  In this work, we generate gauge configurations with $N_f=2$ dynamical charm
quarks on anisotropic lattices. The mass shift of $1S$ and $1P$ charmonia owing
to the charm quark annihilation effect can be investigated directly in a manner
of unitary theory. The distillation method is adopted to treat the charm quark
annihilation diagrams at a very precise level. For $1S$ charmonia, the charm
quark annihilation effect almost does not change the $J/\psi$ mass, but lifts
the $\eta_c$ mass by approximately 3-4 MeV. For $1P$ charmonia, this effect
results in positive mass shifts of approximately 1 MeV for $\chi_{c1}$ and
$h_c$, but decreases the $\chi_{c2}$ mass by approximately 3 MeV. We have not
obtain a reliable result for the mass shift of $\chi_{c0}$. In addition, it is
observed that the spin averaged mass of the spin-triplet $1P$ charmonia is in a
good agreement with the $h_c$, as expected by the non-relativistic quark model
and measured by experiments.
