Presented by

Building Nuclei from the Ground Up: Nuclear Coupled-cluster Theory

David J. DeanOak Ridge National Laboratory

Nuclear Coupled-cluster Collaboration:T. Papenbrock, K. Roche, Oak Ridge National LaboratoryP. Piecuch, M. Wloch, J. Gour, Michigan State University

M. Hjorth-Jensen, Oslo

A. Schwenk, Triumf

Funding: DOE-NP, SciDAC, DOE-ASCR

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?”

SupernovaSupernova

E0102-72.3E0102-72.3

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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 effortNuclear DFT effortThermal properties regionsThermal properties regions

Nuclear Coupled Cluster effortNuclear Coupled Cluster effort

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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 interactionsderived from QCD-based

effective theories

Real three-body interactionsderived 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

Depends on spin, angular momentum, and nucleon(proton and neutron) quantum numbers. Complicated interactions

NNNji

jiAi

ii

VrrVM

H ++∇−

= ∑∑<=

),(2,1

22h

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• 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: 40Ca 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 functionCorrelationoperator

Reference Slaterdeterminant

EnergyEnergy

Amplitude equationsAmplitude equations

… 321 TTTT

∑

∑

><

++

><

+

=

=

f

f

f

f

abij

ijbaabij

ai

iaai

aaaatT

aatT

εε

εε

2

1 THTE exp)exp(

0exp)exp( HTHT ab…ij…

ab…ij…

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exp(T)

T = T1 + T2 + T3 + …

R = excitation operator

POLYNOMIAL SCALING!! (good)

Early results

4 6 8 infty

Number of Oscillator Shells

-130

-125

-120

-115

-110

-105

-100

E =

(M

eV

)Wolch et al PRL 94, 24501 (2005)

E* (0+) = 19.8 MeV

E* (3-) = 12.0 MeV

*Eg.s. = –120.5 MeV

CCSDCR-CCSD(T)

Coupled cluster theory for nuclei

E = H = e-THeT

ab…H = 0ij…

RH = E*R

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Ab initio in medium mass nuclei

Fast convergence w

ith cluster rank

4HeCCSDCCSD(T)FY

-28

E(4 H

e) [

MeV

]

-30

-26

-24

-22

-20

-18

-16

2 4 5 6 10 12 14 16N=2n+1

16O

N = 6

N = 7

N = 8N = 9

N = 9N = 11N = 12

N = 13

-134

-136

-138

-140

-142

-144

E CC

SD(16

O) [

MeV

]

12 14 16 18 20 22 24ħ [MeV]

4He 16O 40Ca

E0

ECCSD

ECCSD(T)

-11.8

-17.1

-0.3

-60.2

-82.6

-5.4

-347.5

-143.7

-11.7

ECCSD(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)

Hagen et al., Phys. Rev. C 76,044305 (2007)

40Ca

N = 3

N = 4

N = 5

N = 6N = 7

E CC

SD(40

Ca)

[MeV

] -400

-420

-440

-460

-480

-500

380

ħ [MeV]16 18 20 22 24 26 28 30 32

N = 8

1063 many-bodybasis states

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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?

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

1 2 3 4 5

/

CC

SD

10-4

10-3

10-2

100

10-1

estimated triples corrections

N

Inclusion of full TNF in CCSD: F-Y comparisons in 4He

<E>=-28.24 MeV +/- 0.1MeV (sys)

-28

-27

-26

-25

-24

-23

-22

EC

CS

D(T

) (M

eV

)

3 4 5 6N

+ +0 =

+ ++

+ ++

+ ++

+ ++

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Introduction of continuum basis states (Gamow, Berggren)

CorrelationdominatedSn=0

Sn

n ~ n

Open QSOpen QS

Closed QSClosed QS

Neutron number

En

erg

yCoupling of nuclear structure and reaction theory (microscopic treatment of open channels)

capturing states decaying states

L-

lm(k)

k1

k2 k3 k4

Re(k)

boun

d st

ates

antib

ound

sta

tes L+

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Ab initio weakly bound and unbound nuclei

Single-particle basis includesbound, resonant, non-resonantcontinuum, and scattering statesENORMOUS SPACES….almost 1k orbitals.1022 many-body basis states in 10He

He Chain Results(Hagen et al)

[feature article in Physics Today (November 2007)]

Challenge: Include 3-body forceChallenge: Include 3-body force

Gamov states capture the halo structure of drip-line nuclei

4 5 6 7 8

r [fm]

10-2

10-1

100

r2

(r)

[fm

-1]

G-HF basis

HO-HF basis

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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:Basic numerical operation:

tnew (ab, ij) = V (kl, cd)told (cd, ij)told (ab, kl)k,l=1, n

c,d=n+1,N

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Partial sumt2 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 sumV(ab, c2, d1)

t2 partial sum

V(ab, c1, d1)

t2 partial sumV(ab, c1, d2)

t2 partial sumV(a, b, c2, d2)

t2 partial sumV(ab, c2, d1)

t2 partial sum

Code parallelism

Global reduce (sum) t2, distribute

t2(ab, ij) = V (kl,cd)tij tklkl <fcd >f

cd ab

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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.

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Contact

David J. DeanPhysical Sciences DirectorateNuclear Theory(865) 576-5229deanj@ornl.gov

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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)