A Pictorial Visualization of Normal Mode Vibrations of the Fullerene

In the Classroom

A Pictorial Visualization of Normal Mode Vibrations of the Fullerene (C60) Molecule in Terms of Vibrations of a Hollow Sphere Janette L. Dunn School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, U.K. [email protected]

A normal mode vibration of a molecule is one in which all atoms in the molecule move with the same vibrational frequency and simultaneously pass through their equilibrium positions with the center of mass remaining in the same position. Each atom can move in three directions, but motion representing an overall rotation or translation of the molecule is not included. It is important to understand the concept of normal modes in order to interpret vibrational spectra. Methods of calculating normal modes of molecules are described in many textbooks, where it is shown that group theory can be used to help simplify the problem and classify the normal modes in terms of irreducible representations (irreps) (1, 2). Pictures of normal modes can be represented in static images by depicting motion of molecules using arrows (3). However, these can be difficult to visualize for all but the simplest molecules. Web-based (4) or other molecular modeling software that can show 3D animations of the normal modes improves the situation, but for molecules in which a large number of atoms are moving in 3D, even these results can be difficult to understand. An important molecule in many areas of chemistry and physics is the fullerene molecule C60. When at rest, its 60 carbon atoms are at the corners of a truncated icosahedron, in a pattern commonly recognized as a soccer ball. A truncated icosahedron is formed by removing the corners from a regular icosahedron, as shown in Figure 1. The geometry is such that all of the carbon atoms lie on the surface of a sphere. In general, a molecule with N atoms will have 3N - 6 normal modes (after subtracting three possible molecular rotations and three translations), meaning that C60 has 174 normal modes. It is impossible to draw meaningful pictures of the normal modes using arrows because of the complex 3D nature of the problem. Even in animations, it is difficult for the eye to follow the simultaneous radial and transverse movements of 60 atoms in 3D. While normal mode problems in chemistry are usually concerned with vibrations of atoms in a molecule, physicists are more familiar with normal modes of systems, such as a vibrating wire or circular drum membrane. A thin hollow sphere can be analyzed in the same way (5). In this article, we will show how images that superimpose vibrations of a thin spherical shell on to the normal modes of C60 provide a guide to the eye that makes the overall pattern of the normal mode motions become much clearer. Following contours on the sphere makes it easier to see how much of the motion is radial and how much is tangential. Motion toward and away from the viewer may not be very obvious in usual visual representations of the 3D motion, but

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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 labels of irreps from group theory. The labels A, E, T, G, and H are used to label sets of 1, 2, 3, 4, and 5 normal modes, respectively, where the individual components of each set have the same frequency. Additional labels g and u are used to distinguish between irreps that have even and odd natures, respectively (from the German gerade and ungerade). Additional numerical labels (1, 2, etc.) are used to distinguish between irreps that would otherwise have the same labels. The C60 molecule has icosahedral (Ih) symmetry and its 174 normal modes can be classified as 2Ag þ 3T1g þ 4T2g þ 6Gg þ 8Hg þ Au þ 4T1u þ 5T2u þ 6Gu þ 7Hu. For all irreps except A, the designation of the normal modes is not unique. Any linear combination of the components of a particular irrep that have the same frequency will also be normal modes. If we define x, y, and z to be twofold axes of an icosahedron as in Figure 1, then we can define particular linear combinations of T modes such that the displacements of one component with respect to the x axis are the same as those of a second component with respect to the y axis and a third component with respect to the z axis. We can understand this in terms of the hydrogen-like dyz, dzx, and dxy orbitals, where the shape of the orbitals is the same but their orientation is about three mutually orthogonal axes. Three components of each G and H mode can be defined in a similar manner. For the G modes, these components are labeled {x, y, z}, whereas for the H mode, we use the labels { yz, zx, xy} because of their correspondence to dyz, dzx, and dxy orbitals. With the definitions described above, it is only necessary to consider one component of each T mode, two components of each G mode, and three components of each H mode. This reduces the number of unique normal mode displacement patterns to consider from 174 to 88. Another effect of using this definition is that the results will also apply to situations in which the symmetry is reduced from Ih to one of the subgroups Th, D2h, or C2h. This is because the defined components map directly into irreps of these subgroups. This could be useful for describing molecules of a lower symmetry than Ih, or in understanding the Jahn-Teller effect in C60 and other icosahedral molecules, where interactions between the electronic motion and the vibrations may produce a distortion to lower symmetry. By convention, the remaining components of the G and H modes are labeled a and {θ, ε}, respectively. We will define

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r 2010 American Chemical Society and Division of Chemical Education, Inc. pubs.acs.org/jchemeduc Vol. 87 No. 8 August 2010 10.1021/ed100263f Published on Web 06/10/2010

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In the Classroom

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

the θ and ε components such that components of the Hg modes correspond to the linear combinations pffiffiffi pffiffiffi pffiffiffi ð 3d3z2 - r2 þ 5dx2 - y2 Þ= 8 and

pffiffiffi pffiffiffi pffiffiffi ð 3dx2 - y2 - 5d3z2 - r2 Þ= 8

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 normal modes of C60 can be obtained. All we need is knowledge of the positions of the carbon atoms in the different normal modes. These can be obtained from a force-constant model, in which interactions between neighboring atoms are described in terms of changes in bond lengths and bond angles, and numerical values for the force constants in the model can be determined by matching the vibrational frequencies predicted by the model to spectroscopic measurements. Alternatively, molecular modeling software using various ab initio methods can be used. We use the results of the force field model defined in ref 7, although the results of other approaches could be used equally well. The normal modes of a thin spherical shell can be written in terms of vector forms of spherical harmonics (5), as introduced by Stone (8). These are extensions of the usual spherical harmonics Ylm(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 numbers as usually defined for angular momentum functions, namely, l is an integer g0 and m is an integer between þl and -l.) Tangential displacements are a combination of even and odd harmonics φ θ ðθ, φÞe^θ þ Vlm ðθ, φÞe^φ Vlm φ θ V~lm ðθ, φÞe^θ þ V~lm ðθ, φÞe^φ

respectively, where êθ and êφ are the usual unit vectors of a spherical coordinate system (see Figure 1 of ref 5) and ∂ φ θ Ylm ðθ, φÞ ðθ, φÞ ¼ V~lm ðθ, φÞ ¼ Vlm ∂θ 1 ∂ φ Ylm ðθ, φÞ Vlm ðθ, φÞ ¼ - V~ θlm ðθ, φÞ ¼ sin θ ∂φ

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

We now need to determine combinations of the normal modes of a thin spherical shell such that each of the atoms in a normal mode of C60 lies on the surface of the distorted shell. There are an infinite number of ways of doing this. The surfaces for the different ways will differ at points in between the carbon atoms, but these are only included as a guide to the eye and do not have a physical significance. As higher values of angular momentum l tend to have more nodes than lower values, we will choose a set of distortions involving low values of l as they produce the simplest 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 and including some of the states with l = 8, as detailed in Table III of ref 5. Most of the required results are tabulated in ref 9, and the remainder can be generated from lower values of l using tabulated values of Clebsch-Gordan coefficients in ref 6. Results We have generated an interactive demonstration that can show any of the 88 unique normal modes of C60 (10) along with distortions of a thin spherical shell. A screenshot of the demonstration is shown in Figure 2. The demonstration has been written in Mathematica and published as a Wolfram Demonstrations Project. In the demonstration, each mode is labeled by its irrep and the transformation properties of its components. Following convention, modes of the same symmetry are numbered in order of increasing frequency. There are options to view just the atoms and bonds, the atoms and bonds joined by polygons (equivalent to a deforming soccer ball), and the atoms and bonds with distortions of the underlying sphere. The maximum displacement can be chosen, the image set to vibrate, and the orientation of the vibration changed. In addition to the interactive demonstration, we have also obtained animations of all 174 modes with fixed orientations,

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r 2010 American Chemical Society and Division of Chemical Education, Inc.

In the Classroom

Figure 3. Plots of the x-component of the T1u(1) mode with displacements of opposite relative signs: (A) representation with atoms and bonds only; (B) as (A) but showing distortions of the underlying sphere.

which are collected together in the supporting information and on Web pages (11). The results can be viewed either with all modes having the same symmetry transformation properties grouped together or with all modes having the same frequency grouped together. The frequencies have been taken from Schettino et al. (12). Our results have the same qualitative form as results in the literature (13, 14), although a quantitative comparison is not possible, as the components of irreps in these papers do not have defined transformation properties. We will now examine some of the normal modes in more detail. We will start by examining the two Ag modes. It is well known that the lower-frequency mode is the breathing mode, where the motion is entirely radial, and the higher-frequency mode is the so-called pentagonal pinch mode, where the pentagons change size with no radial motion (5, 13, 15, 16). Our images are in agreement with this result. As these modes are simple 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 is valid. Images of the distorting sphere are most useful in distinguishing between radial and tangential motion of individual atoms. In previous papers, it has been noted that there is a progression from radial to tangential motion for all symmetries when moving from lower to higher frequencies (13). This is because there is no central atom and the carbon-carbon bonds are strong, which favors a high-frequency tangential motion (13). For the modes with predominantly radial motion, images including the sphere are particularly useful in discerning the overall shape of the modes. Consider as an example the lowestfrequency T1u mode, which we label T1u(1). Static views of the x-component of this mode are shown in Figure 3. Figure 3A shows distortions of opposite signs to each other with representations of atoms and bonds only. Figure 3B shows the same but with the distortions of the underlying sphere. In Figure 3A, it is not clear that the atom in the center of the picture is moving between a larger and a smaller radius than the neighboring atoms because in a flat 2D image, it is not possible to distinguish radial


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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 in which the z axis (origin of contours) is drawn vertically, while the images on the right are the same but reoriented to show that the atomic displacements are the same in all three cases.

motion toward and away from the viewer. The radial motion becomes clearer in Figure 3B, where the contours indicate that this atom moves between lying on a peak and lying in a valley. Also, the overall shape of the mode is clearer. It should be noted that the overall magnitude of the displacements is exaggerated from that of the real molecule in order to emphasize the motion involved. For modes in which the motion is predominantly tangential, images with the sphere are useful in showing small radial components that are otherwise difficult to observe. For example, it is difficult to determine whether there is any radial motion in the x-component of the T1u(3) mode from images without the sphere. When the sphere is included, it is obvious that there is a small amount of radial motion from small protrusions to the sphere and lateral movement on the contours (10, 11). We have already discussed how it is only necessary to consider 88 of the 174 normal modes as the remaining 86 modes can be obtained by appropriate rotation about the x, y, and z axes. However, this is not obvious at first sight. The equivalence between components is illustrated in Figure 4 for the Hg(2) mode. The images on the left show the displacements with a common viewpoint, and the images on the right show the same displacements but with the viewpoint changed to illustrate that the pattern of atomic displacements is in fact the same in all three cases. The same can be done for all other T, G, and H modes. The reader can verify this by rotating images in the interactive demonstration (10). Neither the a component of the G modes


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or the θ and ε components of the H mode can be obtained from their other components by a simple rotation of axes. However, they do share visual similarities, showing similar degrees of radial and tangential displacements to the other components of the same mode. This can also be seen using the interactive demonstration (10) and in the supporting information organized by frequency (11). For example, all components of the Hg(1) mode can be seen to be approximate ellipsoids. Concluding Remarks We have produced an interactive demonstration giving a visual representation of all of the normal modes of the C60 molecule that includes distortions of the sphere upon which the carbon atoms line in the undistorted molecule (10). This provides a useful guide to the eye that aids interpretation of the complicated simultaneous 3D movements of 60 atoms. The images can be animated to show the vibration, and rotated to change the orientation of the vibration. We have also generated animations with fixed orientations in which modes of the same symmetry can be viewed simultaneously (11). In all cases, we have chosen components of each mode that have specific transformation properties. It also means that there are only 88 unique vibrational displacements, from which the remaining 86 modes can be obtained by rotation about the x, y, and z axes. The images we have obtained illustrate the molecular motion of the fascinating C60 molecule. It is not currently possible to discern directly this motion in experiments, although technological developments may make this possible in the future. It is already possible to see signatures of atomic arrangements in scanning tunneling microscopy images of C60 molecules on surfaces (17). Images including distortions of a thin spherical shell are not only useful in visualizing vibrations of the C60 molecule, but they can also help visualize the structure of other molecules in which the positions of the atoms are nearly spherical. For example, possible structures for Si60 and Ge60 clusters having Th symmetry have been proposed in which the 60 atoms are arranged into pentagons and hexagons as in C60 but in which the bond lengths are no longer equal. More specifically, the pentagons in the proposed structures have either one or two different bond lengths, and the hexagons have two or three different bond lengths (18). Studying the images of the normal mode vibrations of C60, we can see that such structures can be generated from combinations of the a components of the Gg modes (which transform as Ag in Th symmetry). Later papers have suggested that these clusters deform to a lower symmetry than Th (19, 20). Images of these structures including a distorted spherical shell could also be obtained if sufficient information were known about the positions of the atoms in these structures. The method could also be used, for example, to probe chlorinestabilized isomers of C60 as an alternative to the usual icosahedral structure (21), or to explore structures of the hypothetical boron buckyball, B80 (22). It would be interesting to see whether the method could be used to help understand the normal mode vibrations of icosahedral viruses,

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where the large number of atoms involved creates a problem that is very complex indeed (23). Literature Cited 1. Vincent, A. Molecular Symmetry and Group Theory; Wiley: Chichester, 2001. 2. Jacobs, P. Group Theory with Applications in Chemical Physics; Cambridge University Press: Cambridge, U.K., 2005. 3. Merlin, J.-C.; Cornard, J.-P. J. Chem. Educ. 2006, 83, 1393. 4. Charistos, N. D.; Tsipis, C. A.; Sigalas, M. P. J. Chem. Educ. 2004, 81, 1231. 5. Ceulemans, A.; Fowler, P. W.; Vos, I. J. Chem. Phys. 1994, 100, 5491. 6. Fowler, P. W.; Ceulemans, A. Mol. Phys. 1985, 54, 767. 7. Hands, I.; Dunn, J. L.; Bates, C. A. J. Chem. Phys. 2004, 120, 6912. 8. Stone, A. J. Mol. Phys. 1980, 41, 1339. 9. Qiu, Q. C.; Ceulemans, A. Mol. Phys. 2002, 100, 255. 10. Normal Mode Vibrations of Buckminsterfullerene (C60), Wolfram Demonstrations Project. http://demonstrations.wolfram.com/ NormalModeVibrationsOfBuckminsterfullereneC60/ (accessed Jun 2010). 11. The University of Nottingham Web page of the Fullerene Theory Group, Normal Modes of the Fullerene Molecule (C60). http:// www.nottingham.ac.uk/~ppzjld/Visualise_vibration/ (accessed Jun 2010). 12. Schettino, V.; Pagliai, M.; Ciabini, L.; Cardini, G. J. Phys. Chem. A 2001, 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. 22. Gopakumara, G.; Nguyena, M. T.; Ceulemans, A. Chem. Phys. Lett. 2008, 450, 175. 23. van Vlijmen, H. W. T. In Normal Mode Analysis. Theory and Applications to Biological and Chemical Systems, Cui, Q., Bahar, I., Eds.; Chapman & Hall/CRC: Boca Raton, FL, 2006; published as part 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 available via the Internet at http://pubs.acs.org.


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A Pictorial Visualization of Normal Mode Vibrations of the Fullerene

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