Benchmarks of the full configuration interaction, Monte Carlo shell model, and no-core full configuration methods

We report no-core solutions for properties of light nuclei with three different approaches in order to assess the accuracy and convergence rates of each method. Full configuration interaction (FCI), Monte Carlo shell model (MCSM), and no core full configuration (NCFC) approaches are solved separately for the ground state energy and other properties of seven light nuclei using the realistic JISP16 nucleon-nucleon interaction. The results are consistent among the different approaches. The methods differ significantly in how the required computational resources scale with increasing particle number for a given accuracy.

Figure 1

Overview of the basis spaces covered with the many-body methods discussed here for the case of ${}^{12}$C. $N_{\max }$ is defined as the number of oscillator quanta above lowest possible number of quanta. $N_{\mathrm{shell}}$ is the number of oscillator shells counting the $0s$ shell as the first shell. The MCSM incorporates an FCI space. That is, all single-particle states in the included shells are available to all particles without additional restrictions except for symmetry constraints.

Figure 2

Comparisons of the energies between the MCSM and FCI along with the fully converged NCFC results where available. The NCFC result for the ${}^{10}$B($1^{+}$) state has a large uncertainty indicated by the grey band. The MCSM (FCI) results are shown as the solid (dashed) lines that nearly coincide where both are available. The extrapolated MCSM results are illustrated by bands. From top to bottom, the truncation of the model space is $N_{\mathrm{shell}}=2$ (red), 3 (green), 4 (blue), and 5 (purple). Note that the MCSM results are extrapolated by the energy variance with the second-order polynomials [17]. Also note that all of the results of ${}^{10}$B and ${}^{12}$C at $N_{\mathrm{shell}}=4$ were obtained only with MCSM.

Figure 3

Comparisons of the ground state energies of ${}^4$He (left) and ${}^6$Li (right) between the FCI with $N_{\mathrm{shell}}$ truncation and NCFC with $N_{\max }$ truncation as a function of the HO frequecy $\hslash \omega$. The FCI (NCFC with cutoff) results are shown as the solid (dashed) lines. From top to bottom, the truncation of the model space increments by unity for $N_{\mathrm{shell}}$ up to $N_{\mathrm{shell}}=7$ (cyan) for ${}^4$He up to $N_{\mathrm{shell}}=5$ (orange) for ${}^6$Li, and by two for $N_{\max }$ up to $N_{\max }=14$ (purple) for both ${}^4$He and ${}^6$Li.

Figure 4

The convergence of the MCSM ground state energy of ${}^4$He to the FCI result for several $N_{\mathrm{shell}}$ values as function of the number of Monte Carlo basis states, $N_b$. Top: ground state energy; Bottom: relative difference between MCSM and FCI calculations of the energy (black) and rms matter radius (red).

Figure 5

The relative difference between MCSM and FCI calculations of the energy (black), rms matter radius (red), and magnetic moment (blue) for several $N_{\mathrm{shell}}$ values as function of the number of MC basis states, $N_b$. Top to bottom on the left: ${}^6$He($J^{\pi }=0^{+}$), ${}^6$Li($1^{+}$), ${}^7$Li(${\frac{1}{2}}^{-}$), and ${}^7$Li(${\frac{3}{2}}^{-}$); Top to bottom on the right: ${}^8$Be($0^{+}$), ${}^{10}$B($1^{+}$), ${}^{10}$B($3^{+}$), ${}^{12}$C($0^{+}$).

Figure 6

Number of MC basis states, $N_b$, required for a given accuracy of the MCSM energy for (a) $N_{\mathrm{shell}}=3$ and (b) $N_{\mathrm{shell}}=4$ for ${}^4$He($J^{\pi }=0^{+}$), ${}^6$He($0^{+}$), ${}^8$Be($0^{+}$), and ${}^{12}$C($0^{+}$).

Figure 7

The convergence of the MCSM ground state energy of ${}^6$Li to the FCI result; circles (red), triangles (green), squares (blue), and diamonds (purple) indicate the results at $N_{\mathrm{shell}}=$ 2, 3, 4, and 5, respectively. (a) Ground state energy as function of the number of Monte Carlo basis states, $N_b$; (b) Energy variance $\Delta E_2$ and extrapolation to zero energy variance.

Figure 8

Comparisons of the point-particle rms matter radii between the MCSM and FCI. Circles (red), triangles (green), squares (blue) and diamonds (purple) indicate the results at $N_{\mathrm{shell}}=$ 2, 3, 4 and 5, respectively; solid (open) symbols stand for the MCSM (FCI) results. The MCSM results are those after extrapolation as described in the text. Note that the ${}^{10}$B and ${}^{12}$C results at $N_{\mathrm{shell}}=4$ were obtained only within the MCSM.

Figure 9

The convergence of the MCSM magnetic moment of ${}^6$Li to the FCI result; circles (red), triangles (green), squares (blue), and diamonds (purple) indicate the results at $N_{\mathrm{shell}}=$ 2, 3, 4, and 5, respectively. (a) Magnetic moment as function of the number of Monte Carlo basis states, $N_b$; (b) Energy variance and extrapolation to zero energy variance. For comparison, the dashed lines (top) and open symbols at $\Delta E_2=0$ (bottom) are the results of our (exact) FCI calculations.

Figure 10

Comparisons between the extrapolated MCSM and FCI results for (a) the magnetic dipole moments, and (b) electric quadrupole moments. The conventions for the symbols are same as in Fig. 8; crosses indicate the experimental values for the ground states from Ref. [23].

Figure 11

The energy variance and extrapolation to the FCI result for the energies. Circles (red), triangles (green), squares (blue), and diamonds (purple) indicate the results at $N_{\mathrm{shell}}=2$, 3, 4 and 5, respectively; the open symbols at $\Delta E_2=0$ are the exact FCI energies. Top to bottom on the left: ${}^6$He($J^{\pi }=0^{+}$), ${}^6$Li($1^{+}$), ${}^7$Li(${\frac{1}{2}}^{-}$), and ${}^7$Li(${\frac{3}{2}}^{-}$); Top to bottom on the right: ${}^8$Be($0^{+}$), ${}^{10}$B($1^{+}$), ${}^{10}$B($3^{+}$), and ${}^{12}$C($0^{+}$).

Figure 12

The energy variance and extrapolation to the FCI result for the rms radii. Symbols are the same as in Fig. 11. Top to bottom on the left: ${}^6$He($J^{\pi }=0^{+}$), ${}^6$Li($1^{+}$), ${}^7$Li(${\frac{1}{2}}^{-}$), and ${}^7$Li(${\frac{3}{2}}^{-}$); Top to bottom on the right: ${}^8$Be($0^{+}$), ${}^{10}$B($1^{+}$), ${}^{10}$B($3^{+}$), and ${}^{12}$C($0^{+}$).

Figure 13

The energy variance and extrapolation to the FCI result for the quadrupole moments. Symbols are the same as in Fig. 11. Top and bottom on the left: ${}^6$Li($1^{+}$) and ${}^7$Li(${\frac{3}{2}}^{-}$); Top and bottom on the right: ${}^{10}$B($1^{+}$) and ${}^{10}$B($3^{+}$).

Figure 14

The energy variance and extrapolation to the FCI result for the dipole moments. Symbols are the same as in Fig. 11. Top and bottom on the left: ${}^7$Li(${\frac{1}{2}}^{-}$) and ${}^7$Li(${\frac{3}{2}}^{-}$); Top and bottom on the right: ${}^{10}$B($1^{+}$) and ${}^{10}$B($3^{+}$).