Convergence properties of {\itshape ab initio} calculations of light nuclei in a harmonic oscillator basis

TL;DR

This paper analyzes the convergence of ab initio calculations of light nuclei using harmonic oscillator basis model spaces, focusing on ultraviolet and infrared momentum regulators to improve extrapolation accuracy.

Contribution

It introduces practical procedures for handling UV and IR regulators in harmonic oscillator basis calculations of light nuclei, enhancing convergence and extrapolation methods.

Findings

Established methods to extend UV cutoff to infinity and IR cutoff to zero.

Analyzed convergence behaviors for nuclei like H-2, H-3, He-4, and He-6.

Compared results using different NN interactions, Idaho N3LO and JISP16.

Abstract

We study recently proposed ultraviolet and infrared momentum regulators of the model spaces formed by construction of a variational trial wavefunction 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 ($\hbar\omega$) of the HO basis; in short by the ordered pair ($\mathcal{N},\hbar\omega$). 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 ultraviolet (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 infrared (ir) momentum cutoff inherent in a finite-dimensional HO basis. One definition is based upon the lowest momentum difference given by $\hbar\omega$ 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" $NN$ 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 ($\mathcal{N},\hbar\omega$).