Peculiarity of the $${}^{{12}}$$C(0$${{}^{+}}$$) and $${}^{{12}}$$C(2$${{}^{+}}$$) Energy Spectrum in a 3$$\alpha$$ Model

Lowest energy spectrum of the $^{12}$C nucleus is analyzed in 3$\alpha$ cluster model with a deep $\alpha\alpha$-potential Buck, Friedrich and Wheatley Pauli forbidden states $S$ $D$ waves. The direct orthogonalization method applied for elimination 3$\alpha$-Pauli states. effects possible first order quantum phase transition are shown lowest $^{12}$C($0_1^+)$ $^{12}$C($2_1^+)$ from weakly bound to phase. ground $2^+$ created by critical eigen projector $0^+$ three-alpha functional spaces, respectively.