Space group constraints

Only 230 different arrangements of these symmetry operations are possible in three-dimensional space and the properties of the corresponding 230 space groups are listed in International Tables for Crystallography, Vol. A (1996) supplemented by the information given in Appendix 2. Every crystalline solid must necessarily satisfy the constraints imposed by one of these space groups even if it is only PI, the space group that has no internal symmetry. 

The principle of maximum symmetry requires that the crystal structure adopted by a given compound be the most symmetric that can satisfy the chemical constraints. We therefore expect to find high-symmetry environments around atoms wherever possible, but such environments are subject to constraints such as the relationship between site symmetry and multiplicity (eqn (10.2)) and the constraint that each atom will inherit certain symmetries from its bonded neighbours. The problems that arise when we try to match the symmetry that is inherent in the bond graph with the symmetry allowed by the different space groups are discussed in Section 11.2.2.4. 

Any 6- or 12-coordinate ion in the graph is initially assumed to have the site symmetry m3m (Oh) if all the ligands are equivalent in the bond graph. If they are not all equivalent, then one must choose a lower site symmetry that is compatible with this inequivalence. Similarly an ion with four equivalent ligands is assumed to be tetrahedrally coordinated with site symmetry 43m (T ). The constraints 1-3 above are then examined to ensure that all have been satisfied. If they are, then one can look in Appendix 2 to find a matching space group using the procedure described below. If they are not, the symmetry of one or more atoms must be lowered until all the constraints are satisfied. 

The initial description of the model is simple, as shown in Figure 3. The atomic coordinates of any suitable structure can serve as the input trial structure, even including a wrong monomer residue. The polar coordinates are calculated from the trial structure, adjusted and modified as necessary, and then subjected to refinement in accordance with the selected list of variables, limits and constraints. Any set of standard values and nonbonded potential function parameters can be used. Hydrogen bonds can be defined as desired, variables can be coupled, and the positions of solvent molecules can be individually refined. Single and multiple helices are equally easily handled, as are a variety of space groups. 

Crystals of Xe(OSeFs)2 are rhombohedral, space group R3m. At 23.5 C the hexagonal axes are fl - 6 = 8.588 (3) and c - 11.918 (3) A Z = 3, da cd = 3.345 g cm"3, and V = 761.23 AL The molecule lies on a threefold axis, and there is orientational disorder of the oxygen and fluorine positions. X-ray diffraction data obtained with an automatic diffractometer were analyzed on the basis of a molecular model with some constraints based on chemical considerations to reduce the number of independent parameters of the poorly resolved oxygen and fluorine atoms. For 122 unique reflections with > a(F ) and with anisotropic thermal parameters, R = 0.064. Bond distances are Xe-0 2.12 (5), Se-0 1.53 (5), and Se-F = 1.70 (2) A (uncorrected) and Se-F = 1.77 A (corrected for thermal motion). 

A successfid indexing of the powder diffraction pattern, which can often be done automatically, yields the unit cell dimensions and information on possible space groups. The chemical analysis and sorption data indicate the framework density, or number of tetrahedra per unit cell. The challenge is then to position these tetrahedra within the unit cell such that (1) they fully interconnect in a sensible manner and (2) the necessary analytical data are reproduced. These structural constraints are quantified in an energy expression and simulated annealing [33,34] is employed as the global optimization approach. 

For example, a racemic mixture of (l-cyanoethyl)(piperidine)-cobaloxime, which contains the chiral 1-cyanoethyl group, crystallizes in the chiral space group 212121 with R and S enantiomers in each asymmetric unit. Irradiation with X rays at 343 K converts the 5 enantiomers to R while the R enantiomer is not affected. This is due to different constraints in the two crystallographic environments. The respective volumes for the two enantiomers are 7.49 and 11.57 A respectively. The critical volume for such racemization is 11.5 A. In agreement with this estimate it is found that the molecule with the small cavity does not show evidence of racemization, while the one with the large reaction cavity does, and a disordered cyanoethyl group is observed as a peak in two positions in the electron-density map. This reaction is illustrated in Figure 18.6 which shows the crystal structure at 293 K (where no reaction occurs) and at 333 K (where reaction is only observed in one of the two cavities). 

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 ... 

As illustrated in Figure 7.18, the pore structure, represented as four cylinders called forbidden zones, is first defined in a unit cell, and then atoms are placed outside the forbidden zones on the basis of specified symmetry and distance constraints. Two constraint conditions must be satisfied when placing the atoms (i) no T atom is allowed inside a forbidden zone, and (ii) the distance between any two T atoms should not be less than 3.0 A (Si—Si distance). This method allows a user to specify the pore size, the number and site symmetry of unique atoms, the unit cell, and the space group. 

A crystal structure is described by a collection of parameters that give the arrangement of the atoms, their motions and the probability that each atom occupies a given location. These parameters are the atomic fractional coordinates, atomic displacement or thermal parameters, and occupancy factors. A scale factor then relates the calculated structure factors to the observed values. This is the suite of parameters usually encountered in a single crystal structure refinement. In the case of a Rietveld refinement an additional set of parameters describes the powder diffraction profile via lattice parameters, profile parameters and background coefficients. Occasionally other parameters are used these describe preferred orientation or texture, absorption and other effects. These parameters may be directly related to other parameters via space group symmetry or by relations that are presumed to hold by the experimenter. These relations can be described in the refinement as constraints and as they relate the shifts, Ap,-, in the parameters, they can be represented by... 

In their simplest form, these constraints can be used for atomic sites on special positions (e.g. x,x,x positions in cubic space groups) where there are special relationships between the individual atomic coordinates and also among the individual anisotropic thermal motion parameters (e.g. Ui i = U22 = U33 and Ui2 = Ui3 = U23 for cubic x,x,x sites). 

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