We report on the results of the calculations of the low energy excitation patterns for three Zirconium isotopes, viz. $^{80}$Zr$_{40}$, $^{96}$Zr$_{56}$ and $^{110}$Zr$_{70}$, reported by other authors to be doubly-magic tetrahedral nuclei (with tetr
ahedral magic numbers $Z$=40 and $N$=40, 56 and 70). We employ the realistic Gogny effective interactions using three variants of their parametrisation and the particle-number, parity and the angular-momentum projection techniques. We confirm quantitatively that the resulting spectra directly follow the pattern expected from the group theory considerations for the tetrahedral symmetric quantum objects. We also find out that, for all the nuclei studied, the correlation energy obtained after the angular momentum projection is very large for the tetrahedral deformation as well as other octupole deformations. The lowering of the energies of the resulting configurations is considerable, i.e. by about 10 MeV or even more, once again confirming the significance of the angular-momentum projections techniques in the mean-field nuclear structure calculations.
We have developed an efficient method for quantum number projection from most general HFB type mean-field states, where all the symmetries like axial symmetry, number conservation, parity and time-reversal invariance are broken. Applying the method,
we have microscopically calculated, for the first time, the energy spectra based on the exotic tetrahedral deformation in $^{108,110}$Zr. The nice low-lying rotational spectra, which have all characteristic features of the molecular tetrahedral rotor, are obtained for large tetrahedral deformation, $alpha_{32} gtsim 0.25$, while the spectra are of transitional nature between vibrational and rotational with rather high excitation energies for $alpha_{32}approx 0.1-0.2$
