Online material

Appendix A: Details on C₆₀⁺ spectroscopy

Neutral C₆₀ has I_h symmetry, its highest occupied molecular orbital (HOMO) is five-fold degenerate, it has h_u symmetry and is fully occupied, resulting is a closed-shell ground state that is totally symmetric and nondegenerate. Upon ionization, the hole in the h_u HOMO yields a five-fold degenerate h_u overall electronic state. This undergoes spontaneous symmetry-breaking due to the Jahn-Teller (JT) effect (Chancey & O’Brien 1997; Bersuker 2006). The degeneracy of the electronic state is lifted by distorting the molecule to a lower symmetry along some of its normal modes, which are determined by symmetry and called JT-active. For C, JT-active modes are those of H_g, G_g, and A_g symmetry. The A_g modes only shift the total energy, without reducing symmetry. The H_g, G_g modes instead break the I_h symmetry, and produce a multisheet adiabatic potential energy surface, with a conical intersection in the symmetric geometry and extrema in configurations of D_5v and D_3v lowered symmetry. DFT predicts that the D_5v geometries should be the absolute minima, with the D_3v ones being shallow transition states (Saito 2002). However, high-resolution photo-electron spectroscopy seems to hint that the D_3v geometry could be the real minimum (Canton et al. 2002). When JT-distorted minima are deep with respect to vibrational energy (static JT-effect), the adiabatic approximation holds in its vicinity, and standard harmonic vibrational analysis is applicable. Conversely, if equivalent minima are separated by negligible potential barriers, the molecule can tunnel among equivalent minima, mixing the near-degenerate electronic states (dynamic JT-effect), and the adiabatic approximation is not applicable. The resulting vibronic states recover the full initial symmetry of the problem. In this case, a much more complex calculation, dropping the adiabatic approximation, is needed for

accurate results. A comparable, but much simpler case of dynamical JT effect is the cation of corannulene, C₂₀H, which can be regarded as a fragment of C₆₀ with peripheral bonds saturated by H atoms. This was studied by Galué et al. (2011), who compared experimental infrared, multiphoton dissociation (IRMPD) spectra with a plain DFT harmonic vibrational analysis at the distorted geometry of minimum energy. This was expected to be the worst possible comparison, since in IRMPD experiments, vibrational energy is increased until the dissociation threshold is reached. This corresponds to energy values that are much higher than all barriers among equivalent minima, thereby maximizing dynamical JT effects. Despite this, the experimental and theoretical spectra do qualitatively agree, allowing for accurate band identification. Bands that are most significantly mispredicted, in position and intensity (but still identifiable in the laboratory spectrum), are those whose normal modes imply displacement along JT-active modes, i.e. those that move the molecule from a minimum in the direction of another one, or to the conical intersection.

In the light of this, we did a similar analysis for C, finding the distorted geometry of minimum energy and computing harmonic vibrational spectra there, thereby neglecting dynamical JT effects. We performed DFT calculations using both the Gaussian version 03.d2 and NWChem version 6.1 codes, and obtained very nearly identical results. Optimization led to the D_5v geometry. We also optimized the geometry of C with the constraint of I_h symmetry, obtaining a JT stabilization energy of  ~ 70 meV, which is consistent with previous calculations (Saito 2002). The D_5v distorted geometry, when compared with the symmetric one, appears to be distorted almost exclusively along normal modes of H_g symmetry, with changes in bond lengths of a few mÅ and bond angles by less than a degree (maximum).

© ESO, 2013