Nakamoto and McKinney (1) provided a symmetry analysis of the vibrational modes of C60 and other fullerenes. I would like to supplement their presentation with another approach, in widespread use, which records the number of atoms that are unmoved by each symmetry operation, yielding the reducible representation Γuma(2). This is particularly easy to do for C60 because only the identity operation and the 15 symmetry planes leave atoms unmoved, 60 and 4, respectively, as is shown in Table 1 of the Supplementary Materials.
Multiplication of Γuma by the representation for translation in the x, y, and z directions (T1u in Ih symmetry) yields, Γtot, the reducible representation for the 180 degrees of freedom of the C60 molecule. Γto is then decomposed into the equivalent linear combination of Ih irreducible representations by the usual method as shown in Table 1 of the Supplementary Materials. The symmetry of the vibrational modes is found by subtracting Γtrans and Γtot from Γtot.
Γvib = 2Ag + 3T1g + 4T2g + 6Gg + 8Hg
+ Au + 4T1u + 5T2u + 6Gu + 7Hu
IR-active modes have the same symmetry as x, y, and z. Raman active modes have the symmetry of the quadratic forms, x2, xy, etc. Thus, we find four IR-active modes and ten Raman-active modes in agreement with experimental spectroscopic evidence (1). The symmetry of the stretching modes can be determined by examining the behavior of the carbon–carbon bonds, Γbonds, under the symmetry operations of the Ih group. In addition, the symmetry of the π-electron density can be easily studied because Γπ = Γuma. The combination of knowing the irreducible representations contributing to Γvib and Γπ allows one to do an in-depth analysis of electronic spectrum of C60 utilizing the mechanism of vibronic coupling (3).
It has been shown that this method of recording the number of unmoved atoms is ideally suited for computer programming environments that have matrix and vector algebra capability (4). It is easy, for example, to prepare a Mathcad worksheet for any finite point group, which can then serve as a template for any molecule with that symmetry. The only thing that changes from one molecule to another within the same point group is the vector representing Γuma.
Supplemental Material
A table synopsizing the values obtained for Ih irreducible representations of C60 is available.
Harris, D. C.; Bertolucci, M. D. Symmetry and Spectroscopy: An Introduction to Vibrational and Electronic Spectroscopy; Dover Publications, Inc.: New York, 1989; p 141.
Leach, S.; Vervloet, M.; Despres, A.; Breheret, E.; Hare, J. P.; Dennis, T. J.; Kroto, H. W.; Taylor, R.; Walton, D. R. M. Chem. Phys. 1992, 160, 451–456.
Rioux, F. J. Chem. Educ. Soft.1998, Issue 9801MW.
Supplement
A table synopsizing the values obtained for Ih irreducible representations of C60 is available.
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