Convergence properties of ab initio calculations of light nuclei in a harmonic oscillator basis

We study recently proposed ultraviolet and infrared momentum regulators of the model spaces formed by construction of a variational trial wave function which uses a complete set of many-body basis states based upon three-dimensional harmonic oscillator (HO) functions. These model spaces are defined by a truncation of the expansion characterized by a counting number ($\mathcal{N}$) and by the intrinsic scale ($\hslash \omega$) of the HO basis—in short by the ordered pair ($\mathcal{N},\hslash \omega$). In this study we choose for $\mathcal{N}$ the truncation parameter $N_{\max }$ related to the maximum number of oscillator quanta, above the minimum configuration, kept in the model space. The uv momentum cutoff of the continuum is readily mapped onto a defined uv cutoff in this finite model space, but there are two proposed definitions of the ir momentum cutoff inherent in a finite-dimensional HO basis. One definition is based upon the lowest momentum difference given by $\hslash \omega$ itself and the other upon the infrared momentum which corresponds to the maximal radial extent used to encompass the many-body system in coordinate space. Extending both the uv cutoff to infinity and the ir cutoff to zero is prescribed for a converged calculation. We calculate the ground-state energy of light nuclei with “bare” and “soft” nucleon-nucleon ($NN$) interactions. By doing so, we investigate the behaviors of the uv and ir regulators of model spaces used to describe ${}^2$H, ${}^3$H, ${}^4$He, and ${}^6$He with $NN$ potentials Idaho N${}^3$LO and JISP16. We establish practical procedures which utilize these regulators to obtain the extrapolated result from sequences of calculations with model spaces characterized by ($\mathcal{N},\hslash \omega$).

Figure 1

Schematic view of a finite model space (limited by the basis truncation parameter $N$ as described in the text), in which the uv and ir momentum cutoffs are arbitrary. To reach the full many-body Hilbert space, symbolized by the complete oval, one needs to let the uv cutoff $\to \infty$ and the ir cutoff $\to 0$.

Figure 2

Dependence of the ground-state energy of ${}^3$H (compared to a converged value; see text) upon the uv momentum cutoff $\Lambda =\sqrt{m_N(N+3/2)\hslash \omega }$ for different fixed $\lambda =\sqrt{m_N\hslash \omega }$. The curves are fit to the calculated points.

Figure 3

Dependence of the ground-state energy of ${}^3$H (compared to a converged value; see text) upon the ir momentum cutoff $\lambda =\sqrt{m_N\hslash \omega }$ for fixed $\Lambda =\sqrt{m_N(N+3/2)\hslash \omega }$.

Figure 4

Dependence of the ground-state energy of ${}^3$H (compared to a converged value; see text) upon the uv momentum cutoff $\Lambda =\sqrt{m_N(N+3/2)\hslash \omega }$ for different values of the ir momentum cutoff ${\lambda }_{sc}=\sqrt{\left(m_N\hslash \omega \right)/(N+3/2)}$. Curves are not fits but simple point-to-point line segments to guide the eye.

Figure 5

Dependence of the ground-state energy of ${}^3$H (compared to a converged value; see text) upon the ir momentum cutoff ${\lambda }_{sc}=\sqrt{\left(m_N\hslash \omega \right)/(N+3/2)}$ for fixed $\Lambda =\sqrt{m_N(N+3/2)\hslash \omega }$.

Figure 6

Dependence of the ground-state energy of three $s$-shell nuclei (compared to a converged value; see text) upon the ir momentum cutoff ${\lambda }_{sc}=\sqrt{\left(m_N\hslash \omega \right)/(N+3/2)}$ for $\Lambda =\sqrt{m_N(N+3/2)\hslash \omega }$ above the ${\Lambda }^{NN}\approx 780$ MeV/$c$ set by the $NN$ potential. The solid curves are fits to the points as described in the text.

Figure 7

The ground-state energy of ${}^3$H calculated at five fixed values of $\Lambda =\sqrt{m_N(N+3/2)\hslash \omega }$ and variable ${\lambda }_{sc}=\sqrt{\left(m_N\hslash \omega \right)/(N+3/2)}$. The curves are fits to the points and the functions fitted are used to extrapolate to the ir limit ${\lambda }_{sc}=0$.

Figure 8

Dependence of the ground-state energy of three $s$-shell nuclei (compared to a converged value; see text) upon the uv momentum cutoff $\Lambda =\sqrt{m_N(N+3/2)\hslash \omega }$ for ${\lambda }_{sc}=\sqrt{\left(m_N\hslash \omega \right)/(N+3/2)}$ above the ${\lambda }_{sc}^{NN}\approx 36$ MeV/$c$ set by the $NN$ potential. Curves are not fits but spline interpolations to guide the eye.

Figure 9

Dependence of the ground-state energy of three $s$-shell nuclei (compared to a converged value; see text) upon the uv momentum cutoff $\Lambda =\sqrt{m_N(N+3/2)\hslash \omega }$ for ${\lambda }_{sc}=\sqrt{\left(m_N\hslash \omega \right)/(N+3/2)}$ below the ${\lambda }_{sc}^{NN}\approx 36$ MeV/$c$ set by the $NN$ potential. The data are fit to Gaussians.

Figure 10

The ground-state energy of ${}^6$He calculated at five approximate values of $\Lambda =\sqrt{m_N(N+3/2)\hslash \omega }$ (see text) and integer values of $\hslash \omega$. The curves are fits to the points and the functions fitted are used to extrapolate to the ir limit $\lambda =\sqrt{m_N\hslash \omega }=0$ with fixed $\Lambda$ as in Fig. 7.

Figure 11

The ground-state energy of ${}^6$He calculated at all values of $\Lambda \ge 510$ MeV/$c$ [$\Lambda =\sqrt{m_N(N+3/2)\hslash \omega }$] and variable ${\lambda }_{sc}=\sqrt{\left(m_N\hslash \omega \right)/(N+3/2)}$. The curve is a fit to the points and the function fitted is used to extrapolate to the ir limit ${\lambda }_{sc}=0$.