*3.1. Neutron-Rich Oxygen and Fluorine Isotopes*

Neutron-rich oxygen isotopes form a particularly interesting chain for experimental and theoretical research. Firstly, the proton number *Z* = 8 shows magical properties for the neutron-rich oxygen isotopes, which provide a good laboratory to perform configuration interaction (shell-model) calculations [22,38,68,69]. Secondly, the nuclei 22O and 24O exhibit doubly magicity at the neutron number *N* = 14 and 16, respectively, [70–73]. Thirdly, experiments have shown that the 25O and 26O are unbound and decay by one- and twoneutron emission, respectively, [8,74]. Experimental studies suggest that 24O is the heaviest bound isotope of the oxygen chain [8,74]. However, the loosely unbound property of 26O, which is only −18 keV unbound [8], is a strong incentive to investigate the bound or unbound character of 28O, which should have a magicity of *N* = 20. Consequently, the neutron-rich oxygen isotopes provide an ideal laboratory to study many-body correlation, continuum coupling, and single-particle structure. By adding one valence proton to the neutron-rich oxygen isotopes, one obtains the fluorine isotopes at the neutron drip line, which can sustain six additional neutrons after 25F, hence, up to 31F, which is suspected to be at the neutron drip line of the fluorine chain [75]. This dramatic change is called an "oxygen anomaly". Moreover, many exotic properties develop at the neutron drip line for fluorine isotopes, such as halos in 29F [76] and 28F within the island of inversion around *N* = 20 [77], and thus fluorine isotopes provide a very interesting ground for theoretical studies.

#### 3.1.1. Realistic Gamow Shell Model Calculations

We have developed realistic GSM with the Berggren basis using a WS potential, while the realistic effective Hamiltonian is constructed within the model space using a nondegenerate *Q*ˆ-box folded-diagram method [38]. We first employed it to investigate the neutron-rich oxygen isotopes up to and beyond the neutron drip line [38]. In our calculations, the realistic CD-Bonn potential [55] was used. To speed up the convergence of many-body calculations, the bare force is usually softened to remove the strong short-range repulsive core. The *V*low−*<sup>k</sup>* method [59] is used for that matter in Ref. [38].

**Figure 2.** Calculated spectrum of 24,25,26O, compared with available experimental data [8,10,74]. The resonances are indicated by shades, and their widths (in MeV) are given by the number below or above the levels. The light blue shade indicates the 3/2<sup>+</sup> many-body scattering states (with permissions from Ref. [38]). The "CGSM" stays for "core Gamow shell model".

Figure 2 shows the calculated low-lying states of <sup>24</sup>−26O, along with experimental data [8,10,74]. Our realistic GSM calculations [38] reproduce the experimental excitedstate spectrum well, including the observed resonance widths. The ground state energies and one-neutron separation energies *Sn* of the neutron-rich oxygen isotopes are also calculated [38] (see Figure 3) and compared to the experimental data [8,10,74,78]. The WS parameters used, taken from Ref. [38], reproduce the experimental 1*s*1/2 and 0*d*3/2 single-particle energies well, including the decay width of the 0*d*3/2 state, but give the 0*d*5/2 energy as lower than the experimental data, at about 1.17 MeV [10]. The results presented in Figure 3 show that adopting the experimental 0*d*5/2 energy can dramatically improve calculations. Overbinding in the GSM calculations of oxygen isotopes after 24O is obtained in Ref. [38], which is caused by the absence of the three-nucleon force (3NF).

**Figure 3.** Calculated ground state energies of oxygen isotopes with respect to the 16O core (upper panel) and associated neutron separation energies *Sn* (lower panel) as a function of atomic number compared with experimental data [8,74,78]. "GSM with WS SPE" indicates that the calculations were performed with Woods-Saxon (WS) single-particle energies (SPE), whereas "GSM with optimized SPE" means that the calculations were performed with the 0*d*5/2 SPE replaced by its experimental value (with permissions from Ref. [38]).

#### 3.1.2. *Ab-initio* Realistic GSM Calculations within GHF Basis

GSM is usually performed using a basis generated by a WS potential [5,19,36–39], whose parameters must be determined by fitting experimental single-particle energies and resonance widths. However, the single-particle energies and resonance widths in the multishell case are sometimes difficult to assess due to the lack of experimental data for that matter [10]. We then developed an ab initio realistic GSM approach by introducing the GHF basis as the Berggren basis [41]. The GHF basis is obtained by using the same interaction as the one used in the construction of the effective SM Hamiltonian [41], and thus there is no parameter introduced in the GHF Berggren basis. Starting from the chiral next-to-next-toleading-order (NNLOopt) force [79], we perform a nondegenerate *Q*ˆ-box folded-diagram

calculation [38,62] in the GHF basis in order to construct a complex effective Hamiltonian. The energies and widths of single-particle orbitals can also be obtained self-consistently using the nondegenerate *S*ˆ-box folded-diagram method [41]. The neutron-rich fluorine isotopes have been extended to the *p f*-shell, using a cross-shell effective Hamiltonian with the following model space : {1*s*1/2, 0*d*5/2, 0*d*3/2} for the valence proton, and {1*s*1/2, 0*d*3/2 + *d*3/2 scattering states, 1*p*3/2 + *p*3/2 scattering states, 1*p*1/2 + *p*1/2 scattering states, *f*7/2 scattering states} for valence neutrons. More details can be found in Ref. [41]. The constructed effective Hamiltonian was employed to study neutron-rich oxygen and fluorine drip line nuclei.

Figure 4 shows the calculated ground-state energies and neutron separation energies *Sn* of oxygen and fluorine isotopic chains, as well as comparisons with experimental data [78] and other theoretical calculations [31,68,79–83]. The GSM calculations using a GHF basis and based on the NNLOopt [79] provide the correct location of the neutron drip line of oxygen isotopes and a good description of the unbound nuclei 25,26O, which lie beyond the neutron drip line (see the left panel of Figure 4). Note that, when using the standard SM calculations with the USDB interaction [68], conventional SM calculations with NN + 3NF [82], or valence–space IMSRG (VS-IMSRG) calculations with NN + 3NF [81], the resonance and continuum couplings are absent. Complex CC [31] and GSM calculations [80] are displayed in Figure 4 for comparison.

**Figure 4.** Calculated ground-state energies (**upper panel**) with respect to the 22O core and associated neutron separation energies *Sn* (**lower panel**) for oxygen and fluorine isotopes, compared with experimental data [78] (the AME2016 extrapolated values are taken for 27,28O and 30,31F) and theoretical calculations from other groups: complex coupled-cluster (CC) with next-to-next-to-next-to-leading-order nucleon-nucleon CC with NNLOopt interaction [79], GSM [80], –space in-medium similarity renormalization group (VS-IMSRG) [81], SM with NN+3NF [82], SM with USDB [68], and SM with SDPF-M [83] (with permissions from Ref. [41]).

The results of fluorine isotopes are shown in the right panels of Figure 4. For comparison, standard SM calculations using USDB [68] and SDPF-M [83] effective interactions are also presented. All ground-state energies in Figure 4 are given with respect to the ground state of 22O. Experiments revealed that 31F is a neutron drip line nucleus [75]. Although our GSM calculations provide a lower energy of 31F compared to that of 30F, 31F is still unbound compared to 29F. However, our GSM calculations provide good descriptions of ground-state energies for <sup>23</sup>−29F.
