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10.1021/ed100263f Published on Web 06/10/2010

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A Pictorial Visualization of Normal ModeVibrations of the Fullerene (C60) Moleculein Terms of Vibrations of a Hollow SphereJanette L. DunnSchool of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, U.K.janette.dunn@nottingham.ac.uk

A normal mode vibration of a molecule is one in which allatoms in the molecule move with the same vibrational frequencyand simultaneously pass through their equilibrium positionswith the center of mass remaining in the same position. Eachatom can move in three directions, but motion representing anoverall rotation or translation of the molecule is not included.It is important to understand the concept of normal modes inorder to interpret vibrational spectra.

Methods of calculating normal modes of molecules aredescribed in many textbooks, where it is shown that grouptheory can be used to help simplify the problem and classifythe normal modes in terms of irreducible representations(irreps) (1, 2). Pictures of normal modes can be represented instatic images by depicting motion of molecules using arrows (3).However, these can be difficult to visualize for all but the simplestmolecules. Web-based (4) or other molecular modeling soft-ware that can show 3D animations of the normal modes improvesthe situation, but for molecules in which a large number ofatoms are moving in 3D, even these results can be difficult tounderstand.

An important molecule in many areas of chemistry andphysics is the fullerene molecule C60. When at rest, its 60 carbonatoms are at the corners of a truncated icosahedron, in a patterncommonly recognized as a soccer ball. A truncated icosahedron isformed by removing the corners from a regular icosahedron, asshown in Figure 1. The geometry is such that all of the carbonatoms lie on the surface of a sphere. In general, a molecule withN atoms will have 3N- 6 normal modes (after subtracting threepossible molecular rotations and three translations), meaningthat C60 has 174 normal modes. It is impossible to drawmeaningful pictures of the normal modes using arrows becauseof the complex 3D nature of the problem. Even in animations, itis difficult for the eye to follow the simultaneous radial andtransverse movements of 60 atoms in 3D.

While normal mode problems in chemistry are usuallyconcerned with vibrations of atoms in a molecule, physicistsare more familiar with normal modes of systems, such as avibrating wire or circular drum membrane. A thin hollow spherecan be analyzed in the same way (5). In this article, we will showhow images that superimpose vibrations of a thin spherical shellon to the normal modes of C60 provide a guide to the eye thatmakes the overall pattern of the normal mode motions becomemuch clearer. Following contours on the sphere makes it easier tosee howmuch of themotion is radial and howmuch is tangential.Motion toward and away from the viewer may not be veryobvious in usual visual representations of the 3D motion, but

including the sphere allows such motion to be seen more clearly,such as changes in the contours from hills to valleys.

Group Theoretical Considerations

It is usual to identify normal modes of molecules using labelsof irreps from group theory. The labelsA, E,T,G, andH are usedto label sets of 1, 2, 3, 4, and 5 normal modes, respectively, wherethe individual components of each set have the same frequency.Additional labels g and u are used to distinguish between irrepsthat have even and odd natures, respectively (from the Germangerade and ungerade). Additional numerical labels (1, 2, etc.) areused to distinguish between irreps that would otherwise have thesame labels. The C60 molecule has icosahedral (Ih) symmetry andits 174 normal modes can be classified as 2Ag þ 3T1g þ 4T2g þ6Gg þ 8Hg þ Au þ 4T1u þ 5T2u þ 6Gu þ 7Hu.

For all irreps except A, the designation of the normal modesis not unique. Any linear combination of the components of aparticular irrep that have the same frequency will also be normalmodes. If we define x, y, and z to be twofold axes of anicosahedron as in Figure 1, then we can define particular linearcombinations of T modes such that the displacements of onecomponent with respect to the x axis are the same as those of asecond component with respect to the y axis and a thirdcomponent with respect to the z axis. We can understand thisin terms of the hydrogen-like dyz, dzx, and dxy orbitals, where theshape of the orbitals is the same but their orientation is aboutthree mutually orthogonal axes. Three components of eachG and H mode can be defined in a similar manner. For theGmodes, these components are labeled {x, y, z}, whereas for theH mode, we use the labels { yz, zx, xy} because of theircorrespondence to dyz, dzx, and dxy orbitals.

With the definitions described above, it is only necessary toconsider one component of each T mode, two components ofeach G mode, and three components of each H mode. Thisreduces the number of unique normal mode displacementpatterns to consider from 174 to 88. Another effect of usingthis definition is that the results will also apply to situations inwhich the symmetry is reduced from Ih to one of the subgroupsTh, D2h, or C2h. This is because the defined components mapdirectly into irreps of these subgroups. This could be useful fordescribing molecules of a lower symmetry than Ih, or in under-standing the Jahn-Teller effect in C60 and other icosahedralmolecules, where interactions between the electronicmotion andthe vibrations may produce a distortion to lower symmetry.

By convention, the remaining components of the G andH modes are labeled a and {θ, ε}, respectively. We will define

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the θ and ε components such that components of theHgmodescorrespond to the linear combinations

ð ffiffiffi

3p

d3z2 - r2 þffiffiffi

5p

dx2 - y2Þ=ffiffiffi

8p

and

ð ffiffiffi

3p

dx2 - y2 -ffiffiffi

5p

d3z2 - r2Þ=ffiffiffi

8p

of the hydrogen-like d3z2-r2 and dx2-y2 orbitals, respectively (6).

Normal Mode Analysis

The purpose of this article is not to show how the normalmodes of C60 can be obtained. All we need is knowledge of thepositions of the carbon atoms in the different normal modes.These can be obtained from a force-constant model, in whichinteractions between neighboring atoms are described in terms ofchanges in bond lengths and bond angles, and numerical valuesfor the force constants in the model can be determined bymatching the vibrational frequencies predicted by the model tospectroscopic measurements. Alternatively, molecular modelingsoftware using various ab initio methods can be used. We use theresults of the force field model defined in ref 7, although theresults of other approaches could be used equally well.

The normal modes of a thin spherical shell can be written interms of vector forms of spherical harmonics (5), as introduced byStone (8). These are extensions of the usual spherical harmonicsYlm(θ,j) that represent eigenfunctions of angular momentum.They are defined such that the radial displacements at a point(θ,j) are proportional to the Yl,m. (l and m are quantum numbersas usually defined for angular momentum functions, namely, l is aninteger g0 and m is an integer between þl and -l.) Tangentialdisplacements are a combination of even and odd harmonics

V θlmðθ,φÞeθ þ V φ

lmðθ,φÞeφ~V θlmðθ,φÞeθ þ ~V φ

lmðθ,φÞeφrespectively, where eθ and eφ are the usual unit vectors of aspherical coordinate system (see Figure 1 of ref 5) and

V θlmðθ,φÞ ¼ ~V φ

lmðθ,φÞ ¼ ∂

∂θYlmðθ,φÞ

V φ

lmðθ,φÞ ¼ - ~V θlmðθ,φÞ ¼ 1

sin θ

∂

∂φYlmðθ,φÞ

We now need to determine combinations of the normalmodes of a thin spherical shell such that each of the atoms in anormal mode of C60 lies on the surface of the distorted shell.There are an infinite number of ways of doing this. The surfacesfor the different ways will differ at points in between the carbonatoms, but these are only included as a guide to the eye and do nothave a physical significance. As higher values of angular momen-tum l tend to have more nodes than lower values, we will choose aset of distortions involving low values of l as they produce thesimplest set of images.

We can produce results for all of the normal modes of C60

by including normal modes of a spherical shell with l up to andincluding some of the states with l = 8, as detailed in Table III ofref 5. Most of the required results are tabulated in ref 9, and theremainder can be generated from lower values of l using tabulatedvalues of Clebsch-Gordan coefficients in ref 6.

Results

We have generated an interactive demonstration that canshow any of the 88 unique normal modes of C60 (10) along withdistortions of a thin spherical shell. A screenshot of the demon-stration is shown in Figure 2. The demonstration has beenwritten in Mathematica and published as a Wolfram Demon-strations Project. In the demonstration, each mode is labeled byits irrep and the transformation properties of its components.Following convention, modes of the same symmetry are num-bered in order of increasing frequency. There are options to viewjust the atoms and bonds, the atoms and bonds joined bypolygons (equivalent to a deforming soccer ball), and the atomsand bonds with distortions of the underlying sphere. Themaximum displacement can be chosen, the image set to vibrate,and the orientation of the vibration changed.

In addition to the interactive demonstration, we have alsoobtained animations of all 174 modes with fixed orientations,

Figure 1. A truncated icosahedron is formed by slicing the corners off aregular icosahedron. Twofold x, y, and z axes are also indicated.

Figure 2. A screenshot of the Wolfram Demonstrations Project of thenormal mode vibrations of C60.

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which are collected together in the supporting information andon Web pages (11). The results can be viewed either with allmodes having the same symmetry transformation properties groupedtogether or with all modes having the same frequency groupedtogether. The frequencies have been taken from Schettinoet al. (12). Our results have the same qualitative form as resultsin the literature (13, 14), although a quantitative comparison isnot possible, as the components of irreps in these papers do nothave defined transformation properties.

We will now examine some of the normal modes in moredetail. We will start by examining the two Ag modes. It is wellknown that the lower-frequency mode is the breathing mode,where the motion is entirely radial, and the higher-frequencymode is the so-called pentagonal pinch mode, where the penta-gons change size with no radial motion (5, 13, 15, 16). Ourimages are in agreement with this result. As these modes aresimple to interpret and the sphere does not distort in either case,there is no advantage in obtaining images with the sphere.However, it does help verify that the method of calculation isvalid.

Images of the distorting sphere are most useful in distin-guishing between radial and tangential motion of individualatoms. In previous papers, it has been noted that there is aprogression from radial to tangential motion for all symmetrieswhen moving from lower to higher frequencies (13). This isbecause there is no central atom and the carbon-carbon bondsare strong, which favors a high-frequency tangential motion (13).For the modes with predominantly radial motion, imagesincluding the sphere are particularly useful in discerning theoverall shape of the modes. Consider as an example the lowest-frequency T1u mode, which we label T1u(1). Static views of thex-component of this mode are shown in Figure 3. Figure 3Ashows distortions of opposite signs to each other with represen-tations of atoms and bonds only. Figure 3B shows the same butwith the distortions of the underlying sphere. In Figure 3A, it isnot clear that the atom in the center of the picture is movingbetween a larger and a smaller radius than the neighboring atomsbecause in a flat 2D image, it is not possible to distinguish radial

motion toward and away from the viewer. The radial motionbecomes clearer in Figure 3B, where the contours indicate thatthis atom moves between lying on a peak and lying in a valley.Also, the overall shape of the mode is clearer. It should be notedthat the overall magnitude of the displacements is exaggeratedfrom that of the real molecule in order to emphasize the motioninvolved.

For modes in which the motion is predominantly tangen-tial, images with the sphere are useful in showing small radialcomponents that are otherwise difficult to observe. For example,it is difficult to determine whether there is any radial motion inthe x-component of the T1u(3) mode from images without thesphere. When the sphere is included, it is obvious that there is asmall amount of radial motion from small protrusions to thesphere and lateral movement on the contours (10, 11).

We have already discussed how it is only necessary toconsider 88 of the 174 normal modes as the remaining 86 modescan be obtained by appropriate rotation about the x, y, and z axes.However, this is not obvious at first sight. The equivalencebetween components is illustrated in Figure 4 for the Hg(2)mode. The images on the left show the displacements with acommon viewpoint, and the images on the right show the samedisplacements but with the viewpoint changed to illustrate thatthe pattern of atomic displacements is in fact the same in all threecases. The same can be done for all otherT,G, andHmodes. Thereader can verify this by rotating images in the interactivedemonstration (10). Neither the a component of the G modes

Figure 3. Plots of the x-component of the T1u(1) mode with displace-ments of opposite relative signs: (A) representationwith atoms and bondsonly; (B) as (A) but showing distortions of the underlying sphere.

Figure 4. Plots of the components of the Hg(2) mode labeled (A) yz;(B) zx; and (C) xy. The images on the left have a common orientation inwhich the z axis (origin of contours) is drawn vertically, while the imageson the right are the same but reoriented to show that the atomic displace-ments are the same in all three cases.

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or the θ and ε components of theHmode can be obtained fromtheir other components by a simple rotation of axes. However,they do share visual similarities, showing similar degrees of radialand tangential displacements to the other components of thesame mode. This can also be seen using the interactive demon-stration (10) and in the supporting information organized byfrequency (11). For example, all components of theHg(1) modecan be seen to be approximate ellipsoids.

Concluding Remarks

We have produced an interactive demonstration giving avisual representation of all of the normal modes of the C60

molecule that includes distortions of the sphere upon which thecarbon atoms line in the undistorted molecule (10). Thisprovides a useful guide to the eye that aids interpretation ofthe complicated simultaneous 3D movements of 60 atoms. Theimages can be animated to show the vibration, and rotated tochange the orientation of the vibration. We have also generatedanimations with fixed orientations in which modes of the samesymmetry can be viewed simultaneously (11). In all cases, we havechosen components of each mode that have specific transforma-tion properties. It also means that there are only 88 uniquevibrational displacements, from which the remaining 86 modescan be obtained by rotation about the x, y, and z axes.

The images we have obtained illustrate the molecularmotion of the fascinating C60 molecule. It is not currently pos-sible to discern directly this motion in experiments, althoughtechnological developments may make this possible in the future.It is already possible to see signatures of atomic arrangements inscanning tunneling microscopy images of C60 molecules onsurfaces (17).

Images including distortions of a thin spherical shell are notonly useful in visualizing vibrations of the C60 molecule, but theycan also help visualize the structure of other molecules in whichthe positions of the atoms are nearly spherical. For example,possible structures for Si60 and Ge60 clusters havingTh symmetryhave been proposed in which the 60 atoms are arrangedinto pentagons and hexagons as in C60 but in which the bondlengths are no longer equal. More specifically, the pentagons inthe proposed structures have either one or two different bondlengths, and the hexagons have two or three different bondlengths (18). Studying the images of the normal mode vibrationsof C60, we can see that such structures can be generated fromcombinations of the a components of the Gg modes (whichtransform as Ag in Th symmetry). Later papers have suggestedthat these clusters deform to a lower symmetry than Th (19, 20).Images of these structures including a distorted spherical shellcould also be obtained if sufficient information were knownabout the positions of the atoms in these structures. Themethod could also be used, for example, to probe chlorine-stabilized isomers of C60 as an alternative to the usualicosahedral structure (21), or to explore structures of thehypothetical boron buckyball, B80 (22). It would be interest-ing to see whether the method could be used to help under-stand the normal mode vibrations of icosahedral viruses,

where the large number of atoms involved creates a problem thatis very complex indeed (23).

Literature Cited

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81, 1231.5. Ceulemans, A.; Fowler, P. W.; Vos, I. J. Chem. Phys. 1994, 100,

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10. Normal Mode Vibrations of Buckminsterfullerene (C60), WolframDemonstrations Project. http://demonstrations.wolfram.com/NormalModeVibrationsOfBuckminsterfullereneC60/ (accessedJun 2010).

11. The University of Nottingham Web page of the Fullerene TheoryGroup, Normal Modes of the Fullerene Molecule (C60). http://www.nottingham.ac.uk/~ppzjld/Visualise_vibration/ (accessedJun 2010).

12. Schettino, V.; Pagliai, M.; Ciabini, L.; Cardini, G. J. Phys. Chem. A2001, 105, 11192.

13. Weeks, D. E.; Harter, W. G. J. Chem. Phys. 1989, 90, 4744.14. Heid, R.; Pintschovius, L.; Godard, J. M. Phys. Rev. B 1997, 56,

5925.15. Adams, G. B.; Page, J. B.; Sankey, O. F.; O'Keeffe, M. Phys. Rev. B

1994, 50, 17471.16. van Vlijmen, H. W. T.; Karplus, M. J. Chem. Phys. 2001, 115, 691.17. Wachowiak, A.; Yamachika, R.; Khoo, K.H.;Wang, Y.; Grobis,M.;

Lee, D.-H.; Louie, S. G.; Crommie, M. F. Science 2005, 310, 468.18. Li, B. X.; Jiang, M.; Cao, P. L. J. Phys.: Condens. Matter 1999, 11,

8517.19. Han, J. G.; Ren, Z. Y.; Sheng, L. S.; Zhang, Y. W.; Morales, J. A.;

Hagelberg, F. J. Mol. Struct. 2003, 625, 47.20. Chen, Z. F.; Jiao, H. J.; Seifert, G.; Horn, A. H. C.; Yu, D. K.; Clark,

T.; Thiel, W.; Schleyer, P. V. J. Comput. Chem. 2003, 24, 948.21. Tan, Y. Z.; Liao, Z. J.; Qian, Z. Z.; Chen, R. T.; Wu, X.; Liang, H.;

Han, X.; Zhu, F.; Zhou, S. J.; Zheng, Z. P.; Lu, X.; Xie, S.-Y.; Huang,R.-B.; Zheng, L.-S. Nat. Mater. 2008, 7, 790.

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23. van Vlijmen, H. W. T. In Normal Mode Analysis. Theory andApplications to Biological and Chemical Systems, Cui, Q., Bahar, I.,Eds.; Chapman & Hall/CRC: Boca Raton, FL, 2006; published aspart of the Mathematical and Computational Biology Series.

Supporting Information Available

Images of all 174 normal modes of C60 in a common orientation,each of which can be animated, are available. This material is availablevia the Internet at http://pubs.acs.org.