The Franck-Condon (FC) approximation is used to obtain analytical values for the second-order factors describing vibronic reduction for the case of the strongly coupled icosahedral T1u⊗hg Jahn-Teller (JT) system in which an electronic triplet ground state is coupled to vibrations of fivefold degeneracy. This is believed to be applicable to the ground state of the C60- ion and so the calculations should help in providing valuable physical information on this fullerene ion. The procedures for determining vibronic reduction effects depend critically on the particular JT system and upon the vibronic coupling strength. In the FC approximation, the major contributions to these reduction factors originate from virtual vertical transitions between the ground electronic adiabatic state and excited vibronic states associated with upper sheets in the adiabatic potential energy surface. At strong coupling, other transitions may be neglected. The FC results are compared with those obtained from an alternative approach involving a shift transformation. Details are given for perturbations containing orbital operators of threefold (T1) and fivefold (H) symmetry. The discussion emphasizes the inherent high symmetry of the system.

First- and second-order vibronic reduction factors are calculated analytically for the H⊗(g⊕h) Jahn-Teller system in Ih symmetry. Results are given as a function of the strength of the coupling of the H orbital to the vibrations of h and g symmetries. As the product systems H⊗g and H⊗h contain repeated representations, the calculations of many of the reduction factors are more complicated than in other systems. These complications and their implications are analysed in detail. This system models the ground state of hole-doped C60 material, which has possible applications for high-temperature superconductivity.

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Analytical expressions for the vibronic states and energy spectrum of the icosahedral G ⊗ (g ⊕ h) Jahn-Teller system are derived. From these states, expressions for first-and second-order vibronic reduction factors are determined as a function of the strengths of the coupling of the G orbital to the g and h modes of vibration. The possibility of the vibronic ground state being a singlet A state, rather than the G state that would be expected in the absence of vibronic coupling, is explored. The reduction factors obtained provide a convenient basis for the modelling of spectra involving some of the excited states of the fullerene molecule C 60 and related ions.