J e r r y Donohue University of Pennsylvania Philadelphia 19104
A Key to Point Group Classification
The recent presentation by Carter1 of a flow-chart for point group classification has prompted the construction of a dichotomous key for this classification. In such a key the student is given a yes-or-no choice which directs him to subsequent choices, and eventually to the correct point group. As Carter pointed out, several introductions to group theory likely to be encountered by the beginner either present no systematic method for classification or present a system which is unnecessarily cumber~ome.~Carter's system, based on the steps given by CottonS(also given in more compart form by Drago4) and presented in the form of an "inverted tree," unfortunately omits a number of point groups. Furthermore, at one branch of the tree, the student, on the basis of the presence of a rather loosely defined property-"special groupsv-arrives at a cluster of no less than five point groups, the differences among which may not be obvious to the beginner.5 In the key given below the student is led to the correct point group on the basis of a maximum of seven choices. Both Schoenflies6 and Hermann-Mauguin symbols have been used: the former are older, and are favored by spectroscopists, organic chemists, etc.; they
are not as informative, per se, as the latter, which are favored by crystallographers. Unfortunately, therefore, it is desirable for students to know both sets of notations. If,for some reason, the inverted tree format is considered preferable, the way of constructing such a tree from the key below is obvious.
' CAETER,R . L., J . CHEM.EDUC.,45, 44 (19611). l F o r these references, see ref. ( 1 ) . WOWON, F. A., "Chemical Applications of Gronp Theory," Interscience (division of John Wilev Q Sons. Inc.). New York, 1963, pp. 38-39. DRAW,R.. S., "Physicd Methods in Inarganio Chemistry," Reinhold Publishing Corp., New York, 1965, p. 110. 6 1 f these differences are obviou to a person, then he is not a beginner and has no need for either a tree or a key. Spectroscopists, organic chemists, and others who currently use the Schoenflies symbols exclusively, also use the symbol o to denote a. mil.ror plane. This symbol, which apparently was arrived a t via the German "S~iezel" for minor, thence to the . Greek "on far "s" seems to us rather contrived, and we can see no compelling reason for using it in place of the HermannMauguin s p b o l m far the same operation, vir., reflection. Furthermore, to a large number of chemists, the symbol a has quite a different meaning. ~~
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A Key t o t h e Classification of t h e Point Groups
Has more than one rotation axis of 2 order hieher than 2 Does not Ikve 8 Hay planes of symmetry Does not have Has 5-fold axes Does not have Has 4-fold axes Does not have %fold axes lying in the planes of symmetry 3-fold axes not lying in the planes of symmetry 1-236 Has 5-fold axes 7 Does not have 0 4 3 2 Has &fold axes Does not have T-23 Has s t lemt one rotation axis of order 9 higher than 1 16 Does not have Has 2-fold axes perpendicular t o the principal rotation axis 10 Does not have 12 At 8: including rn rotation axis. At 10: Dnh: n even = Nlmmm, n = N; n odd = m m . 2n = N. At 11: Dns: n even = m m , 2n = N; n odd = Rm, n = N.
10. Has a. plane of symmetry perpendicular. to the principal ~ ~ t r t t i oaxis n D.L-N/mmm. mZm Does nothave 11. Has planes of symmetry parallel to the principal rotation axis Does not have
12. Has plane of symmetry perpendicular to the principal rotation axis Does not have 13. Has planes of symmetry parallel to the principal rotation axis Does not have 14. Has a center of symmetry Does not have 15. Rotation axis proper Itotation axis improper 16. Has a plane of symmetry Does not have 17. Has a center of symmetry Does not have At 11: At 12: At 13: At 14: At 16:
= N22, n = N ; n odd = N2, n = N. n even = Nlm, n = N; n odd = 2n = N . C : n even = Nmm, n = N: n odd = Nm. n = N. C.,: n odd = R, n = N. n = N. S,: n = 0 (mod. 4) =
D : n even
m,
C,n:
m,
Volume 46, Number 1, January ,1969
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