  We compute, for the first time using lattice QCD methods, radiative
transition rates involving excited charmonium states, states of high spin and
exotics. Utilizing a large basis of interpolating fields we are able to project
out various excited state contributions to three-point correlators computed on
quenched anisotropic lattices. In the first lattice QCD calculation of the
exotic 1-+ eta_c1 radiative decay, we find a large partial width Gamma(eta_c1
-> J/psi gamma) ~ 100 keV. We find clear signals for electric dipole and
magnetic quadrupole transition form factors in chi_c2 -> J/psi gamma,
calculated for the first time in this framework, and study transitions
involving excited psi and chi_c1,2 states. We calculate hindered magnetic
dipole transition widths without the sensitivity to assumptions made in model
studies and find statistically significant signals, including a non-exotic
vector hybrid candidate Y_hyb? -> eta_c gamma. As well as comparison to
experimental data, we discuss in some detail the phenomenology suggested by our
results and the extent to which it mirrors that of quark potential models and
make suggestions for the interpretation of our results involving exotic quantum
numbered states.
