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1 SUPPLEMENTARY INFORMATION doi: /nature11188 In addition to the figures shown in the article, we include here further results for isovector and isoscalar quadrupole deformations and neutron skin using other energy functionals. The supplementary methods section contains details of high-performance computations. Tables including explicit two-proton and two-neutron drip lines for all of the energy functionals used are also presented. The supplementary discussion includes drip-line results from other state-of-theart models (shown in supplementary figures), a discussion of the possible dependence of the drip lines on the size of the harmonic oscillator basis used, and a comparison with former results. The last section contains the equations used and definitions. This supplementary information contains Supplementary figures Supplementary methods Supplementary tables Supplementary discussion Supplementary equations 1

2 Supplementary Figures and Legends RESEARCH SUPPLEMENTARY INFORMATION Supplementary Figure 1 Nuclear landscape. Similar to Fig. 1 but includes predictions of FRDM18 and HFB-2117 models. The λn = Δn and λp = Δp one-nucleon drip-line trajectories are added to indicate the extent of odd-n and odd-z systems, respectively (see Supplementary Discussion). 2 2 W W W. N A T U R E. C O M / N A T U R E

3 SUPPLEMENTARY INFORMATION RESEARCH Supplementary Figure 2 Predicted isoscalar quadrupole deformations. Survey of mass (isoscalar) quadrupole deformations calculated with six EDFs: SkM*, SkP, UNEDF1, SLy4, UNEDF0, and SV-min. 3 W W W. N A T U R E. C O M / N A T U R E 3

4 RESEARCH SUPPLEMENTARY INFORMATION Supplementary Figure 3 Predicted isovector quadrupole deformations. Same as in supplementary Fig. 2 but for isovector quadrupole deformations. 4 4 W W W. N A T U R E. C O M / N A T U R E

5 SUPPLEMENTARY INFORMATION RESEARCH Supplementary Figure 4 Predicted neutron skin radii. Same as in supplementary Fig. 2 but for neutron skin radii. 5 W W W. N A T U R E. C O M / N A T U R E 5

6 RESEARCH SUPPLEMENTARY INFORMATION Supplementary Figure 5 Basis-size dependence of calculated separation energies. (a) Difference ΔS 2n between S 2n values of even-even erbium isotopes calculated with UNEDF0 using the harmonic oscillator basis corresponding to N sh =20 major shells with respect to N sh =25 (gray dashed line) and N sh =30 (orange solid line) results. (b) Even-even drip line nuclei predicted with UNEDF0 that have S 2n in the range of ±100 kev. The results for λ n > 0 should be considered as rough estimates as the HFB theory does not guarantee that the nucleonic densities and fields are localized in this regime. 25,32,33 6

7 SUPPLEMENTARY INFORMATION RESEARCH Supplementary Methods To include all even-even particle-bound nuclei with Z 120 for a given EDF parameterization, mass table calculations were done in a wide range of nuclei and shape deformations. 31 In our study, we consider only axial and parity-conserving intrinsic shapes as most nuclei are expected to have axial and reflection symmetric ground states. 30 However, since the ground-state shape is not known a priori, one still has to explore configurations associated with spherical, prolate, and oblate shapes. To this end, we scan the nuclear energy surface of each isotope in a wide range of shape deformations to identify local energy minima. The ground-state energy is obtained by taking the lowest-energy solution obtained by means of unconstrained minimization. Our mass tables were obtained using the JAGUAR and KRAKEN Cray XT5 supercomputers housed at Oak Ridge National Laboratory's Leadership Computing Facility and the University of Tennessee's National Institute for Computational Sciences. Using 9,060 processors of JAGUAR, computation of the entire even-even mass table took about 2 hours. As one HFBTHO run representing the single nuclear configuration can be performed on a single processor, the mass table calculations scale with the number of processors. To optimize performance, the HFBTHO solver has been extended with a minimal MPI communication to run in a parallel regime across the nodes. 7

8 in a parallel regime across the nodes. RESEARCH SUPPLEMENTARY INFORMATION Supplementary Tables Supplementary Tables Supplementary Table 1 Two-nucleon drip lines predicted by the nuclear energy density Supplementary Table 1 Two-nucleon drip lines predicted by the nuclear energy density functionals employed in this work. For each even-even element Z (first column) given are functionals employed in this work. For each even-even element Z (first column) given are neutron numbers N corresponding to the two-proton (white columns) and two-neutron (green neutron numbers N corresponding to the two-proton (white columns) and two-neutron (green columns) drip line. Asterisks mark the cases where the binding re-entrance (cf. inset in Fig. 1) is expected. Z SkM* SkP SLy4 7 SV- min UNEDF0 UNEDF Z 6 4 SkM* 16 4 SkP 16 4 SLy SV- min 16 4 UNEDF UNEDF * * * 36* * * * * * * * * * * * * * * * * * * * * * *

9 SUPPLEMENTARY INFORMATION RESEARCH Z SkM* SkM* SkP SkP SLy4 SLy4 SV- min SV- min UNEDF0 UNEDF0 UNEDF1 UNEDF * 170* * * * * * * * * * * * * * * * * * * * * 226* * 220* * * * 234* * * 226* * * * 246* * 228* * 232* * * * * 230* * 236* * * * * * * 244* * 232* * * * * * * * * * * 280* * * 258* * * * * 260* * 260* * * * * * * * * * * * * Supplementary Discussion Supplementary Discussion In the following, we expand on several points made in the manuscript. In the following, we expand on several points made in the manuscript. The drip line is reached when S n 0 (one-neutron drip line) or S 2n 0 (two-neutron drip line). The drip line is reached when S n 0 (one-neutron drip line) or S 2n 0 (two-neutron drip line). The particle drip lines have been obtained from calculated binding energies B(Z,N) of even-even nuclei. The particle The two-neutron drip lines have separation been obtained energy is from given calculated by S 2n (Z,N) binding = B(Z,N 2) energies B(Z,N). To of even-even estimate nuclei. The S 1n for two-neutron odd-n nuclei, separation we compute energy the is chemical given by potential S 2n (Z,N) λ= n (N) B(Z,N 2) and pairing B(Z,N). gap n To (N) by estimate S 1n for odd-n nuclei, we compute the chemical potential λ n (N) and pairing gap n (N) by 9 9

10 RESEARCH SUPPLEMENTARY INFORMATION taking an average over the HFB values obtained for even-even nuclei with neutron numbers N 1 and N+1. The one-neutron separation energy in a Z-even, N-odd nucleus can be estimated as S 1n λ n Δ n. A one-neutron drip line is reached when λ n Δ n. To estimate the position of a oneneutron drip line in an odd-z system, we take an average over neighbouring even-z elements. Analogous relationships apply to protons. The positions of one-nucleon drip lines estimated in such way are shown in supplementary Fig. 1. To check the accuracy of our two-neutron drip-line estimate for even-even nuclei, we computed the two-neutron drip line from the HFB chemical potential, λ n 0, and obtained the same trajectory as given by the condition of S 2n 0. This means that the canonical relation between the chemical potential and binding energy, λ n de/dn ½[B(Z,N-1) B(Z,N+1)], is well obeyed by the self-consistent DFT calculations. The application of modern optimization and statistical methods, together with highperformance computing, has revolutionized nuclear DFT during recent years. Statistical methods of linear-regression and error analysis have been recently applied to determine the correlations between EDF parameters, parameter uncertainties, and theoretical errors of calculated observables. The use of advanced covariance analysis has provided many new insights ,34,35 Many previous limitations have been removed with the help of modern resources that allow rapid computation of the properties of millions of nuclear configurations throughout the nuclear landscape (see Refs. 19, 20, 22, 23, and references quoted therein). The mean neutron and proton drip lines and associated systematic uncertainties have been obtained by averaging the predictions of individual models (given in tabular form in Supplementary Information, SI). 10

11 SUPPLEMENTARY INFORMATION RESEARCH Two-nucleon drip lines predicted by the nuclear energy density functionals employed in this work are listed in supplementary Table 1. The results for SLy4 can be compared to a survey of drip lines in Ref. 16. Their results were based on a volume contact pairing, in contrast to the mixed type pairing used here. In spite of this difference, we exactly reproduce the two-proton drip line of Ref. 16. The only exceptions are Sr and Hg, but those are special cases where λ p < 0 and S 2p < A comparison of two-neutron drip lines shows that the results differ for only five elements, excluding the nuclei with λ n < 0 and S 2n < 0 exhibiting the coexistence effects and those for which Ref. 16 provides the secondary drip-line results. As illustrated in Fig. 1 of SI, the predictions of FRDM and HFB-21 models fall generally within our uncertainty band. Supplementary Fig. 1 compares our drip-line estimates with results of HFB-21 and FRDM mass models. As already seen in Fig. 2 for the example of the erbium isotopes, both models are generally consistent with our calculated uncertainty band for the neutron drip line and show excellent agreement with the predicted proton drip line. The astrophysical r-process is expected to proceed along a path of constant neutron separation energies, fairly close to the neutron drip line. 9 Depending on the temperature and neutron densities, the r-process is expected to run along trajectories of constant neutron separation energies. More discussion on this point can be found in Refs. 9 and The trajectory corresponding to S 2n =2 MeV marks the drip-line r-process path where the uncertainties are the largest. The values corresponding to other separation energies can be provided, if requested. 11

12 RESEARCH SUPPLEMENTARY INFORMATION How model-dependent are DFT extrapolations when it comes to observables other than the separation energy? Supplementary Figs. 2 4 display, respectively, systematics of isoscalar and isovector quadrupole deformations, and neutron skin radii calculated with the energy functionals SkM*, SkP, UNEDF1, SLy4, UNEDF0, and SV-min. Isoscalar quadrupole deformations displayed in supplementary Fig. 2 show the familiar pattern of large deformation in mid-shell nuclei and spherical shapes around magic gaps 20, 28, 50, 82, and, as well as the predicted regions of stability in superheavy nuclei 39 around N=184 and 258 (see also systematics in Refs ). The notable exceptions are the very neutron rich Ca, Sn, and Pb isotopes, as well as the neutronrich N=28, 50, and 82 isotones, which are systematically predicted to be deformed. Isovector quadrupole deformations measure the difference between proton and neutron quadrupoledeformed shapes. There is a renewed interest in those deformations in weakly bound nuclei While the deformations of protons and neutrons are usually predicted to be similar, as evidenced by very small values of β IV shown in supplementary Fig. 3, for the majority of light- and medium-mass nuclei and at shell closures, β IV < 0 for heavy nuclei above tin (Z=50), primarily because of large rms neutron radii. Systematics of neutron skin radii displayed in supplementary Fig. 4 show a smooth transition from proton skins (ΔR np <0) in proton-rich nuclei to pronounced neutron skins (ΔR np >0) in neutron-rich nuclei (cf. Refs. 43 and 44 for more discussion). As seen in supplementary Figs. 2 4, while the results obtained with different effective interactions can differ in detail, the overall systematics are indeed very robust. According to our Skyrme-DFT mass tables, the number of particle-bound even-even nuclei with 2 Z 120 is 2333 in the SkM* model, 2042 in SkP, 1928 in SLy4, 2116 in SVmin, 2209 in UNEDF0, and 2219 in UNEDF1. Adding the odd-mass and odd-odd neighbours, we predict that 6900±500 syst nuclei with Z 120 are bound to proton and neutron emission. 12

13 SUPPLEMENTARY INFORMATION RESEARCH The number of particle-bound odd-a and odd-odd nuclei can be estimated from the HFB conditions λ n Δ n and λ p Δ p for one-particle drip lines. By adding this estimate to the number of predicted even-even nuclides, we obtain 7512 particle-bound nuclei in the SkM* model, 6575 in SkP, 6235 in SLy4, 6734 in SV-min, 7400 in UNEDF0, and 7163 in UNEDF1. Therefore, the total number of bound nuclei predicted by the Skyrme-DFT approach is 6900±500 syst. To put things in perspective, the number of bound nuclei reported in the literature is indeed very uncertain, with the estimates ranging between 5,000 and 12,000 isotopes. 11,45,46 In the long term, of particular importance is the development of novel nuclear energy density functionals that reproduce both bulk nuclear properties and spectroscopic data. Work along these lines is in progress. As discussed in Refs. 15 and 22, in addition to imposing new constraints on pseudodata on neutron drips, we intend to improve the spectroscopic quality of EDFs by considering the experimental data on single-particle states. We shall also improve the density dependence of the kinetic term by adding new constraints on giant resonances. As far as the form of the EDF development is concerned, novel functionals can be constructed from nucleon-nucleon interactions by using effective field theory and the density matrix expansion technique 47 or by enriching density dependence and adding higher gradient terms in a systematic way. 48 The single-particle basis consisted of the HO states originating from the 20 major oscillator shells (see supplementary Fig. 5 and SI for more discussion). Our calculations were performed using a single-particle basis of N sh =20 major oscillator shells. To assess the error on S 2n and drip-line positions caused by the size of the basis, supplementary Fig. 5(a) displays two-neutron separation energies of erbium isotopes calculated with N sh =20, 25,

14 RESEARCH SUPPLEMENTARY INFORMATION and 30 major oscillator shells. The difference between N sh =20 and N sh =30 results does not exceed 100 kev for the most neutron-rich nuclei. Generally, the increase of basis size lowers the binding energy, most notably in very neutron rich nuclei; hence, it tends to increase S 2n values in nuclei close to the drip line. However, as seen in supplementary Fig. 5(b), this has a very minor influence on the position of the two-neutron drip line by shifting it in a few cases by at most two neutrons. Since the systematical uncertainty in the two-neutron drip-line position in heavy nuclei is about 12, the numerical error associated with the basis size is expected to hardly impact the error budget. Supplementary Equations The dimensionless quadrupole deformations for neutrons (n), protons (p), and the total (t) mass distribution are respectively defined as β τ = 4π 5 Q 20,τ N τ r 2 τ, where τ=n,p or t; N n =N, N p =Z, N t =A; and Q 20 = r 2 Y 20 is the quadropole moment operator. The isovector and isoscalar quadrupole deformations, and the neutron skin radii are: β IV = β n β p, β IS = β t, and ΔR np = R n R p. Supplementary Notes References for Supplementary Information (Numbering continued from article.) 31. Erler, J. et al. Microscopic nuclear mass table with high-performance computing. Proceedings of CCP J. Phys: Conf. Ser. (2012). 32. Bulgac, A. Hartree-Fock-Bogoliubov approximation for finite systems. Preprint FT , CIP-IPNE, Bucharest, Romania, 1980; arxiv:nucl-th/ v

15 SUPPLEMENTARY INFORMATION RESEARCH 33. Dobaczewski, J. et al. Mean-field description of ground-state properties of drip-line nuclei: pairing and continuum effects, Phys. Rev. C 53, (1996). 34. Bertsch, G.F., Sabbey, B. & Uusnäkki, M. Fitting theories of nuclear binding energies. Phys. Rev. C 71, (2005). 35. Toivanen, J., Dobaczewski, J., Kortelainen, M. & Mizuyama, K. Error analysis of nuclear mass fits. Phys. Rev. C 78, (2008). 36. Kratz, K.L. et al. Isotopic r-process abundances and nuclear-structure far from stability implications for the r-process mechanism. Ap. J. 403, (1993). 37. Surman, R. et al. Source of the rare-earth element peak in r-process nucleosynthesis. Phys. Rev. Lett. 79, (1997). 38. Freiburghaus, C. et al. The astrophysical r-process: a comparison of calculations following adiabatic expansion with classical calculations based on neutron densities and temperatures. Ap. J. 516, (1999). 39. Bender, M., Nazarewicz, W. & Reinhard, P.-G. Shell stabilization of super- and hyperheavy nuclei without magic gaps. Phys. Lett. B 515, (2001). 40. Severyukhin, A.P., Bender, M., Flocard, H. & Heenen, P.-H. Large-amplitude Q n -Q p collectivity in the neutron-rich oxygen isotope 20 O. Phys. Rev. C 75, (2007). 41. Misu, T., Nazarewicz, W. & Åberg, S. Deformed nuclear halos. Nucl. Phys. A 614, (1997). 42. Li, Lulu et al. Deformed relativistic Hartree-Bogoliubov theory in continuum. Phys. Rev. C 85, (2012). 43. Mizutori, S., Dobaczewski, J., Lalazissis, G. A., Nazarewicz, W., and Reinhard, P.-G. Nuclear skins and halos in the mean-field theory. Phys. Rev. C 61, (2000). 44. Rotival, V., Bennaceur, K. & Duguet, T. Halo phenomenon in finite many-fermion systems: atom-positron complexes and large-scale study of atomic nuclei. Phys. Rev. C 79, (2009). 15

16 RESEARCH SUPPLEMENTARY INFORMATION 45. Vretenar, D. Nuclear structure far from stability. Nucl. Phys. A 751, (2005). 46. The Scientific Objectives of the SPIRAL 2 Project ( 47. Stoitsov, M. et al. Microscopically based energy density functionals for nuclei using the density matrix expansion: Implementation and pre-optimization. Phys. Rev. C 82, (2010). 48. Raimondi, F., Carlsson, B.G. & Dobaczewski, J. Effective pseudopotential for energy density functionals with higher-order derivatives. Phys. Rev. C 83, (2011). Acknowledgements with grant numbers This work was supported by the U.S. Department of Energy under Contract Nos. DE- FG02-96ER40963 (University of Tennessee) and DE-FC02-09ER41583 (UNEDF SciDAC Collaboration) and by the Academy of Finland under the Centre of Excellence Programme (Nuclear and Accelerator Based Physics Programme at JYFL) and FIDIPRO programme. Computational resources were provided through an INCITE award Computational Nuclear Structure'' by the National Center for Computational Sciences (NCCS) and the National Institute for Computational Sciences (NICS) at Oak Ridge National Laboratory. 16