Ab initio no-core full configuration calculations of light nuclei

We perform no-core full configuration calculations for a set of light nuclei including ${}^{16}\mathrm{O}$ with a realistic $\mathit{NN}$ interaction, JISP16. We obtain ground-state energies and their uncertainties through exponential extrapolations that we demonstrate are reliable in ${}^2\mathrm{H}$, ${}^3\mathrm{H}$, and ${}^4\mathrm{He}$ test cases where fully converged results are obtained directly. We find that ${}^6\mathrm{He}$, ${}^6\mathrm{Li}$, and ${}^8\mathrm{He}$ are underbound by about 600 keV, 560 keV, and $1.7$ MeV, respectively. ${}^{12}\mathrm{C}$ is overbound by about $1.7$ MeV and ${}^{16}\mathrm{O}$ is overbound by about 16 MeV. The first excited $0^{+}$ states in ${}^{12}\mathrm{C}$ and ${}^{16}\mathrm{O}$ are also evaluated but their uncertainties are significantly larger than the uncertainties for the ground states.

Figure 1

Representative Hamiltonian matrix dimensions for total magnetic projection $M=0$ states in the single-particle $m$-scheme as a function of the maximum total oscillator quanta of excitations specified by $N_{\max }$. The natural-parity matrix dimensions are represented by the specific points while unnatural-parity matrix dimensions would lie close to the interpolating lines at odd values of $N_{\max }$.

Figure 2

Number of nonvanishing many-body Hamiltonian matrix elements for representative light nuclei as a function of the basis space dimension. The points represent sample cases that have been solved and correspond to those indicated in the legend. The curves approximate a $D^{3/2}$ power law where $D$ is the basis space dimension. The vertical arrow measures a factor of 30 between the two-body Hamiltonian (rank $=$ 2) and three-body Hamiltonian (rank $=$ 3) cases for ${}^{12}\mathrm{C}$ at the same dimension corresponding to $N_{\max }=6$. Note the logarithmic scales.

Figure 3

Calculated ground-state energy of ${}^4\mathrm{He}$ as a function of $N_{\max }$ at various values of the oscillator energy, $\hslash \Omega$, as indicated in the legend. The results are connected by straight-line segments to guide the eye. The results with chiral N3LO are from Ref. [9]. The results for JISP16 are closer to convergence even in rather modest basis spaces. No extrapolations are needed in these cases as converged results are obtained directly.

Figure 4

Calculated ground-state energy of ${}^{12}\mathrm{C}$ in the ESPM as a function of $\hslash \Omega$ and $N_{\max }$, the maximum value of the HO states’ quanta, $2n+l$, used in expanding the Saxon-Woods single-particle states. The curve closest to convergence corresponds to the value $N_{\max }=20$ and successively higher curves are obtained with $N_{\max }$ decreased by 2 units for each curve.

Figure 5

Calculated ground-state energy of ${}^{12}\mathrm{C}$ in the ESPM as a function of $N_{\max }$ for selected values of $\hslash \Omega$ used in defining the basis states. The points correspond to $\hslash \Omega$ values ranging from 5 to 30 MeV in 2.5-MeV increments. The curves represent an exponential plus a constant fit to each set of results at fixed $\hslash \Omega$, excluding the $N_{\max }=0$ result. Each point carries equal weight.

Figure 6

Calculated ground-state energy of ${}^2\mathrm{H}$ as a function of the oscillator energy, $\hslash \Omega$, for selected values of $N_{\max }$ used in defining the basis states. The curve closest to experiment corresponds to the value $N_{\max }=20$ and successively higher curves are obtained with $N_{\max }$ decreased by two units for each curve. The curves are formed by straight-line segments joining calculated results.

Figure 7

Calculated ground-state energy of ${}^2\mathrm{H}$ as a function of $N_{\max }$ at $\hslash \Omega$ values that bracket the best upper bound as described in the text. The smooth curves are fits by Eq. (1) to the four data points shown at each value of $\hslash \Omega$ with each point weighted equally.

Figure 8

Extrapolated ground-state energies and variational upper bounds from each set of four (extrapolation A) or three (extrapolation B) successive $N_{\max }$ values as a function of the largest value of $N_{\max }$ in each set. Error bars are dominated by the uncertainties in the extrapolations and are obtained as described in the text. Note the expanded scale and the reasonable consistency of the extrapolated results: for $N_{\max }⩾10$ all but one are within their uncertainty range of the exact answer.

Figure 9

Calculated ground-state energy of ${}^2\mathrm{H}$ for $N_{\max }=8$, $10$, $12$, and the extrapolated ground-state energy using method B, as function of the oscillator energy, $\hslash \Omega$. Error bars are obtained from the difference with the extrapolation using $N_{\max }=6$, $8$, and $10$ calculations.

Figure 10

Calculated ground-state energy of ${}^3\mathrm{H}$ as a function of the oscillator energy, $\hslash \Omega$, for selected values of $N_{\max }$. The curve closest to experiment corresponds to the value $N_{\max }=20$ and successively higher curves are obtained with $N_{\max }$ decreased by two units for each curve.

Figure 11

Calculated ground-state energy of ${}^3\mathrm{H}$ as a function of $N_{\max }$ at $\hslash \Omega$ values that bracket the best upper bound. The smooth curves are fits of an exponential plus a constant [see Eq. (1)] to the four data points shown at each value of $\hslash \Omega$. There is a 5-keV spread in these asymptotes that is used to assign the uncertainty to the asymptote derived from the constrained fit as described in the text. The asymptote of the constrained global fit (extrapolation A) is quoted in the figure and the experimental result is shown for comparison.

Figure 12

Extrapolated ground-state energies and variational upper bounds from each set of four (extrapolation A) or three (extrapolation B) successive $N_{\max }$ values as a function of the largest value of $N_{\max }$ in each set. Error bars are dominated by the uncertainties in the extrapolations and obtained as described in the text. Note the consistency of the extrapolated results.

Figure 13

Calculated ground-state energy of ${}^4\mathrm{He}$ as function of the oscillator energy, $\hslash \Omega$, for a sequence of $N_{\max }$ values. The curve closest to experiment corresponds to the value $N_{\max }=16$ and successively higher curves are obtained with $N_{\max }$ decreased by 2 units for each curve.

Figure 14

Calculated ground-state energy of ${}^4\mathrm{He}$ for $N_{\max }=2-8$ at selected values of $\hslash \Omega$. Each set of four points is fit, using equal weights, with an exponential plus constant [see Eq. (1)] producing the solid curves. Half the resulting spread in the asymptotic values is used to determine the uncertainty assigned to the first point in Fig. 17 for extrapolation A as described in the text.

Figure 15

Calculated ground-state energy of ${}^4\mathrm{He}$ for $N_{\max }=8$, $10$, $12$, and the extrapolated ground-state energy using method B, as function of the oscillator energy, $\hslash \Omega$. Error bars are obtained from the difference with the extrapolation using $N_{\max }=6$, $8$, and $10$ calculations.

Figure 16

Calculated ground-state energy of ${}^4\mathrm{He}$ for $N_{\max }=2-16$ for JISP16 at selected values of $\hslash \Omega$. Each set of eight points at fixed $\hslash \Omega$ is fit by Eq. (1) producing the solid curves. Each point is a true upper bound to the exact answer. The asymptotes $E_{\mathrm{gs}}(\infty )$ are the same to within 35 keV of their average value and they span the experimental ground-state energy.

Figure 17

Extrapolated ground-state energies and variational upper bounds from each set of four (extrapolation A) or three (extrapolation B) successive $N_{\max }$ values as a function of the largest value of $N_{\max }$ in each set. Error bars represent the assessed uncertainties in the extrapolations and are obtained as described in the text. Note the consistency of the extrapolations: the exact answer is well within the uncertainty range of the extrapolations for all $N_{\max }$ points, with the uncertainty diminishing with increasing $N_{\max }$.

Figure 18

Calculated ground-state energy of ${}^6\mathrm{He}$ as function of the oscillator energy, $\hslash \Omega$, for selected values of $N_{\max }$. The figure displays also the experimental result, the common asymptote from the global extrapolation (A), and the experimental threshold for the $\alpha +2n$ breakup. The curve closest to experiment corresponds to the value $N_{\max }=14$ and successively higher curves are obtained with $N_{\max }$ decreased by two units for each curve.

Figure 19

Calculated ground-state energy of ${}^6\mathrm{He}$ for $N_{\max }=2-14$ for JISP16 at selected values of $\hslash \Omega$. Each set of points at fixed $\hslash \Omega$ is fit by Eq. (1) using equal weights producing the solid curves. Each point is a true upper bound to the exact answer. The resulting asymptotes $E_{\mathrm{gs}}(\infty )$ are the same to within 600 keV of their average value. The figure displays the experimental result and the common asymptote from the global extrapolation (A) as described in the text.

Figure 20

Extrapolated ground-state energies and upper bounds from sets of successive $N_{\max }$ values as a function of the largest value of $N_{\max }$ in each set. Uncertainties are determined for each value of $\hslash \Omega$ as described in the text. Note the consistency of the extrapolated results as they fall well within their uncertainty ranges along the paths of converging sequences.

Figure 21

Calculated ground-state energy of ${}^6\mathrm{Li}$ as function of the oscillator energy, $\hslash \Omega$, for selected values of $N_{\max }$. The curve closest to experiment corresponds to the value $N_{\max }=14$ and successively higher curves are obtained with $N_{\max }$ decreased by two units for each curve. The figure displays also the global extrapolation (A) and the threshold for the $\alpha +d$ breakup.

Figure 22

Calculated ground-state energy of ${}^6\mathrm{Li}$ for $N_{\max }=8-14$ at values of $\hslash \Omega$ that span the minimum at $N_{\max }=14$. Curves define the fits using the global extrapolation (A) that produces a common constant, the asymptote, labeled by “Extrap”.

Figure 23

Upper bounds and extracted asymptotes for the ground-state energy of ${}^6\mathrm{Li}$ from each set of successive $N_{\max }$ values as a function of the largest value of $N_{\max }$ in each set. Uncertainties in the asymptotes are determined as described in the text. Note the consistency of the extrapolated results as they fall well within their uncertainty ranges along the paths of converging sequences.

Figure 24

Calculated ground-state energy of ${}^4\mathrm{He}$ for $N_{\max }=6$, $8$, $10$, $12$, and the extrapolated ground-state energies using method B, as well as the result from extrapolation method A, with error bars. For comparison, we also show the experimental value, and the threshold for $\alpha +4n$ breakup.

Figure 25

Upper bounds and extrapolated ground-state energies of ${}^8\mathrm{He}$ from each set of successive $N_{\max }$ values as a function of the largest value of $N_{\max }$ in each set. Uncertainties in the asymptotes are determined as described in the text.

Figure 26

Calculated ground-state energy of ${}^{12}\mathrm{C}$ as function of the oscillator energy, $\hslash \Omega$, for selected values of $N_{\max }$. The curve closest to experiment corresponds to the value $N_{\max }=8$ and successively higher curves are obtained with $N_{\max }$ decreased by 2 units for each curve.

Figure 27

Calculated ground-state energy of ${}^{12}\mathrm{C}$ for $N_{\max }=2-8$ at selected values of $\hslash \Omega$ as described in the text. For each $\hslash \Omega$ the data are fit by Eq. (1). These independent asymptotes $E_{\mathrm{gs}}(\infty )$ provide a measure of our uncertainty within the global extrapolation (A). The figure displays the experimental ground-state energy and the common asymptote obtained in the global extrapolation.

Figure 28

Calculated energy of ${}^{12}\mathrm{C}$ first excited $0^{+}$ state (Hoyle state) for $N_{\max }=2-8$ at selected values of $\hslash \Omega$ as described in the text. For each $\hslash \Omega$ the data are fit by Eq. (1). These independent asymptotes $E_{\mathrm{gs}}(\infty )$ provide a measure of our uncertainty within the global extrapolation (A). The figure displays the experimental energy and the common asymptote of the global extrapolation.

Figure 29

Calculated ground-state energy of ${}^{16}\mathrm{O}$ as function of the oscillator energy, $\hslash \Omega$, for selected values of $N_{\max }$. The curve closest to experiment corresponds to the value $N_{\max }=8$ and successively higher curves are obtained with $N_{\max }$ decreased by 2 units for each curve.

Figure 30

Calculated ground-state energy of ${}^{16}\mathrm{O}$ for $N_{\max }=2-8$ at selected values of $\hslash \Omega$ as described in the text. For each $\hslash \Omega$, the data are fit to an exponential plus a constant, the asymptote. The figure displays the experimental ground-state energy and the common asymptote obtained in the global extrapolation (A).

Figure 31

Calculated first excited $0^{+}$ state energy of ${}^{16}\mathrm{O}$ as function of the oscillator energy, $\hslash \Omega$, for selected values of $N_{\max }$. The curve closest to experiment corresponds to the value $N_{\max }=8$ and successively higher curves are obtained with $N_{\max }$ decreased by 2 units for each curve. The figure displays the experimental excited state energy and the common asymptote obtained in the global extrapolation (A).