M. J. Downward Deportment of Crystallography Birkbeck College London WCI, England
A System for the Description of Point Groups
The point group notations most familiar to the chemist are those of Schoenflies and Hermann-Mauguin, the former used by the spectroscopist, the latter by the crystallographer. However, the chemist often finds it difficult to relate these notations to the descriptions of abstract groups in standard algebra texts, and because of this he is often limited in the application of group theory to chemical problems. From the chemical education standpoint a solution may lie in introducing group theory to the chemist using the mathematical notations, first applied to twodimensional soaces then to three-dimensional soaces. The three-dimensional point group notations can be simplified in a wav exolained later so that i t is as comoact as either ~choeniiiesor Hermann-Mauguin. ~urthermore,it can be shown that the point group notation so developed gives greater insight into both the structure of the groups and the physical nature of the molecules or crystals they represent. For a point monp in three dimensions the only operations necksarffor a complete description are the s&ple rotation and the rotation-inversion. Rotation-inversion consists of rotation about an axis followed by inversion through the center of the system. If the rotation is through zero degrees then a central inversion results, hut if it is through 180" then i t is equivalent to a mirror plane of symmetry. The simple rotation is said to be a proper operation and the rotation-inversion is said to be an improper operation. Generally a proper operation is one which could actually be performed in space. Familiarity with the basic conditions of group theory is assumed, particularly that the elements are closed and associative under multiplication and that the group contains identity and inverses for all elements. The three dimensional point groups can then he classified into three classes 1) The groups which are described completely by proper operations we call the enantiomorphic class. These consist of the cyclic series having a single n-fold rotation Cn and the dihedral series D, containing the C, group as a sub-group hut adding n two-fold rotation axes a t right angles to the Cn axis. In addition to the cyclic and dihedral series there exist five special cases, the so called regular bodies. These are the tetrahedron of group A4, the cube and the octahedron of group 84, and the icosohedron and dodecahedron of group As. The letters S and A stand for symmetric and alternating as derived from the permutation groups in mathematics. The cube and the octahedron or the icosohedron and dodecahedron are each described by one point group and are said in each case to he dual. 2) The centrosvmmetric moups. For each enantiomorphic point group there will be aaorresponding centrosymmetric moup. The elements of the centrosvmmetric moup . . are obtained simply by taking each element of the enantiomorphic group and comhining with i t each element of itself multiplied by the central inversion. It is a starting premise that each group must contain the identity element I, so that the new group must contain an element I X i = i, that is, it is centrosymmetric.
3) Product groups of the type S'S", where S' is a subgroup index two of a group S and S' consists of all the elements in S that are not contained in S' each multiplied by the central inversion. Such a group is best shown by the notation SS." The condition that S ' b e an index 2 subgroup of S limits consideration to the following product groups: C d , , DL,,D d L , and &A4.
We must now develop a more compact notation, especially for the product groups. The enantiomorphic class can he denoted by the straight mathematical notation, which is virtually the same as Schoenflies' notation. The centrosymmetric groups can he represented by adding an i suffix after the corresponding enantiomorphic group notation. The product groups may he shown very simply because of the condition that S" must contain exactly half the elements of S , making it only necessary to specify whether S" is cyclic, dihedral, or alternating. For example, if S = Dr and S" = C4 then the point group may be shown asD4Cleaving no ambiguity in the notation. Several series of point groups are evolved then in each of which the values of n may run from 1 to infinity, giving Table 1. The suggested Notation Compared to Schoenflies and Hermann-Mauguin Notations
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an infinite number of point groups. But the incorporation of these point groups into a space lattice restricts the rotation axes to 1, 2, 3, 4 , and 6-fold rotations cutting down the number of point groups under consideration to the familiar 32 crystallographic point groups. These 32 point groups appear in Table 1 in the suggested notation compared to the Schoenflies and Hermann-Mauguin notations. It can be seen that eleven of the 32 crystallographic point groups belong to the enantiomorphic class, eleven to the centrosymmetric class while the remaining ten have rotation-inversion axes. The allocation of a molecule to its point group is initiated by looking for the highest rotation axis regardless of whether it is proper or improper. Having found the highest rotation axis look for the two-folds at right angles to that axis or if it is improper for the proper rotation on the same axis. It is often particularly easy to pick out a center of symmetry. The classification of the tetrahedral and octahedral symmetries to the A4 and S4 groups requires a little familiarity with the geometry of these bodies. It is often necessary to know the order of a point group, that is the number of elements in the group; in spectroscopy a matrix may have to be set up for each element while in crystallography it will indicate the number of general equivalent positions in the cell. Using the suggested notation the order of each point group is immediately obvious if a few simple rules are remembered. The order of any point group which begins Cn is equal to n except for the centrosymmetric cases where it is 2n. The order of any point group which begins D, is equal to 2n except for the centrosymmetric cases where it is 2 X 2n. For groups beginning Sp i t is 24 and for A4 i t is 12; again these values are doubled in centrosymmetric cases. In addition it is often very useful to he aware of the sub-groups or super-groups of the point group with which one is a t present involved. In the suggested notation the sub-groups and super-groups are easily observed without recourse to tables. Centrosymmetry, although a most important characteristic, is not immediately obvious in the Schoenflies notation; for example it is not clear to the investigator that Schoenflies Dm is centrosyrnmetric but that D3h is not. With the suggested notation this is not in doubt, it is in fact part of the specification. Only molecules or crystals that are completely described by proper operations can he optically active in a unique sense. This means tbat any assemblage of molecules which displays optical activity must belong to the enantiomorphic class. In terms of the notation any point group without a letter suffix could represent an optically active material. The application of mechanical pressure to a crystal will sometimes produce a charge across the crystal. This occurs only in crystals belonging to non-centrosymmetric classes.
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Table 2. The Laue Groups. Point Groups on the Same Row Have the Same Laue Grouo Crystal Svstern
enantiomomhic
Point Groups omduct
centrosvmmetric
Trielinic Monoclinic
Orthorhornbic
Trigonal
Tetragonal Hexagonal Cubic
The ten polar classes are those which contain a unique polar direction; tbat is, a polar direction whose ends are not related by the symmetry of the point group. They consist of the cyclic groups CI, Cz, C3, C4. C6 and the dihedral-cyclic product groups Dl,, Dz,, D3e, D4c, Dm. The reader may protest that Dl, does not appear in the list of point groups but he should also observe that neither does DI itself. DI is in fact exactly the same group as C2 SO that Dl, appears in the table as Cze. A moment's thought will show that a two-fold axis with no rotations a t right angles to it is identical to a one-fold axis with one twofold axis a t right angles to it. Cn and D,, are in general then the polar molecules. Pyroelectricity in crystals will of course he limited to polar classes within the crystallographic point groups. The process of taking an X-ray photograph of a crystal adds a center of symmetry to whatever point group symmetry the crystal may already have. Because of this it is usual to divide the 32 crystallographic point groups into eleven groups called the Laue groups-groups which have the same diffraction symmetry. These eleven groups are the centrosymmetric groups listed; each one is a supergroup of point groups whose notation starts in the same way, because of which the Laue groups are obvious by inspection. D4, D I ~ ,D4d, and Dlt belong to the D4 Laue group, the last letter being added as investigation proceeds (Table 2 ) . The principle of the method may be extended to cover the 230 infinite order space groups or to the 122 finite order groups obtained when time inversion is admitted as an operation. References Cdton, F. C., "Chemical Application8 of Group Theory," (2nd Ed.) Wiley-totorscience. NeaYork. 1970. Weyl. H.. ''Theory of Gmups and Quantum Meehanies." D o v e t h s s . NewYork. 1931. Ncring, E. D.."Linear Algebra and Matrix Theory.'' (2nd Ed). John Wiley & Sons. Ine.. NevYork 1970. Phillips. F. 6.:'lnVoduetia, to Cryitallography:'(3rd Ed), Longrna.~, London. I Y Q .
