Nonlocalized motion in a two-dimensional container of α particles in 3− and 4− states of C12

The first ${3}^{\ensuremath{-}}$ and ${4}^{\ensuremath{-}}$ states of $^{12}\mathrm{C}$ are studied in the present container model, in which the shift parameter is introduced to break the parity symmetry for projecting out the negative-parity states. Taking the limit as the shift parameter approaches zero and by variational calculations for one-deformed size parameter, the local energy minima are obtained for the ${3}^{\ensuremath{-}}$ and ${4}^{\ensuremath{-}}$ states. It is found that the obtained single Tohsaki-Horiuchi-Schuck-R\"opke (THSR) wave functions for ${3}^{\ensuremath{-}}$ and ${4}^{\ensuremath{-}}$ states are 96% and 92% equivalent to the corresponding generator coordinate method (GCM) wave functions, respectively. The calculated intrinsic densities further show that these negative-parity states of three clusters, different from the traditional understanding of a rigid triangle structure, are found to have nonlocalized clustering structure in the two-dimensional container picture.