Geometric phase symmetry approach

Once the atomic coordinates have been determined, often before any refinement, it is usual to calculate interatomic distances to 3.5 or 4 A, in order to check the connectivity of atoms, that is, to determine which atoms are bonded to which. This will show whether the chemical formula is correct. The connectivity also demonstrates whether or not the experimental atomic positions are for atoms connected in one molecule, or for atoms which are in different molecules (related by space-group symmetry). Finally, if the connectivity calculation includes longer distances, it will show how the molecules or ions pack with respect to each other in the crystal. Information on the different intermolecular interactions, such as hydrogen bonds present in the crystal, are found in this way. If any intermolecular distance is substantially less than the expected value (see Table 11.4, for example), implying that molecules approach each other too closely, the reported crystal structure may not be correct. The derived set of relative phases (see Chapter 8) should be scrutinized, as there may be another more suitable set. Any truly unusual geometrical features in a structure determination should be carefully analyzed before being accepted as experimental evidence. 

A cycloaddition reaction that forms a four-, five-, or six-membered ring must involve suprafacial bond formation. The geometric constraints of these small rings make the antarafacial approach highly unlikely even if it is symmetry-allowed. (Remember that symmetry-allowed means the overlapping orbitals are in-phase.) Antarafacial bond formation is more likely in cycloaddition reactions that form larger rings. 

Of course not aU of these mesophase have to appear in a single liquid crystalline system. For very few liquid crystals, exceptions from this sequence rule are known to exist. In these liquid crystals a mesophase with a higher symmetry reappears on cooling, even though a less symmetric mesophase has already formed at higher temperatures. Such phases are called re-entrant and are indicated with a subscript RE . Re-entrant behavior was first observed for a N-SmA-NRE-Cr phase sequence [39], but it was also found for other types of mesophases [40, 41]. It is not completely clarified when and why re-entrant phases appear. Different approaches to explain the re-entrant behavior were made, e.g. on the basis of frustration, geometric complexity or competing fluctuations [42, 43]. 

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