Crystal coordinates

The comparison with experiment can be made at several levels. The first, and most common, is in the comparison of derived quantities that are not directly measurable, for example, a set of average crystal coordinates or a diffusion constant. A comparison at this level is convenient in that the quantities involved describe directly the structure and dynamics of the system. However, the obtainment of these quantities, from experiment and/or simulation, may require approximation and model-dependent data analysis. For example, to obtain experimentally a set of average crystallographic coordinates, a physical model to interpret an electron density map must be imposed. To avoid these problems the comparison can be made at the level of the measured quantities themselves, such as diffraction intensities or dynamic structure factors. A comparison at this level still involves some approximation. For example, background corrections have to made in the experimental data reduction. However, fewer approximations are necessary for the structure and dynamics of the sample itself, and comparison with experiment is normally more direct. This approach requires a little more work on the part of the computer simulation team, because methods for calculating experimental intensities from simulation configurations must be developed. The comparisons made here are of experimentally measurable quantities. 

Figure 6-3. Top Structure of the T6 single crystal unit cell. The a, b, and c crystallographic axes are indicated. Molecule 1 is arbitrarily chosen, whilst the numbering of the other molecules follows the application of the factor group symmetry operations as discussed in the text. Bottom direction cosines between the molecular axes L, M, N and the orthogonal crystal coordinate system a, b, c. The a axis is orthogonal to the b monoclinic axis.

A sinusoidal plot of grf>2 vs.

crystal plane gives another set of Ks that depend on other combinations of the gy, eventually enough data are obtained to determine the six independent values of gy (g is a symmetric matrix so that gy = gy,). The g-matrix is then diagonalized to obtain the principal values and the transformation matrix, elements of which are the direction cosines of the g-matrix principal axes relative to the crystal axes. An analogous treatment of the effective hyperfine coupling constants leads to the principal values of the A2-matrix and the orientation of its principal axes in the crystal coordinate system. 

A more complex notation is needed for non-stoichiometric phases. Selected simple examples are given below, and more detailed information will be reported when discussing crystal coordination formulae ... 

Coordination and dimensionality symbols in the crystal coordination formula... 

In the description of crystal coordination formulae a coordination number was introduced, defined as the NNN of atoms X around the atom Y under consideration. 

The multipole formalism described by Stewart (1976) deviates from Eq. (3.35) in several respects. It is a deformation density formalism in which the deformation from the IAM density is described by multipole functions with Slater-type radial dependence, without the K-type expansion and contraction of the valence shell. While Eq. (3.35) is commonly applied using local atomic coordinate systems to facilitate the introduction of chemical constraints (chapter 4), Stewart s formalism has been encoded using a single crystal-coordinate system. 

In typical organic crystals, molecular pairs are easily sorted out and ab initio methods that work for gas-phase dimers can be applied to the analysis of molecular dimers in the crystal coordination sphere. The entire lattice energy can then be approximated as a sum of pairwise molecule-molecule interactions examples are crystals of benzene [40], alloxan [41], and of more complex aziridine molecules [42]. This obviously neglects cooperative and, in general, many-body effects, which seem less important in hard closed-shell systems. The positive side of this approach is that molecular coordination spheres in crystals can be dissected and bonding factors can be better analyzed, as examples in the next few sections will show. 

Wireko et al. (1991) used HINT on crystal coordinates of haemoglobin... 

Figure 5. The relationship between molecular and crystal coordinate systems for a unit cell containing two molecules.

Perspective view of the asymmetric unit t crystal coordinates)... 

Coordinates of molecules may be represented in a global or in an internal coordinate system. In a global coordinate system each atom is defined with a triplet of numbers. These might be the three distances x,, y,-, z, in a crystal coordinate system defined by the three vectors a, b, c and the three angles a, / , y or by a, b, c, a, P, y with dimensions of 1,1,1,90°, 90°, 90° in a cartesian, i. e. an orthonormalized coordinate system. Other common global coordinate systems are cylindrical coordinates (Fig. 3.1) with the coordinate triples r, 6, z and spherical coordinates (Fig. 3.2) with the triples p, 9, . 

Fig. 4.2. Diagrammatic representation of the experimental crystal and beam geometries for p-polarized radiation incident upon the (111) crystal face as viewed (a) from the side and (b) from the top including the second atomic plane ( ). The crystal coordinates are labeled x, y, z with the Z direction along the [211] crystal direction. The beam coordinates are labeled s, k, z. From Ref. 122.

Fig. 6. Molecular packing in [Mn(taa)] crystal. Coordination octahedra are colored according to four sublattices discriminated by mapping a body-diagonal of the unit cell to C3 axes of member molecules. Darken side of each octahedron faces to the C3 axis.

This is a simplification as the catalyst is a solid and the active Ti atom almost certainly Ti(l II) rather than Ti(IV) as we have shown here. The third Cl ligand is in fact shared with other Ti atoms in the crystal. Coordination of the active Ti(ill) atom must be such that each o complex is a 16-electron species while the jc complexes are lB-electron species. 

Pfeiffer regarded sodium chloride as a highmolecu-lar-weight coordination compound (NaCl) constructed of equal numbers of [NaCle] and [ClNa ] units. The difference between primary valences and secondary valences disappears in crystals of symmetrical compounds. He demonstrated that, in other crystals, coordination centers can be groups of atoms as well as single atoms, and that coordination numbers as high as 12 must sometimes be considered. He proposed that, in crystals of simple organic molecular compounds of type AB, each constituent acts as a coordination center ABe and BA6 units interpenetrate exactly as in sodium chloride. 

Three additional concepts may be introduced by means of Fig. 2A. First, all measurements of the diffracted beams are made relative to the incident x-ray beam so that the diffracted beam leaves the crystal at an angle 20, as illustrated. Thus, while the diffraction condition is determined by the angle between the incident and diffracted beams relative to a lattice plane, the measured position of the diffracted beam is determined relative to the incident beam. Second, there is a reciprocal relation between the spacing between lattice planes and the positions of the diffraction spots small lattice spacings give diffraction spots with large values of 20. This leads to two types of space. The crystal coordinate system is in real space, whereas the diffrac-... 

FIGURE 11.4. From crystal coordinates to orthogonal (Cartesian) coordinates (.4 ... 

Fig. 13. A stereodiagram of the ligands of CRBP, IFABP, MFB2, and P2. These structures were aligned as described in Fig. 5. ALBP and HFABP bind their ligands in the same manner as P2, and CRBPIl binds its ligand in the same manner as CRBP. The alignment of the crystal coordinates is based only on protein atoms. The striking similarity in the location of the ligands in the crystals of the holo forms is due to the homology in the iLBP family.

The fully refined model of the crystal (coordinates, population and individual isotropic displacement parameters of atoms) and interatomic distances in the crystal structure of Mn7(0H)3(V04)4 is found in Table 7.25 and it is shown in Figure 7.37 as the arrangement of the coordination polyhedra of the Mn and V atoms. 

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