Natural orbitals for the ab initio no-core configuration interaction approach

Ab initio no-core configuration interaction (NCCI) calculations for the nuclear many-body problem have traditionally relied upon an antisymmetrized product (Slater determinant) basis built from harmonic oscillator orbitals. The accuracy of such calculations is limited by the finite dimensions which are computationally feasible for the truncated many-body space. We therefore seek to improve the accuracy obtained for a given basis size by optimizing the choice of single-particle orbitals. Natural orbitals, which diagonalize the one-body density matrix, provide a basis which maximizes the occupation of low-lying orbitals, thus accelerating convergence in a configuration-interaction basis, while also possibly providing physical insight into the single-particle structure of the many-body wave function. We describe the implementation of natural orbitals in the NCCI framework and examine the nature of the natural orbitals thus obtained, the properties of the resulting many-body wave functions, and the convergence of observables. After taking ${}^3\mathrm{He}$ as an illustrative testbed, we explore aspects of NCCI calculations with natural orbitals for the ground state of the $p$-shell neutron halo nucleus ${}^6\mathrm{He}$.

Figure 1

Dimension of the NCCI many-body space as a function of the number of oscillator excitations $N_{\text{max}}$ included in the basis, including for ${}^{3,6}\text{He}$ (highlighted). The dimension of the FCI space constructed from the same orbitals is also shown for ${}^3\mathrm{He}$ (dotted gray line). Dimensions are those obtained with $M$-scheme bases ($M=0$ for even-mass nuclei, or $M=1/2$ for odd-mass nuclei) for the normal-parity space.

Figure 2

Convergence of ${}^3\mathrm{He}$ ground-state energy, as calculated in (a) oscillator (solid circles) and (b) natural-orbital (open squares) bases, shown also (c) on a logarithmic scale as the residual $E-E_{\mathrm{ref}}$ with respect to the true “full-space” value. Calculated values are shown as functions of the basis parameter $\hslash \omega$, for successive even value of $N_{\text{max}}$, from $N_{\text{max}}=8$ (dotted lines) to 16 (solid lines, highlighted). The experimental binding energy (solid diamond) [80] is also shown.

Figure 3

Comparison of ${}^3\mathrm{He}$ ground-state energies as calculated in spaces defined by $N_{\text{max}}$-truncated bases—oscillator (solid circles) or natural-orbital (open squares)—and the corresponding enveloping FCI space (solid triangles). Energies are shown as residuals, as in Fig. 2. Calculated values are shown as functions of the basis parameter $\hslash \omega$, for $N_{\text{max}}=4$ (dotted lines) and 10 (solid lines, highlighted).

Figure 4

Convergence of ${}^3\mathrm{He}$ ground-state point-proton r.m.s. radius, as calculated in oscillator (solid circles) and natural-orbital (open squares) bases. Calculated values are shown as functions of the basis parameter $\hslash \omega$, for successive even value of $N_{\text{max}}$, from $N_{\text{max}}=8$ (dotted lines) to 16 (solid lines, highlighted). The value deduced from the experimental charge radius [83] is also shown (filled diamond).

Figure 5

Radial wave functions obtained for the ${}^3\mathrm{He}$ proton $0s_{1/2}$ natural orbital, from different underlying oscillator-basis calculations, plotted as the radial probability density $P\left(r\right)=r^2{\left|\psi \left(r\right)\right|}^2$. Results are shown as obtained from underlying oscillator-basis calculations with (a) $\hslash \omega =9\mathrm{MeV}$, (b) $\hslash \omega =15\mathrm{MeV}$, and (c) $\hslash \omega =25\mathrm{MeV}$. Radial wave functions are shown for $N_{\text{max}}=2$ (dotted lines) through $N_{\text{max}}=16$ (solid lines, highlighted), with the oscillator $0s$ function for the given $\hslash \omega$ (thick gray lines) shown for comparison. The locations of the peaks of the underlying harmonic-oscillator orbital and $N_{\text{max}}=16$ natural orbital are marked with dashed vertical lines.

Figure 6

Radial wave functions obtained for the ${}^3\mathrm{He}$ proton $0s_{1/2}$ natural orbital, from different underlying oscillator-basis calculations, plotted as the radial probability density $P\left(r\right)$, as in Fig. 5, but now on a logarithmic scale.

Figure 7

Radial wave functions for the ${}^3\mathrm{He}s$-, $p$-, and $sd$-shell natural orbitals, for both protons (short dashed lines) and neutrons (long dashed lines), plotted as the radial probability density $P\left(r\right)$. These are obtained from the underlying oscillator-basis calculation near the variational minimum ($\hslash \omega =15\mathrm{MeV}$) and at high $N_{\text{max}}$ ($N_{\text{max}}=16$). The corresponding oscillator radial functions for $\hslash \omega =15\mathrm{MeV}$ (thick gray lines) are shown for comparison. The mean occupation $n_a$ for each natural orbital, from the corresponding eigenvalue of the scalar density matrix, is indicated by the filling of the bar at top right (upper bar for protons, lower bar for neutrons).

Figure 8

Dependence of $\langle N_{\text{c.m.}}\rangle$ on $\hslash {\omega }_{\text{c.m.}}$, for ${}^3\mathrm{He}$ ground state wave functions obtained in calculations with a natural-orbital basis, derived from underlying oscillator-basis calculations with (a) $\hslash \omega =9\mathrm{MeV}$, (b) $\hslash \omega =15\mathrm{MeV}$, and (c) $\hslash \omega =25\mathrm{MeV}$. Results are shown for calculations with $N_{\text{max}}=4$ (short-dashed lines) through $N_{\text{max}}=16$ (solid lines, highlighted), with the curve obtained for an oscillator $0s$ wave function with $\hslash {\omega }_{\text{c.m.}}=\hslash \omega$ (thick gray lines)—or, equivalently, the calculation in an $N_{\text{max}}=0$ natural-orbital basis—shown for comparison. The underlying oscillator basis $\hslash \omega$ is indicated (dotted vertical line), as are the minimal $\hslash {\tilde{\omega }}_{\text{c.m.}}$ and ${\tilde{N}}_{\text{c.m.}}$ for each curve (dots, with dotted vertical line at highest $N_{\max }$).

Figure 9

Dependence of the approximate $0s$ center-of-mass motion of the calculated ${}^3\mathrm{He}$ ground state (and its degree of contamination) on the $\hslash \omega$ of the underlying oscillator basis, in calculations with a natural-orbital basis, as measured by (a) $\hslash {\tilde{\omega }}_{\text{c.m.}}$ and (b) ${\tilde{N}}_{\text{c.m.}}$. Results are shown for calculations with $N_{\text{max}}=4$ (dotted lines) through $N_{\text{max}}=16$ (solid lines, highlighted).

Figure 10

The ${}^6\mathrm{He}$ ground-state energy, as calculated in oscillator (solid circles) and natural orbital (open squares) bases. Calculated values are shown as functions of the basis parameter $\hslash \omega$, for successive even value of $N_{\text{max}}$, from $N_{\text{max}}=8$ (dotted lines) to 14 (solid lines, highlighted). The experimental binding energy [80] is also shown (filled diamond).

Figure 11

Differences of calculated ${}^6\mathrm{He}$ ground-state energies obtained for successive $N_{\text{max}}$, as obtained for oscillator (solid circles) and natural orbital (open squares) bases, shown on a logarithmic scale.

Figure 12

The ${}^6\mathrm{He}$ ground-state point-proton and point-neutron r.m.s. radii, as calculated in oscillator (solid circles) and natural orbital (open squares) bases. Calculated values are shown as functions of the basis parameter $\hslash \omega$, for successive even value of $N_{\text{max}}$, from $N_{\text{max}}=8$ (dotted lines) to 14 (solid lines, highlighted). The value deduced from the experimental charge radius [83] is also shown (filled diamond).

Figure 13

Radial wave functions for the ${}^6\mathrm{He}s$-, $p$-, and $sd$-shell natural orbitals, for both protons (short dashed lines) and neutrons (long dashed lines), plotted as the radial probability density $P\left(r\right)$. These are obtained from the underlying oscillator-basis calculation near the variational minimum ($\hslash \omega =15\mathrm{MeV}$) and at high $N_{\text{max}}$ ($N_{\text{max}}=14$). The corresponding oscillator radial functions for $\hslash \omega =15\mathrm{MeV}$ (thick gray lines) are shown for comparison. The mean occupation $n_a$ for each natural orbital, from the corresponding eigenvalue of the scalar density matrix, is indicated by the filling of the bar at top right (upper bar for protons, lower bar for neutrons).

Figure 14

Radial wave functions for the ${}^6\mathrm{He}s$-, $p$-, and $sd$-shell natural orbitals, plotted as the radial probability density $P\left(r\right)$, as in Fig. 13, but now on a logarithmic scale.

Figure 15

Radial wave functions obtained for the ${}^6\mathrm{He}$ proton $0s_{1/2}$ natural orbital, from different underlying oscillator-basis calculations, plotted as the radial probability density $P\left(r\right)$. Results are shown as obtained from underlying oscillator-basis calculations with (a) $\hslash \omega =9\mathrm{MeV}$, (b) $\hslash \omega =15\mathrm{MeV}$, and (c) $\hslash \omega =25\mathrm{MeV}$. Radial wave functions are shown for $N_{\text{max}}=2$ (dotted lines) through $N_{\text{max}}=14$ (solid lines, highlighted), with the oscillator $0s$ function for the given $\hslash \omega$ (thick gray lines) shown for comparison. The locations of the peaks of the underlying harmonic-oscillator orbital and $N_{\text{max}}=14$ natural orbital are marked with dashed vertical lines.

Figure 16

Radial wave functions obtained for the ${}^6\mathrm{He}$ neutron $0p_{3/2}$ natural orbital, from different underlying oscillator-basis calculations, plotted as the radial probability density $P\left(r\right)$, on a logarithmic scale. Results are shown as obtained from underlying oscillator-basis calculations with (a) $\hslash \omega =9\mathrm{MeV}$, (b) $\hslash \omega =15\mathrm{MeV}$, and (c) $\hslash \omega =25\mathrm{MeV}$. Radial wave functions are shown for $N_{\text{max}}=2$ (dotted lines) through $N_{\text{max}}=14$ (solid lines, highlighted), with the oscillator $0s$ function for the given $\hslash \omega$ (thick gray lines) shown for comparison.

Figure 17

Dependence of the approximate $0s$ center-of-mass motion of the calculated ${}^6\mathrm{He}$ ground state (and its degree of contamination) on the $\hslash \omega$ of the underlying oscillator basis, in calculations with a natural-orbital basis, as measured by (a) $\hslash {\tilde{\omega }}_{\text{c.m.}}$ and (b) ${\tilde{N}}_{\text{c.m.}}$. Results are shown for calculations with $N_{\text{max}}=4$ (dotted lines) through $N_{\text{max}}=14$ (solid lines, highlighted).