Variational methods for the few-body bound state in a harmonic oscillator basis

Coon, S. A.; Portilho, O.

doi:10.1007/3-540-17647-0_31

Cited by 3 publications

(11 citation statements)

References 26 publications

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“…Such examinations (albeit for rather small basis size compared to those used in this study) are displayed in Ref. [15].…”

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confidence: 98%

“…[2]. Slow convergence hinders progress either because the amount of computation needed to reach a desired accuracy is prohibitive, or because too many arithmetic operations cause excessive round off error [10,15,19]. Indeed, the slow convergence of systematic expansions was likely a contributing factor to the replacement of variational methods by finite difference methods (based upon the Faddeev decomposition) in the 1970s to treat the three-nucleon bound state problem.…”

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confidence: 99%

“…The asymptotic convergence of the nuclear structure problem is not changed by including Jastrow type two-body correlation functions in the trial wavefunction [15]. The onset of asymptotic convergence occurs, however, at a much smaller value of an N than for the case without correlation functions and the convergence is to the same final value of the lowest variational energy [7,10,15]. Therefore much less computational resources are required to get the answer.…”

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confidence: 99%

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

et al. 2012

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We study recently proposed ultraviolet and infrared momentum regulators of the model spaces formed by construction of a variational trial wavefunction 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 (N ) and by the intrinsic scale ( ω) of the HO basis; in short by the ordered pair (N , ω). In this study we choose for N the truncation parameter N max related to the maximum number of oscillator quanta, above the minimum configuration, kept in the model space. The ultraviolet (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 infrared (ir) momentum cutoff inherent in a finite-dimensional HO basis.One definition is based upon the lowest momentum difference given by ω 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" N N 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 N N 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 (N , ω).

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“…Such examinations (albeit for rather small basis size compared to those used in this study) are displayed in Ref. [15].…”

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confidence: 98%

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confidence: 99%

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confidence: 99%

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

et al. 2012

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“…Notice that the fixed N max curves do not appear to depend strongly upon the HO energy ω for the largest N max and the minimum of this curve is often chosen as that point at which one makes the extrapolation in N max . [10][11][12] However, as shown in Figure 1, the extrapolated ground state energy does depend rather strongly upon ω, everywhere in the range displayed. This strong dependence is true for the values ω = 32 − 34 MeV at the minimum of the lowest fixed N max curve, values which would be the traditional choice for the extrapolated result.…”

Section: 13mentioning

confidence: 79%

“…In an early example, ω is simply fixed at a value which gives the fastest convergence in N . 8 Later, for each N the non-linear parameter ω is varied to obtain the minimal energy 10,12 for a fixed N and then the convergence with N is examined at that fixed value of ω. Other extrapolation schemes have been proposed and used.…”

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confidence: 99%

Properties of infrared extrapolations in a harmonic oscillator basis

Coon

Kruse

2016

Int. J. Mod. Phys. E

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The success and utility of effective field theory (EFT) in explaining the structure and reactions of few-nucleon systems has prompted the initiation of EFT-inspired extrapolations to larger model spaces in ab initio methods such as the no-core shell model (NCSM). In this contribution, we review and continue our studies of infrared (ir) and ultraviolet (uv) regulators of NCSM calculations in which the input is phenomenological N N and N N N interactions fitted to data. We extend our previous findings that an extrapolation in the ir cutoff with the uv cutoff above the intrinsic uv scale of the interaction is quite successful, not only for the eigenstates of the Hamiltonian but also for expectation values of operators, such as r 2 , considered long range. The latter results are obtained with Hamiltonians transformed by the similarity renormalization group (SRG) evolution. On the other hand, a possible extrapolation of ground state energies in the uv cutoff when the ir cutoff is below the intrinsic ir scale is not robust and does not agree with the ir extrapolation of the same data or with independent calculations using other methods.

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Infrared and ultraviolet cutoffs in variational calculations with a harmonic oscillator basis

Coon¹

2013

Preprint

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I abstract from a recent publication [1] the motivations for, analysis in and conclusions of a study of the ultraviolet and infrared momentum regulators induced by the necessary truncation of the model spaces formed by a variational trial wave function. This trial function is built systematically from a complete set of many-body basis states based upon three-dimensional harmonic oscillator (HO) functions. Each model space is defined by a truncation of the expansion characterized by a counting number (N ) and by the intrinsic scale ( ω) of the HO basis. Extending both the uv cutoff to infinity and the ir cutoff to zero is prescribed for a converged calculation. In [1] we established practical procedures which utilize these regulators to obtain the extrapolated result from sequences of calculations with model spaces. Finally, I update this subject by mentioning recent work on our extrapolation prescriptions which have appeared since the submission of [1]. The numerical example chosen for this contribution consists of calculations of the ground state energy of the triton with the "bare" and "soft" Idaho N 3 LO nucleon-nucleon (N N ) interaction.

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Variational methods for the few-body bound state in a harmonic oscillator basis

Cited by 3 publications

References 26 publications

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

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

Properties of infrared extrapolations in a harmonic oscillator basis

Infrared and ultraviolet cutoffs in variational calculations with a harmonic oscillator basis

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