1. Presented by Building Nuclei from the Ground Up: Nuclear Coupled-cluster Theory David J. Dean Oak Ridge National Laboratory Nuclear Coupled-cluster Collaboration: T. Papenbrock, K. Roche, Oak Ridge National Laboratory P. Piecuch, M. Wloch, J. Gour, Michigan State University M. Hjorth-Jensen, Oslo A. Schwenk, Triumf Funding: DOE-NP, SciDAC, DOE-ASCR

2. 2 Dean_NCCT_SC07 How do we describe nuclei we cannot measure? Robust, predictive nuclear theory exists for structure and reactions. Nuclear data needed to constrain theory. Goal is the Hamiltonian and nuclear properties: – Bare intra-nucleon Hamiltonian. – Energy density functional. Mission relevant to NP, NNSA. Half of all elements heavier than iron produced in r-process where limited (or no) experimental information exits. Nuclear reaction information relevant to NNSA and AFCI. “Given a lump of nuclear material, what are its properties, and how does it interact?” Supernova E0102-72.3

3. 3 Dean_NCCT_SC07 The Leadership Computing Facility effort will Enlarge ab-initio square to mass 100 Enable initial global DFT calculations with restored symmetries Pushing the nuclear boundaries All Regions : Nuclear cross-section efforts (NNSA, SC/NP, Nuclear Energy) Nuclear DFT effort Thermal properties regions Nuclear Coupled Cluster effort

4. 4 Dean_NCCT_SC07 Solved up to mass 12 with GFMC, converged mass 8 with diagonalization. We want to go much further! Nuclear interactions: Cornerstone of the entire theoretical edifice Real three-body interactions derived from QCD-based effective theories Method of Solution: Nuclear Coupled-Cluster Theory Method of Solution: Nuclear Coupled-Cluster Theory Depends on spin, angular momentum, and nucleon (proton and neutron) quantum numbers. Complicated interactions   

5. 5 Dean_NCCT_SC07 It boils down to a set of coupled, nonlinear algebraic equations (odd-shaped tensor-tensor multiply). Storage of both amplitudes and interactions is an issue as problems scale up. Largest problem so far: 40 Ca with 10 million unknowns, 7 peta-ops to solve once (up to 10 runs per publishable result). Breakthrough science: Inclusion of 3-body force into CC formalism (6-D tensor) weakly bound and unbound nuclei. Coupled-cluster theory: Ab initio in medium mass nuclei   exp   Correlated ground-state wave function Correlation operator Reference Slater determinant Energy Amplitude equations …  321 TTTT   THTEexp)exp(  0exp)exp(  HTHT ab… ij… ab… ij…

6. 6 Dean_NCCT_SC07   exp ( T )  T = T 1 + T 2 + T 3 + … R = excitation operator POLYNOMIAL SCALING!! (good) Early results 468infty Number of Oscillator Shells -130 -125 -120 -115 -110 -105 -100 E = (MeV) Wolch et al PRL 94, 24501 (2005) E* (0 + ) = 19.8 MeV E* (3 - ) = 12.0 MeV *E g.s. = –120.5 MeV CCSD CR-CCSD(T) Coupled cluster theory for nuclei E =  H  =  e -T He T   ab…  H  = 0 ij… RH  = E*R 

7. 7 Dean_NCCT_SC07 Ab initio in medium mass nuclei Fast convergence with cluster rank 4 He CCSD CCSD(T) FY -28 E( 4 He) [ MeV] -30 -26 -24 -22 -20 -18 -16 245610121416 N=2n+1 16 O N = 6 N = 7 N = 8 N = 9 N = 11 N = 12 N = 13 -134 -136 -138 -140 -142 -144 E CCSD ( 16 O) [MeV] 12141618202224 ħ  [MeV] 4 He 16 O 40 Ca E0  E CCSD  E CCSD(T) -11.8 -17.1 -0.3 -60.2 -82.6 -5.4 -347.5 -143.7 -11.7  E CCSD(T) -29.2-148.2-502.9 Exact (FY)-29.19(5) Error estimate:1%<< 1%< 1% Hagen et al., Phys. Rev. C 76, 044305 (2007) 40 Ca N = 3 N = 4 N = 5 N = 6 N = 7 E CCSD ( 40 Ca) [MeV] -400 -420 -440 -460 -480 -500 380 ħ  [MeV] 16 18 20 222426283032 N = 8 10 63 many-body basis states

8. 8 Dean_NCCT_SC07 Hagen, Papenbrock, Dean, Schwenk, Nogga, Wloch, Piecuch, Phys. Rev. C 76, 034302 (2007) Solution at CCSD and CCSD(T) levels involve roughly 67 more diagrams… Challenge: Do we really need the full 3-body force, or just its density dependent terms? 2-body only 0-body 3NF 1-body 3NF 2-body 3NF Residual 3NF 12345  /  CCSD  10 -4 10 -3 10 -2 10 0 10 -1 estimated triples corrections N Inclusion of full TNF in CCSD: F-Y comparisons in 4 He =-28.24 MeV +/- 0.1MeV (sys) -28 -27 -26 -25 -24 -23 -22 E CCSD(T) (MeV) 3456 N ++ 0 = +++ +++ +++ +++

9. 9 Dean_NCCT_SC07 Introduction of continuum basis states (Gamow, Berggren) Correlation dominated S n =0 SnSn n ~  n Open QS Closed QS Neutron number Energy Coupling of nuclear structure and reaction theory (microscopic treatment of open channels) capturing statesdecaying states L-L- lm(k) k1k1 k2k2 k3k3 k4k4 Re(k) bound states antibound states L+L+

10. 10 Dean_NCCT_SC07 Ab initio weakly bound and unbound nuclei Single-particle basis includes bound, resonant, non-resonant continuum, and scattering states ENORMOUS SPACES….almost 1k orbitals. 10 22 many-body basis states in 10 He He Chain Results (Hagen et al) [feature article in Physics Today (November 2007)] Challenge: Include 3-body force Gamov states capture the halo structure of drip-line nuclei 45678 r [fm] 10 -2 10 -1 10 0 r 2  (r) [fm -1 ] G-HF basis HO-HF basis

11. 11 Dean_NCCT_SC07  System of non-linear coupled algebraic equations  solve by iteration  n = number of neutrons and protons  N = number of basis states  Solution tensor memory  (N-n) ** 2 * n ** 2  Interaction tensor memory  N ** 4  Operations count scaling  O(n ** 2 * N ** 4)  O(n ** 4 * N ** 4) with 3-body  O(n ** 3 * N ** 5) at CCSDT Solution of coupled-cluster equation Many such terms exist. Cast into a matrix-matrix multiply algorithm. Parallel issue: block sizes of V and t. Many such terms exist. Cast into a matrix-matrix multiply algorithm. Parallel issue: block sizes of V and t. Basic numerical operation: t new (ab, ij) =  V (kl, cd)t old (cd, ij)t old (ab, kl) k,l=1, n c,d=n+1,N

12. 12 Dean_NCCT_SC07 Partial sum t 2 reside on each processor Memory distribution across processors V(ab, c1, d1) t2 partial sum V(ab, c1, d2) t2 partial sum  V(ab, c2, d2) t2 partial sum V(ab, c2, d1) t2 partial sum  V(ab, c1, d1) t2 partial sum V(ab, c1, d2) t2 partial sum V(a, b, c2, d2) t2 partial sum V(ab, c2, d1) t2 partial sum Code parallelism Global reduce (sum) t2, distribute t 2 (ab, ij) =  V (kl,cd)t ij t kl kl <  f cd >  f cd ab

13. 13 Dean_NCCT_SC07 Future direction  Current algorithm scales to 1K processors with about 20% efficiency. Attacking problems in mass 40 region is doable with current code.  Develop algorithm that spreads both the 2-body matrix elements and the CC amplitudes (in collaboration with Ken Roche)  Enables nuclei in the mass 100 region and should scale to 100K processors (under way).  Designing further parallel algorithms that calculate nuclear properties to calculate densities and electromagnetic transition amplitudes.  Eventual time-dependent CC for fission dynamics.

14. 14 Dean_NCCT_SC07 Contact David J. Dean Physical Sciences Directorate Nuclear Theory (865) 576-5229 deanj@ornl.gov 14 Dean_NCCT_SC07 References: Dean and Hjorth-Jensen, PRC 69, 054320 (2004); Kowalski, Dean, Hjorth-Jensen, Papenbrock, Piecuch, PRL 92, 132501 (2004); Wloch, Dean, Gour, Hjorth-Jensen, Papenbrock, Piecuch, PRL 94, 21501 (2005); Gour, Piecuch, Wloc, Hjorth-Jensen, Dean, PRC (2006); Hagen, Dean, Hjorth-Jensen, Papenbrock, PLB (2007); Hagen, Dean, Hjorth-Jensen, Papenbrock, Schwenk, PRC 76, 044305 (2007); Hagen, Papenbrock, Dean, Schwenk, Nogga, Wloch, Piecuch, PRC 76, 034302(2007); Dean, Physics Today (November 2007)