Spherical coupled-cluster theory for open-shell nuclei

Background: A microscopic description of nuclei is important to understand the nuclear shell model from fundamental principles. This is difficult to achieve for more than the lightest nuclei without an effective approximation scheme.

Purpose: Define and evaluate an approximation scheme that can be used to study nuclei that are described as two particles attached to a closed (sub-)shell nucleus.

Methods: The equation-of-motion coupled-cluster formalism has been used to obtain ground- and excited-state energies. This method is based on the diagonalization of a non-Hermitian matrix obtained from a similarity transformation of the many-body nuclear Hamiltonian. A chiral interaction at the next-to-next-to-next-to leading order (N${}^3$LO) using a cutoff at 500 MeV was used.

Results: The ground-state energies of ${}^6$Li and ${}^6$He were in good agreement with a no-core shell-model calculation using the same interaction. Several excited states were also produced with overall good agreement. Only the $J^{\pi }=3^{+}$ excited state in ${}^6$Li showed a sizable deviation. The ground-state energies of ${}^{18}$O, ${}^{18}$F, and ${}^{18}$Ne were converged but underbound compared to experiment. Moreover, the calculated spectra were converged and comparable to both experiment and shell-model studies in this region. Some excited states in ${}^{18}$O were high or missing in the spectrum. It was also shown that the wave function for both ground and excited states separates into an intrinsic part and a Gaussian for the center-of-mass coordinate. Spurious center-of-mass excitations are clearly identified.

Conclusions: Results are converged with respect to the size of the model space and the method can be used to describe nuclear states with simple structure. Especially the ground-state energies were very close to what has been achieved by exact diagonalization. To obtain a closer match with experimental data, effects of three-nucleon forces, the scattering continuum, as well as additional configurations in the coupled-cluster approximations are necessary.

Figure 1

Expectation value of the center-of-mass Hamiltonian (71) at the frequency $\hslash \tilde{\omega }=\hslash \omega$. Three different states are shown: the $J^{\pi }=0^{+}$ ground state of ${}^6$He, the $J^{\pi }=1^{-}$ excited state of ${}^6$He, and the $J^{\pi }=3^{+}$ excited state of ${}^6$Li. All states are assumed to be the lowest eigenstates of the center-of-mass Hamiltonian in Eq. (71) with $n=0$.

Figure 2

Expectation value of the center-of-mass Hamiltonian (71) at the frequency $\hslash \tilde{\omega }=\hslash \omega$. Three different states are shown: the $J^{\pi }=0^{+}$ ground state of ${}^6$He, the $J^{\pi }=1^{-}$ excited state in ${}^6$He, and the $J^{\pi }=3^{+}$ excited state in ${}^6$Li. The positive-parity states are assumed to be the lowest eigenstates of the center-of-mass Hamiltonian (71) with $n=0$, while the negative-parity state is assumed to be the first excited state of the center-of-mass Hamiltonian (71) with $n=1$.

Figure 3

Center-of-mass frequency as calculated by the description in Hagen et al. [19, 30] as a function of the oscillator parameter $\hslash \omega$. Three different states are shown: the $J^{\pi }=0+$ ground state of ${}^6$He, the $J^{\pi }=1^{-}$ excited state in ${}^6$He, and the $J^{\pi }=3^{+}$ excited state in ${}^6$Li. The positive-parity states are assumed to be the lowest eigenstates of the center-of-mass Hamiltonian (71) with $n=0$, while the negative-parity state is assumed to be the first excited state of the center-of-mass Hamiltonian with $n=1$.

Figure 4

The expectation value of the center-of-mass Hamiltonian (71) is calculated at the center-of-mass frequency $\hslash \tilde{\omega }$ for the $J^{\pi }=0+$ ground state of ${}^6$He, the $J^{\pi }=1^{-}$ excited state in ${}^6$He, and the $J^{\pi }=3^{+}$ excited state in ${}^6$Li. The positive-parity states are assumed to be the lowest eigenstates of the center-of-mass Hamiltonian (71) with $n=0$, while the negative-parity state is assumed to be the first excited state of the center-of-mass Hamiltonian with $n=1$.

Figure 5

Ground-state energy of ${}^6$Li as a function of the oscillator parameter $\hslash \omega$ and the size of the model space $N_{\max }$ (see text for details).

Figure 6

The total energy of the first excited $J^{\pi }=3^{+}$ state of ${}^6$Li as a function of the oscillator parameter $\hslash \omega$ and the size of the model space $N_{\max }$ (see text for details).

Figure 7

Excitation energy for selected states of ${}^6$Li, as a function of the size of the model space defined by $N_{\max }$. The rightmost column shows the experimental values from Tilley et al. [31].

Figure 8

Excitation energy for the first excited $J^{\pi }=2^{+}$ state of of ${}^6$He, as a function of the size of the model space defined by $N_{\max }$. The rightmost column shows the experimental value from Tilley et al. [31].

Figure 9

Excitation levels of selected states in ${}^6$Li and ${}^6$He, calculated using 2PA-EOM-CC (this work) and NCSM [35], compared to experimental [31] values.

Figure 10

The ground-state energy of ${}^{18}$O as a function of the oscillator parameter, $\hslash \omega$. Different lines correspond to different model spaces, parametrized by the variable $N_{\max }$ (69).

Figure 11

The total energy of the $J^{\pi }=3+$ state in ${}^{18}$O as a function of the oscillator parameter, $\hslash \omega$. Different lines correspond to different model spaces, parametrized by the variable $N_{\max }$ (69).

Figure 12

The excitation energies of the first $J^{\pi }=2^{+}$, $3^{+}$, and $4^{+}$ excited states in ${}^{18}$O where the best values for $\hslash \omega$ has been chosen for each state. The different columns represent different model spaces parametrized by the variable $N_{\max }$ (69). The rightmost column contains experimental values from Tilley et al. [41].

Figure 13

The excitation energies of the first $J^{\pi }=2^{+}$, $3^{+}$, and $4^{+}$ excited states in ${}^{18}$O for $\hslash \omega =32$ MeV. The different columns represent different model spaces parametrized by the variable $N_{\max }$ (69). The rightmost column contains experimental values from Tilley et al. [41].

Figure 14

The excitation energies of the first $J^{\pi }=1^{-}$, $2^{-}$, and $3^{-}$ excited states in ${}^{18}$O for $\hslash \omega =32$ MeV. The different columns represents different model spaces parametrized by the variable $N_{\max }$ (69). The rightmost column contains experimental values from Tilley et al. [41].

Figure 15

The excitation energies of the first few $J^{\pi }=0+$ and $2+$ states in ${}^{18}$O for $\hslash \omega =32$ MeV. The different columns represents different model spaces parametrized by the variable $N_{\max }$ (69). The rightmost column contains experimental values from Tilley et al. [41].

Figure 16

The excitation energies of the first $J^{\pi }=0^{+}$, $2^{+}$, $3^{+}$, $4^{+}$, and $5^{+}$ states in ${}^{18}$F, calculated at $\hslash \omega =30$ MeV. The different columns represents different model spaces parametrized by the variable $N_{\max }$ (69). The rightmost column contains experimental values from Tilley et al. [41].

Figure 17

Excitation energies for selected states in ${}^{18}$O, ${}^{18}$F, and ${}^{18}$Ne compared to experimental values from Tilley et al. [41].