Anisotropic atomic positions

A modification of the united-atom approach, called the anisotropic united-atom (AUA) model was the focus of extensive work by Karabomi et al. [362-365]. As in the other models of hydrocarbon chains described so far, the AUA approach to monolayers was preceded by work on alkanes [367]. hi the AUA model the interaction site is located at the geometrical mean of the valence electrons of the atoms it represents, while the pseudoatom itself is located at the carbon atom position. The movement of each interaction center depends on the conformation of the molecule as a whole. 

The structure was refined by block-diagonal least squares in which carbon and oxygen atoms were modeled with isotropic and then anisotropic thermal parameters. Although many of the hydrogen atom positions were available from difference electron density maps, they were all placed in ideal locations. Final refinement with all hydrogen atoms fixed converged at crystallographic residuals of R=0.061 and R =0.075. 

The diffraction lines due to the crystalline phases in the samples are modeled using the unit cell symmetry and size, in order to determine the Bragg peak positions 0q. Peak intensities (peak areas) are calculated according to the structure factors Fo (which depend on the unit cell composition, the atomic positions and the thermal factors). Peak shapes are described by some profile functions 0(2fi—2fio) (usually pseudo-Voigt and Pearson VII). Effects due to instrumental aberrations, uniform strain and preferred orientations and anisotropic broadening can be taken into account. 

Carbon and oxygen atom positions were refined with anisotropic thermal parameters. Hydrogen atoms were not located. Figure 3 shows the molecular structure and the atom numbering scheme utilized for the x-ray data presented in the supplementary material for heritiana acetate. Coordinates, bond length, and bond angles for compound III acetate are available as supplementary material. 

With small molecules, it is usually possible to obtain anisotropic temperature factors during refinement, giving a picture of the preferred directions of vibration for each atom. But a description of anisotropic vibration requires six parameters per atom, vastly increasing the computational task. In many cases, the total number of parameters sought, including three atomic coordinates, one occupancy, and six thermal parameters per atom, approaches or exceeds the number of measured reflections. As mentioned earlier, for refinement to succeed, observations (measured reflections and constraints such as bond lengths) must outnumber the desired parameters, so that least-squares solutions are adequately overdetermined. For this reason, anisotropic temperature factors for proteins have not usually been obtained. The increased resolution possible with synchrotron sources and cryocrystallography will make their determination more common. With this development, it will become possible to obtain better estimates of uncertainties in atom positions than those provided by the Luzzati method. 

A preliminary knowledge of the crystal structure is important prior to a detailed charge density analysis. Direct methods are commonly used to solve structures in the spherical atom approximation. The most popular code is the Shelx from Sheldrick [26] which provides excellent graphical tools for visualization. The refinement of the atom positional parameters and anisotropic temperature factors are carried out by applying the full-matrix least-squares method on a data corrected if found necessary, for absorption and diffuse scattering. Hydrogen atoms are either fixed at idealized positions or located using the difference Fourier technique. 

ORTEP were used to draw the structures. The final refinement included atomic positions for all the atoms, anisotropic thermal parameters for all the non-hydrogen atoms, and isotropic thermal parameters for the hydrogen atoms. The aromatic hydrogen atoms for I—III, methylene hydrogen atoms of the DABCO unit (for IV), and ethylenediammonium ion (for V) were introduced in the... 

The only remaining degree of freedom in this crystal structure is to refine the displacement parameters of all atoms in the anisotropic approximation (the presence of preferred orientation is quite imlikely since the used powder was spherical and we leave it up to the reader to verify its absence by trying to refine the texture using available experimental data). As noted in Chapter 2, special positions usually mandate certain relationships between the anisotropic atomic displacement parameters of the corresponding atoms. In the space group P6/mmm, the relevant constraints are as follows ... 

Displacement parameters of atoms are also expected to be different as the temperature of the powder diffraction experiment varies. Furthermore, it is also feasible that atomic positions may change due to generally anisotropic thermal expansion of crystal lattices. These considerations are in addition to the most obvious cause (different lattice parameters) preventing combined refinement using powder diffraction data collected at different temperatures. In general, material may also be polymorphic but this is not the case here, as was established in Chapter 6, sections 6.10 and 6.11. 

Notably, the anisotropic thermal displacement factors form the elements of a 3x3 symmetric matrix. The physically meaningful form of this matrix when it is positive-definite is that of an ellipsoidal probability surface centered at the equilibrium atom position. An alternative form for Equation (22) frequently used in crystallography ... 

Parameters in the structural model, and other experiment-dependent parameters, are allowed to vary until a best-fit of the PDF calculated from the model and the data derived PDF is obtained, using a least-squares approach. The sample dependent parameters thus derived include the unit cell parameters (unit cell lengths and angles), atomic positions in the unit cell expressed in fractional coordinates, anisotropic thermal ellipsoids for each atom and the average atomic occupancy of each site. 

Figure 11, Atomic positions and ellipsoids of anisotropic thermal vibrations for residues 7-10 of avian pancreatic polypeptide (from 1. Glover, Ph.D. Thesis, 1984, University of London, see [194]). The side chain of Tyr7 stacks above Gly9. There are indications of concerted thermal motion for these residues, with the largest vibrations in approximately the vertical direction of the page. There is least motion along the bond directions. Atoms at the end of side chains have greater anisotropic motion than main chain atoms.

Anisotropic displacement parameters are a measure of the uncertainty of the atomic position and this arises fi om two sources the vibrational motion of the atoms (both internal and external vibrations, also called... 

Once least-squares methods came into general use it became standard practice to refine not only atomic positional parameters but also the anisotropic thermal parameters or displacement parameters (ADPs), as they are now called [22]. These quantities are calculated routinely for thousands of crystal structures each year, but they do not always get the attention they merit. It is true that much of the ADP information is of poor quality, but it is also true that ADPs from reasonably careful routine analyses based on modem point-by-point or area diffractometer measurements can yield physically significant information about atomic motions in solids. We may tend to think of crystal structures as static, but in reality the molecules undergo translational and rotational vibrations about their equilibrium positions and orientations, as well as internal motions. Cruickshank taught us in 1956 how analysis of ADPs can yield information about the molecular rigid-body motion [23], and many improvements and modifications have been introduced since then. In particular, various computer programs are available to estimate the amplitudes of simple postulated types of internal molecular motion e.g., torsion-... 

Fig. 5.1 Anisotropic displacement parameters of a disordered ethyl group on the left without and on the right with modeling of the disorder (empty hnes for the minor component). If the disorder is ignored, the refinement program tries to describe both atom positions with one eUipsoid, giving rise to a cigar-shaped probability eUipsoid.

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