Benchmark calculations for ${}^3\mathrm{H}$, ${}^4\mathrm{He}$, ${}^{16}\mathrm{O}$, and ${}^{40}\mathrm{Ca}$ with ab initio coupled-cluster theory

We present ab initio calculations for ${}^3\mathrm{H}$, ${}^4\mathrm{He}$, ${}^{16}\mathrm{O}$, and ${}^{40}\mathrm{Ca}$ based on two-nucleon low-momentum interactions $V_{\mathrm{low}k}$ within coupled-cluster theory. For ${}^3\mathrm{H}$ and ${}^4\mathrm{He}$, our results are within 70 and 10 keV of the corresponding Faddeev and Faddeev-Yakubovsky energies. We study in detail the convergence with respect to the size of the model space and the single-particle basis. For the heavier nuclei, we report practically converged binding energies and compare with other approaches.

Figure 1

CCSD and CCSD(T) energies for ${}^3\mathrm{H}$ using a model space with fixed maximum $N=2n+l=12$ and fixed $\hslash \omega =14\mathrm{MeV}$ as a function of the maximum orbital angular momentum $l$. For comparison, we also show the exact Faddeev result of Ref. [34].

Figure 2

CCSD and CCSD(T) energies for ${}^4\mathrm{He}$ as a function of the maximum orbital angular momentum $l$ and the exact Faddeev-Yakubovsky (FY) result of Ref. [34]. For details, see the caption to Fig. 1.

Figure 3

CCSD and CCSD(T) results for the ground-state energy of ${}^3\mathrm{H}$ as a function of the model-space size $N=2n+l$, with fixed $l⩽5$ and fixed $\hslash \omega =14\mathrm{MeV}$. For comparison, we also show the exact Faddeev result of Ref. [34].

Figure 4

CCSD and CCSD(T) results for the ground-state energy of ${}^4\mathrm{He}$ as a function of the model-space size $N=2n+l$ and the exact Faddeev-Yakubovsky (FY) result of Ref. [34]. For details, see the caption to Fig. 3.

Figure 5

CCSD and various approximate CCSDT energies for ${}^4\mathrm{He}$ as a function of the oscillator frequency $\hslash \omega$, for fixed $N=12$ and $l⩽5$. For comparison, we also show the exact Faddeev-Yakubovsky (FY) result.

Figure 6

CCSD and CCSD(T) results for the binding energy of ${}^{16}\mathrm{O}$ using a model space with fixed maximum $N=2n+l=10$ and fixed $\hslash \omega =20\mathrm{MeV}$ as a function of the maximum orbital angular momentum $l$.

Figure 7

CCSD results for the binding energy of ${}^{16}\mathrm{O}$ as a function of the oscillator frequency $\hslash \omega$ for increasing sizes of the model space $N=2n+l$ with fixed $l⩽7$.

Figure 8

CCSD results (taken at the $\hslash \omega$ minima) for the binding energy of ${}^{16}\mathrm{O}$ as a function of the model-space size $N=2n+l$ at fixed $l⩽7$ and exponential fit (solid line).

Figure 9

CCSD(T) and CCSDT-1 results for the binding energy of ${}^{16}\mathrm{O}$ as a function of the oscillator frequency $\hslash \omega$ for increasing sizes of the model space $N=2n+l$ ($l⩽7$).

Figure 10

Contributions to the binding energy of ${}^{16}\mathrm{O}$ from triples corrections CCSD(T) and CCSDT-1 as a function of the model-space size $N=2n+l$ at fixed $\hslash \omega =22$ MeV and $l⩽7$. The dashed and dotted lines are exponential fits and yield the extrapolated energy corrections.

Figure 11

CCSD results for the binding energy of ${}^{40}\mathrm{Ca}$ as a function of the oscillator frequency $\hslash \omega$ for increasing sizes of the model space $N=2n+l$. (Note that there is no restriction on $l$ for these model spaces.)

Figure 12

CCSD results (taken at the $\hslash \omega$ minima) for the binding energy of ${}^{40}\mathrm{Ca}$ as a function of the model-space size $N=2n+l$ (without restriction in $l$) and exponential fit (solid line).

Figure 13

Contributions to the binding energy of ${}^{40}\mathrm{Ca}$ from triples corrections CCSD(T) as a function of the model-space size $N=2n+l$ at fixed $\hslash \omega =22\mathrm{MeV}$ (without restriction in $l$) and exponential fit (dashed line).