The Electron Density Equation

To this point we have focused single-mindedly on understanding and writing an expression that describes the diffraction from a crystal, its Fourier transform, if we know the atomic coordinates xj, yj, zj in the crystallographic unit cell. The reader may be impatient by now as the real life objective is to do the opposite, to define the xj, yj, zj coordinates of the atoms when we can measure the F making up the diffraction pattern. It is now time to go the other way, but understanding the meaning of the F makes that task easier. 

In deriving the structure of a molecule, or distribution of atoms using X-ray crystallography, we do not directly obtain the x, y, z coordinates of the atoms. That is, we don t solve some system of linear equations whose solution is the set of numerical values for jc, y, z. We employ a Fourier transform equation that incorporates the diffraction data, the structure factors, and yields the value of the electron density p(x, y, z) at any and all points x, y, z within the crystallographic unit cell. From the peaks and features of this continuous electron density distribution in the unit cell we then infer the locations of the atoms, and hence their coordinates. This will be described as it is done in practice, in Chapter 10. Following this, the coordinates are improved by applying refinement procedures, as also outlined in Chapter 10. Here, however, our objective is to understand this Fourier transform equation, namely the electron density equation. 

In the case of a periodic, three-dimensional function of x, y, z, that is, a crystal, the spectral components are the families of two-dimensional planes, each identifiable by its Miller indexes hkl. Their transforms correspond to lattice points in reciprocal space. In a sense, the planes define electron density waves in the crystal that travel in the directions of their plane normals, with frequencies inversely related to their interplanar spacings. 

We have seen that the diffracted waves Fhki, from a particular family of planes hkl, when Bragg s law is satisfied, depends only on the perpendicular distances of all of the atoms from those hkl planes, which are h xj for all atoms j. Therefore each Fhki carries information regarding atomic positions with respect to a particular family hkl, and the collection of Fhki for all families of planes hkl constitutes the diffraction pattern, or Fourier transform of the crystal. If we calculate the Fourier transform of the diffraction pattern (each of whose components Fhki contain information about the spatial distribution of the atoms), we should see an image of the atomic structure (spatial distribution of electron density in the crystal). What, then, is the mathematical expression that we must use to sum and transform the diffraction pattern (reciprocal space) back into the electron density in the crystal (real space)  

In the structure factor equation the coefficients in the summation were all nonzero electron densities fj occurring at Xj, yj, Zj, which is really p(xj, yj, zj), and Fhki were the entities being calculated. Hence the coefficients in the electron density equation yielding p(x, y, z) must be the reciprocal space entities Fhki Finally, to keep units consistent, and the mathematics consistent with Monsieur Fourier, the sign of the imaginary term must be changed to minus, and the constant V must be inverted to 1/V, the volume of the reciprocal unit cell. Thus the electron density equation assumes the form. 

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The value of at zero temperature can be estimated from the electron density ( equation Al.3.26). Typical values of the Femii energy range from about 1.6 eV for Cs to 14.1 eV for Be. In temis of temperature (Jp = p//r), the range is approxunately 2000-16,000 K. As a consequence, the Femii energy is a very weak ftuiction of temperature under ambient conditions. The electronic contribution to the heat capacity, C, can be detemiined from... 

To determine the structure, we have to locate the atoms, which are given by the electron density equation ... 

From the intensity, the value for the observed F is obtained. This is substituted into the electron density equation for locating the atoms that determine the structure of the protein. Through an iterative process, the observed and calculated F values are compared to determine the goodness of fit and hence the quality of the structure. 

Substituting this expression for Fhkl in Eq. 5.18, the electron-density equation (remembering that a. is the phase a.hkl of a specific reflection), gives... 

The same argument can be applied to the electron density (Equation 1.100), which satisfies the conservation relation ... 

The situation, in truth, is somewhat more involved than this explanation would suggest. The individual reflections of the diffraction pattern are the interference sum of the waves scattered by all of the atoms in the crystal in a particular direction and, therefore, are themselves waves. Being waves they have not only an amplitude, but also a unique phase angle associated with each of them. This too depends on the distribution of the atoms, their xj, yj, Zj. The phase angle is independent of the amplitude of the reflection, but most important, it is an essential part of the individual terms that contribute to the Fourier synthesis, the electron density equation. Unfortunately, the phase angle of areflection cannot be recorded, as we record the intensity. In fact we have no practical way (and rather few impractical ways either) to directly measure it at all. But, without the phase information, no Fourier summation can be computed. In the 1950s, however, it became possible, with persistence, skill, and patience (and luck), to recover this elusive phase information for... 

Remember from Chapter 4 that the periods and frequencies of waves are reciprocally related.) Exactly those properties are expressed by their reciprocal lattice vectors h. The amplitudes of these electron density waves vary according to the distribution of atoms about the planes. Although the electron density waves in the crystal cannot be observed directly, radiation diffracted by the planes (the Fourier transforms of the electron density waves) can. Thus, while we cannot recombine directly the spectral components of the electron density in real space, the Bragg planes, we can Fourier transform the scattering functions of the planes, the Fhki, and simultaneously combine them in such a way that the end result is the same, the electron density in the unit cell. In other words, each Fhki in reciprocal, or diffraction space is the Fourier transform of one family of planes, hkl. With the electron density equation, we both add these individual Fourier transforms together in reciprocal space, and simultaneously Fourier transform the result of that summation back into real space to create the electron density. 

The electron density equation, as presented here, is formally correct, but it does lend itself to some simplification, which ultimately makes it both more realistic and more computationally efficient. Before embarking on this bit of refinement, it is useful to once again recall two important corollaries ... 

At this point the most problematic feature of the process emerges. Inspection of the electron density equation as it was initially stated shows that the coefficient of each term in the summation for p(x, y, z) at any value of x, y, z is T ki The structure factor Ff,u is, as we have seen, a wave. It is a complex number it has an amplitude and a phase. In the final form of the equation we see that this feature persists in the form of the phase angle for each structure factor that must be included in the kernal. To calculate p(x, y, z), then,... 

Remembering that Fflkl is simply the measured intensity / <, a scaler quantity, then the expression can be recognized as simply the electron density equation (see Chapter 5) with squared coefficients and all phases 4>hkt set equal to zero. The normalization constant is here l/V2 because of the squared coefficient, where V is the volume of the unit cell. The units it implies for the function, something per volume squared, immediately indicates that P(u, v, w) is not electron density but some other spatial function. Because the equation yields something other than electron density, existing in some unique space, we cannot denote it by p(x, y, z) in jc, y, z (real space) we must designate it by P(u, v, w) in some alternative coordinate space whose variables are u, v, w. Otherwise, P(u, v, w) is the equation for aperiodic function in u, v, w space. The Patterson function, or Patterson wave... 

The electron density equation very simple structures such as NaCl can be solved by comparison of the relative intensities of the diffraction spots. For more complicated structures, the power of Fourier transform methods was soon appreciated [27]. In order to produce an image of the structure, the diffracted rays must be combined. In the light microscope this is achieved by the focussing power of the objective lens (Fig. 3b). For X-rays the refractive index of almost all substances is close to 1 and it is not possible to construct a lens. The diffracted rays must be combined mathematically. This is achieved with the electron density equation. 

Using the LCGTO expansions for the Kohn-Sham orbitals (Equation (2)) and the electronic density (Equation (3)), the Kohn-Sham SCF energy expression can be expressed as ... 

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 ... 

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