Keywords: 
• 柯西交错定理 /
• 投影后变分 /
• 低激发态 /
• 壳模型 /
• USDB相互作用

Abstract: We present a comprehensive introduction in our newly developed Variation After Projection (VAP) calculations for the low-lying nuclear states. First, we discussed the VAP calculation with a fully JTA-projected wavefunction for the ground state in even-even nucleus. This leads to the conclusion that the spin projection plays a key role in obtaining a good shell model approximation. With this conclusion, we simplified the VAP with a time-odd Hartree-Fock mean field, on which only spin projection is required. Due to the time reversal symmetry breaking, this VAP now can be applied to the yrast states in all kinds of nuclei. It turns out that our VAP yrast energies as well as the corresponding VAP wavefunctions are very close the exact ones from the full shell model calculations. Such good approximation encourages us to extend the VAP calculations further to the non-yrast nuclear states. For this purpose, we proposed a new algorithm in our VAP based on the Cauchy's interlacing theorem. This theorem ensures that the sum of the calculated lowest projected energies with the same quantum numbers can be safely minimized. After minimization, all the calculated states can be determined simultaneously. Again, all the calculated VAP energies are very close to the exact shell model results. Recently, we have added the parity projection into the VAP, and the yrast states with both parity in 12C have been calculated in the psd model space. This time, we still have good shell model approximation for both parity states. Finally, we should point out that the present algorithm should be applicable to the low-lying states in different quantum many-body systems.