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

Abstract

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).

Infrared extrapolations for atomic nuclei

“…The leading-order estimates for these cutoffs are based on L ≈ √ 2N b and Λ UV ≈ √ 2N /b, respectively [16,17,18]. Clearly, L and Λ UV are semiclassical estimates for the extent of the oscillator basis in position and momentum space, respectively.…”

Convergence properties of the effective theory for trapped bosons

“…(33) then imply that the leading corrections to the energies E (2) i for contact interactions are linear in the effective range r e .…”

“…Due to their singularity, we expect the contact interactions to behave quite differently, but it is interesting to compare our results for smeared interactions to Refs. [33,39]. However, one has to keep in mind that we consider a physical trapping potential instead of a mere basis expansion.…”

“…In the literature, there are two definitions of the IR-cutoff for an oscillator basis. Both the expression β/ N + (A − 1)3/2 [31,32], which quantifies the maximum size of structures that can be captured in the given basis, or the oscillator length β itself [33] have been interpreted as an IR-cutoff. In our calculations, the harmonic oscillator represents a physical trap and thus β is a physical parameter.…”

In-medium similarity renormalization group for closed and open-shell nuclei

“…While mature techniques to extrapolate results to infinite HO bases are available [142][143][144][145][146][147][148], we limit ourselves to finite bases here, using sufficiently large e max values to eliminate the single-particle basis truncation as a relevant source of uncertainty, typically up to e max = 14 (15 major HO shells). An additional truncation is necessary to manage the enormous memory requirements of 3N interaction matrix elements.…”