Coulomb-Sturmian basis for the nuclear many-body problem

Calculations in ab initio no-core configuration interaction (NCCI) approaches, such as the no-core shell model or no-core full configuration methods, have conventionally been carried out using the harmonic-oscillator many-body basis. However, the rapid falloff (Gaussian asymptotics) of the oscillator functions at large radius makes them poorly suited for the description of the asymptotic properties of the nuclear wave function. We establish the foundations for carrying out NCCI calculations with an alternative many-body basis built from Coulomb-Sturmian functions. These provide a complete, discrete set of functions with a realistic exponential falloff. We present illustrative NCCI calculations for ${}^6\mathrm{Li}$ with a Coulomb-Sturmian basis and investigate the center-of-mass separation and spurious excitations.

Figure 1

Radial basis functions (shown as squared amplitudes), with $0\le n\le 3$, for $l=0$ (left), $l=2$ (center), and $l=8$ (right): (a)–(c) Coulomb-Sturmian functions $S_{nl}\left(r\right)$, with fixed length parameter $b=1$, (d)–(f) rescaled Coulomb-Sturmian functions $S_{nl}(b_l;r)$, with $l$-dependent length parameter $b_l$ given by the prescription (38), and (g)–(i) harmonic-oscillator functions $R_{nl}\left(r\right)$, with fixed length parameter $b=1[\equiv b_{\mathrm{HO}}]$. The dots mark the location of the node of the $n=1$ function in each panel, and the connector lines highlight the shift in this node between the top and middle rows.

Figure 2

Integrand $R_{nl}(b_{\mathrm{HO}};r)S_{\overline{n}l}(b_l;r)$ for the overlap integral (40), taken for a representative case ($l=0$, $\overline{n}=5$, and $n=5$). For this plot, $b_l/b_{\mathrm{HO}}$ is given by the prescription (38), and $b_{\mathrm{HO}}$ is taken to be unity.

Figure 3

Probability decomposition of the Coulomb-Sturmian radial functions $S_{\overline{n}l}(b_l;r)$ with respect to the basis of harmonic-oscillator radial functions $R_{nl}(b_{\mathrm{HO}};r)$. Results are shown for Coulomb-Sturmian functions with $0\le n\le 3$ (top to bottom) and for $l=0$ (left), $l=2$ (center), and $l=8$ (right), with $b_l/b_{\mathrm{HO}}$ given by the node-matching prescription (38). The histogram bars indicate squared amplitudes ${\langle R_{nl}(b_{\mathrm{HO}};r)|S_{\overline{n}l}(b_l;r)\rangle }^2$ with respect to individual oscillator basis functions. The dashed curve indicates accumulated probability, i.e., for all oscillator basis functions of lesser or equal $n$. The vertical dotted line indicates the truncation of the radial basis in effect if the oscillator functions are limited to 14 major shells ($N_{\mathrm{cut}}=13$), as in the least-truncated calculations of Sec. 4.

Figure 4

The ${}^6\mathrm{Li}$ $1^{+}$ ground-state energy (top) and $3^{+}$ excited-state energy (bottom), calculated using the conventional harmonic-oscillator basis (left) and the Coulomb-Sturmian basis (right). Calculated energies are plotted as a function of the basis $\hslash \Omega$ parameter, for $N_{\max }=4$, 6, 8, and 10 (successive curves, as labeled). For the Coulomb-Sturmian basis, calculations are shown variously for truncations $N_{\mathrm{cut}}=9$ (dotted curves), $N_{\mathrm{cut}}=11$ (dashed curves), and $N_{\mathrm{cut}}=13$ (solid curves) in the change-of-basis transformation of two-body matrix elements. Exponentially extrapolated values (based on the $N_{\mathrm{cut}}=13$ calculations in the case of the Coulomb-Sturmian basis) are indicated by crosses ($\times$). The best extrapolated values from the large-basis calculations of Ref. [35] are shown as horizontal dashed lines.

Figure 5

The ${}^6\mathrm{Li}$ $1^{+}$ ground-state energy, calculated using the Coulomb-Sturmian basis, but without making use of the separable method of Sec. 3d for the two-body matrix elements of the $T_{\mathrm{rel}}$ operator, for comparison with Fig. 4b. That is, the entire Hamiltonian, including $T_{\mathrm{rel}}$, is transformed from the oscillator basis following the approach of Sec. 3c. See the caption to Fig. 4 for further explanation of curves and symbols.

Figure 6

The ${}^6\mathrm{Li}$ $1^{+}$ ground-state RMS radius, calculated using the conventional harmonic-oscillator basis (left) and the Coulomb-Sturmian basis (right). Calculated energies are plotted as a function of the basis $\hslash \Omega$ parameter, for $N_{\max }=4$, 6, 8, and 10 (successive curves, as labeled). For the Coulomb-Sturmian basis, calculations are shown variously for truncations $N_{\mathrm{cut}}=9$ (dotted curves), $N_{\mathrm{cut}}=11$ (dashed curves), and $N_{\mathrm{cut}}=13$ (solid curves) in the change-of-basis transformation of two-body matrix elements. Exponentially extrapolated values (based on the $N_{\mathrm{cut}}=13$ calculations in the case of the Coulomb-Sturmian basis) are indicated by crosses ($\times$). The best estimated value from the large-basis calculations of Ref. [35] is shown as a horizontal dashed line.

Figure 7

Expectation value of the number operator $N_{\mathrm{c}.\mathrm{m}.}^{\tilde{\Omega }}$ for center-of-mass oscillator quanta, as a function of oscillator energy $\hslash \tilde{\Omega }$. These calculations are for the ${}^6\mathrm{Li}$ $1^{+}$ ground state, using the Coulomb-Sturmian basis, with $\hslash \Omega =30\mathrm{MeV}$. Calculations are shown for $N_{\max }=4$, 6, 8, and 10 (successive curves, top to bottom) and for truncations $N_{\mathrm{cut}}=9$ (dotted curves), $N_{\mathrm{cut}}=11$ (dashed curves), and $N_{\mathrm{cut}}=13$ (solid curves) in the change-of-basis transformation of two-body matrix elements. The analogous curve expected for a pure harmonic-oscillator $0s$ function, with $\hslash \Omega =17.5\mathrm{MeV}$, is also shown for comparison (dotted curve, labeled).

Figure 8

Level spectrum for ${}^6\mathrm{Li}$, including spurious states, calculated using the conventional harmonic-oscillator basis. Calculations (left to right) are for $N_{\max }=4$, 6, and 8 and then shown again for $N_{\max }=8$ with addition of a Lawson term, of sufficient strength to shift the spurious states above the energy range displayed in this plot. For each level, the angular momentum $J$ is indicated on the left, and $\langle N_{\mathrm{c}.\mathrm{m}.}^{}\rangle$ is indicated on the right. For degenerate multiplets of spurious states, the thickness of the line is proportional to the number of states. These calculation are for $\hslash \Omega =20\mathrm{MeV}$.

Figure 9

Level spectrum for ${}^6\mathrm{Li}$, calculated using the Coulomb-Sturmian basis. Calculations (left to right) are for $N_{\max }=4$, 6, and 8 and then again for $N_{\max }=8$ with addition of a Lawson term $a{N}_{\mathrm{c}.\mathrm{m}.}^{{\Omega }_L}$ of strength $a=2\mathrm{MeV}$. Energies corrected by $-a\langle N_{\mathrm{c}.\mathrm{m}.}^{{\Omega }_L}\rangle$ are shown at the far right. For each level, the angular momentum $J$ is indicated on the left, and $\langle N_{\mathrm{c}.\mathrm{m}.}^{\tilde{\Omega }}\rangle$ is indicated on the right, as a measure of the number of center-of-mass oscillator quanta. Approximately degenerate multiplets of spurious states are marked by brackets and connected to the state at lower $N_{\max }$ to which they are approximately related by coupling to two center-of-mass quanta. The dashed lines trace the change in level energy induced by the Lawson term. For $N_{\max }=8$, an arrow connects the two $J=2$ levels, which may be described (see text) as admixtures of a nonspurious and spurious level. These calculation are for $\hslash \Omega =20\mathrm{MeV}$, with $N_{\mathrm{cut}}=13$. The quantity $\langle N_{\mathrm{c}.\mathrm{m}.}^{\tilde{\Omega }}\rangle$ is evaluated for $\hslash \tilde{\Omega }=20\mathrm{MeV}$, and the Lawson term is defined for $\hslash {\Omega }_L=20\mathrm{MeV}$ as well.

Figure 10

The ${}^6\mathrm{Li}$ $3^{+}$ excited-state energy, calculated using the Coulomb-Sturmian basis, as in Fig. 4d, but now including a Lawson term, with strength $a=2\mathrm{MeV}$ and Lawson term oscillator energy $\hslash {\Omega }_L$ chosen equal to the basis $\hslash \Omega$. Energies are corrected by subtracting $a\langle N_{\mathrm{c}.\mathrm{m}.}^{{\Omega }_L}\rangle$. See the caption to Fig. 4 for further explanation of curves and symbols.