Extrapolation uncertainties in the importance-truncated no-core shell model

Abstract

Background: The importance-truncated no-core shell model (IT-NCSM) has recently been shown to extend theoretical nuclear structure calculations of $p$-shell nuclei to larger model (${N}_{\mathrm{max}}$) spaces. The importance truncation procedure selects only relatively few of the many basis states present in a ``large'' ${N}_{\mathrm{max}}$ basis space, thus making the calculation tractable and reasonably quick to perform. Initial results indicate that the procedure agrees well with the NCSM, in which a complete basis is constructed for a given ${N}_{\mathrm{max}}$.Purpose: An analysis of uncertainties in IT-NCSM such as those generated from the extrapolations to the complete ${N}_{\mathrm{max}}$ space have not been fully discussed. We present a method for estimating the uncertainty when extrapolating to the complete ${N}_{\mathrm{max}}$ space and demonstrate the method by comparing extrapolated IT-NCSM to full NCSM calculations up to ${N}_{\mathrm{max}}=14$. Furthermore, we study the result of extrapolating IT-NCSM ground-state energies to ${N}_{\mathrm{max}}=\ensuremath{\infty}$ and compare the results to similarly extrapolated NCSM calculations. A procedure is formulated to assign uncertainties for ${N}_{\mathrm{max}}=\ensuremath{\infty}$ extrapolations.Method: We report on ${}^{6}\mathrm{Li}$ calculations performed with the IT-NCSM and compare them to full NCSM calculations. We employ the Entem and Machleidt chiral two-body next-to-next-to-next leading order (N3LO) interaction (regulated at 500 MeV/$c$), which has been modified to a phase-shift equivalent potential by the similarity renormalization group (SRG) procedure. We investigate the dependence of the procedure on the technique employed to extrapolate to the complete ${N}_{\mathrm{max}}$ space, the harmonic oscillator energy ($\ensuremath{\hbar}\ensuremath{\Omega}$), and investigate the dependence on the momentum-decoupling scale ($\ensuremath{\lambda}$) used in the SRG. We also investigate the use of one or several reference states from which the truncated basis is constructed.Results: We find that the uncertainties generated from various extrapolating functions used to extrapolate to the complete ${N}_{\mathrm{max}}$ space increase as ${N}_{\mathrm{max}}$ increases. The extrapolation uncertainties range from a few keV for the smallest ${N}_{\mathrm{max}}$ spaces to about 50 keV for the largest ${N}_{\mathrm{max}}$ spaces. We note that the difference between extrapolated IT-NCSM and NCSM ground-state energies, however, can be as large as 100--250 keV depending on the chosen harmonic oscillator energy ($\ensuremath{\hbar}\ensuremath{\Omega}$). IT-NCSM performs equally well for various SRG momentum-decoupling scales, $\ensuremath{\lambda}=2.02$ fm${}^{\ensuremath{-}1}$ and $\ensuremath{\lambda}=1.50$ fm${}^{\ensuremath{-}1}$.Conclusions: In the case of ${}^{6}$Li, when using the softened chiral nucleon-nucleon N3LO interaction, we have determined the difference between extrapolated ${N}_{\mathrm{max}}=\ensuremath{\infty}$ IT-NCSM and full NCSM calculations to be about 100--300 keV. As $\ensuremath{\hbar}\ensuremath{\Omega}$ increases, we find that the agreement with NCSM deteriorates, indicating that the procedure used to choose the basis states in IT-NCSM depends on $\ensuremath{\hbar}\ensuremath{\Omega}$. We also find that using multiple reference states leads to a better ground-state description than using only a single reference state.