Symmetry of Non-rigid Molecules

Research: Science and Education

258 Journal of Chemical Education Vol. 79 No. 2 February 2002 JChemEd.chem.wisc.edu

Although the primary concern of molecular symmetryas presented in most texts is the static symmetry of moleculesand crystals, dynamic aspects are never far away. If moleculeswere solely rigid entities, then, for example, chemical reactionswould never occur, chemical kinetics would be a nonexistentsubject, and the idea of NMR (and other) time scales wouldhave no relevance. Chemistry, as we know it, would scarcelyexist. To some, it will come as a surprise to learn that thereexists a group theory of nonrigid floppy moleculesyet this isa subject that clearly has a potential relevance to the topicsjust mentioned.

Although the genesis of some of the ideas can be foundin earlier papers, there is no doubt that the most importantpublication on the subject of the group theory of nonrigidmolecules is to be found in the work of Longuet-Higgins (1).However, even though his ideas have been in the literature formany years, it is probably true to say that they have yet tohave a real impact on chemistry apart from specialized areas.

No doubt there are many reasons for thisin part becausesome aspects of his work proved to be the subject of debate,but perhaps, equally, because some of his ideas superficiallyappeared to be self-contradictory. The object of the presentcommunication is to present a part of the original Longuet-Higgins paper in such a way that the source of the apparentself-contradiction is revealed and therefore removed. I hopethat an obstacle to the fuller (and surely timely) incorporationof these ideas into the subject will thus also be removed (2).

The Case of BF2CH3

This communication is concerned with two moleculesthat are discussed in some detail in the Longuet-Higgins paper.The first is BF2CH3. Regarded as a static entity, the highestsymmetry that this molecule can have is Cs (Fig. 1). How-ever, the arrangement shown in Figure 1 cannot be regardedas stable; the slightest quiverthe zero point vibrations are

The Symmetry Groups of Two Nonrigid MoleculesSidney F. A. KettleSchool of Chemical Sciences, University of East Anglia, Norwich NR4 7TJ, UK; [email protected]

Advanced Chemistry Classroom and Laboratoryedited by

Joseph J. BelBrunoDartmouth College

Hanover, NH 03755

Figure 2. An arrangement of theCH3BF2 molecule with no apparentsymmetry. It may be thought of as be-ing obtained by slightly rotating theCH3 group of the upper part of Figure1 so that H-1 moves out of the mirrorplane. In this Figure are shown thenumber labels that will used in thefollowing discussion (but where theywill be without the element symbols).

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E (45)

(123) (123)(45)

(132) (132)(45)

Figure 3. Six iso-energetic arrangements of the CH3BF2 moleculeshown in Figure 2. The three in the left-hand column differ only inthe arrangement of the hydrogen atoms. Those in the right-handcolumn differ from their left-hand column partner by an interchangeof the fluorine atoms.

Figure 1. The structural arrangement in which the CH3BF2 moleculepossesses Cs symmetry; although the two diagrams appear differ-ent they are, in fact, the same. The Cs symmetry means that thetwo halves of the molecule are interconverted by reflection in themirror plane shown.

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JChemEd.chem.wisc.edu Vol. 79 No. 2 February 2002 Journal of Chemical Education 259

enoughand the molecular symmetry reduces to C1. Howcan it be then, that Longuet-Higgins treats this molecule ashaving a symmetry akin to D3h?

We need a new concept, the concept of structures that,although different, have precisely identical energies. Takethe BF2CH3 molecule in one of its lowest symmetry, C1,structures. Such an arrangement is shown in Figure 2. (Onecould think of an infinity of equally acceptable alternatives.)Are there any other structures that have precisely the sameenergy? Well, if we are prepared to label the atoms, as hasbeen done for the structure of Figure 2drawn from a dif-ferent viewpointat the top left-hand corner of Figure 3, thenthe answer clearly is yes.1 We can readily find 11 more struc-tures that are isoenergetic with that of Figure 2 but withthe atoms labeled differently; five of them are shown in Fig-ure 3. For a start, we can cyclically permute the hydrogen at-oms labeled 1, 2, and 3. We might think of this as a rotation

about a pseudo-threefold axis coincident with the CB bond,but it is not a good idea to do this. There is no need to invent amechanism; the three arrangements are energetically equivalent,whatever their mechanism of interconversion.

Having thus learned how to play the game with the threehydrogens, it is natural to turn next to the two fluorines. Theytoo can be interchanged, by some unspecified mechanism, togive isoenergetic structures. Each of the two fluorine arrange-ments can be combined with any of the three hydrogen arrange-ments to give a total of six isoenergetic atomic arrangementsin BF2CH3. All of these are shown in Figure 3, the hydrogen-atom-only permutations in the left-hand column and thesame permutations with the addition of the fluorine inter-change in the right-hand column.

But we have not reached the end. What is wrong withthe arrangement shown in Figure 4? In it, the CH3 grouphas been rotated to a quite new position relative to the CBF2plane. If one invents a mirror plane containing the CB axisand perpendicular to the CBF2 plane, the new hydrogenpositions are the mirror images of the old in this plane (butthe mirror image only applies to the atomic positions, notto their labels). The molecule at the top left-hand corner ofFigure 5 results. Immediately, we are into a new world. The newmolecule cannot be superimposed on the old but it certainlyis isoenergetic with it. The new arrangement of hydrogens isone that can cyclically permute and, in each of the permutedarrangements, can be combined with each of the alternativefluorine arrangements. We arrive at Figure 5, which containssix more isoenergetic arrangements of the atoms in BF2CH3,a total of 12 (Figs. 3 and 5, combined).

What have we done? We have discovered something aboutthe potential energy surface (a multidimensional surfacemanyquite independent atomic movements can cause a BF2CH3molecule to change its position on the surface) of the BF2CH3molecule. To proceed, we have to retrace our steps. We haveemphatically eschewed all mention of mechanism. But, with-out mentioning mechanism, we ask a related question. Is thisinterconversion feasible? Can we envisage it as occurring, by somemechanism or other? To put this into context, the intercon-version of C with B is not feasible because it would involvenuclear changes of a magnitude that would not be accessedexcept under very rare conditions. On the other hand, it isentirely reasonable that the arrangements shown in Figures3 and 5 can be interconverted. The operations that inter-convert them are feasible, even if unspecified; to interchangethe top left-hand corner diagrams of Figures 3 and 5 a verysmall rotation about the BC axis is all that is needed (not areflection in a mirror plane; that was invoked because it isunambiguous).

With this background, it is clear that the arrangementsshown in Figures 3 and 5 collectively define all feasibleisoenergetic arrangements of the atoms of BF2CH3. Note thatthe cyclic ordering of the hydrogens 1, 2, and 3 never changes.Of course, we can envisage the possibility that just two of them1 and 2, sayinterchange so that the ordering changes from123 to 213. If we were to believe that this hydrogen atominterchange is a feasible process then 12 more isoenergeticstructures immediately become accessible, a total of 24. It isthe belief that the hydrogen atom interchange is not a feasibleprocess, which leads us to restrict our discussion to the 12structures of Figures 3 and 5.

Figure 4. This arrangement is relatedto that in Figure 2 in that it representsthe same angular rotation out of themirror plane of Figure 1 but it is inthe opposite direction. The structuresof Figures 2 and 4 are therefore iso-energetic.

Figure 5. The arrangement of Figure 4 is shown at the top left-handcorner with labels to match that diagram in the same position ofFigure 3. The three molecules in the left-hand column differ only inthe arrangement of the hydrogen atoms. Those in the right-handcolumn differ from their left-hand column partner by an interchangeof the fluorine atoms.

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(23)(45)* (23)*

(31)(45)* (31)*

(12)(45)* (12)*

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So far, so good. Now comes the tricky part. Can we inventa group theory that interrelates the 12 structures shown inFigures 3 and 5? If we can, we will have invented the grouptheory of nonrigid molecules. Of course it can be done, andthis is just what Longuet-Higgins showed in his seminal paper.It can be donebut only by apparently contradicting thefundamentals of the development so far. In the discussionabove we replaced the concept of mechanism by that offeasibility. To be included, it has to be feasible that the targetstructure should be accessible. To develop the group theory weare seeking, Longuet-Higgins introduced a strange operation,the inversion of the position of every particle, nuclei andelectrons, in the center of mass of the molecule. Can onethink of a less feasible operation? Some might nominate timeinversion as even less feasible, but others might point outthat this is merely a trick that serves, for example, to changea clockwise motion into an anticlockwise one and vice versa.Perhaps the inversion of every particle in the center of mass ofthe molecule is a similar trick and we should not allow ourselvesto be too baffled by its apparent contradiction with the centraltenet of our approach.

To have a group, a variety of conditions has to be satisfied(they are invariably listed somewhere in all texts on theapplication of group theory to chemistry). For instance, wemust be able to combine any two elements of the group, theoutcome being an element of the group (not necessarilydifferent from the two that we started with). In the case ofBF2CH3, it seems reasonable that the group elements shouldbe those operations that serve to interrelate the isoenergeticstructures of Figures 3 and 5. Within either of these figures theproblem is not severe. The operations serve simply to permutethe labels on some of the atoms. Some sort of permutationoperations is involved. But how do we get from the arrange-ments of Figure 3 to those of Figure 5? While finding an actualphysical motion that will do the job is no great problem, as wesaw above, we are interested in the group theory. How do wedefine an operation that will bring about these interconversionsno matter what the actual disposition of the atoms in space?

One could say something like reflect in a mirror planecontaining the CB axis and perpendicular to the CBF2 planeand then interchange any two of the hydrogens (without thislast step we would have changed from the 123 pattern to the213). The theory could have been developed invoking such anoperation, but it was not. The reason is that this operationlacks generality. It is fine to invoke it for simple, low-symmetrymolecules (such as the two of this paper) but what of morecomplicated, perhaps inherently high-symmetry molecules? Wecannot guarantee in advance that a suitable unique plane willexist. It would be preferable to have an operation that is ofgeneral applicability. And here the strange operation of inver-sion of all particles in the center of mass of the molecule comesto our rescue. We will follow the notation used by Longuet-Higgins and represent this operation by a superscript asterisk, *.

This is a convenient point at which to complete thediscussion of notation. A cyclic permutation of the hydrogenatoms 1, 2, and 3 is represented as (123) and interchange ofthe fluorines 4 and 5 by (45). The symbol (123) can meaneither 1 is replaced by 2, 2 is replaced by 3, and 3 is replacedby 1 or 1 goes to the position previously occupied by 2, 2goes to the position previously occupied by 3, and 3 goes tothe position previously occupied by 1; take your pick, but

be consistent. In this paper we follow Longuet-Higgins andadopt the former definition.

We now return to a discussion of the * operation, or E *, asit is more convenient to label it when it stands alone. ConsiderFigure 6, which shows the molecule and labeling of Figure 2but now with the effect of the operation of inversion in thecenter of mass, E *, symbolically added. The outcome of thisoperation is pictorially something we have not seen before,but that does not mean that the outcome is a pattern that wehave not previously recognized. As Figure 6 shows, after a bit ofmanipulation, the molecule we have produced has the labelsappropriate to one of the 12 that we have chosen to excludefrom our discussion because it involved a nonfeasible interchangeof the (12) type; the hydrogens have the wrong labelingpattern. So, the operation E * will not appear in our list of fea-sible operations of CH3BF2. It will only appear in conjunctionwith an operation that corrects the wrong pattern of labelingof the hydrogens to which it leads. Such an operation is (12),so (12)* has the right arrangement of hydrogens. Boththe operations (12) and * are themselves nonfeasible but theircombination (12)* is a feasible operation, as we will see shortly.

Let us summarize the position reached so far. It is possibleto interconvert the 12 isoenergetic arrangements of BF2CH3by use of just three types of operations: a cyclic permutationof the hydrogens, an interchange of the fluorines, and, ofcourse, the operation of inversion in the center of mass ofthe molecule combined with the interchange of two of the

Figure 6. The top two diagrams show the arrangement at the top leftin Figure 3 (the identity arrangement) and the effect of E* on it. Theresult is a molecule arranged in space in a different way from thosewe have met so far. To get it back to a familiar arrangement, rotate it180 about an axis in the direction of the BC bond (see the right-hand side of the figure). This axis is shown dotted and is labeledC2. However, this is a physical rotation in space, not a symmetryoperation. Follow this (bottom) with a further physical rotation of 180about a vertical axis. The result is an arrangement akin to those inFigure 5, but it is not to be found there because the hydrogen atomarrangement is wrong: it is of the 132 type, not 123.




hydrogens. But the problem is not yet fully solved. For theoperations to form a group, it must be possible to select anyone of the arrangements in Figures 3 and 5 and, by applicationof each operation of the group in turn, to obtain each of theother 11 arrangements. That is, we must be working with agroup with 12 elements because each element (operation) mustlead to a different final arrangement (the identity operationprojects the chosen arrangement onto itself ). A glance atFigures 3 and 5 indicates the sort of answer we must expect.Some arrangements differ by a cyclic permutation of thehydrogen atoms, so the cyclic permutation operations mustbe operations of the group. Some differ by fluorine atominterchanges; they must also be operations of the group. Somediffer by both types of rearrangements, so combinations of thetwo must be included in the list of the operations of the group.However, all of these operations keep us within either Figure 3or Figure 5. To connect arrangements in different figures weneed to invoke the E * operation combined with a hydrogenatom interchange. These operations can either stand alone orbe combined with an interchange of the two fluorines.

With this background, we now have to compile a com-plete list of the operations of the nonrigid group of CH3BF2.

The simplest way of doing this is to play the game backwards:to select each arrangement in Figures 3 and 5 and ask, whatoperations connect it with the arrangement we have chosen asthe starting arrangement? The answer is given in the legendsto the individual arrangements shown in Figures 3 and 5, whichshould be examined rather carefully. In Figure 7 is given a break-down of the application of one of the difficult operations(12)(45)* was selectedand bulk rotation of the consequentarrangement back to a reference-frame position.

From Figures 3 and 5 we conclude that the 12 operationsof the nonrigid group of CH3BF2 are:

E(123), (132)

(45)(123)(45), (132)(45)

(23)*, (31)*, (12)*

(23)(45)*, (31)(45)*, (12)(45)*

Of these, the first six connect our reference molecule witharrangements in Figure 3 and the final six with arrangementsof Figure 5. In this listing we take an additional step andgive the class structure of the operations. To derive this classstructure we would first have to compile the multiplicationtable of the group elements; but this construction follows thestandard, if tedious, procedure and so will not be given here.With the class structure in place, we can obtain the charactertable for the group and this is given in Table 1; the charactertable is isomorphous to that of D3h. Examples of the use ofthis character table have been given by Longuet-Higgins (1).

The example just considered was a molecule having noinherent symmetry of the conventional type. Our secondexample, also taken from the Longuet-Higgins paper, showshow conventional symmetry operations can easily be incor-porated into the approach but warns that the outcome maycontain surprises.

The Case of N2H4

The example is hydrazine, N2H4, taken to have inherentC2 symmetry. The nonrigidity of the molecule is associated withinterchange between the two hydrogens attached to a givennitrogen. Interchange between the hydrogens on differentnitrogens is not feasible and so is not included. Interchangeof the two NH2 groups, as units, is associated with the C2operation. The E * operation, inversion of the positions of allparticles in the center of mass of the molecule, also occurs inthe nonrigid group of N2H4 in a way similar to its occurrence

Figure 7. The operation (12)(45)* acting on the identity arrange-ment. The first operation is (12), shown at the top. This is followedby the interchange of fluorines, (45), shown at the right. This isfollowed (center) by the E* operation, giving the molecule in anunfamiliar position, just as in Figure 6. This time, however, just asingle physical rotation of 180 (shown as C2 and with a dottedline to indicate the direction of the axis) is needed to give the finalarrangement (bottom). This arrangement, of course, is to be foundin Figure 5.

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in CH3BF2. In both, it connects two arrangements that cannotbe interconverted by simple cyclic permutations. In the case ofN2H4, however, E * is an acceptable operation of the nonrigidgroup. Here we see why we were careful earlier to reject anyrequired connection between the group theoretical symbolfor an operation of the group of a nonrigid molecule and thephysical process to which it corresponds, seductive though itoften is to make such a connection. In N2H4, for every initialatomic arrangement there exists a corresponding feasible atomicarrangement that is related to it by the E * operation. However,one can be confident that the actual physical process that inter-relates them does not involve the E * operation in its formaldefinition (one gets to the same end but in a feasible manner)!

In Figure 8 are given the eight isoenergetic arrangementsthat result from the (feasible) interchange of the hydrogens ateach nitrogen, both as separate pairs and combined, along withthe interchange of the two NH2 groups, which is inherent inthe existence of a C2 rotation axis. Following Longuet-Higgins,the two nitrogen atoms are labeled a and b; the hydrogens on aare labeled 1 and 2 and those on b are 3 and 4. These labels areused in Figures 8 and 9. From Figure 8 it is evident that theC2 rotation corresponds to the permutation (ab)(14)(23).

In Figure 9 are given the eight arrangements that can bereached by inclusion of the E * operation. In contrast to thepatterns in Figures 3 and 5, where quite different operationswere involved, in Figures 8 and 9 the operations are in evidentcorrespondence. Those involved in the arrangements inFigure 9 are the ones in Figure 8 augmented by the additionof the * . (But the correspondence does not mean that theyoccupy the same positions in the two Figures. As a detailedstudy will show, a sort of center of inversion operates.) Theclass structure of the 16 operations is as follows:

E(12), (34)(12)(34)

(ab)(13)(24), (ab)(14)(23)(ab)(1324), (ab)(1432)

E *(12), (34)*(12)(34)*

(ab)(13)(24)*, (ab)(14)(23)*(ab)(1324)*, (ab)(1432)*

Several points are immediately apparent from this list. Whenthe nitrogens remain unpermuted the only possible hydrogenpermutations are between those on the same nitrogen; reason-ably, (12) and (34) fall into the same class; (12)(34) is in aclass of its own. When the nitrogens permute it is inevitablethat the hydrogens also permute. One consequence is thatthe C2 rotation of the rigid group of the N2H4 molecule losesits uniqueness; the operation to which it corresponds,(ab)(14)(23), is paired with (ab)(13)(24) in the same class.Whereas for the rigid molecule there is only one way in whichthe hydrogens can interchange under a C2 rotation, in the non-rigid molecule there are two possible patterns of hydrogen inter-change. Cyclic permutations of the hydrogens are impossiblein the absence of nitrogen interchange; the two possible cyclicpermutations, (ab)(1324) and (ab)(1432), fall in the sameclass. It is evident from the class structure that the group is adirect product group of the group of the first five classes andone that comprises E and E *. This direct product relation-

ship is, of course, also seen in the character table, which isisomorphous to D4h. Both this table and applications of itare given by Longuet-Higgins.

Finally, the groups of nonrigid molecules of the sort thatare the subject of this paper have been given several names,of which the molecular symmetry group is one. But perhapstalking about the dynamic symmetry group of a moleculeis better. The dynamic symmetry groups relate points of thesame energy and topology on the (multidimensional) potentialenergy surface of a molecule.

Note that not all molecules have dynamic symmetrygroups. A triatomic cannot, for instance. In the water moleculethe two hydrogens can be permuted, of course, but the resultcan be superimposed on the starting arrangement. A tetratomiccan. Think of the hypothetical molecule N2H2, hydrazine withone hydrogen atom removed from each nitrogen. Considera plane defined by N2H; the remaining hydrogen can be innon-superimposable equivalent positions on either side of this

Figure 8. The simple permutation operations of the hydrazine mol-ecule. The left-hand column shows permutations of the hydrogens.The top right-hand diagram is derived from the top left by a (genuine)twofold rotation (shown dotted and indicated as C2). However, thisoperation is not indicated as such but is described by the permutationthat it produces. Each diagram in the right-hand column is derivedfrom its left-hand partner by this C2 rotation.




plane. A similar result holds if both hydrogens were removedfrom the same nitrogen. With increase of molecular size andnumber of chemically equivalent groups the size of the dynamicsymmetry group can increase rapidly. So, as Longuet-Higginspointed out, the molecule B(CH3)3 has a dynamic symmetrygroup with 324 operations. For C(CH3)4 the number is 1944.But take care. For Fe(C5H5)2 it is only 20.

Although it has had little impact on student tuition, thework of Longuet-Higgins has spawned an extensive literature.One can find books, conferences, and reviews based on it (3).The apparently subjective nature of the choice of feasibleoperations has been the subject of some controversy. Analternative approach circumventing this concept was discussedby Altmann, in particular (4). There are several papers fromothers who have sought to define more closely the differences(not just of approach but also of applicability) of the variousmodels advanced (5). Many applications have been described.

Figure 9. The permutations of the hydrazine molecule involving theE* operation. The diagram in the top left-hand corner is derivedfrom that in the same position in Figure 8. Those below it are de-rived from it by simple permutation operations. Each diagram inthe right-hand column is derived from its left-hand partner by theC2 rotation that is shown (note the change in position of this axiscompared with Figure 8).

For instance, the application to the vibrational symmetry ofnonrigid molecules was discussed by Turrell (6 ). However,despite the improved understanding that has resulted with thepassage of time, the work of Longuet-Higgins remains keyand an understanding of it is essential for even the most cursoryentry into the field. It is hoped that the present contributionwill facilitate this entry.

Postscript

In the first of the examples in this paper, CH3BF2, theinterchange of two hydrogens, and thus a change from a 123 toa 132 ordering, was excluded as a feasible mechanism. Yet justsuch an interchange was regarded as feasible for the secondexample, N2H4. The energies implicit in the sort of inter-changes discussed in this paper are small, perhaps in the micro-wave region, yet one wonders whether there is a qualitativeexplanation for this difference between the two examples. Is itthe difference between a nitrogen lone pair and a CH bond?Is it the different polarities of the central bond in the twomolecules? Or a combination, or something else? Answerson a postcard, please.

Note

1. A worthwhile exercise is to compare carefully diagrams atthe top left-hand corner of related figures. They are always relatedin a very simple way, even if the legends might lead you to thinkotherwise. Similarly, there is a simple (and, within any one figureand for related figures, constant) relationship between diagramsdrawn side by side.

Literature Cited

1. Longuet-Higgins, H. C. Mol. Phy. 1963, 6, 445460.2. I am aware of only oneindirectreference to the work of

Longuet-Higgins in the student textbook literature. This is in adiscussion of the vibrations of already vibrating molecules (wherethe instantaneous symmetry is usually much lower than themolecular symmetry) to be found in Kettle, S. F. A. Symmetryand Structure, Readable Group Theory for Chemists; Wiley:Chichester and New York, 1995.

3. See, for example, Ezra, G. S. Symmetry Properties of Molecules;Springer: New York, 1982. Bocras, J. The Permutational Approachto Dynamic Stereochemistry; Cambridge University Press: NewYork, 1983. Gielen, M. F. Advances in Dynamic Stereochemistry;Freund: London, 1988. Smeyers, Y. G. Adv. Quantum Chem.1992, 24, 177. Frei, H.; Bauder, A.; Gnthard H. H. Top.Curr. Chem. 1979, 81, 197. Studies in Physical and Theoreti-cal Chemistry, Vol. 23, Symmetries and Properties of Non-Rigid Molecules; Maruani, J., Serre, J., Eds.; Elsevier: New York,1983; proceedings of an international symposium, Paris, 17Jul 1982.

4. Altmann, S. L. Proc. R. Soc. London 1967, 298, 184; Mol. Phys.1971, 21, 587607.

5. See, for example, Dalton, B. J. Mol. Phys. 1966, 11 265-285;Woodman C. M., Molecular Physics 1970 19, 753780.Watson J. K. G. Mol. Phys. 1971, 11, 577585. Ezra, G. S.Mol. Phys. 1981, 43, 773783.

6. Turrell, G. J. Mol. Struct. 1970, 5, 245252.


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Symmetry of Non-rigid Molecules