Implementation of the variation-after-projection approach in calculations with a time-odd Hartree-Fock mean field

We have implemented a new variation-after-projection (VAP) calculation based on a time-odd Hartree-Fock (HF) mean field. The exact Hessian matrix of the projected energy has been successively evaluated for the first time, which makes the VAP calculation much more stable. With the time-odd mean field, the present VAP can be applied to the yrast states not only in the even-even nuclei, but also in the odd-$A$ and odd-odd ones. The VAP energies throughout all spins for some calculated $sd$-shell nuclei are very close to the corresponding shell model ones. Our calculations clearly show that the spin projection is very important in achieving a good approximation to the full shell model.

Figure 1

Calculated VAP energies $E_{\mathrm{VAP}}$ and the shell model energies $E_{\mathrm{SM}}$, with the USDB interaction along the yrast line for (a) ${}^{24}\mathrm{Mg}$, (b) ${}^{25}\mathrm{Mg}$, (c) ${}^{26}\mathrm{Al}$, and (d) ${}^{26}\mathrm{Mg}$.

Figure 2

The energy differences between the shell model energies $E_{\mathrm{SM}}$, and the present VAP energies $E_{\mathrm{VAP}}$.

Figure 3

The $J$-scheme shell model dimension as a function of the spin, $J$, for the calculated nuclei in $sd$ model space.

Figure 4

The relative energies of $E_{\mathrm{VAP}}$ (present), $E_{\mathrm{JTA}}$ [5], and $E_{\mathrm{PHF}}$ [5] to the SM energies $E_{\mathrm{SM}}$ for the ground states of even-even $sd$ nuclei.

Figure 5

Eigenvalues of the Hessian matrix of the projected energy for the ground state of ${}^{24}\mathrm{Mg}$. The black filled rectangles show the ones at an initial HF vacuum $|{\mathrm{\Phi }}_0\rangle$. The red filled dots show the ones at the converged HF vacuum. The total number of the independent VAP parameters is $2D_{\mathrm{VAP}}=128$.

Figure 6

Calculated $B(E2:I\to I-2)$ values from VAP (red filled circles) and SM (black filled squares) with the wave functions corresponding to Fig. 1. The effective charges are taken as $e_n=0.5e,e_p=1.5e$, and the oscillator value $\hslash \omega =45A^{-1/3}-25A^{-2/3}\mathrm{MeV}$.