Robert L. Carter University of Kansas Lawrence, Konsas 66044
A Flow-Chart Approach to Point Group Classifitation
The classification of molecular symmetry into the appropriate Schoenflies point group is basic to chemical applications of group theory. Yet introductions to group theory likely to be encountered by the beginner often present no systematic method for making this classification1 or present a system which is unnecessarily cumber~ome.~Cottona presents perhaps the most practical system, and this is even more succinctly stated by drag^.^ The student begins by determining if the molecule belongs to one of the Oh,Ih) m d then proceeds "special groups" (C,,, dm&,Ta, by looking for specific symmetry elements in a definite order. The existence or nonexistence of a particular element determines for which element the student should look next. I n this manner the number of possible point groups to which the molecule may belong is systematically reduced until the proper classification is the only one remaining. The logical process involved in this system is entirely analogous to such systems as the qualitative analysis scheme for cations, familiar to most freshman chemistry students. There the student performs a sequential set of operations which systematically reduce the number of possible ions to those which alone are present. Here the same process is used to determine the point group. Just as the qualitative analysis scheme lends itself to simplified representation through a "flow chart" or "inverted tree chart," so too does the systematic classification of point groups. Such a chart, based on the steps given by Cotton, is shown below. The student examines the molecular structure for the particular symmetry property designated to the right of the vertical lines on the chart. If this property is present ("Yes") he then follows the right horizontal branch; if not ("No") he follows the left branch. This is continued until a branch terminates in the 'WILSON,E.B.,D ~ c m s J. , C., AND CROSS,P. C., "Molecular Vibrations," McGmv-Hill Co., New York, 1955. HOCHSTRLSSER, R. M.. "Molecular Asoects of Svmmetrv." .. W. A. Benjamin, Inc., New kork, 1966. SCHONLAND, D. S., "Molecular ~ymmitry," D. Van Nastrand Co., Ltd., London, 1965. SJnmO, H. H., AND ORCHIN,M., "Symmetry in Chemistry," John Wiley & Sons, Ino., New York, 1965, pp. 52-53. F. A,, "Chemical Applications of Group Theory," COTTON, Interscience (a division of John Wiley & Sons, Inc.), New York, 1963, pp. 38-39. 'DRAW, R. S., "Physical Methods in Inorganic Chemistry," Rheinhold Publishing Corp., New York, 1965, p. 110.
44 / journal of Chemical Education
Schoedies symbol for the group. Note that the value of n is taken to be the order of the n-fold proper axis of highest order, except in the case of the S , groups, where n is always an even number equal to twice the order of the highest value n-fold proper axis. Presenting the classification procedure in this form has several advantages. Perhaps the least of these is that the simplistic format of the flow chart speeds the classification process itself. However, it is of greater significance that the beginning student immediately sees the similarities and differences between the various groups in a form which can be most easily digested. Longer and more thorough discussions of the individual group peculiarities usually do not provide this. Furthermore, since a flow chart serves as a mnemonic device, the beginner very quickly acquires a feeling for molecular symmetry classification. A Flow Chart for Classifying Molecular Symmetry Into Point Groups
'Special Grriops"
&,or d,and i only, eolinear with unique or highest order C.
pDn No Yes nod's
D*
D"d
