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

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_max) spaces. The importance truncation procedure selects only relatively few of the many basis states present in a "large" N_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_max. Purpose: An analysis of uncertainties in IT-NCSM such as those generated from the extrapolations to the complete N_max space have not been fully discussed. We present a method for estimating the uncertainty when extrapolating to the complete N_max space and demonstrate the method by comparing extrapolated IT-NCSM to full NCSM calculations up to N_max=14. Furthermore, we study the result of extrapolating IT-NCSM ground-state energies to N_max=∞ and compare the results to similarly extrapolated NCSM calculations. A procedure is formulated to assign uncertainties for N_max=∞ extrapolations. Method: We report on 6Li 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 _max space, the harmonic oscillator energy (âΩ), and investigate the dependence on the momentum-decoupling scale (λ) 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_max space increase as N_max increases. The extrapolation uncertainties range from a few keV for the smallest N _max spaces to about 50 keV for the largest N_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 (âΩ). IT-NCSM performs equally well for various SRG momentum-decoupling scales, λ=2.02 fm^-1 and λ=1.50 fm^-1. Conclusions: In the case of 6Li, when using the softened chiral nucleon-nucleon N3LO interaction, we have determined the difference between extrapolated N_max=∞ IT-NCSM and full NCSM calculations to be about 100-300 keV. As âΩ increases, we find that the agreement with NCSM deteriorates, indicating that the procedure used to choose the basis states in IT-NCSM depends on âΩ. We also find that using multiple reference states leads to a better ground-state description than using only a single reference state.