently been applied to the electron density in inorganic crystals by Pendas et al. (1997, 1998) and Luana et al. (1997). While this analysis holds great promise, the bond paths of the electron density in inorganic solids are not the same as the more traditional chemical bonds and, for reasons discussed in Section 14.8, the electron density model is difficult to compare with the traditional chemical bond models. 

In this case, the phase problem reduces to a question of the sign of the structure factor. Since the sign of the structure factor can be evaluated more reliably than the phase angle, the electron density in a centrosymmetric crystal can be evaluated more accurately than that in an acentric crystal (93), Equation (2) implies that total, time-averaged electron density can be determined by the diffraction method. 

X-ray crystallography seeks to obtain the best model to describe the periodic electron density in a crystal by a least squares fit of the parameters of the model (used to calculate structure factors) against the observed structure factors derived from the diffraction experiment. All models used are atomic in nature, but vary in the complexity of the description of the atomic electron density. 

Each electron-density wave provides a component for summation to give the electron-density map, shown in Figure 6.11. If the electron density of a crystal could be described precisely by a single cosine wave that repeats three times in the unit cell dimension d, then the electron density has a periodicity of d/3 and the diffraction pattern will have intensity only in the third order (only one diffracted beam, 3 0 0). This is the electron-density wave that is used in the summation that gives an electron-density map if there is only one term because only one Bragg reflection is ob-... 

Electron-density map A contour representation of electron density in a crystal structure. Peaks appear at atomic positions. The map is computed by a Fourier synthesis, that is, the summation of waves of known amplitude, periodicity, and relative phase. The electron density is expressed in electrons per cubic A. 

This chapter is concerned with the precision of the atomic arrangement that we derive from a crystal structure analysis. Therefore it is necessary at this point to pause and discuss errors and estimated standard deviations, and the methods used to assign a confidence or weight to a measurement. Precision, which should not be confused with accuracy, is a measure of how closely a series of measurements of the same quantity agree with each other.Accuracy is a measure of how close a measurement is to the true value. If rulers have become distorted with time, measurements made with them may be highly precise but are unlikely to be accurate. The true electron density in a crystal is not yet attainable by theory or experiment. As detection and theoretical methods improve, we hope that the data on the geometry of molecular structures, measured in various ways, will converge and approach those of the true molecular structure. 

Fig. 5. The arrangement of the electron density in a tetragonal crystal of human serum albumin. Prominent features of the molecular packing arrangement are large (90 x 90 A) solvent channels (shown in white) that pass through the crystal parallel to the crystallographic c axis. The unit cell and symmetry operations parallel to the c axis are illustrated. Reproduced with permission from Carter et al. (1989) American Association for the Advancement of Science (AAAS).

To this point we have been interested in the scattered waves, or X rays from atoms that combine to yield the observed diffraction from a crystal. Because the waves all have the same wavelength, we could ignore frequency in our discussions. In X-ray crystallography, however, we are equally interested in understanding how the waves diffracted by a crystal can be transformed and summed, in a symmetrical process, to produce the electron density in a unit cell. 

Single crystal neutron diffraction is in many ways a complementary technique to X-ray diffraction. In neutron diffraction scattering by the atomic nuclei rather than the electron density gives rise to diffraction. However, neutrons have a spin and polarisation of the neutron beam can be used to undertake diffraction experiments to map the distribution of unpaired electrons (the spin density) in a crystal. 

The three-dimensional periodic electron density in a crystal can be represented with the three-dimensional Fourier series... 

The one-dimensional equation can be generalised to three dimensions and the electron density of a crystal at any point in the unit cell x, y, z, is given by ... 

We should mention that in the few cases in which the variation in electron density in a crystal has been accurately determined (e.g. NaCl), the minimum electron density does not in fact occur at distances from the nuclei indicated by the ionic radii in general use e.g. in LiF and NaCl, the minima are found at 92 and 118 pm from the nucleus of the cation, whereas tabulated values of / l + and rj4a+ are 76 and 102 pm, respectively. Such data make it clear that discussing lattice structures in terms of the ratio of the ionic radii is, at best, only a rough guide. For this reason, we restrict our discussion of radius ratio rules to that in Box 5.4. 

Gatti, C., Saunders, V.R. and Roetti, G. (1994) Crystal field effects on the topological properties of the electron density in molecular crystals the case of urea, /. Chem. Phys., 101, 10686-10696. Tsirelson, V.G., Zou, P.F. and Bader, R.F.W. (1995) Topological definition of crystal structure determination of the bonded interactions in solid molecular chlorine. Cryst., A51, 143-153. Platts, J.A. and Howard, S.T. (1996) Periodic Hartree-Fock calculations on crystalline HCN,/. Chem. Phys., 105,4668-4674,... 

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