Diffraction and the Electrostatic Potential

This chapter deals with the evaluation of the electrostatic potential and its derivatives by X-ray diffraction. This may be achieved either directly from the structure factors, or indirectly from the experimental electron density as described by the multipole formalism. The former method evaluates the properties in the crystal as a whole, while the latter gives the values for a molecule or fragment lifted out of the crystal. 

Like other properties derived from the charge distribution, the experimental electrostatic potential will be affected by the finite resolution of the experimental data set. But as the contribution of a structure factor F(H) to the potential is proportional to H 2, as shown below, convergence is readily achieved. A summary of the dependence of electrostatic properties of the magnitude of the scattering 

Diamagnetic shielding tensor Second rank tensor -1 -2 

Electric field gradient Traceless Second-rank tensor -3 0 

Gradient of field gradient Third rank tensor -4 1 

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Another study focusing on the comparison between theoretical and experimental densities is that of Tsirelson el al. on MgO.133 Here precise X-ray and high-energy transmission electron diffraction methods were used in the exploration of p and the electrostatic potential. The structure amplitudes were determined and their accuracy estimated using ab initio Hartree-Fock structure amplitudes. The model of electron density was adjusted to X-ray experimental structure amplitudes and those calculated by the Hartree-Fock model. The electrostatic potential, deformation density and V2p were calculated with this model. The CPs in both experimental and theoretical model electron densities were found and compared with those of procrystals from spherical atoms and ions. A disagreement concerning the type of CP at ( , 0) in the area of low,... 

Interaction of incident electrons with the electrostatic potential (ESP) gives the possibility to reconstmct the potential from transmission electron diffraction (ED) experiments. ESP and the electron density determine all physical properties of crystals (e.g. energy of electrostatic interaction, characteristics of the electrostatic field in crystals, dipole, quadmple and other momentum of nuclear, diamagnetic susceptibility. 

In contrast to Fourier synthesis, which yields with electron diffraction data high electrostatic potential at the positions of the atoms, the maps obtained from Patterson synthesis show peaks at the tips of vectors. The length of each vector (drawn from the origin of the Patterson map) corresponds always to the distances between pairs of atoms and the direction each vector points... 

The electrostatic potential y(r) is a physical observable, which can be determined experimentally by diffraction methods as well as computationally. It directly reflects the distribution in space of the positive (nuclear) and the negative (electronic) charge in a system. V (r) can also be related rigorously to its energy and its chemical potential, and further provides a means for defining covalent and ionic radii" ... 

Here a and d are the number of atoms in the acceptor and the donor, respectively, Ry is the distance between atoms i and j and and are the van der Waals and electrostatic potentials, respectively. The van der Waals potential is often represented by a Lennard-Jones potential (Eq. 8) or by a Buckingham potential (Eq. 9). The parameters a, fi, y and o are obtained from solid-state crystal data. The leading term in the electrostatic potential is the Coulomb interaction (first term in Eq. 10), where D is the effective dielectric constant (usually < D <2). Other terms may be added to represent induced polarization, etc. [40]. The geometries of the two components of the cluster are obtained from microwave or electron diffraction data or from quantum chemical calculations. It is assumed that these geometries do not change upon adduct formation. An initial guess is made for the structure of the adduct, and then the relative positions of the two (or more) components are varied until a local energy minimum is obtained. 

The other two types of radiation that can diffract fi om crystals are neutron and electron beams. Unlike x-rays, neutrons are scattered on the nuclei, while electrons, which have electric charge, interact with the electrostatic potential. Nuclei, their electronic shells (i.e. core electron density), and electrostatic potentials, are all distributed similarly in the same crystal and their distribution is established by the crystal structure of the material. Thus, assuming a constant wavelength, the differences in the diffraction patterns when using various kinds of radiation are mainly in the intensities of the diffracted beams. The latter occurs because various types of radiation interact in their own way with different scattering centers. The x-rays are the simplest, most accessible and by far the most commonly used waves in powder diffraction. 

The diffraction of electrons can also be regarded as a special case of the electro magnetic wave equation, where the electrostatic potential of the atom replaces electron density ep (X-ray diffraction). This potential is composed of the effect of the atomic nucleus and that of the electron cloud. The spatial distribution of the potential corresponds approximately to that of the electron density, but falls off less rapidly. The maxima of the Fourier series, however, correspond in both cases to the position of the atomic nucleus. 

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