  The benzene cation (C6H6+) has a doublet (e_{1g}) ground state in hexagonal
ring (D_{6h}) geometry. Therefore a Jahn-Teller (JT) distortion will lower the
energy. The present theoretical study yields a model Huckel-type Hamiltonian
that includes the JT coupling of the e_{1g} electronic ground state with the
two e_{2g} vibrational modes: in-plane ring-bending and C-C bond-stretching. We
obtain the JT couplings from density functional theory (DFT), which gives a JT
energy lowering of 970 cm^{-1} in agreement with previous quantum chemistry
calculations. We find a non-adiabatic solution for vibrational spectra and
predict frequencies shifts of both the benzene cation and anion, and give a
reinterpretation of the available experimental data.
