Accurate nuclear radii and binding energies from a chiral interaction

Accurate nuclear radii and binding energies from a chiral interaction

A. Ekström Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, USA Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA    G. R. Jansen Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, USA    K. A. Wendt Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, USA Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA    G. Hagen Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, USA    T. Papenbrock Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, USA Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA    B. D. Carlsson Department of Fundamental Physics, Chalmers University of Technology, SE-412 96 Göteborg, Sweden    C. Forssén Department of Fundamental Physics, Chalmers University of Technology, SE-412 96 Göteborg, Sweden Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, USA Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA    M. Hjorth-Jensen Department of Physics and Astronomy and NSCL/FRIB Laboratory, Michigan State University, East Lansing, MI 48824, USA Department of Physics, University of Oslo, N-0316 Oslo, Norway    P. Navrátil TRIUMF, 4004 Wesbrook Mall, Vancouver, British Columbia, V6T 2A3 Canada    W. Nazarewicz Department of Physics and Astronomy and NSCL/FRIB Laboratory, Michigan State University, East Lansing, MI 48824, USA Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland
Abstract

With the goal of developing predictive ab-initio capability for light and medium-mass nuclei, two-nucleon and three-nucleon forces from chiral effective field theory are optimized simultaneously to low-energy nucleon-nucleon scattering data, as well as binding energies and radii of few-nucleon systems and selected isotopes of carbon and oxygen. Coupled-cluster calculations based on this interaction, named NNLO, yield accurate binding energies and radii of nuclei up to Ca, and are consistent with the empirical saturation point of symmetric nuclear matter. In addition, the low-lying collective states in O and Ca are described accurately, while spectra for selected - and -shell nuclei are in reasonable agreement with experiment.

pacs:
21.30.-x, 21.10.-k, 21.45.-v, 21.60.De

Introduction – Interactions from chiral effective field theory (EFT) Bedaque and van Kolck (2002); Epelbaum et al. (2009a); Machleidt and Entem (2011); Hammer et al. (2013) and modern applications of renormalization group techniques Bogner et al. (2003); Navrátil et al. (2009); Bogner et al. (2010); Furnstahl and Hebeler (2013) have opened the door for a description of atomic nuclei consistent with the underlying symmetries of quantum chromodynamics, the theory of the strong interaction. Chiral nuclear forces can be constructed systematically from long-range pion physics augmented by contact interactions. Over the past decade, the renaissance of nuclear theory based on realistic nuclear forces and powerful computational methods has pushed the frontier of ab initio calculations from few-body systems and light nuclei Pieper and Wiringa (2001); Navrátil et al. (2009); Barrett et al. (2013) to medium-mass nuclei Mihaila and Heisenberg (2000); Dean and Hjorth-Jensen (2004); Hagen et al. (2008); Barbieri and Hjorth-Jensen (2009); Hagen et al. (2010); Epelbaum et al. (2010); Lähde et al. (2014); Somà et al. (2013); Hergert et al. (2013).

One of the main challenges in ab-initio calculations is the accurate 111In this Rapid Communication accuracy refers to an agreement with data at the precision one would expect from the model and method. reproduction of binding energies and radii of finite nuclei simultaneously with the empirical nuclear matter saturation point (binding energy per nucleon  MeV at Fermi momentum  fm) and incompressibility ( MeV Stone et al. (2014)). For instance, lattice EFT calculations at next-to-next-to leading order (NNLO) employ a phenomenological four-nucleon contact force (a correction beyond NNLO) to counter the overbinding in nuclei heavier than Lähde et al. (2014), while the radii of C and O are still too small Epelbaum et al. (2012, 2014). Ab initio calculations overbind medium-mass and heavy nuclei by about 1 MeV per nucleon, underestimate charge radii Binder et al. (2014), and yield too large separation energies Hergert et al. (2014). The status of chiral-force predictions for binding energies and charge radii in finite nuclei is summarized in Fig. 1, with dark grey symbols representing the predictions of various state-of-the-art calculations. This is a serious shortcoming of current chiral Hamiltonians as it prevents theory from making accurate predictions when extrapolating to higher masses. The problem with the reproduction of nuclear matter saturation properties has been discussed extensively in the literature Müther and Polls (2000); Heiselberg and Hjorth-Jensen (2000); Dewulf et al. (2003); Dickhoff and Barbieri (2004); Sammarruca (2010); van Dalen and Müther (2010); Krewald et al. (2010); Hebeler et al. (2011); Tsang et al. (2012); Baardsen et al. (2013); Carbone et al. (2013); Kohno (2013); Hebeler et al. (2013); Shirokov et al. (2014), and various solutions have been proposed, ranging from short-range correlations and Pauli blocking effects to the inclusion of many-body forces.

Figure 1: (Color online) Ground-state energy (negative of binding energy) per nucleon (top), and residuals (differences between computed and experimental values) of charge radii (bottom) for selected nuclei computed with chiral interactions. In most cases, theory predicts too small radii and too large binding energies. References:  Navrátil et al. (2007); Jurgenson et al. (2011),  Binder et al. (2014),  Epelbaum et al. (2014),  Epelbaum et al. (2012),  Maris et al. (2014),  Włoch et al. (2005),  Hagen et al. (2012),  Bacca et al. (2014),  Maris et al. (2011). The red diamonds are NNLO results obtained in this work.

We start from the optimization of the chiral interaction at NNLO. Traditionally, one takes the pion-nucleon coupling constants ’s either from pion-nucleon scattering Epelbaum et al. (2000); Büttiker and Meißner (2000) or from peripheral partial waves in the nucleon-nucleon () sector Entem and Machleidt (2003); Ekström et al. (2013), while the remaining coupling constants (denoted as low-energy constants (LECs)) are adjusted in the sector. The corresponding optimizations consider scattering data up to laboratory energies of  MeV. In a subsequent step, the remaining LECs of the leading three-nucleon () forces van Kolck (1994); Epelbaum et al. (2002); Navrátil (2007) are adjusted to data on systems Navrátil et al. (2007); Epelbaum et al. (2009b); Gazit et al. (2009). For details, we refer the reader to Refs. Epelbaum et al. (2009a); Machleidt and Entem (2011); Hammer et al. (2013). Hitherto such a strategy has not produced interactions that simultaneously describe bulk properties of both nuclei and nuclear matter  Hagen et al. (2014a).

Our optimization strategy is based on a different approach. Most importantly, we optimize forces together with forces. This is consistent with the idea of EFT that improvements are made order by order and not nucleon by nucleon. The simultaneous optimization of and forces is important because the long-range contributions of the force contain LECs from pion-nucleon vertices that also enter the force. Moreover, in addition to low-energy data and the binding energies and charge radii of H, He, our set of fit-observables also contains data on heavier nuclei; namely, binding energies and radii of carbon and oxygen isotopes. This is a major departure from the traditional approach that seeks to adjust LECs to data on few-body systems with and then attempts to extrapolate to nuclei with and to infinite nuclear matter. The following arguments motivate the strategy of including heavier nuclei into the optimization: First, no reliable experimental data constrain the isospin components of the force in nuclei with mass numbers (see Refs. Lazauskas (2009); Viviani et al. (2011) for more discussion and prospects). Second, since our goal is to describe nuclear properties at low energies, LECs are adjusted to low-energy observables (as opposed to the traditional adjustment of two-nucleon forces to scattering data at higher energies). Third, the impact of many-body effects entering at higher orders (e.g., higher-rank forces) might be reduced if heavier systems, in which those effects are stronger, are included in the optimization.

Besides these theoretical arguments, there is also one practical reason for a paradigm shift: predictive power and large extrapolations do not go together. In traditional approaches, where interactions are optimized for , small uncertainties in few-body systems (e.g., by forcing a rather precise reproduction of the sectors at a rather low order in the chiral power counting) get magnified tremendously in heavy nuclei, see for example Ref. Binder et al. (2014). Consequently, when aiming at reliable predictions for heavy nuclei, it is advisable to use a model that performs well for light- and medium-mass systems. In our approach, light nuclei are reached by interpolation while medium-mass nuclei by a modest extrapolation. In this context, it is worth noting that the most accurate calculations for light nuclei with  Wiringa and Pieper (2002) employ forces adjusted to 17 states in nuclei with  Pieper et al. (2001). Finally, we point out that nuclear saturation can be viewed as an emergent phenomenon. Indeed, little in the chiral EFT of nuclear forces suggest that nuclei are self-bound systems with a central density (or Fermi momentum) that is practically independent of mass number. This viewpoint makes it prudent to include the emergent momentum scale into the optimization, which is done in our case by the inclusion of charge radii for H, He, C, and O. This is similar in spirit to nuclear mean-field calculations Gogny et al. (1970) and nuclear density functional theory Bender et al. (2003); Kortelainen et al. (2010) where masses and radii provide key constraints on the parameters of the employed models.

Optimization protocol and model details – We seek to minimize an objective function to determine the optimal set of coupling constants of the chiral + interaction at NNLO. Our dataset of fit-observables includes the binding energies and charge radii of H, He, C, and O, as well as binding energies of O as summarized in Table 1. To obtain charge radii from computed poin-proton radii we use the standard expression Friar and Negele (1975): , where fm (Darwin-Foldy correction), fm Angeli and Marinova (2013) and fm Mohr et al. (2012). In this work we ignore the spin-orbit contribution to charge radii Ong et al. (2010). From the sector, the objective function includes proton-proton and neutron-proton scattering observables from the SM99 database Arndt et al. () up to 35 MeV scattering energy in the laboratory system as well as effective range parameters, and deuteron properties (see Table 2). The maximum scattering energy was chosen such that an acceptable fit to both scattering data and many-body observables could be achieved.

Exp. Wang et al. (2012) Exp. Angeli and Marinova (2013); Mohr et al. (2012)
H
He
He
C
O
O
O
O
Table 1: Binding energies (in MeV) and charge radii (in fm) for H, He, C and O employed in the optimization of NNLO.
NNLO NLO Entem and Machleidt (2003) Exp. Ref.
Bergervoet et al. (1988)
Bergervoet et al. (1988)
Chen et al. (2008)
Miller et al. (1990)
Machleidt (2001)
Machleidt (2001)
Wang et al. (2012)
Huber et al. (1998)
Machleidt (2001)
Table 2: Low-energy data included in the optimization. The scattering lengths and effective ranges are in units of fm. The proton-proton observables with superscript include the Coulomb force. The deuteron binding energy (, in MeV), structure radius (, in fm), and quadrupole moment (, in fm) are calculated without meson-exchange currents or relativistic corrections. The computed -state probability of the deuteron is 3.46%.

In the present optimization protocol, the NNLO chiral force is tuned to low-energy observables. The comparison with the high-precision chiral interaction NLO Entem and Machleidt (2003) and experimental data presented in Table 2 demonstrates the quality of NNLO at low energies.

The results for H and He (and Li) were computed with the no-core shell model (NCSM) Navrátil et al. (2009); Barrett et al. (2013) accompanied by infrared extrapolations More et al. (2013). The force of NNLO yields about 2 MeV of binding energy for He. Heavier nuclei are computed with the coupled-cluster method (see Ref. Hagen et al. (2014b) and the discussion below).

A total of 16 LECs determine the strengths of the contact potential, the potential in the + sector, and the contacts. The LECs are constrained simultaneously by the optimization algorithm POUNDerS Kortelainen et al. (2010). We employ standard nonlocal regulators in the construction of the potential, see e.g., Refs Epelbaum et al. (2002); Entem and Machleidt (2003) for details. This type of regulator improves the convergence of nuclear matter calculations Hagen et al. (2014a). In detail, the regulator functions consist of exponentiated Jacobi momenta divided by a cutoff value , i.e., . For the present work, we set and MeV. Furthermore, the subleading two-pion exchange in the interaction is regularized using spectral function regularization with a cutoff  MeV. The details of this procedure can be found in Refs. Epelbaum et al. (2004); Epelbaum (2006).

The objective function is numerically expensive, requiring us to adopt some approximations when computing nuclei with . In the optimization, we employed a model space of 9 oscillator shells for the interaction, the energy cutoff for the forces, and the coupled-cluster method in its singles and doubles approximation (CCSD). We use nucleus-dependent estimates for larger model spaces and triples-cluster corrections based on Ref. Hagen et al. (2010). During the optimization, we verified that these estimates were accurate by performing converged calculations. In our final computation of the objective function and for the results presented in this paper, we employ much larger model spaces and coupled-cluster methods with higher precision.

The coupled-cluster calculations are based on the intrinsic Hamiltonian to minimize spurious center-of-mass effects Hagen et al. (2009, 2010); Jansen (2013). For the binding energies presented in this paper we employ the -CCSD(T) approximation Taube and Bartlett (2008); Hagen et al. (2010); Binder et al. (2013) in a model space consisting of 15 oscillator shells with  MeV. The forces are limited to an energy cutoff , and truncated at the normal-ordered two-body level in the Hartree-Fock basis Hagen et al. (2007); Roth et al. (2012). We also include the leading-order residual contribution to the total energy as a second-order perturbative energy correction Hagen et al. (2014a), computed with .

To compute excited states in, and around, nuclei with closed shells, we employ equation-of-motion coupled-cluster methods Stanton and Bartlett (1993); Gour et al. (2006); Jansen et al. (2011); Jansen (2013); Shen and Piecuch (2013); Ekström et al. (2014); these are accurate for excited states that are generalized particle-hole excitations of low rank. For instance, N is computed with the charge-symmetry breaking equation-of-motion method from the closed sub-shell nucleus C, see  Ekström et al. (2014). Similar comments apply to F. The intrinsic charge radii are computed from the two-body density matrix (2BDM) in the CCSD approximation Shavitt and Bartlett (2009). Benchmark calculations of the He charge radius shows that the 2BDM result is 1% larger than the NCSM result. Intrinsic charge densities are computed using the one-body density matrix and correcting for the Gaussian center-of-mass wave-function Kanungo et al. (2011); Hagen et al. (2014b). In the case of O, this approach has been validated against 2BDM to four significant digits.

The values for the LECs that result from the optimization and define the chiral potential NNLO are listed in Table 3.

LEC Value LEC Value LEC Value
-1.12152120 -3.92500586 3.76568716
-0.15814938 -0.15982245 -0.15915027
2.53936779 1.00289267 -0.17767436
0.55595877 1.39836559 -1.13609526
0.60071605 -0.80230030 0.81680589
-0.03957471
Table 3: The values of the LECs for the NNLO interaction. The constants , , and are in units of GeV,  GeV, and  GeV, respectively.

We note that the pion-nucleon LECs and are in the range of the published values Büttiker and Meißner (2000); Entem and Machleidt (2003); Krebs et al. (2012). Following Ref. Entem and Machleidt (2003), we set the pion-decay constant  MeV and the axial-vector coupling constant . The value for is greater than the experimental estimate Liu et al. (2010) to account for the Goldberger-Treiman discrepancy. We use the following neutron, proton, and nucleon masses:  MeV,  MeV, and  MeV, respectively. For the pion masses we used  MeV and  MeV. For the scattering data up to 35 MeV a total was reached. Representative phase shifts are shown in Fig. 2. The phase shifts at higher scattering energies, demonstrates that NNLO is at the limits of expectations one can have for an interaction at this chiral order. Furthermore, the accuracy of NNLO in the few-body sector is similar to other chiral interactions at order NNLO Epelbaum et al. (2002); Gezerlis et al. (2014).

Figure 2: (Color online) Selected neutron-proton scattering phase-shifts as a function of the laboratory scattering energy . (Top) NNLO prediction (solid lines) compared to the Nijmegen phase shift analysis Stoks et al. (1993) (symbols) at low energies  MeV. Note the two vertical scales. (Bottom) Neutron-proton scattering phase shifts from NNLO (red diamonds) compared to the Nijmegen phase shift analysis (black squares) and the NNLO potentials (green) from Ref. Epelbaum et al. (2004).

Predictions – We begin with predictions for the -decay half-life of H. The reduced matrix element compares well to the corresponding experimental value of  Akulov and Mamyrin (2005); Gazit et al. (2009). Figure 1 shows that binding energies and charge radii of the -shell nuclei He, C, and O are in good agreement with experiment. For He the computed binding energy and charge radius are 30.9 MeV and 1.91 fm, respectively, and in good agreement with the experimental binding energy 31.5 MeV  Wang et al. (2012) and experimental charge radius 1.959(16) fm  Brodeur et al. (2012). For Li we compute a binding energy of 32.4(4) MeV and 43.9 MeV, respectively, which compare well with experiment (32.0 MeV and 45.34 MeV  Wang et al. (2012)). The charge radius of Li with NNLO is 2.22 fm, also consistent with the measured value of 2.217(35) fm Sánchez et al. (2006). We now discuss results for excited states in Li, C, N, and O, see Fig. 3. The nucleus Li is difficult to compute because it is bound by only 1.5 MeV relative to the threshold for deuteron emission. Effects of continuum is expected to lower the resonances significantly Hupin et al. (2014), thus we conclude that our results are in reasonable agreement with experiment. We also compared the spectra computed in the NCSM and agreement with the coupled-cluster prediction is good. The binding energy computed from two-particle attached equation-of-motion method Jansen et al. (2011); Jansen (2013) is 30.9 MeV and in reasonable agreement with the NCSM extrapolated result 32.4(4) MeV. Our predictions for the excited states of C and N agree with experiment except for the state in N.

Figure 3: (Color online) Energies (in MeV) of selected excited states for various nuclei using NNLO. For Li we also include spectra from the NCSM (dotted lines), and isospin quantum numbers are also given. The NCSM results were obtained with and  MeV. Parenthesis denote tentative spins assignments for experimental levels. Data are from Refs. Ajzenberg-Selove (1991); Firestone (2005, 2007); Tshoo et al. (2012).

The ab-initio computation of negative-parity states in O, particularly the state at 6.13 MeV Emrich and Zabolitzky (1981); Barbieri and Dickhoff (2002); Włoch et al. (2005); Epelbaum et al. (2014) has been a long-standing theoretical challenge. We computed this state, dominated by about 90% of - () excitations, at 6.34 MeV. The energy of the state is strongly correlated with the charge radius of O, with smaller charge radii leading to higher excitation energies. For - excited states, the excitation energy depends on the particle-hole gap and therefore on one-nucleon separation energies of the and systems. The charge radius depends also on the proton separation energy . For O we find  MeV and the neutron separation energy  MeV, in an acceptable agreement with the experimental values of 12.12 MeV and 4.14 MeV, respectively. For F we find  MeV, to be compared with the experimental threshold at 0.6 MeV.

The inset of Fig. 4 shows that the state in O also comes out well, suggesting a - nature. However, the state is about 1.5 MeV too high compared to experiment. This state is dominated by - excitations from the occupied to the unoccupied orbitals. In the state is computed at an excitation energy of 2.2 MeV, which is about 1.4 MeV too high. This probably explains the discrepancy observed for the state in O.

Figure 4 shows that the experimental charge-density of O is well reproduced with NNLO, and our charge form factor is, for momenta up to the second diffraction maximum, similar in quality to what Mihaila and Heisenberg (2000) achieved with the Av18 + UIX potential. For the heavier isotopes O and F Fig. 3 shows good agreement between theory and experiment for excited states. For F our computed spin assignments agree with results from shell-model Hamiltonians Brown and Richter (2006) and with recent ab initio results Ekström et al. (2014). The binding energies for N, F are 103.7 MeV, 163 MeV and 175.1 MeV, respectively, in good agreement with data (104.7 MeV, 167.7 MeV and 179.1 MeV). We also computed the intrinsic charge (matter) radii of O and obtained 2.72 fm (2.80 fm) and 2.76 fm (2.95 fm), respectively. The matter radius of O agrees with the experimental result from  Kanungo et al. (2011). We note that the computed spectra in O is too compressed compared to experiment (theory yields 0.7 MeV compared to 1.9 MeV for the first excited state), possibly due to the too high excited state in O. In general, the quality of our spectra for -shell nuclei is comparable to those of recent state-of-the-art calculations with chiral Hamiltonians Hagen et al. (2012); Bogner et al. (2014); Jansen et al. (2014); Caceres et al. (2015), while radii are much improved.

Figure 4: (Color online) Charge density in O computed as in Ref. Reinhard et al. (2013) compared to the experimental charge density DeVries et al. (1987). The inset compares computed low-lying negative-parity states with experiment.

For Ca the computed binding energy  MeV, charge radius  fm, and  MeV all agree well with the experimental values of 342 MeV,  fm Angeli and Marinova (2013), and 3.736 MeV respectively. We checked that our energies for the states in O and Ca are practically free from spurious center-of-mass effects. The results for Ca illustrate the predictive power of NNLO when extrapolating to medium-mass nuclei.

Finally, we present predictions for infinite nuclear matter. The accurate reproduction of the saturation point and incompressibility of symmetric nuclear matter has been a challenge for ab initio approaches, with representative results from chiral interactions shown in Fig. 5. The solid line shows the equation of state for NNLO. Its saturation point is close to the empirical point, and its incompressibility lies within the accepted empirical range Stone et al. (2014). At saturation density, coupled-cluster with doubles yields about 6 MeV per particle in correlation energy, while triples corrections (and residual forces beyond the normal-ordered two-body approximation) yield another 1.5 MeV.

Figure 5: (Color online) Equation of state for symmetric nuclear matter from chiral interactions. Solid red line: prediction of NNLO. Blue dashed-dotted and black dashed lines: Ref. Hagen et al. (2014a). Symbols (red diamond, blue circle, black square) mark the corresponding saturation points. Triangles: saturation points from other models (upward triangles Hebeler et al. (2011), rightward triangles Coraggio et al. (2014), downward triangles Carbone et al. (2013)). The corresponding incompressibilities (in MeV) are indicated by numbers. Green box: empirical saturation point.

Let us briefly discuss the saturation mechanism. Similar to potentials Bogner et al. (2003), the interaction of NNLO is soft and yields nuclei with too large binding energies and too small radii. The interactions of NNLO are essential to arrive at physical nuclei, similarly to the role of forces in the saturation of nuclear matter with low-momentum potentials Hebeler et al. (2011). This situation is reminiscent of the role the three-body terms play in nuclear density functional theory Vautherin and Brink (1972).

Summary – We have developed a consistently optimized interaction from chiral EFT at NNLO that can be applied to nuclei and infinite nuclear matter. Our guideline has been the simultaneous optimization of and forces to experimental data, including two-body and few-body data, as well as properties of selected light nuclei such as carbon and oxygen isotopes. The optimization is based on low-energy observables including binding energies and radii. The predictions made with the new interaction NNLO include accurate charge radii and binding energies. Spectra for Ca and selected isotopes of lithium, nitrogen, oxygen and fluorine isotopes are well reproduced, as well as the energies of excitations in O and Ca. To our knowledge, NNLO is currently the only microscopically-founded interaction that allows for a good description of nuclei (including their masses and radii) in a wide mass-range from few-body systems to medium-mass.

Acknowledgements.
We thank K. Hebeler and E. Epelbaum for providing the matrix elements of the non-local threebody interaction. This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under Award Numbers DEFG02-96ER40963 (University of Tennessee), DE-SC0008499 and DE-SC0008511 (NUCLEI SciDAC collaboration), the Field Work Proposal ERKBP57 at Oak Ridge National Laboratory and the National Science Foundation with award number 1404159. It was also supported by the Swedish Foundation for International Cooperation in Research and Higher Education (STINT, IG2012-5158), by the European Research Council (ERC-StG-240603), by the Research Council of Norway under contract ISP-Fysikk/216699, and by NSERC Grant No. 401945-2011. TRIUMF receives funding via a contribution through the National Research Council Canada. Computer time was provided by the Innovative and Novel Computational Impact on Theory and Experiment (INCITE) program. This research used resources of the Oak Ridge Leadership Computing Facility located in the Oak Ridge National Laboratory, which is supported by the Office of Science of the Department of Energy under Contract No. DE-AC05-00OR22725, and used computational resources of the National Center for Computational Sciences, the National Institute for Computational Sciences, the Swedish National Infrastructure for Computing (SNIC), and the Notur project in Norway.

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