Convergence properties ofab initiocalculations of light nuclei in a harmonic oscillator basis

Abstract

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 ($\ensuremath{\hbar}\ensuremath{\omega}$) of the HO basis---in short by the ordered pair ($\mathcal{N},\ensuremath{\hbar}\ensuremath{\omega}$). In this study we choose for $\mathcal{N}$ the truncation parameter ${N}_{\mathrm{max}}$ 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 $\ensuremath{\hbar}\ensuremath{\omega}$ 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 ($\mathcal{N},\ensuremath{\hbar}\ensuremath{\omega}$).