Atomic nuclei: Computational challenges in an ab initio approachp g pp

David J. DeanOak Ridge National Laboratory

IGCMS SeminariApril 2009

Collaboration: G. Hagen (ORNL), T. Papenbrock (ORNL/UT)M. Hjorth-Jensen (Oslo), P. Piecuch (MSU); K. Roche (ORNL)

Funding: gDOE/SC Nuclear Physics; ORNL LDRD (FY02/FY03) & SEED (FY05/FY06) SciDAC-II; ASCR/NNSAComputing: NERSC/LBL and NCCS/ORNL

The Physics

QCD

EFT}}

Nuclear Structure

Applications in astrophysics, defense, energy, and medicine

Landscape and consequences

Present and next Generation Radioactive Ion Beam facilities(multi $100M investments world wide)

“[C]ountries throughout theworld are aggressively pursuingrare-isotope science, often as their highest priority in nucleartheir highest priority in nuclearscience, attesting to the significance accorded internationallyto this exciting area of research”NAS RISAC ReportNAS RISAC Report

Future U.S. FRIB basedon a heavy-ion linac driver

hi h i ita high priority-- NAS RISAC Report-- NSAC RIBT Report-- NSAC 2007 LRP

Key nuclear physics questions

• What is the nature of the nuclear force that bindsprotons and neutrons into stable nuclei and rareisotopes?Wh t i th i i f i l tt i l l i?• What is the origin of simple patterns in complex nuclei?

• What is the nature of neutron stars and dense nuclear matter?• What is the origin of the elements in the cosmos?• What are the nuclear reactions that drive stars and

How are we going to describe nuclei that we cannot measure?

stellar explosions?

cannot measure?Robust and predictive nuclear theoryNeed for nuclear data to constrain theoryWe are after the Hamiltonian

bare intra-nucleon Hamiltonianenergy density functional

This talk:This talk: Walk through some of thecomputational aspects of the problem…

What’s the vision?Calculate at least to mass 100+ nuclei with ab initio techniques.

WHY?WHY? Nuclear Discovery: 60 years

Old Moore’s law figure

Moore’s lawMoore’s Law

• Discovery of new nuclei is slow (ND law)• Discovery of new nuclei is slow (ND law)• Computational capability follows a power law (Moore’s law)

• TODAY: 1.3 Pflops (sustained)!!• ab initio (from V2+V3+ forces) gives us a viable platform toab initio (from V2+V3+…forces) gives us a viable platform to make predictions.

• exa-scale computing is in the DOE/SC 10 year plan.

Physics of nuclei and computing

• 1980s: Time-dependent Hartree FockKoonin, Strayer, Davies, Bonche,…Umar

• Early 1990s: Auxilliary Field Monte CarloKoonin Dean Langanke Ormand Alhassid JohnsonKoonin, Dean, Langanke, Ormand, Alhassid, Johnson

• 2000s to present: ab inito takes hold• GFMC (Pieper, Wiringa, Calson, Pandharipande, Lusk)• NCSM (Barrett, Vary, Ormand, Navratil, Ng)• Coupled-clusters (Dean, Hjorth-Jensen, Hagen, Piecuch,

Papenbrock, Bernholdt)Papenbrock, Bernholdt)

• Late 2000s: quest for Universal Nuclear Energy Density Functional• Nazarewicz, More’, Fann,… (UNEDF)

With Skyrme’s force, each iteration of the Hartree-Fock equations for 208Pb takes lessthan 30 seconds on the Univac 1008. Vautherin & Brink, PLB32, 149 (1970)

Begin with a NN (+3N) Hamiltonian

From the interaction to solving the nuclear many-body problem

Begin with a NN (+3N) Hamiltonian

cmkji

kjiNji

jiN

A

i i

i TrrrVrrVm

H

,,61,

21

2 321

2 Bare (GFMC)(Local only, Av18plus adjusted 3-body)

B i i

kjijii im 622 1 plus adjusted 3 body)

Basis expansion(explore forces)Basis expansions:

• Determine the appropriate basis• Generate Heff in that basis• Use many-body technique to solve problem

( p )

• Use many-body technique to solve problem

NucleusNucleus 4 shells 4 shells 7 shells7 shells

4He4He 4E44E4 9E69E6Oscillatorsingle-particle

Substantial progress inmany-body developments• GFMC; AFDMC

N C h ll d l4He4He 4E44E4 9E69E6

8B8B 4E84E8 5E135E13

12C12C 6E116E11 4E194E19

basis states

Many body

• No Core shell model (not a model)

• Coupled-cluster theory• UCOM,…

16O16O 3E143E14 9E249E24

Many-body basis states

,• AFMC

Green’s Function Monte Carlo

Idea:1. Determine accurate approximate wave function via variation of the

energy (The high-dimensional integrals are done via Monte Carlo integration).

2. Refine wave function and energy via projection with Green’s functionfunction

Vi t ll t th d Virtually exact method. Limited to certain forms of Hamiltonians. Computational expense increases dramatically with A due to

sampling of spin/isospin sampling. Possible extensions to heavy nuclei via AFMDC

Basis expansion techniques• Generate a single-particle basis (e.g., spherical oscillator)• Transform problem into a large, sparse eigenvalue problem• Worry about ‘down-folding’ of the interaction• Solve with Krylov space techniques

CMkji

kjiNji

jiN

A

i i

i TrrrVrrVm

H

,,61,

21

2 321

2

)()(),()()(),(|| 21212*

1*

2121 rrrrVrrrdrdrsrrVpqrspq srqp

pqrs

rsqppq

qp aaaarsVpqaaqtpH41

aaaarsVpqaaqtpH ff1

pqrsrsqp

pqqp aaaarsVpqaaqtpH eff4

The a+ and a are creation and anihilation operators

General many-body problem for fermions(basis expansions)

particles are spin ½ fermionsmany-body wave function is fully anti-symmetricy y y y certain quantum numbers will be conserved

for nuclei: total angular momentumtotal paritytotal parity‘isospin’ (analogous to spin)‘isospin projection, Tz= (N-Z)/2

H ilt i ill b l ti i ti ( ll ) Hamiltonian will be non-relativistic (usually)We (usually) work in second quantization

particlesandstatesparticlesinglewithspaceFock AN01010010

particles.andstatesparticlesingle with spaceFock

aaaaaaaaaaaa

aaaaAN

1000011021

Aaaa

aaaaaaaaaaaa

0 3

Specific example: 2 particles in 4 states

113

012

10012

10101

11000

aaI

aaI0

1

3

323

214

01014

01103

10012

I

aaI

aaI1 4

534

424

00115

01014

aaI

aaI2 5

!states particle-single ofnumber

particles;ofnumber

NNn

Scaling: Number of basis states 13107.1)100,10(!!

!,

xCnnN

NnNC

Oops. These are HUGE numbers

139106100,1000 xC

PROBLEM : How to deal with such large dimensions???

Correlated wave function representation

W h l t t f t t th t t t d Hilb tWe have a complete set of states that span our truncated Hilbert space:

IJ

N

JIII

;11

“mean field” Uncorrelated state of lowest energy

I0

mean field Uncorrelated state of lowest energy.

11000

0 jibaabij

iaai aaaabaabb

1p-1h 2p-2h … np-nh1p 1h 2p 2h … np nh(implicit summation assumed)

;11N

Problem II: How do we solve for the correlated many-body wave function?

;10

Diagonal contributions to the Hamiltonian matrix

Here we apply Wick’s theorem to the one body termHere we apply Wick s theorem to the one-body termand the diagonal contributions of the two-body term.

1222212121121111 aaaaaaaaaaaaH

11121212122121121211 4

141 VaaaaaaaaVH

121221 4

1 V

23

41

31

43

23

43123412432134212143124321341234 VVVVVVVV

Two-body contributions to the Hamiltonian matrix

123434342121123416 41

41 VaaaaaaaaVH Hamiltonian matrix now

‘mixes’ bare eigenstates

H0

41

12342143123461 aaaaaaaaVH

mixes bare eigenstates

1313311312

12341213121221

1141

41

21

VV

VVV

341212124343341261 4

141 VaaaaaaaaVH

2134141441

1313311312

1141

21

24

VV

VV

231414143232231443 41

41 VaaaaaaaaVH

343443

2323322314

21

21

41

V

VV

142323234141142334 41

41 VaaaaaaaaVH

3434433412 2

141 VV

121313134121121312 41

41 VaaaaaaaaVH

Solve the eigen problem

• Generate the Hamiltonian matrix and diagonalize (Lanczos)• Yields eigenvalues and eigenvectors of the problem

IJ HUIHJU

I

I IU

0aaaabaabb jibaabij

iaai

surfaceFermitheaboverunsurface Fermi thebelowrun ,

baji

j

surfaceFermitheaboverun ,ba

Theorists agree with each other

No core shell model

Idea: Solve the A-body problem in a harmonic oscillator basis.1. Take K single particle orbitals2. Construct a basis of Slater determinants3. Express Hamiltonian in this basis4. Find low-lying states via diagonalization

Get eigenstates and energies Symmetries like center-of-mass treated exactly No restrictions regarding Hamiltonian No restrictions regarding Hamiltonian

Number of configurations and resulting matrix very large: There are

ways to distribute A nucleons over K single-particle orbitals.ways to distribute A nucleons over K single particle orbitals.

Progress: from structure to reactions in the same framework

n- scattering

Nollett et al., PRL 99, 022502 (2007)

NCSM clustersNCSM clusters

Navratil et al., PRC73, 065801 (2006)

Coupled Cluster Theory: ab initio in medium mass nuclei

TexpCorrelated Ground-State Correlation Reference Slater

wave function operator determinant

TTTT Energy 321 TTTT

f

ai

iaai aatT

1 THTE exp)exp(

Energy

Amplitude equations

f

f

f

abij

ijbaabij

a

aaaatT

2 0exp)exp( HTHT abij

abij

Amplitude equations

fab

• Nomenclature• Coupled-clusters in singles and doubles (CCSD)• with triples corrections CCSD(T);…with triples corrections CCSD(T);

Dean & Hjorth-Jensen, PRC69, 054320 (2004); Kowalski et al., PRL 92, 132501 (2004); Wloch et al., PRL94, 212501 (2005) Gour et al., PRC (2006); Hagen et al, PLB (2006); PRC 2007a, 2007b

Am

The many-body wave function in cluster amplitudes

AA

kk

AT TTe1

)(,)(

kji

abcijk

abcijk

ji

abij

abij

i

ai

ai tTtTtT 321 ,,

cbabaa

a,b,…

i,j,…

22422

theoryeapproximat ,;ryexact theo ,

A

A

CCSDTTT

NmNm

3353

321

224221

3

2

uouoA

uouoA

nnnnCCSDTTTTTm

nnnnCCSDTTTm

Relationship between shell model and CC amplitudes

2

11

1 TTB

TB

3

2122

121 TTB

CCSD

422

311233

11161 TTTTB CR-CCSD(T)

41

212

221344 24

121

21 TTTTTTTB

“Disconnected quadruples”

“Connected quadruples”

These equations are sort of messy in their full blown formulation

CCSD T1, T2 amplitudesCCSD T1, T2 amplitudes

Non-linearCoupledCoupledAlgebraicTensor-tensor multiplies

Solution of Coupled-Cluster equation

System of non-linear coupled algebraic equations solve by iteration Basic Numerical Operation:

n=number of neutrons and protons

N=number of basis states

oldold

new

klabtijcdtcdklVijabt

),(),(),(),(

Solution tensor memory (N-n)**2*n**2

Interaction tensor memory

Nndc

nlkoldold j

,1,,1,

),(),(),(

Interaction tensor memory N**4

Operations count scaling

• Many terms like this• Cast into a matrix-matrix

multiply algorithm O(n**2*N**4) O(n**4*N**4) with 3-body O(n**3*N**5) at CCSDT

multiply algorithm• Parallel Issue: block sizes of V and t• Petascale target problem

100 N 1000( )

• n=100; N=1000

Verification and Validation (V&V)

Doing the problem right. – VerifyDoing the right problem. – Validate

Ab initio in medium mass nuclei4HeHe

16OHagen, Dean, Hjorth-Jensen,Papenbrock, Schwenk, PRC76, 044305 (2007)

Error estimate: << 1% < 1% 1%

p ( )

40CaCa

1063 many-body basis states

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

0 Solution at CCSD and CCSD(T) levelsSolution at CCSD and CCSD(T) levels involve roughly 67 more diagrams…..

E 28 24 M V -1

100

|

2-body only

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

10-2

10-1

E /

EC

CSD

| 0-body 3NF

1-body 3NF

estimated triples corrections

10-3

10

| ΔE

/

2-body 3NFestimated triples corrections

Challenge: do we really need the full(1) (2) (3) (4) (5)10

-4residual 3NF

Challenge: do we really need the full 3-body force, or just its density dependent terms? Hagen, Papenbrock, Dean, Schwenk, Nogga, Wloch, Piecuch

PRC76, 034302 (2007)

Utilize the nuclear total spin symmetry to push further

Implemented a CCSD J-coupled (Hagen) code for heavier nuclei:• Scaling at CCSD goes from O(no

2nu4) to O(no

4/3nu8/3)

• Can do up to 14 complete major shells on a single node• Can do up to 14 complete major shells on a single node. • CCSDT Gold standard for these heavier nuclei (developing)• Enables specified calculations for heavy nuclei• The large model spaces mean that we can approach BARE interactions!The large model spaces mean that we can approach BARE interactions!

• Must start with a spherical HF basis

• Is it technically feasible to go further? (YES)• Does size extensivity work in the nuclear case? (YES)• Opens some interesting doors for future research….

Ca isotopes from bare chiral NN potential at NCa isotopes from bare chiral NN potential at N33LO (no 3LO (no 3--body yet)body yet)

1081 Mb b i !!1081 Mb basis states!!

3Chiral NN potential at N3LO underbinds by ~1MeV/nucleon. (Size extensivity at its best.)

NucleusNucleus E / A [MeV]E / A [MeV]44HeHe 1.08 1.08 (0.73(0.73FYFY))1616OO 1.251.2540404040CaCa 0.840.844848CaCa 1.271.274848NiNi 1.211.21

Ca isotopes from bare chiral NN potential at NCa isotopes from bare chiral NN potential at N33LO (no 3LO (no 3--body yet)body yet)

1081 Mb b i !!1081 Mb basis states!!

Chiral NN potential at N3LO underbinds by ~1MeV/nucleon. (Size extensivity at its best.)

Including triples corrections

CCSD(T) h t 4H b b t 0 2 K VCCSD(T) over shoots 4He by about 0.2 KeVCCSD(T) close to the F-Y result for 4He (E/A = 0.7 MeV)

Triples corrected results

E ti t b t 0 3 d 0 4 M V/A ill b i i fEstimate: between 0.3 and 0.4 MeV/A will be missing from binding energies across medium mass nuclei

There is some room for the 3-body force (perturbative?) 40Ca may be slightly over bound with just 2-body interaction

Oxygen chain results: Is 28O bound??

28O b d i h• 28O bound with respectto 4n by about 1 MeV

• Results not fully converged(N=13)(N=13)

• Experiment would be difficult!

• Systematic deviation fromSystematic deviation fromdata (but fairly constant)

PRELIMINARY

Nuclear ‘scale separation’in weakly bound nuclei

TRIUMF/GSI (2006)T1/2 ≈ 8.6 ms

8He

ANL (2004) ANL/GANIL (2007)

d f 11Li / 7 10 8… and mass of 11Li: m/m=7·10-8

nuclear radius (fm)

gyCoupling of nuclear structure and reaction theory

(microscopic treatment of open channels)

Open QSOpen QS

Ene

rg

Important interdisciplinaryaspects…(see recent ECT*workshop on subject)

Correlation

p j )

Correlationdominated n ~ n

Sn=0

S

Closed QS

Sn

Closed QSClosed QSNeutron number

Introduction of Continuum basis states (Gamow, Berggren)Continuum shell models

(many including: Michel, Rotureau, Volya, Ploszajczak, Liotta, Nazarewicz,…)

Progress: ab initio weakly bound and unbound nuclei

CCSD He Chain ResultsCCSD He Chain Resultsp

N3LO Vlowk (=1.9 fm-1)

n

6He gs spin (smaller space)6He gs spin (smaller space)Naïve filling =1.4CCSD =0.6CCSD(T) =0.6CCSDT 1 2 3 =0 2

Challenge: include 3-body forceHagen, Dean, Hjorth-Jensen, Papenbrock, Phys. Lett. B 656, 169 (2007)

CCSDT-1,2,3 =0.2CCSDT =0.04

Strategy:Strategy:Resources:Resources:Opportunity:Opportunity:

Nuclear Coupled Cluster Theory Perspectives

Strategy: Develop and deploy all

necessary elements to calculate nuclear masses, excitation energies and

Strategy: Develop and deploy all

necessary elements to calculate nuclear masses, excitation energies and

Resources: CC theory implementations

described in this talk: CCSD, CCSD(T), CCSDT-1, V3-CCSD(T) Gamow-based

Resources: CC theory implementations

described in this talk: CCSD, CCSD(T), CCSDT-1, V3-CCSD(T) Gamow-based

Opportunity: Capitalize on CC

developments to realize an ab initio foundation for nuclear structure (including

Opportunity: Capitalize on CC

developments to realize an ab initio foundation for nuclear structure (including excitation energies, and

transition properties Develop ‘shell-model’ like

effective interactions from CC theory

excitation energies, and transition properties

Develop ‘shell-model’ like effective interactions from CC theory

V3 CCSD(T), Gamow based CCSD(T), CCSDT, (some under development in J-scheme)

Excited state calculations

V3 CCSD(T), Gamow based CCSD(T), CCSDT, (some under development in J-scheme)

Excited state calculations

nuclear structure (including the shell model) and reactions with calculations reaching into heavy nuclei

Develop the CC technology

nuclear structure (including the shell model) and reactions with calculations reaching into heavy nuclei

Develop the CC technology y Leverage ultrascale

computing and nuclear theory expertise to solve a wide range interesting

y Leverage ultrascale

computing and nuclear theory expertise to solve a wide range interesting

Effective (and now BARE) interaction expertise

Software development at scale

Effective (and now BARE) interaction expertise

Software development at scale

p gyto include powerful tools for investigating the relationship between ab inito approaches and DFT

p gyto include powerful tools for investigating the relationship between ab inito approaches and DFT

problems within the CC framework

Develop ‘one-off’ problems for nurturing of post-docs and students (NRC grant)

problems within the CC framework

Develop ‘one-off’ problems for nurturing of post-docs and students (NRC grant)

Dynamic and promising interface of nuclear theory and computational science

Broad collaboration base

Dynamic and promising interface of nuclear theory and computational science

Broad collaboration base

Enable future experimental directions through ab initio predictions of nuclear properties in uncharted regions

Enable future experimental directions through ab initio predictions of nuclear properties in uncharted regions and students (NRC grant)and students (NRC grant)

Longstanding partnerships with DOE (NP and ASCR), Oslo/CMA among others

Longstanding partnerships with DOE (NP and ASCR), Oslo/CMA among others

regionsregions

Outcome: A cross-cutting theoryfor understanding and building nuclei from the ground up