The parameters of the Hamiltonian $ H $ (see Eq. (1)) are firstly fitted at the level of HO-SM on the low-lying spectrum of 17O and 17F. The parameters of the WS potential of the core common to all partial waves read $ d $ = 0.65 fm for diffuseness, $ R_0 = 3.2 $ fm for radius, and $ V_{\rm so} $ = 7.5 MeV fm2 for spin-orbit strength. The WS central depth, denoted as $ V_o $, is $ \ell $-dependent and reads for protons: 50.022 MeV ($ \ell $ = 0), 37.214 MeV ($ \ell = 1 $), 47.800 MeV ($ \ell $ = 2), 39.705 MeV ($ \ell = 3 $), and for neutrons: 49.902 MeV ($ \ell $ = 0), 36.438 MeV ($ \ell $ = 1), 48.603 MeV ($ \ell = 2 $) and 38.998 MeV ($ \ell = 3 $).
The FHT interaction is the sum of three Gaussian terms, depicting the central, spin-orbit and tensor components of the nucleon-nucleon interaction[30-31]. The parameters of the FHT interaction were fitted in Ref. [29] for $ A \leqslant 9 $ nuclei. One could see therein that four parameters were very constrained, whereas the remaining three parameters bore large statistical errors[29]. Consequently, one fitted only the first four parameters, namely the central terms $ V_c^{00},V_c^{10},V_c^{01} $ and the tensor term $ V_T^{10} $, where the two superscripts equal to 0 or 1 refer to the spin $ S $ and isospin $ T $ of the two nucleons[29]. The other three parameters are put to zero. Note that this choice is not arbitrary, as neglected parameters are of higher order compared to well constrained parameters[35]. The fitted parameters read: $ V_c^{00} = $14.351 MeV, $ V_c^{10} = - $8.263 MeV, $ V_c^{01} = - $8.723 MeV and $ V_T^{10} = - $0.089 MeV fm-2.
The precision of the fit at HO-SM level is rather low, of the order of 500 keV (see Figs. (1, 2, 3, 4). Continuum coupling is indeed necessary to reproduce experimental data. It is included in GSM-CC, where one uses Eq. (5) to generate many-body nuclear states. For this, all the eigenstates of 16O of excitation energy smaller than 10 MeV are used as target pseudo-states in Eq. (3). As model spaces are not identical at HO-SM and GSM-CC levels, one multiplied the two-body nucleon-nucleon interaction of Eq. (1) by a corrective factor depending on the angular momentum $ J $ and parity $ \pi $ of the many-body state in GSM-CC. Moreover, in order to remove spurious energy dependence in the calculated cross sections as much as possible, corrective factors have been devised so that the low-lying eigenstates of 17F reproduce experimental data (see Figs. (1, 2). The corrective factors therein are of the order of 2%~4%, nevertheless, so that the overall features of the Hamiltonian fitted at HO-SM level still remain in GSM-CC.
Let us consider the particle-emission widths of the eigenstates of 17F (see Figs. (1, 2). One can see that they are well reproduced in general, as they typically differ by a factor at most equal to three. However, discrepancies arise in the $ 1/2^+_2 $, $ 1/2^-_1 $ and, especially $ 3/2^-_1 $, eigenstates. Indeed, the experimental width of the $ 3/2^-_1 $, eigenstate is equal to 225 keV, while its theoretical value is 1 keV. The widths of the $ 1/2^+_2 $ and $ 1/2^-_1 $ eigenstates differ by about a factor 7~10 from experimental widths, which is, however, not excessive due to their rather small values. Otherwise, the energies of the high-lying excited states of 17F usually differ by a few hundreds of keV, with the exception of the $ 3/2^-_2 $ eigenstate, differing by almost 2 MeV.
The low-lying spectrum of 17O is depicted in Figs. (3, 4). All the 17O states mirroring the fitted eigenstates of 17F fairly reproduce experimental data, by 200~300 keV at most. In particular, the $ 1/2^+_1 $ state of 17O is about 1.1 MeV above the $ 5/2^+_1 $ ground state of 17O, in good agreement with the experimental value of 0.87 MeV. The Thomas-Ehrmann shift is indeed rather large therein, as the $ 1/2^+_1 $ state of 17F has an excitation energy of 0.495 MeV with respect to the ground state of 17F. One can also note that the neutron-emission width of the s.p. $ 3/2^+_1 $ state of 17O, of 63 keV, is very close to the experimental value of 96 keV. Thus, continuum coupling, which is included in GSM-CC, accounts for the Thomas-Ehrmann shift occurring in 17O and 17F. Otherwise, one can notice that the $ 3/2^-_2 $ eigenstate is too high in energy, as was the case with 17F, and that the $ 7/2^-_2 $ eigenstate of 17O is too low by about 500 keV compared to experimental data. While the widths of the positive-parity states of 17O compare well with experiment, by a factor 10 at most, the widths of negative-parity states are all of the order of 1 keV, when experimental widths can reach 40 keV.
As the main discrepancies with experimental data in the low-lying spectra of 17O and 17F arise above 5 MeV, one can consider that the devised Hamiltonian can be expected to provide sound reaction observables in this region. Moreover, no fine tuning of the Hamiltonian parameters has been effected from experimental cross sections, so that the following results are fully predictive. One calculated excitation functions of the 16O(p,p) elastic scattering reaction at the angles $ \theta = 140.76^\circ $ and $ \theta = 166.75^\circ $ [see Figs. (5, 6)].
The excitation function calculated with GSM-CC at $ \theta = 140.76^\circ $ is very close to experimental data (see Fig. 5). The only noticeable difference is that of the maximum of the cross section, which differs from its experimental value by about 300 keV. Nevertheless, the 16O(p,p) elastic scattering reaction at $ \theta = 140.76^\circ $ is not very sensitive to the details of nuclear structure, as the peak in Fig. 5 is almost entirely due to the s.p. $ 3/2^+_1 $, eigenstate of 17F, so that its consideration is not sufficient to fully assess the ability of the Hamiltonian to describe reaction observables.
Hence, we will study the 16O(p,p) excitation function for $ \theta = 166.75^\circ $ (see Fig. 6). This angle is indeed sufficiently large for the excitation function to have a stringent dependence on the nuclear structure of 17F. The excitation function calculated at $ \theta = 166.75^\circ $ with GSM-CC bears an overall good agreement with the experimental cross section, as the magnitude of the theoretical excitation function is only about 50~100 mb/sr too large compared the experimental excitation function (see Fig. 6). However, one can see that it is not well reproduced around 4 MeV. Note that the sudden decrease of the excitation function around 4 MeV is difficult to obtain at theoretical level, as it is due to 17F negative-parity states, whose theoretical energy and widths must be very close to experimental data for the excitation function to be well described therein. Indeed, the same excitation function calculated in Ref. [9] in the SMEC model, where the most advanced shell model interactions were used for the resonant part of the Hamiltonian is similar to ours. In fact, the experimental excitation function at $ \theta = 166.75^\circ $ could be well reproduced only in Ref. [10], where the Zuker-Buck-Mc Grory (ZBM) shell model interaction[37], fitted from nuclei in the vicinity of 16O, had to be used for the resonant part of the Hamiltonian. The origin of this discrepancy is clearly due to the width of the $ 3/2^-_1 $ eigenstate, which is almost zero in our calculations, whereas it is 225 keV experimentally. This width was indeed sufficiently large in the SMEC calculations of Ref. [10], as it was around 100 keV. The energy of the $ 3/2^-_1 $ eigenstate plays no role therein, as it is well reproduced with both GSM-CC (see Fig. 2) and SMEC in Ref. [10].
Note also that the decrease of the theoretical excitation function obtained in GSM-CC at $ E = 4.5 $ MeV, which is too steep, seems to arise from the $ 3/2^-_2 $ eigenstate, which is too high in energy by about 2 MeV in GSM-CC compared to experimental data. Indeed, the SMEC excitation function fairly reproduces experimental data around this energy[10], which is probably due to the fact that the $ 3/2^-_2 $ eigenstate calculated with SMEC has an energy close to its experimental value[10].