  We calculate the production of $\chi_{c}$ and $\eta_{c}$ by the semi-coherent
and coherent two-photon interaction in ultra-peripheral heavy ion collisions at
Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC)
energies. The differential cross section of transverse momentum distribution
and rapidity distribution for $AB\stackrel{\gamma\gamma}{\longrightarrow}AHB$
(H=$\chi_{c}$ and $\eta_{c}$), are estimated by using the equivalent photon
approximation in ultra-peripheral nucleus-nucleus collisions. The numerical
results demonstrate that the experimental study of $\chi_{c}$ and $\eta_{c}$ in
ultra-peripheral nucleus-nucleus collisions is feasible at RHIC and LHC
energies.
