Properties of the Projected Spectra for Finite Nuclei

The general features of the expressions useful for numerical calculations of the projected deformed Hartree-Fock (HF) spectra for finite nuclei are investigated. It is proved that the projected deformed HF wave-function gives the possible nuclear spins as $I=0,2,4,\dots ,I_{max}$ for a $K=0\mathrm{}$ band and $I=K,K+1,\dots ,I_{max}$ for a $K\ne 0$ band. It is further proved that if the energy $E_{I=K}$ of the projected $I=K$ state is greater (less) than the HF energy $E_K^{}{}_{}{}^{\mathrm{HF}}$, then $E_I$ is greater (less) than $E_I^{}{}_{}{}^{\prime }$ for $I<\mathrm{}{I}^{\prime }$. A plausible reason why one should use the deformed HF state rather than any other deformed state is also pointed out.