Enantiomorphic pair space groups

Chirality in Crystals. When chiral molecules form crystals the space group symmetry must conform with the chirality of the molecules. In the case of racemic mixtures there are two possibilities. By far the commonest is that the racemic mixture persists in each crystal, where there are then pairs of opposite enantiomorphs related by inversion centers or mirror planes. In rare cases, spontaneous resolution occurs and each crystal contains only R or only S molecules. In that event or, obviously, when a resolved optically active compound crystallizes, the space group must be one that has no rotoinversion axis. According to our earlier discussion (page 34) the chiral molecule cannot itself reside on such an axis. Neither can it reside elsewhere in the unit cell unless its enantiomorph is also present. 

It is important to note that not all space groups that can accommodate chiral molecules are necessarily chiral. For example, it is clearly possible to place 2 nonchiral molecules in a monoclinic unit cell in, say P2, and have a nonchiral crystal. On the other hand there are 11 enantiomorphous pairs of space groups that must give chiral crystals because they are inherently chiral, regardless of what is in them. These are the following, which are all based on screw axes, and the pairs simply have axes of the same type spiraling in opposite directions ... 

While a collection of molecules that are all of the same chirality (e.g., a D- or L-amino acid or a naturally occurring protein) must form a chiral crystal, inherently nonchiral molecules are not barred from doing so, if they crystallize in one of the 11 pairs of enantiomorphous space groups. In that event, which is rather rare, there will, of course, be an equal probability of forming either enantiomorph and a batch of crystals will normally contain both. A couple of real examples are (NH4)3Tc2C18 3H20 (P3,21 and P3 >21) and SntTa Cl (P6 22 and P6522). 

Space groups (or enantiomorphous pairs) that are uniquely determined from the symmetry of the diffraction pattern and systematic absences are shown in boldface type. 

Some space groups are enantiomers of others. There are 11 such pairs, listed in Table 4.6 (Chapter 4) and Table 14.2. If the (+) isomer of a chiral molecule crystallizes in one of these space groups, the (-) isomer will crystallize in the enantiomorphous space group. The systematic absences in the Bragg reflections are the same for both members of these pairs of space groups, but anomalous dispersion can aid in distinguishing between them. Several proteins have crystallized in enantiomorphous space groups. 

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