Ab initio effective interactions for $sd$-shell valence nucleons

We perform ab initio no-core shell-model calculations for $A=18$ and 19 nuclei in a $4\hslash \mathrm{\Omega }$, or $N_{\max }=4$, model space by using the effective JISP16 and chiral N3LO nucleon-nucleon potentials and transform the many-body effective Hamiltonians into the $0\hslash \mathrm{\Omega }$ model space to construct the $A$-body effective Hamiltonians in the $sd$ shell. We separate the $A$-body effective Hamiltonians with $A=18$ and $A=19$ into inert core, one-, and two-body components. Then we use these core, one-, and two-body components to perform standard shell-model calculations for the $A=18$ and $A=19$ systems with valence nucleons restricted to the $sd$ shell. Finally, we compare the standard shell-model results in the $0\hslash \mathrm{\Omega }$ model space with the exact no-core shell-model results in the $4\hslash \mathrm{\Omega }$ model space for the $A=18$ and $A=19$ systems and find good agreement.

Figure 1

Flow of renormalizations adopted to derive an effective interaction for valence nucleons. The OLS procedure is first applied to derive a NCSM effective interaction for the full $A$-nucleon system resulting in the “primary” effective Hamiltonian $P{H}_{\mathrm{eff}}P$ for the chosen no-core basis space (the “$P$-space”) indicated on the large square on the right of the figure in its upper-left corner. The many-body truncation is indicated by $N_{\max }$, the total number of HO quanta above the minimum for that system. The OLS procedure is applied again by using the results of the NCSM calculation to derive the “secondary” effective Hamiltonian $P^{\prime }{H}_{\mathrm{eff}}^{\prime }{P}^{\prime }$ for the chosen valence space (the $P^{\prime }$-space with the smaller many-body cutoff $N_{\max }^{\prime }$) indicated on the square in the upper right of the figure.

Figure 2

Correlation between chiral N3LO and JISP16 TBMEs plotted in units of MeV. The 63 TBMEs are derived for $A=18$ with the methods described in the text and are presented in the eighth columns of Tables 4 and 5. Red circles (blue squares) represent $T=0\left(1\right)$ matrix elements. The diagonal dashed line is the reference for equal matrix elements. The solid red line is a linear fit to the correlation points with the result $y=0.981x$. The root-mean-square deviation between the two sets of TBMEs is 0.203 MeV. A plot of the $A=19$ results in the tenth columns of Tables 4 and 5 would be nearly indistinguishable from this plot.

Figure 3

The ground-state energy (in MeV) and low-lying excited-state energies of ${}^{18}\mathrm{F}$ and ${}^{19}\mathrm{F}$ obtained by the NCSM and SSM calculations using the effective JISP16 interaction. The tags $A=18$ and $A=19$ at the bottom of each column refer to the effective JISP16 interaction obtained with the 2BVC approximation for general $A$. That is, the tags $A=18$ and $A=19$ represent nucleus $A$ used for deriving the primary effective Hamiltonian. In addition, we retain only effective core, one-body, and two-body terms for the secondary effective Hamiltonian.

Figure 4

The ground-state energy (in MeV) and low-lying excited-state energies of ${}^{18}\mathrm{F}$ and ${}^{19}\mathrm{F}$ obtained by the NCSM and SSM calculations using the effective chiral N3LO interaction. The tags $A=18$ and $A=19$ at the bottom of each column refer to the effective JISP16 interaction obtained with the 2BVC approximation for general $A$. That is, the tags $A=18$ and $A=19$ represent nucleus $A$ used for deriving the primary effective Hamiltonian. In addition, we retain only effective core, one-body, and two-body terms for the secondary effective Hamiltonian.