Resource Pa pers-IX Prepared under the sponsorship of The A d v i s o r y Council o n College Chemistry
Milton M. Orchin ond H. H. J a f f e University of Cincinnati Cincinnati, Ohio 45221
Symmetry, Point Groups, and Character Tables 11, Classification o f molecules into point groups
After the introduction, in part I, to point symmetry operations and their applications to determining optical activity, classificat,ion of isomers, nmr studies, and dipole moments, we are ready to take a brief look at translational symmetry and then move on to a general discussion of point groups and molecular point group classification. §Translational S y m m e t r y
We have thus far concerned ourselves with point symmetry hecause we have been discussing the symmetry properties of individual molecules. If we wish to consider the fixed arrangement of molecules with respect to one another (crystallography), then we must consider space symmetry. The conventions used to denote various symmetry elements and operations which we have considered earlier are C,, S,, and i. I n discussing space symmetry, these point symmetry operations are indicated by other symbols, the so-called ITermann-Mauguin (H-M) symbols instead of the point symmetry symbols used so far. The symmetry operations a.nd the symbols for them are shown in Table 1. From this table i t can be seen that what we have earlier designated as C, and c in the paint symmetry notation are p and m in H-M nomenclature. Also, it should he noted that in H-M nomenclature there is no i. One of the big differences hetween the H-M nomenclature aud the point symmetry notation described ezrlier lies in the handling of improper axes. In t,he H-M notat,ian.. the imnroner rotation is defined ss a rotation-invernper%~iuni n plwe i r f the rl,lalit,lr-r~crtint~S, trpcl.nr i m $MYI in the point s\.nisne
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Special Groups*
Figure 4.
A Row chart forslarrifying molecular symmetryinto point groups.
planes of symmetry (ethylene), and to DW if the molecule has two equal halves twisted 90' with respect to each other (ailene). Three (2%axes and no planes is D*. 2, the (4) If the molecule has only a rotational axis, C,, p molecule helongs to point group C, (trans-1,2-dichlorocyc10propane, CI). If in addition to 8 single C. axis there is one horizontal plane (often the moleculm plane), the molecule belongs to C-A (trans-dichloroethylene,CSA). If there is no horizontal plane hut p vertical planes, the molecule belongs to C,. (ammonia, Ca.). (5) If there is no rotational axis (other than the infinite number of CI axes) but the molecule has a plane of symmetry, it belongs to C.; if it has only a center of symmetry, Ci; and if it is truly asymmetric, it belongs to Ci.
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A systemat.ic flow chart arrangement for classification is shown in Figure 4.
Volume 47, Number 5, May 1970
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