Natural orbitals for ab initio no-core shell model calculations

We explore the impact of optimizations of the single-particle basis on the convergence behavior and robustness of ab initio no-core shell model calculations of nuclei. Our focus is on novel basis sets defined by the natural orbitals of a correlated one-body density matrix that is obtained in second-order many-body perturbation theory. Using a perturbatively improved density matrix as the starting point informs the single-particle basis about the dominant correlation effects on the global structure of the many-body state, while keeping the computational cost for the basis optimization at a minimum. Already the comparison of the radial single-particle wave functions reveals the superiority of the natural-orbital basis compared to a Hartree-Fock or harmonic oscillator basis, and it highlights pathologies of the Hartree-Fock basis. We compare the model-space convergence of energies, root-mean-square radii, and selected electromagnetic observables for all three basis sets for selected $p$-shell nuclei using chiral interactions with explicit three-nucleon terms. In all cases the natural-orbital basis provides the fastest and most robust convergence, making it the most efficient basis for no-core shell model calculations. As an application we present no-core shell model calculations for the ground-state energies of all oxygen isotopes and assess the accuracy of the normal-ordered two-body approximation of the three-nucleon interaction in the natural-orbital basis.

Figure 1

Squared radial wave functions ${\left|u\left(r\right)\right|}^2$ (in units of ${\text{fm}}^{-1})$ of the HO, HF, and NAT bases for ${}^{16}\mathrm{O}$. The different rows display different single-particle orbits: the occupied $0p_{3/2}$ orbit and the unoccupied and HF-unbound $1s_{1/2}$ and $1p_{1/2}$ orbits. The different lines correspond to different values for the oscillator frequency $\hslash \mathrm{\Omega }=16\text{MeV}$ (blue), $20\text{MeV}$ (green), and $24\text{MeV}$ (red). We use the NN+$3N$ interaction with ${\mathrm{\Lambda }}_{3N}=400\text{MeV}/c$ and $\alpha =0.08{\text{fm}}^4$.

Figure 2

Ground-state energies for ${}^4\mathrm{He}$ obtained in the NCSM with the HO, HF, and NAT basis set as functions of the oscillator frequency $\hslash \mathrm{\Omega }$ for $N_{max}=4\left(•\right),6\left(♦\right),8\left(▴\right),10\left(\blacksquare \right)$, and $12\left(+\right)$. All calculations employ the chiral NN+$3N$ interaction $\left({\mathrm{\Lambda }}_{3N}=400\text{MeV}/c\right)$ after an SRG evolution with $\alpha =0.08{\text{fm}}^4$.

Figure 3

Ground-state energies for ${}^{16}\mathrm{O}$ obtained in the NCSM with the HO, HF, and NAT basis sets with $N_{max}=2\left(•\right),4\left(♦\right),6\left(▴\right),8\left(\blacksquare \right)$, and $10\left(+\right)$. The open symbols indicated results obtained with the NO2B approximation. All other parameters as in Fig. 2.

Figure 4

Point-proton radii for the ground state of ${}^4\mathrm{He}$ obtained in the NCSM with the HO, HF, and NAT basis sets as functions of the oscillator frequency $\hslash \mathrm{\Omega }$ for $N_{max}=4\left(•\right),6\left(♦\right),8\left(▴\right),10\left(\blacksquare \right)$, and $12\left(+\right)$. All calculations employ the chiral NN+$3N$ interaction $\left({\mathrm{\Lambda }}_{3N}=400\text{MeV}/c\right)$ after an SRG evolution with $\alpha =0.08{\text{fm}}^4$.

Figure 5

Point-proton radii for the ground state of ${}^{16}\mathrm{O}$ obtained in the NCSM with the HO, HF, and NAT basis sets with $N_{max}=2\left(•\right),4\left(♦\right),6\left(▴\right),8\left(\blacksquare \right)$, and $10\left(+\right)$. All other parameters as in Fig. 4.

Figure 6

Summary of the spectroscopy of ${}^{12}\mathrm{C}$ obtained in NCSM calculation with the HO basis (open symbols) and NAT basis (solid symbols) as functions of $N_{max}$ for $\hslash \mathrm{\Omega }=16\text{MeV}$ (blue), $20\text{MeV}$ (red), and $24\text{MeV}$ (green). The left-hand panels show the ground-state energy and the excitation energies for the first $2^{+}$ and $1^{+}$ states. The right-hand panels show the quadrupole moments of the first $2^{+}$ state, the $B\left(E2\right)$ transition strength from the $2^{+}$ to the ground state, and the $B\left(M1\right)$ strength from the $1^{+}$ to the ground state. All calculations employ the chiral NN+$3N$ interaction $\left({\mathrm{\Lambda }}_{3N}=500\text{MeV}/c\right)$ after an SRG evolution with $\alpha =0.08{\text{fm}}^4$.

Figure 7

Ground-state energies of all oxygen isotopes from ${}^{14}\mathrm{O}$ to ${}^{26}\mathrm{O}$ using the chiral NN+$3N$ interaction $\left({\mathrm{\Lambda }}_{3N}=400\text{MeV}/c\right)$ after an SRG evolution with $\alpha =0.08{\text{fm}}^4$. Shown are NCSM calculations using the NAT basis $\left(\hslash \mathrm{\Omega }=20\text{MeV}\right)$ with the explicit $3N$ interaction $\left(\blacksquare \right)$ and with the NO2B approximation $\left(♦\right)$. For comparison we also plot the HO basis calculations of Ref. [29] with optimized oscillator frequency $\left(•\right)$. The inset shows the $N_{max}$ dependence and extrapolation for selected isotopes with the explicit $3N$ interaction (see text).