  We study the $B_{c}$$\rightarrow$$\psi(2S)$K, $\eta_{c}(2S)$K, $\psi(3770)$K
decays with perturbative QCD approach (pQCD) based on $k_T$ factorization. The
new orbitally excited charmonium distribution amplitudes $\psi(1^{3}D_{1})$
based on the Schr\"{o}dinger wave function of the $n=1$, $l=2$ state for the
harmonic-oscillator potential are employed. By using the corresponding
distribution amplitudes, we calculate the branching ratio of
$B_{c}$$\rightarrow$$\psi(2S)$K, $\eta_{c}(2S)$K, $\psi(3770)$K decays and the
form factors $A_{0,1,2}$ and $V$ for the transition
$B_{c}$$\rightarrow$$\psi(1^{3}D_{1})$. We obtain the branching ratio of both
$B_{c}$$\rightarrow$$\psi(2S)$K and $B_{c}$$\rightarrow$$\eta_{c}(2S)$K are at
the order of $10^{-5}$. The effects of two sets of the S-D mixing angle
$\theta=-12^{\circ}$ and $\theta=27^{\circ}$ for the decay
$B_{c}$$\rightarrow$$\psi(3770)$K are studied firstly in this paper. Our
calculations show that the branching ratio of the decay
$B_{c}$$\rightarrow$$\psi(3770)$K can be raised from the order of $10^{-6}$ to
the order of $10^{-5}$ at the mixing angle $\theta=-12^{\circ}$, which can be
tested by the running LHC-b experiments.
