  Shell model calculations are done to study the structure of neutron-rich
carbon isotopes. For both even-A and odd-A neutron-rich carbon isotopes, the
energy levels are strongly affected by the configuration mixing of valence
neutrons. The calculated energy levels in the nucleus $^{17}$C are
significantly improved compared with experimental values when the model space
of the three valence neutrons is enlarged from pure $\nu(0d_{5/2})^{3}$
configuration to full $sd$ space. We also investigate the configuration mixing
effect on the $B(E2)$ values in even-even nuclei $^{16-20}$C.
