  We systematically investigate the mass spectrum and two-body open-charm
strong decays of charmonium states in a coupled-channel model where the $^3P_0$
quark-antiquark pair creation mechanism is employed. The results of masses,
mass shifts, proportions of the $c\bar{c}$ component, and open-charm decay
widths are provided. The $S$-$D$ wave mixing angles and di-electric decay
widths for vector mesons are also presented. Based on our results, we find that
the $\psi(3770)$, $\psi(4040)$, $\psi(4160)$, $\psi(4360)$, and $\psi(4415)$
can be assigned as the $1^3D_1$-, $3^3S_1$-, $2^3D_1$-, $4^3S_1$-, and
$3^3D_1$-dominated charmonium states, respectively. The $\psi_3(3842)$ is a
good candidate of the $\psi_3(1D)$ charmonium state. The calculated mass and
strong decay width of $\chi_{c1}(2P)$ with significant continuum contribution
($\sim$57\%) favor the charmonium interpretation for the mysterious
$\chi_{c1}(3872)$. When considering the large uncertainty in the observed decay
width, the possibility to assign the $\chi_{c0}(3860)$ as the $\chi_{c0}(2P)$
charmonium state cannot be ruled out. One may describe well the properties of
$\chi_{c2}(3930)$ with the $\chi_{c2}(2P)$ charmonium. The predictions on
properties of other $c\bar{c}$ states can be tested by future experiments.
