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Angular momentum and parity projected multidimensionally constrained relativistic Hartree-Bogoliubov model

The nuclear deformations are of fundamental importance in nuclear physics. Recently we developed a multi-dimensionally constrained relativistic Hartree-Bogoliubov (MDCRHB) model, in which all multipole deformations respecting the $V_4$ symmetry can be considered self-consistently. In this work we extend this model by incorporating the angular momentum projection (AMP) and parity projection (PP) to restore the rotational and parity symmetries broken in the mean-field level. This projected-MDCRHB (p-MDCRHB) model enables us to connect certain nuclear spectra to exotic intrinsic shapes such as triangle or tetrahedron. We present the details of the method and an exemplary calculation for $^{12}$C. We develop a triangular moment constraint to generate the triangular configurations consisting of three $α$ clusters arranged as an equilateral triangle. The resulting $^{12}$C spectra are consistent with that from a triangular rigid rotor for large separations between the $α$ clusters. We also calculate the $B(E2)$ and $B(E3)$ values for low-lying states and find good agreement with the experiments.