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 = infinity and compare the results to similarly extrapolated NCSM calculations. A procedure is formulated to assign uncertainties for N-max = infinity extrapolations. Method: We report on Li-6 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 ((h) over bar Omega), and investigate the dependence on the momentum-decoupling scale (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-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 ((h) over bar Omega). IT-NCSM performs equally well for various SRG momentum-decoupling scales, lambda = 2.02 fm(-1) and lambda = 1.50 fm(-1). Conclusions: In the case of Li-6, when using the softened chiral nucleon-nucleon N3LO interaction, we have determined the difference between extrapolated N-max = infinity IT-NCSM and full NCSM calculations to be about 100-300 keV. As (h) over bar 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 (h) over bar Omega. We also find that using multiple reference states leads to a better ground-state description than using only a single reference state. DOI:10.1103/PhysRevC.87.044301