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

M. K G Kruse, E. D. Jurgenson, P. Navrátil, Bruce R Barrett, W. E. Ormand

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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 (Nmax) spaces. The importance truncation procedure selects only relatively few of the many basis states present in a "large" Nmax 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 Nmax. Purpose: An analysis of uncertainties in IT-NCSM such as those generated from the extrapolations to the complete Nmax space have not been fully discussed. We present a method for estimating the uncertainty when extrapolating to the complete Nmax space and demonstrate the method by comparing extrapolated IT-NCSM to full NCSM calculations up to Nmax=14. Furthermore, we study the result of extrapolating IT-NCSM ground-state energies to Nmax=∞ and compare the results to similarly extrapolated NCSM calculations. A procedure is formulated to assign uncertainties for Nmax=∞ 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 Nmax space increase as Nmax increases. The extrapolation uncertainties range from a few keV for the smallest N max spaces to about 50 keV for the largest Nmax 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 Nmax=∞ 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.

Original languageEnglish (US)
Article number044301
JournalPhysical Review C - Nuclear Physics
Volume87
Issue number4
DOIs
StatePublished - Apr 1 2013

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extrapolation
decoupling
harmonic oscillators
ground state
momentum
energy
nucleon-nucleon interactions
nuclear structure
estimating
phase shift
nuclei
approximation

ASJC Scopus subject areas

  • Nuclear and High Energy Physics

Cite this

Extrapolation uncertainties in the importance-truncated no-core shell model. / Kruse, M. K G; Jurgenson, E. D.; Navrátil, P.; Barrett, Bruce R; Ormand, W. E.

In: Physical Review C - Nuclear Physics, Vol. 87, No. 4, 044301, 01.04.2013.

Research output: Contribution to journalArticle

Kruse, M. K G ; Jurgenson, E. D. ; Navrátil, P. ; Barrett, Bruce R ; Ormand, W. E. / Extrapolation uncertainties in the importance-truncated no-core shell model. In: Physical Review C - Nuclear Physics. 2013 ; Vol. 87, No. 4.
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title = "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 (Nmax) spaces. The importance truncation procedure selects only relatively few of the many basis states present in a {"}large{"} Nmax 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 Nmax. Purpose: An analysis of uncertainties in IT-NCSM such as those generated from the extrapolations to the complete Nmax space have not been fully discussed. We present a method for estimating the uncertainty when extrapolating to the complete Nmax space and demonstrate the method by comparing extrapolated IT-NCSM to full NCSM calculations up to Nmax=14. Furthermore, we study the result of extrapolating IT-NCSM ground-state energies to Nmax=∞ and compare the results to similarly extrapolated NCSM calculations. A procedure is formulated to assign uncertainties for Nmax=∞ 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 ({\^a}Ω), 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 Nmax space increase as Nmax increases. The extrapolation uncertainties range from a few keV for the smallest N max spaces to about 50 keV for the largest Nmax 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 ({\^a}Ω). 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 Nmax=∞ IT-NCSM and full NCSM calculations to be about 100-300 keV. As {\^a}Ω increases, we find that the agreement with NCSM deteriorates, indicating that the procedure used to choose the basis states in IT-NCSM depends on {\^a}Ω. We also find that using multiple reference states leads to a better ground-state description than using only a single reference state.",
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T1 - Extrapolation uncertainties in the importance-truncated no-core shell model

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AU - Barrett, Bruce R

AU - Ormand, W. E.

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N2 - 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 (Nmax) spaces. The importance truncation procedure selects only relatively few of the many basis states present in a "large" Nmax 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 Nmax. Purpose: An analysis of uncertainties in IT-NCSM such as those generated from the extrapolations to the complete Nmax space have not been fully discussed. We present a method for estimating the uncertainty when extrapolating to the complete Nmax space and demonstrate the method by comparing extrapolated IT-NCSM to full NCSM calculations up to Nmax=14. Furthermore, we study the result of extrapolating IT-NCSM ground-state energies to Nmax=∞ and compare the results to similarly extrapolated NCSM calculations. A procedure is formulated to assign uncertainties for Nmax=∞ 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 Nmax space increase as Nmax increases. The extrapolation uncertainties range from a few keV for the smallest N max spaces to about 50 keV for the largest Nmax 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 Nmax=∞ 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.

AB - 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 (Nmax) spaces. The importance truncation procedure selects only relatively few of the many basis states present in a "large" Nmax 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 Nmax. Purpose: An analysis of uncertainties in IT-NCSM such as those generated from the extrapolations to the complete Nmax space have not been fully discussed. We present a method for estimating the uncertainty when extrapolating to the complete Nmax space and demonstrate the method by comparing extrapolated IT-NCSM to full NCSM calculations up to Nmax=14. Furthermore, we study the result of extrapolating IT-NCSM ground-state energies to Nmax=∞ and compare the results to similarly extrapolated NCSM calculations. A procedure is formulated to assign uncertainties for Nmax=∞ 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 Nmax space increase as Nmax increases. The extrapolation uncertainties range from a few keV for the smallest N max spaces to about 50 keV for the largest Nmax 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 Nmax=∞ 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.

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