I investigate ${}^{12}\mathrm{C}$, particularly the four kinds of the $0^{+}$ bands with various types of the $3\alpha$ configurations. These states are obtained in the variation of the multiple bases of the antisymmetrized molecular dynamics, where the bases are optimized simultaneously in the variation of the total energy. In the $0_2^{+}$ Hoyle state and the $0_3^{+}$ linear-chain state, I confirm a mixture of the $\alpha$-cluster breaking of the $s$-wave configuration with contributions from the spin-orbit force, while the $0_4^{+}$ state exhibits a relatively pure $3\alpha$ cluster state characterized by a large radius. The $2_{2-4}^{+}$ and $4_{2-4}^{+}$ states also tend to be the pure $3\alpha$ cluster states. The monopole transitions between the $0_2^{+}$ and $0_4^{+}$ bands exhibit large values, suggesting breathing mode of the $3\alpha$ states in the $0_4^{+}$ band. This conclusion aligns with the predictions of the $3\alpha$ models with an $\alpha$ condensate and also with a neural network, although the order of the $0_3^{+}$ and $0_4^{+}$ bands is reversed in the present results due to the attraction of the spin-orbit force.

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• Collective levels
• Electromagnetic transitions
• Energy levels
• Nuclear structure & decays
• 6 ≤ A ≤ 19
• Cluster models