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We report an investigation of the structure of $^{12}$C nucleus employing anewly developed configuration-mixing method. In the three-dimensionalcoordinate-space representation, we generate a number of Slater determinantswith various correlated structures using the imaginary-time algorithm. We thendiagonalize a many-body Hamiltonian with the Skyrme interaction in the spacespanned by the Slater determinants with parity and angular momentumprojections. Our calculation reasonably describes the ground and excited statesof $^{12}$C nucleus, both for shell-model-like and cluster-like states. Theexcitation energies and transition strengths of the ground-state rotationalband are well reproduced. Negative parity excited states, $1_1^-$, $2_1^-$, and$3_1^-$, are also reasonably described. The second and third $0^+$ states,$0_2^+$ and $0_3^+$, appear at around 8.8 MeV and 15 MeV, respectively. The$0_2^+$ state shows a structure consistent with former results of the\alpha-cluster models, however, the calculated radius of the $0_2^+$ state issmaller than those calculations. The three-{\alpha} linear-chain configurationdominates in the $0_3^+$ state.

Deformation and cluster structures in $^{12}$C studied with configuration mixing using Skyrme interactions