We respond to Comment on our recent letter (Phys.Rev.Lett.99:092501,2007) by
Dean et al (arXiv:0709.0449).Comment: 2 page

Navratil, P.

Roth, R.

English

arXiv

In their comment on our recent Letter [1] Dean et al. [2] criticize the calculations for the ground-state energy of {sup 40}Ca within the importance truncated no-core shell model (NCSM). In particular they address the role of configurations beyond the 3p3h level, which have not been included in the {sup 40}Ca calculations for large N{sub max} {h_bar}{Omega} model spaces. Before responding to this point, the following general statements are in order. For the atomic nucleus as a self-bound system, translational invariance is an important symmetry. The only possibility to preserve translational invariance when working with a Slater determinant basis is to use the harmonic oscillator (HO) basis in conjunction with a basis truncation according to the total HO excitation energy, i.e. N{sub max} {h_bar}{Omega}, as done in the ab initio NCSM. This is important not only for obtaining proper binding or excitation energies, but also for a correct extraction of physical wavefunctions. The spurious center-of-mass components can be exactly removed only if the HO basis and the N{sub max} {h_bar}{Omega} truncation are employed. The minimal violation of the translational invariance was one of the main motivations for developing the importance-truncation scheme introduced in the Letter. In this scheme, we start with the complete N{sub max} {h_bar}{Omega} HO basis space and select important configurations via perturbation theory. All symmetries are under control and our importance-truncated NCSM calculations are completely variational and provide an upper bound of the ground-state energy of the system. The restriction to the 3p3h level, made for computational reasons in the N{sub max} &gt; 8 calculations for {sup 40}Ca, is not inherent to the importance truncation scheme. The explicit inclusion of 4p4h configurations--though computationally more demanding--is straight-forward, even for the largest N{sub max} {h_bar}{Omega} model spaces discussed. To demonstrate this fact we have performed full 4p4h calculations for {sup 40}Ca in a 14{h_bar}{Omega} no-core model space at {h_bar}{Omega} = 24 MeV using the V{sub low k} interaction employed in the Letter. The resulting ground state energy of E{sub 4p4h} = -471.0 MeV can be compared with E{sub 3p3h} = -461.2 MeV for the 3p3h calculation reported in Fig. 5(b) with an uncertainty of typically 1 MeV due to the extrapolation {kappa}{sub min} {yields} 0. Thus the 4p4h configurations change the resulting ground-state energy of {sup 40}Ca by approximately 2%. In addition to the explicit inclusion of 4p4h or higher-order configurations in the importance truncated space, there are various other means to account for their effect on the energy. We are presently developing a perturbative method for the inclusion of up to 6p6h configurations using the ground state obtained from the NCSM diagonalization as unperturbed state. Furthermore, there are even simpler methods, such as multireference Davidson corrections, which are employed successfully in quantum chemistry [3], for estimating the contribution of excluded configurations to the energy. They even restore size extensivity in an approximate way. The coupled-cluster method (CCM) used by the authors of the Comment lacks the above discussed features important for the nuclear many-body problem, in particular it violates translational invariance from the very beginning and it does not fulfill the variational principle. The problem of a spurious center-of-mass contamination of the many-body states in CCM is not resolved and often not even mentioned (see e.g. Ref. [4]). Moreover, non-iterative triples corrections like CCSD(T), as referred to in the comment, tend to overestimate the correlation energy or even collapse. It is claimed that the CCM is very accurate. A closer inspection of recently published nuclear many-body results does not support this statement. In Ref. [5], the {sup 4}He binding energy with the chiral N{sup 3}LO NN potential was determined with an uncertainty of several MeV. The same is obtained in the ab initio NCSM with accuracy of 10 keV [6, 7]. The CCM binding energy results for {sup 16}O with the identical chiral N{sup 3}LO NN potential obtained in Refs. [8] and [9] differ by more than 5 MeV. According to Ref. [9], the CD-Bonn NN potential overbinds {sup 16}O. This is contrary to results obtained in Ref. [10] and also contrary to expectations as the CD-Bonn underbinds lighter nuclei. The CCSD(T) ground state energy for {sup 40}Ca with V{sub low k} reported in Ref. [4] is about 30 MeV lower than our 4p4h-result (CCSD is about 20MeV lower). Taking into account the small 2% difference of our 3p3h and 4p4h results and the violation of the variational principle and the translational invariance in the CCM method, we believe that the CCM result overestimates the exact ground-state energy in this case

Roth, R

Navratil, P

UNT (University of North Texas) Digital Library

LLNL-JRNL-400311Reply to Comment on "Ab Initio Study of40Ca with an Importance TruncatedNo-Core Shell Model"R. Roth, P. NavratilJanuary 8, 2008Physical Review LettersDisclaimer  This document was prepared as an account of work sponsored by an agency of the United States government. Neither the United States government nor Lawrence Livermore National Security, LLC, nor any of their employees makes any warranty, expressed or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States government or Lawrence Livermore National Security, LLC. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States government or Lawrence Livermore National Security, LLC, and shall not be used for advertising or product endorsement purposes.  Reply to Comment on “Ab Initio Study of 40Ca with an Importance Truncated No-Core ShellModel”R. Roth∗Institut fu¨r Kernphysik, TU Darmstadt, Schlossgartenstr. 9, 64289 Darmstadt, GermanyP. Navra´til†Lawrence Livermore National Laboratory, P.O. Box 808, L-414, Livermore, CA 94551, USAIn their comment on our recent Letter [1] Dean et al. [2]criticize the calculations for the ground-state energy of 40Cawithin the importance truncated no-core shell model (NCSM).In particular they address the role of configurations beyond the3p3h level, which have not been included in the 40Ca calcula-tions for large Nmax~Ω model spaces.Before responding to this point, the following general state-ments are in order. For the atomic nucleus as a self-boundsystem, translational invariance is an important symmetry.The only possibility to preserve translational invariance whenworking with a Slater determinant basis is to use the harmonicoscillator (HO) basis in conjunction with a basis truncationaccording to the total HO excitation energy, i.e. Nmax~Ω, asdone in the ab initio NCSM. This is important not only forobtaining proper binding or excitation energies, but also fora correct extraction of physical wavefunctions. The spuriouscenter-of-mass components can be exactly removed only if theHO basis and theNmax~Ω truncation are employed. The min-imal violation of the translational invariance was one of themain motivations for developing the importance-truncationscheme introduced in the Letter. In this scheme, we start withthe complete Nmax~Ω HO basis space and select importantconfigurations via perturbation theory. All symmetries are un-der control and our importance-truncated NCSM calculationsare completely variational and provide an upper bound of theground-state energy of the system.The restriction to the 3p3h level, made for computationalreasons in the Nmax > 8 calculations for 40Ca, is not inher-ent to the importance truncation scheme. The explicit inclu-sion of 4p4h configurations—though computationally moredemanding—is straight-forward, even for the largestNmax~Ωmodel spaces discussed. To demonstrate this fact we haveperformed full 4p4h calculations for 40Ca in a 14~Ω no-coremodel space at ~Ω = 24 MeV using the Vlowk interactionemployed in the Letter. The resulting ground state energyof E4p4h = −471.0 MeV can be compared with E3p3h =−461.2 MeV for the 3p3h calculation reported in Fig. 5(b)with an uncertainty of typically 1 MeV due to the extrapo-lation κmin → 0. Thus the 4p4h configurations change theresulting ground-state energy of 40Ca by approximately 2%.In addition to the explicit inclusion of 4p4h or higher-orderconfigurations in the importance truncated space, there arevarious other means to account for their effect on the energy.We are presently developing a perturbative method for the in-clusion of up to 6p6h configurations using the ground state ob-tained from the NCSM diagonalization as unperturbed state.Furthermore, there are even simpler methods, such as multi-reference Davidson corrections, which are employed success-fully in quantum chemistry [3], for estimating the contributionof excluded configurations to the energy. They even restoresize extensivity in an approximate way.The coupled-cluster method (CCM) used by the authors ofthe Comment lacks the above discussed features important forthe nuclear many-body problem, in particular it violates trans-lational invariance from the very beginning and it does not ful-fill the variational principle. The problem of a spurious center-of-mass contamination of the many-body states in CCM is notresolved and often not even mentioned (see e.g. Ref. [4]).Moreover, non-iterative triples corrections like CCSD(T), asreferred to in the comment, tend to overestimate the correla-tion energy or even collapse. It is claimed that the CCM isvery accurate. A closer inspection of recently published nu-clear many-body results does not support this statement. InRef. [5], the 4He binding energy with the chiral N3LO NNpotential was determined with an uncertainty of several MeV.The same is obtained in the ab initio NCSM with accuracy of10 keV [6, 7]. The CCM binding energy results for 16O withthe identical chiral N3LO NN potential obtained in Refs. [8]and [9] differ by more than 5 MeV. According to Ref. [9], theCD-Bonn NN potential overbinds 16O. This is contrary to re-sults obtained in Ref. [10] and also contrary to expectations asthe CD-Bonn underbinds lighter nuclei.The CCSD(T) ground state energy for 40Ca with Vlowk re-ported in Ref. [4] is about 30 MeV lower than our 4p4h-result(CCSD is about 20MeV lower). Taking into account the small2% difference of our 3p3h and 4p4h results and the violationof the variational principle and the translational invariance inthe CCM method, we believe that the CCM result overesti-mates the exact ground-state energy in this case. Since theCCM results for 40Ca were obtained only after our Letter wassubmitted and posted on the eprint archive we have only com-pared to the CCM result for 16O which was available before-hand as a private communication.In conclusion, the importance truncation provides an ef-ficient way to extend the domain of NCSM calculations tomedium-heavy nuclei while preserving translational invari-ance and the variational principle and allowing for systematicand controlled improvements.Supported by the Deutsche Forschungsgemeinschaftthrough contract SFB 634 and by the Department of Energyunder Grant DE-FC02-07ER41457. This work in part per-formed under the auspices of the U.S. Department of Energyby Lawrence Livermore National Laboratory under ContractDE-AC52-07NA27344.∗ Electronic address: robert.roth@physik.tu-darmstadt.de† Electronic address: navratil1@llnl.gov[1] R. Roth and P. Navra´til, Phys. Rev. Lett. 99, 092501 (2007).[2] D. J. Dean et al, arXiv:0709.0449.[3] K. Jankowski, L. Meissner, and J. Wasilewski, Int. J. QuantumChem. 28, 931 (1985) and references therein.[4] G. Hagen et al, Phys. Rev. C 76, 044305 (2007).[5] D. J. Dean and M. Hjorth-Jensen, Phys. Rev. C 69, 054320(2004).[6] P. Navra´til and E. Caurier, Phys. Rev. C 69, 014311 (2004).[7] S. Quaglioni and P. Navra´til, Phys. Lett. B 652, 370 (2007).[8] M. Wloch et al., Phys. Rev. Lett. 94, 212501 (2005).[9] J. R. Gour et al., Phys. Rev. C 74, 024310 (2006).[10] S. Fujii et al., Phys. Rev. C 69, 034328 (2004).

Reply to Comment on ``Ab Initio Study of 40-Ca with an Importance Truncated No-Core Shell Model''

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