It is well-known that vibronic interactions can be modelled in terms of an effectiveHamiltonian incorporating first and second-order reduction factors (RFs), particularlywhen analysing the spectroscopic properties. Measurements and calculations of the RFs asa function of the strength of vibronic coupling are therefore of much interest. In this paper,we develop a new general method for determining second-order RFs (soRFs) from thestrength of the Jahn–Teller (JT) coupling for systems in which electron orbital degeneracyor pseudo-degeneracy exists. These include in particular the fullerene moleculeC60, pseudo-Jahn–Teller molecules and impurity centres in crystals. In order to calculate theimportant soRFs for intermediate to strong coupling, it is necessary to determinenon-Condon corrections to the strong coupling values obtained using the Franck–Condon(FC) approximation. This gives an additional contribution to the nuclear polarizability ofthe system, thus enabling the electrons to follow the nuclear vibrations. These non-Condoncorrections are derived using perturbation theory and are found to be inverselyproportional to the square of the JT energy. The validity of the approximation is firsttested in the cubic JT system due to its relative simplicity. It is found that the results arecloser to those obtained earlier by numerical methods than the analyticalFC values alone. Results are then presented that are applicable toC60− anions.