Atom independent model

At a certain stage in the refinement, the electron density map is interpreted using a model representation of the charge density distribution to extract the atomic coordinates. A commonly used scattering formalism is the independent-atom model (IAM), in which the total charge density in the crystal is approximated by the superposition... 

The true phases of the structure factors will, in general, be different from the phases calculated with the independent-atom model. In centrosymmetric structures, with phases restricted to 0 or n, only very few weak reflections are affected. In acentric structures, only the reflections of centrosymmetric projections, such as the hkO, hOl, and Okl reflections in the space group P212,21, are invariant. 

When extinction is present in the data set, it must be corrected for before the MEM procedure is started. The structure factors must similarly be corrected for anomalous scattering, if present. Both corrections require a model for their evaluation. The independent-atom model is usually adequate for this purpose. 

The crystal potential for L-alanine calculated with Eq. (8.34) is shown in Fig. 8.1(a). The term Ospherical.atom(r) can be evaluated in direct space by the methods described in the following section. The term 0(0) for the independent-atom model [not exactly equal to the true 0(0)] was evaluated by a summation of the IAM potential over the unit cell. 

Covalent contribution to the bonding is confirmed by analysis of the components of V2p. The average values of 2, and X2, representing the contraction of the density in the directions perpendicular to the bond (chapter 6), are found to be — 9eA"s, compared with —4eA 5 for the independent-atom model (IAM) density and — 6eA"5 for the ionic model. In addition, the density at the bond critical point ph is l.leA"3, higher than 0.85 eA"3 and 0.68 eA"3 calculated for the IAM and ionic models, respectively. Results for the borosilicate danburite CaB2Si208, are similar, with ph(SiO) 0.95 eA-3, and large positive values of V2p for the Si—O bonds (Downs and Swope 1992). 

The conclusions of Ciraci et al. (1990a) can be understood with a simple independent-atom model, that is, by considering the tip as a single atom, and the total force is the sum of the forces on every atom on the sample surface. Assuming that the force between individual pairs of atoms can be represented as a Morse curve, the z component of the force to the nth atom at a point in space, r, is ... 

Consider Fig. 9 showing the diffraction profile for the polymer Lucite (C5H802) taken from Tartari et al. [23], The dashed line represents the function F2 + S calculated from the mixture rule on the basis of the Independent Atom Model (IAM) using the tabulated atom form factors for H, C and O data presented in Fig. 8. The IAM curve is seen to approach the measured profile at high x values. 

Fig. 12. Universal fit (long dashed line) to tabulated total Independent Atom Model (LAM) cross-section (unbroken line) for N. The Gaussian (G) and Lorenzian (L) components of the fit are also shown.

Spatially Oriented Molecules [72-77]. A continuous beam of CH3CI or CH3I molecules is oriented via the hexapole technique before their scattered intensities are measured in the GED unit described before [31,32]. The molecules are oriented preferentially parallel or antiparallel to the electron beam and show oscillating deviations from the scattering pattern of unoriented molecules up to 4% as a function of momentum transfer, quite similar to previous calculations using the independent-atom model (lAM)... 

Direct access to structure determination, i.e., atomic positions is obtained via the independent atom model (IAM, also denoted as the isolated-spherical-atom model). Here, the electron density is written as a sum of free atomic densities... 

Finally, in order to simplify the relation between the differential scattering signal and the nuclear motion, we adopt as in conventional X-ray diffraction theory, the independent atom model (IAM) for the electrons [12-14, 17]. In the IAM, according to (6)... 

We note that the valence electron in Nal undergoes a substantial reorganization -corresponding to electron transfer between Na and I - whenever the intemuclear distance passes the avoided crossing [36]. This, however, does not seriously invalidate the independent atom model used to represent the molecular electron density, since the vast majority of the electrons can be considered as core electrons associated with a specific atom. 

Systematic residuals in scattered intensities were examined in relation to the electron distribution in the molecule (imperfection of the independent-atom model or its modified... 

For anisotropic motions the expressions we have just discussed become more complicated. Note also that while these equations refer to positions and displacements of atoms (i.e. nuclei), the X-rays themselves are actually scattered by electrons. This is a potential problem, because the nature of chemical bonding means that the distribution of electrons is not a simple superposition of spherical atoms. And yet the assumption of exactly spherical atoms, also known as the independent atom model, is the basis of these equations. A more rigorous treatment relates the structure factors to electron density, i.e. the three-dimensional distribution of electrons in space represented by the function p x) with x representing space in three-dimensional coordinates x, y, z. Within this formalism the structure amplitude can then be expressed as... 

The second level of a crystal structure investigation is to go beyond the independent atom model and to resolve more subtle details of the electron density distribution. For such a study one needs to measure... 

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