Norman C. Craig
I
M&cular
I n recent years many chemists have become familiar with the language of symmetry. This development has been stimulated by the appearance of several texts devoted to the application of group theory to chemical problems including those of Cotton ( I ) , Jaffffeand Orchin ( 2 ) ,and Schonland (3). The assimilation process has now gone far enough that the ultimate step, the introduction of symmetry considerations into the general chemistry course, no longer seems remote. I n experimenting with the presentation of the systematics of point group symmetry to college freshmen and high school students, I have found an inductive development of the subject supported by the models described in this article to he quite effective. The general chemistry course, in which molecular structure is a principal theme, is not too early for the chemistry student to begin to replace his intuitive concept of symmetry with a rigorous one. Although the application of the power of group theory to the construction and transformation of atomic and molecular wave functions is best left until the student has studied some linear algebra, he nonetheless can make immediate use of a systematic procedure for recognizing symmetry in molecules. It is likely that he will also be intrigued by the algebraic properties of the sets of symmetry operations which constitute the symmetry point groups. I n the first place, the student who has learned how to recognize symmetry features and how to assign molecules to point groups has a new geometric perspective and thus a heightened awareness of the geometry of molecules. He may notice, for example, that two molecules with rather different shapes such as diazo-
iv -.
methane, CH2=N=N,
I' I
and diazirine, CH2
\, ,
,
are
N symmetrically related, having in this case C2, symmetry. Secondly, discussions of stereoisomerism are placed on a sounder basis. Thus, optical isomerism may be said to be due to the absence of any symmetry elements save rotation axes. For example, trans-1,2difluorocyclopropane, which has only a twofold axis as a symmetry element, has optical isomers whereas cis-1,2difluorocyclopropane, which has a plane of symmetry, is the internally compensated meso form. Fuller discussions of this approach to stere~>isomerismmay he found in the texts by Jaff6and Orchin (2), by Wheatley (4))and by Rlislo!~(5). Thirdly, designations such as a, T, 6, g, and u, for molecular orbitals are clarified. Some students a t this level may even wish to investigate the meaning of orbital designations such as t2, and e, uhich they may encounter in treatments of complex ions. Finally, in the modern general chemistry course
Symmetry Models
the student may well encounter discussions of nuclear magnetic resonance and vibrational spectra to which symmetry considerations are applied. A crucial step in interpreting the nuclear magnetic resonance spectrum of a molecule is the grouping of its atoms into symmetrically equivalent sets. I n vibrational spectra "mutual exclusion" (6) of the activity of vibratioual modes between infrared absorption and Raman scattering for centrosymmetric molecules is a useful, simple selection rule. In my first experiences in presenting discussions of point group symmetry by means of an inductive development based on the geometry of selected molecular examples, I found two persistent areas of difficulty for the students. One was a tendency for them to confuse the related concepts: symmetry elements and symmetry operations. Among textbook authors Cotton (7) has emphasized this distinction. The second area of difficulty mas obtaining a full count of the number of symmetry operations that comprised the point group to which a particular molecule belonged without switching over to a deductive argument based on the postulates of group theory. The symmetry models shown in the photographs in Figures 1-3 were designed to help students avoid these difficulties. The models were inspired by a figure in the text of Wilson, Decius, and Cross (8). They are also, of course, related to stereographic projections, which are discussed, for example, in Chapter 3 in Jaff6 and Orchin's text ( 2 ) . The models have the obvious advantages of being three-dimensional representations of a three-dimensional problem and of showing the atoms of representative molecules in direct relationship to the symmetry features of a given point group. But, more importantly, the models give physical substance to the symmetry elements and the symmetry operations that usually must be seen in the mind's eye. First, we cohsider symmetry elements. Symmetry planes are Lucite sheets; symmetry axes are brass rods. For example, the C2, point group, which is illustrated in Figure 1, is based on two symmetry planes and one symmetry axis. Although the center of symmetry is not illustrated within the three models, it could be designated by a small sphere in the center of a model such as that of the trans-1,2-difluoroethylenemolecule which belongs to the C2hpoint group. Full physical representation of the fourth type of symmetry element, the reflection-rotation axis, is a prohlem to which we shall return later. Not only do the models have physical representations of symmetry elements but they also have physical representations of symmetry operations. Symmetry elements have a static relationship to a molecular structure. I n contrast, symmetry operations may be Volume
46, Number I, January 1969
/
23
t,hought to involve action. The action involved in a symmetry operation is defined with respect to a static symmetry element and is associated u-ith the transformation of a generalized point in space into it symvzetvically equivalent point. To be a generalized point, a point must not lie on any symmetry element. I n the models the arbitrary generalized point and its symmetrically equivalent points are represented as black tips on the small Lucite wedges. Alongside each point on the wedges is printed the symmetry operation which sends t,he generalized point into that equivalent point. In the C2" example, Figure 1, the arbitrary generalized point is labelled with the identity operation, 8. Two of the three symmetrically equivalent points are labelled wit,h the reflection operations a, 'nd a,' as defined by reflection in the mirror planes, and one is labelled with the rotation operation C2 as defined by 180'-rotation around the twofold rotat.ion axis. The models t,hus make clear the distinction between symmetry elements and symmetry operations: The second area of difficulty, the number of operations which make up a given point group, is also removed by the models when they are used in conjunction with t,he concept of the generalized point and its sgmmetrically equivalent point,s. Although the symmetrically equivalent. points me clearly displayed on the models under discussion, they could also easily be visualized on a model which has only the symmetry elements represented. For, in general, each point is located in one of t,he asymmetric units of space into which space is divided by the symmetry elements.' Figure 2 is a model for the Danpoint group associated with a molecule like boron trifluoride or an ion like the carbonate ion. In this model one sees immediately the necessit,y of twelve symmetrically equivalent points and hence recognizes that the D,, point group consists of twelve symmetry operations. For the D,,point group, which is associated with a molecule like allene in Figure
Figure 1. woter.
24
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M o d e l of
B, ~ y m m e t r y in relolion
Journol of Chemical Edukotion
to
molecule like
Figure 2. fluoride.
Dii, symmetry in relation to o molecule like boron tri
3, one sees by the same line of argument that this group
consists of eight symmetry operations. I n this manner a complete count of the symmetry operations of a point group is obtained before all of the types of symmetry elements and operations are fully developed. As was stated above, I have found the models to be of considerable help in an inductive development of the systematics of point group symmetry. I n this type of. development one customarily introduces the three more obvious types of symmetry elements and corresponding operations one-by-one, using for each type a molecule having only the one symmetry element. The mirror plane and associated operation of reflection can be introduced with a molecule like HOC1; the twofold rotation axis and associated operation of 180'-rotation with a molecule like HOOH; and the center of symmetry and associated operation of inversion with the trans conformer of a molecule like CFClHCFClH. The first two examples can also be related, in turn, to the C2. model,.Figure 1, by lowering its symmetry by redefining its suhstituents or by adding structures to the model. If the left-hand hall is considered to he a different color from the right-hand one, the a,' plane and the C2 axis must he disregarded. This change leaves only the a , plane and corresponding reflection operation which characterizes the C , point group. HOOH can also be related to the model if one places on the top edge of the a, plane two slotted balls each fitted with a dowel bearing a ball and pointing down into diagonally opposite quadrants. The three balls embedded in the u , plane are then disregarded as are the a, and a,' planes and corresponding operations. Only the operations E and Cz of the C1 point group remain. If finally the C2"model is restored to its orginal form, 'An additional model having demountable Lucite wedges would help to illustrate this point. For this purpose the DII symmetry associated with cyclobutane might be an interestmg choice since it involves all fonr types of symmetry elements and requires sixteen symmetrically equivdent points.
it is of course an example involving the combinat,ion of the two types of symmetry elements, the mirror plane and the t,wofold rotation axis.2 The fourth type of symmetry operation, reflection-rotat,ion (or its equivalent), cannot be introduced with a simple molecular example involving only t,his one type of operation (unless the inversion operation is replaced with the twofold reflection-rotation operation). Usually the reflection-rotation operation is introduced by appeal to authority or on the basis of a discussion of the multiplication and completeness properties of a group of symmetry operations. With the models one can adhere to the inductive argument. The natural next step in the inductive approach is to examine molecules which have higher symmetry than that of the C2. point group. The Dzh point group, to which the boron trifluoride molecule belongs, seems a t first glance in Figure 2 only to be a more complex combination of mirror planes and rotation axes. It turns out, however, to involve also the less obvious symmetry element, the rotation-reflection axis. We have already seen in our development that completeness of the group of symmetry operations is insured by the concept of symmetrically equivalent points and does not require the completeness postulate of group theory. (In fact, the several postulates of group theory can be abstracted from this inductive development.) One starts out with the arbitrary generalized point, labelled E, and then attempts to verify the symmetry operations required to transform this point into the other eleven symmetrically equivalent points. The five rotation operations CZ1,CaZ,C2, C2', and Cz" and the four reflection operations a,, a,', a,", and ah account for nine of these nperations. Hox~ever,the points labelled with the opera.t,ions S3 and S35 require a new type of operation. They can be reached by introducing the plane perpendicular to this axis. A similar development for the case of the Dza point group, Figure 3, provides anot,her example of the necessity of the reflection-rotation axis, which, in this case, is the fourfold S4 axis. In this way, the student is led, through geometric considerations alone, to the necessity of a reflection-rotation axis (or its equivalent). The presence of the C2-, S3-,and &-related operations in the and Did examples also emphasizes the fact that there is not a one-to-one correspondence between symmet,ry operations and elements, which was implied by t,he simpler examples. And, the Dz6 model immediately makes clear the location of the obscure C2 axes in an allene-like molecule. As was mentioned above the full physical representation of the reflection-rotation axis in these models is a problem. The incomplete representation of the S4 axis in the Dzd model could lead to an erroneous construction. One might mistakenly place the generalized point in the horizontal plane defined by the two C , axes. This plane, which is the unrepresented part of the S 4 On the basis of the foregoing discussion it is apparent that a model of the Cnnpoint group associated with a molecule like tram-1,2-difluoroethylenemight he a useful addition to the set of models. The Cshgroup is a simple example illustrt~tingthe inversion operation as well as those of twofold rutation aud of reflection.
I' Figure 3.
Dg,,symmetry
2
in relation to o molervle lhke ollene.
axis, is omitted since its presence might suggest a an mirror plane. A fuller physical representation of a reflection-rotation axis is unnecessary, however, if the specifications on the location of a generalized point are sufficiently restrictive. The necessary additional restriction is that a generalized point may not be located on a plane defined by two symmetry axes. Also, as in the case of allene, the shape of the molecule usually leads quite naturally to an acceptable location for the generalized point. The flexibility of the three models should not be overlooked. As was described above, each model may be used to discuss point groups of lower symmetry by altering the definition of the color of a ball or by introducing additional ones. I n the first way the DZhor D2, models can be reduced to the C2"symmetry of the simplest model. Or, the Dabmodel can be reduced to Cs. symmetry, e.g., ammonia, by placing an auxiliary ball on the top of the C3axis and disregarding the center ball. In this case the horizontal plane, a,, and the symmetrically equivalent points below this plane must also be disregarded. As examples of representing other molecules of CZasymmetry with the model for water we have C H z N = N which is formed by adding a
N string of two balls onto the C2axis and
\
N
is formed by placing two slotted balls on the top edge of the a, plane. Volume
46, Number I, January 1969 / 25
In a recent article in THIS JOURNAL (9),White has presented an introductory treatment of group theory. He emphasizes the postulates of group theory, the construction of multiplication tables, and representation theory. His article is thus a valuable complement to the emphasis on models in the present article. Acknowledgments
This article has also drawn on the experience of Dr. Martin N. Ackermann who has used these symmetry models as part of a full development of the theory of point groups in his course in advanced inorganic chemistry. The models were constructed by Mr. Harlan A. Hurd. Literature Cited (1) COTTON, F. A,, "Chemical Applications of Group Theory,"
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l o u h a l o f Chemical Education
Interscience (division of John Wiley & Sons, Inc.), New York, 1963. (2) JAFF*, El. H., AND ORCHIN,M., "Symmetry in Chemistry,'' John Wiley & Sons, Inc., New York, 1965. (3) SCHONLAND, D. S., "Molecular Symmetry," D. Van Nostrand Company, Ltd., London, 1965. (4) WHEATLEY, P. J., "The Determination of Molecular Structure," Clarendon Press (Oxford University Press), London, 1959.
(5) MISLOW, K., "Intmdu~tion to Stereochemistry," W. A. Benjamin, Inc., New York, 1966. (6) Ref. (I), pp. 265-6. (7) Ref. ( d ) , pp. 15-6. (8) 3. C.. ANDCROSS.P.C.. "Molec. . WILSON. E. B..JR.., I~ECIUS. ular Vibrations,'' McGraw-Hill Book Co., New York, 1955, p. 84. (9) WHITE,J. E., J. CHEM.EDUC.,44, 128 [1967).
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