Ground-state properties of light $4n$ self-conjugate nuclei in ab initio no-core Monte Carlo shell model calculations with nonlocal $NN$ interactions

We report $J^{\pi }=0^{+}$ ground-state energies and point-proton radii of ${}^4\mathrm{He}$, ${}^8\mathrm{Be}$, ${}^{12}\mathrm{C}$, ${}^{16}\mathrm{O}$, and ${}^{20}\mathrm{Ne}$ nuclei calculated by the ab initio no-core Monte Carlo shell model with the JISP16 and Daejeon16 nonlocal $NN$ interactions. Ground-state energies are obtained in the basis spaces up to seven oscillator shells $(N_{\mathrm{shell}}=7)$ with several oscillator energies $\left(\hslash \omega \right)$ around the optimal oscillator energy for the convergence of ground-state energies. These energy eigenvalues are extrapolated to obtain estimates of converged ground-state energies in each basis space using energy variances of computed energy eigenvalues. We further extrapolate these energy-variance-extrapolated energies obtained in the finite basis spaces to infinite basis-space results with an empirical exponential form. This form features a dependence on the basis-space size but is independent of the value of $\hslash \omega$ used for the harmonic-oscillator basis functions. Point-proton radii for these states of atomic nuclei are also calculated following techniques employed for the energies. From these results, it is found that the Daejeon16 $NN$ interaction provides good agreement with experimental data up to approximately ${}^{16}\mathrm{O}$, while the JISP16 $NN$ interaction provides good agreement with experimental data up to approximately ${}^{12}\mathrm{C}$. Beyond these nuclei, the interactions produce overbinding accompanied by radii that are too small. These findings suggest and encourage further revisions of nonlocal $NN$ interactions towards the investigation of nuclear structure in heavier-mass regions.

Figure 1

$M$-scheme dimensions $D_{\mathrm{M}}$ for light nuclei examined in this work as a function of basis space $N_{\mathrm{shell}}$. The horizontal black line indicates the approximate limit of the dimension for full configuration interaction (FCI) calculations using the Lanczos diagonalization method.

Figure 2

Convergence of ${}^4\mathrm{He}$ ground-state energy with respect to the number of basis states $N_{\mathrm{b}}$ in fixed sizes of basis space (left panels) and that with respect to energy variances $\mathrm{\Delta }{E}^2$ (right panels). The upper panels are the results with the JISP16 $NN$ interaction, and the lower panels are for the Daejeon16 $NN$ interaction. The Coulomb interaction is included perturbatively in the case of JISP16, while directly in the case of Daejeon16. The basis space results in $N_{\mathrm{shell}}=2$ to 7 follow a decreasing sequence from the top to the bottom symbols in each panel. The results are shown with the optimal value of $\hslash \omega$ (25 MeV for the JISP16 and 20 MeV for the Daejeon16) for the convergence of energy in the largest basis space examined in this work ($N_{\mathrm{shell}}=7$). The black curves in the right panels indicate the fit of a second-order polynomial function to the calculated results. See more details in the text.

Figure 3

Convergence of ${}^4\mathrm{He}$ ground-state energy with respect to the size of basis space with the optimal $\hslash \omega$ value obtained by fitting a simple exponential form (6) to results obtained by energy-variance extrapolation within each value of $N_{\mathrm{shell}}$. The left (right) panel is for the JISP16 (Daejeon16) $NN$ interaction. The solid circles indicate the energy-variance-extrapolated energies in each basis-space size. The error bars from the energy-variance extrapolation are smaller than the size of the symbols. The “Extrap.” above three symbols denotes the basis-space-extrapolated energies with the extrapolation errors from the results in $N_{\mathrm{shell}}=2\text{–}7$, 3–7 and 4–7 from left to right, respectively. “NCFC” denotes the ab initio results with the JISP16 [38, 67] and with the Daejeon16 [67]. “Expt.” corresponds to the experimental value taken from the compilation of AME2012 [75].

Figure 4

${}^4\mathrm{He}$ ground-state energy as a function of $\hslash \omega$ at the maximum $N_{\mathrm{b}}$ in each basis space (i.e., without energy-variance extrapolation). Results with the JISP16 $NN$ interaction are shown in the left panel and those with Daejeon16 $NN$ interaction in the right. The numbers in the panels indicate the size $N_{\mathrm{shell}}$ of basis space. The results before the energy-variance extrapolation are presented here.

Figure 5

${}^4\mathrm{He}$ ground-state energy as a function of $\hslash \omega$ including some of the results from Fig. 4 on an enlarged scale. Results with the JISP16 $NN$ interaction are shown in the left panel and those with Daejeon16 $NN$ interaction in the right. The horizontal solid line specifies the experimental value. The numbers in the panels indicate the size of basis space $N_{\mathrm{shell}}$. The results after the energy-variance and infinite basis-space extrapolations (“Extrap.”) are presented here. Note that the errors from the energy-variance extrapolation are smaller than the size of symbols. The black symbols with gray band are the results extrapolated into the infinite basis-space-size limit. The gray bands denote the range of upper and lower edges of three different set of infinite-basis extrapolations. See more details in the text.

Figure 6

Comparison of the extrapolated MCSM results for the energy per nucleon with the JISP16 and Daejeon16 $NN$ interactions to experimental data. The MCSM results are shown for the extrapolations using the $N_{\mathrm{shell}}=4\text{–}7$ results with the respective optimal $\hslash \omega$ values. The experimental data are taken from the 2012 Atomic Mass Evaluation (AME2012) [75].

Figure 7

Point-proton root-mean-square (rms) radius of the ${}^4\mathrm{He}$ ground state as a function of $\hslash \omega$. Results with the JISP16 $NN$ interaction are shown in the left panel and those with Daejeon16 $NN$ in the right panel. The MCSM results before the energy-variance extrapolations are plotted here. The numbers in the leftmost side of figures specify the size $N_{\mathrm{shell}}$ of the basis space. The horizontal black bar with gray band denotes the average value of the rms radius with the uncertainty around the optimal HO frequency range (three successive points with least sensitivity to the HO frequency). The horizontal black line stretching across each full panel is the experimental value taken from the ADNDT2013 [78]. The inset is an enlarged drawing of the average of the MCSM results around the optimal $\hslash \omega$ values.

Figure 8

Comparison of the MCSM results of point-proton rms radius with the JISP16 and Daejeon16 $NN$ interactions to experimental data. The MCSM results with error bars are shown for the averages around the optimal $\hslash \omega$ values (three successive points with the least sensitivity to the HO frequency at $N_{\mathrm{shell}}=7$). The experimental data are taken from the ADNDT2013 [78]. In the case of ${}^8\mathrm{Be}$, the radii for the ground states of the neighboring nuclei, ${}^{7,9}\mathrm{Be}{\frac{3}{2}}^{-}$ states (the left black symbol for ${}^7\mathrm{Be}$ and the right for ${}^9\mathrm{Be}$) are taken as reference radii, because of the missing experimental value of ${}^8\mathrm{Be}$. The result with JISP16 for the ${}^8\mathrm{Be}$ radius are not shown in the figure (see more details in the text and in the caption of Table 2).

Figure 9

Energy-variance extrapolation of point-proton rms radii using the results with the Daejeon16 $NN$ interaction. The calculated radii are from $N_{\mathrm{shell}}=7$ results around the optimal HO frequencies ($\hslash \omega =10$ MeV for ${}^4\mathrm{He}$ and 15 MeV for ${}^8\mathrm{Be}$, ${}^{12}\mathrm{C}$, ${}^{16}\mathrm{O}$, and ${}^{20}\mathrm{Ne}$). The solid lines are the fits to the computed radii by the linear functions. The range of the fit is the region where the points and line overlap.

Figure 10

Convergence of ground-state energies of four $4n$ nuclei with respect to the number of basis states $N_{\mathrm{b}}$ in fixed sizes of basis space (left) and convergence with respect to energy variances (right). The JISP16 two-nucleon interaction is employed. The basis space is taken from $N_{\mathrm{shell}}=2$ to 7. The results are shown with the optimum value of $\hslash \omega$ for energy in the largest basis space examined in this work ($N_{\mathrm{shell}}=7$). The notation is the same as in Fig. 2.

Figure 11

Convergence of ground-state energy of four $4n$ nuclei with respect to the number $N_{\mathrm{b}}$ of basis states in fixed sizes of basis space (left) and convergence with respect to energy variances (right). The Daejeon16 two-nucleon interaction is employed. The basis space is taken from $N_{\mathrm{shell}}=2$ to 7. The results are shown with the optimum value of $\hslash \omega$ for energy in $N_{\mathrm{shell}}=7$. The notation is the same as in Fig. 2.

Figure 12

Ground-state energies at the maximum value of $N_{\mathrm{b}}$ in each space as functions of $\hslash \omega$. Results with the JISP16 $NN$ interaction are shown in the left column and those with Daejeon16 $NN$ interaction are in the right column. The notations are the same as in Fig. 4.

Figure 13

Ground-state energies as functions of $N_{\mathrm{shell}}$. Results with the JISP16 $NN$ interaction are shown in the left column and those with the Daejeon16 $NN$ interaction are in the right column. The no-core MCSM results with optimal $\hslash \omega$ values for each state after the energy-variance extrapolations are presented here as solid round symbols. The dotted, dashed, and solid curves are the fits to results in $N_{\mathrm{shell}}=2\text{–}7$, 3–7, and 4–7, respectively. The open symbols with error bars are for the resultant extrapolated values fit to $N_{\mathrm{shell}}=2\text{–}7$, 3–7, and 4–7, from left to right, respectively. The right-most symbol is the experimental data taken from Ref. [75]. Note that the fits of the results in $N_{\mathrm{shell}}=2\text{–}7$ for ${}^{16}\mathrm{O}$ and ${}^{20}\mathrm{Ne}$ are absent due to the lack of $N_{\mathrm{shell}}=2$ results for those cases. The notation is the same as in Fig. 3.

Figure 14

Ground-state energies with the extrapolations as functions of $\hslash \omega$. Results with the JISP16 $NN$ interaction are shown in the left column and those with Daejeon16 $NN$ interaction are in the right column. The no-core MCSM results after the energy-variance extrapolations are presented here. For additional information, see the caption to Fig. 5.

Figure 15

Convergence of ${}^4\mathrm{He}$ point-proton radius with respect to the number $N_{\mathrm{b}}$ of basis states in the fixed sizes of basis space (left) and that with respect to energy variances (right). In the figure, the results are for the JISP16 in the upper row, whereas results for Daejeon16 are in the lower row. The basis space is taken from $N_{\mathrm{shell}}=2$ to 7. The symbols and colors are the same as in the figures of ground-state energy. The results are shown for the value of $\hslash \omega$ around the point-proton radius inflexion point in the largest basis space examined in this work ($N_{\mathrm{shell}}=7$).

Figure 16

Convergence of point-proton radii for additional $4n$ nuclei with respect to the number $N_{\mathrm{b}}$ of basis states in the fixed sizes of basis space (left) and that with respect to energy variances (right). The JISP16 two-nucleon interaction is employed. The basis space is taken from $N_{\mathrm{shell}}=2$ to 7 for ${}^8\mathrm{Be}$ and ${}^{12}\mathrm{C}$ and from $N_{\mathrm{shell}}=3$ to 7 for ${}^{16}\mathrm{O}$ and ${}^{20}\mathrm{Ne}$. The basis space size is indicated by the number shown in the right panel at the second row. The results are shown with the optimum value of $\hslash \omega$ for the convergence of the radius in the $N_{\mathrm{shell}}=7$ space. The notations and symbols are the same as in Fig. 15.

Figure 17

Convergence of point-proton radii for $4n$ nuclei beyond ${}^4\mathrm{He}$ with respect to the number $N_{\mathrm{b}}$ of basis states in the fixed sizes of basis space (left) and that with respect to energy variances (right). The Daejeon16 two-nucleon interaction is employed. The basis space is taken from $N_{\mathrm{shell}}=2$ to 7 for ${}^8\mathrm{Be}$ and ${}^{12}\mathrm{C}$ and from $N_{\mathrm{shell}}=3$ to 7 for ${}^{16}\mathrm{O}$ and ${}^{20}\mathrm{Ne}$. The symbols and colors are the same as in Fig. 16. The basis-space size is also specified by the numbers in the panel of the lower-right corner. The results are shown with the optimum value of $\hslash \omega$ for the convergence of radius in the largest basis space examined in this work ($N_{\mathrm{shell}}=7$).

Figure 18

Point-proton rms radius of the ground states of $4n$ self-conjugate nuclei from ${}^8\mathrm{Be}$ to ${}^{20}\mathrm{Ne}$ as a function of $\hslash \omega$. Results with the JISP16 $NN$ interaction are shown in the left column and those with the Daejeon16 $NN$ interaction are in the right column. The no-core MCSM results without the energy-variance extrapolation are presented here. The short line, with the gray band for the estimated error, describes the final result averaged around the optimal $\hslash \omega$ value. For the ${}^8\mathrm{Be}$ radius with the JISP16 interaction, we extrapolated the results at fixed $\hslash \omega$ values as in the extrapolation of energy since we did not locate the inflection point. The notations and symbols are the same as in Fig. 7.