Ab Initio Study of ${}^{40}\mathrm{Ca}$ with an Importance-Truncated No-Core Shell Model

We propose an importance truncation scheme for no-core shell model or configuration interaction approaches, which enables converged calculations for nuclei well beyond the $p$ shell. It is based on an a priori measure for the importance of individual basis states constructed by means of many-body perturbation theory. Only the physically relevant states of the no-core model space are considered, which leads to a dramatic reduction of the basis dimension. We analyze the validity and efficiency of this truncation scheme using different realistic nucleon-nucleon interactions and compare to conventional no-core shell model calculations for ${}^4\mathrm{He}$ and ${}^{16}\mathrm{O}$. Then, we present first converged calculations for the ground state of ${}^{40}\mathrm{Ca}$ within no-core model spaces including up to $16\hslash \Omega$ excitations using realistic low-momentum interactions. The scheme is universal and can be easily applied to other quantum many-body problems.

Figure 1

Correlation between the a priori importance weight ${\kappa }_{\nu }$ and the a posteriori amplitudes $C_{\nu }$ resulting from the explicit diagonalization. Shown are the $2p2h$ (a) and $4p4h$ contributions (b) for a $6\hslash \Omega$ calculation for ${}^{16}\mathrm{O}$ with $V_{\mathrm{UCOM}}$ at $\hslash \Omega =20\mathrm{MeV}$. The dashed vertical line marks a typical truncation parameter ${\kappa }_{\min }$.

Figure 2

Dependence of the ground-state energy of ${}^{16}\mathrm{O}$ (a) and ${}^{40}\mathrm{Ca}$ (b) on the truncation parameter ${\kappa }_{\min }$ obtained with $V_{\mathrm{UCOM}}$. The three data sets correspond to model spaces with up to $2p2h$ (●), $3p3h$ (◆), and $4p4h$ states (▪), respectively. The lines show a 5th order polynomial interpolation.

Figure 3

Convergence of the ground-state energy for ${}^4\mathrm{He}$ (a) and ${}^{16}\mathrm{O}$ (b) versus $N_{\max }$ obtained using $V_{\mathrm{UCOM}}$. Shown are three data sets corresponding to model spaces with up to $2p2h$ (●), $3p3h$ (◆), and $4p4h$ states (▪), respectively. Black crosses (+) indicate the results of full NCSM calculations. Lines to guide the eye.

Figure 4

Convergence of the ground-state energy of ${}^{40}\mathrm{Ca}$ as function of model-space size $N_{\max }$ for $\hslash \Omega =17\mathrm{MeV}$ (lower curves, left-hand axis) and 20 MeV (upper curves, right-hand axis) using the $V_{\mathrm{UCOM}}$ interaction. Symbols as described in Fig. 3.

Figure 5

Ground-state energy of ${}^{16}\mathrm{O}$ (a) and ${}^{40}\mathrm{Ca}$ (b) as function of the harmonic-oscillator frequency $\hslash \Omega$ for different model-space sizes up to $N_{\max }=16$ obtained with the $V_{\mathrm{low}k}$ interaction. For ${}^{16}\mathrm{O}$ up to $4p4h$ configurations are included, for ${}^{40}\mathrm{Ca}$ up to $3p3h$. Crosses indicate the results of full NCSM calculations.