A pseudo-Jahn-Teller model describing central atom distortions is proposed for endohedral fullerenes of the form A@C60 where A is either a rare gas or a metal atom. A critical (dimensionless) coupling gc is found, below which the symmetric configuration is stable and above which inversion symmetry is broken. Vibronic parameters are given for selected endohedral fullerenes. 33.20.Wr, 02.20.Df Shortly after it was established that fullerenes are closed-cage structures of carbon, investigators speculated that atoms and clusters might be coaxed to occupy the space inside the cage. The existence of such endohedral fullerenes is now well-established by experiment [1]. Such endohedral doping of C 60 shows some promise as a technique to modify fullerene properties such as electric polarizibility, optical and infrared absorption, and magnetic susceptibility.Several numerical studies [2][3][4][5][6] have concluded that for the case of metal endohedral dopants, such as Na@C 60 , Li@C 60 , and Ca@C 60 , the metal atom transfers its valence electrons to the C 60 cage, and establishes an offcenter equilibrium position, resulting in a net electric dipole moment for the molecule. For the case of rare gas dopants, however, the central atom sits at the cage center.A classical model was used by Erwin [7] to explain the two disparate cases; for metal dopants, the charge transferred to the cage occupies a highly delocalized t 1u orbital. Erwin treats the resulting negatively charged cage as a perfectly conducting sphere. The total energy of the metal cation and the conducting sphere is minimized by placing the cation off-center where electrostatic interaction between the cation and the cage can be made more attractive as the charge on the cage is polarized. The distortion is stabilized in Erwin's model by including a short-ranged repulsive interaction of Lennard-Jones form for the electron overlap between the cation and cage electrons. Such a classical model would predict that for the case of metal dopants, inversion symmetry is broken as the central atom is stabilized in an off-center equilibrium position; while for closed-shell rare gas atoms, there will be no charge transfer and, consequently, no central atom distortion. However, this classical model suffers from two serious problems. First, for the case of singly-ionized alkali atoms, such a classical model would again predict no charge transfer, as such an atom is isoelectronic to a rare gas atom. Thus, charged metallofullerenes such as Li + @C 60 would have the cation symmetrically positioned according to the classical model. This, however, contradicts electronic structure calculations [2,3]. Second, for the distortive case of metallofullerenes, the net dipole moment calculated from the classical model is zero, as the contributions from the cation and the polarized cage exactly cancel.The purpose of this work is two-fold: (1) to remedy the shortcomings of the classical model, and (2) to give insight into the physical mechanisms underlying previous numerical studies. We pre...