The UNEDF Project

facilities and methods

18 Nuclear Physics News, Vol. 21, No. 2, 2011

The UNEDF Project

Overview A new era has dawned for nuclear

structure and reaction theory.Renewed interest in the physics ofnuclei is fueled by experiments at rareisotope beam facilities, which explorenew regions of exotic nuclei; by astro-physical observations and simula-tions, which require controlled extra-polations of the nuclear equation ofstate; by the use of nuclei in experi-ments on fundamental symmetries,which rely on robust nuclear structureinformation; and by nuclear energyand security needs, which include pre-dictions of reaction cross-sections andfission fragment properties. TheUNEDF collaboration of nuclear the-orists, applied mathematicians, and

computer scientists is addressing thisbroad spectrum of physics through acomprehensive study of all nuclei,based on the most accurate knowl-edge of the strong nuclear interaction,the most reliable theoreticalapproaches, the most advanced algo-rithms, and extensive computationalresources.

UNEDF stands for “UniversalNuclear Energy Density Functional.”The mission of this five-yearSciDAC (“Scientific Discovery thro-ugh Advanced Computing”) project isthree-fold [1]:

• First, to find an optimal energydensity functional (EDF) using allour knowledge of the nucleonic

Hamiltonian and basic nuclearproperties.

• Second, to apply the EDF theoryand its extensions to validate thefunctional using all the availablerelevant nuclear structure andreaction data.

• Third, to apply the validated the-ory to properties of interest thatcannot be measured, in particularthe properties needed for reactiontheory.

The long-term vision is to arrive at acomprehensive and quantitativedescription of nuclei and their reac-tions. UNEDF is striving to replacephenomenological models with awell-founded microscopic theory thatdelivers maximum predictive powerwith quantified uncertainties.

By nuclear theory standards,UNEDF is large: it involves over 50researchers from 9 universities and 7national laboratories. Annually, itprovides training to about 30 youngresearchers (postdocs and students).UNEDF is like a large experimentalcollaboration with multiple facets toits research goals.

The SciDAC model (http://www.scidac.gov) has led UNEDF membersto form partnerships in which appliedmathematicians and computer scien-tists work collaboratively with physi-cists to develop the required advancedalgorithms and tools. At the same time,the project has catalyzed new cross-cutting physics collaborations, whichinclude nuclear theorists around theworld as international partners. Thescope of the UNEDF project is large.Here we present brief snapshots thathighlight the collaborative effort,the progress made, and the future

Figure 1. Table of the nuclides and the scope of theoretical methods todescribe it.

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Vol. 21, No. 2, 2011, Nuclear Physics News 19

prospects; more details and referencescan be found at the unedf.org website.

Forging New Connections The playing field for UNEDF is

the nuclear landscape of Figure 1. Thesizable area marked “terra incognita,”which is populated by unstable iso-topes with large neutron fractions, isof particular interest. Above the tableof nuclides are shown three broadclasses of theoretical methods, whichare also used in atomic and condensedmatter physics. The bulk of thenuclides are covered by Density Fun-ctional Theory (DFT), which provides

the theoretical underpinning and com-putational framework for building anuclear EDF. By enhancing andexploiting the overlaps with ab initioand configuration interaction (CI)approaches, we are constructing andvalidating a universal EDF informedby microscopic interactions as well asexperimental data.

The UNEDF components andpaths between them are shown in theUNEDF strategy diagram in Figure 2.New interconnections and collabora-tive efforts within and across theboundaries have become indispensi-ble. By necessity the developmentshave largely occurred in parallel; in

subsequent sections we give examplesof the interplay as UNEDF hasworked to fulfill its mission.

Enhancing ab initio Structure Calculations

Direct connections of the EDF tomicroscopic nuclear interactionsrequire improving the scope and accu-racy of ab initio methods. Majoradvances in ab initio nuclear structurecalculations under UNEDF havefocused on improving the effectiveuse of high performance computing—especially the efficient scaling to verylarge numbers of parallel processors.

A prime example of UNEDF syn-ergy has been the development of theAsynchronous Dynamic Load Bal-ancing (ADLB) software library byusing Green’s Function Monte Carlo(GFMC) calculations as a test bed.Load balancing means ensuring thatevery parallel processor is used effec-tively; it is essential to optimal scal-ing. The ADLB library has enabledGFMC to run efficiently on over100,000 core processors [2].

A major test case is the 12C energyand proton density (Figure 3), whichis also a proof of principle whether amicroscopic nuclear Hamiltonian cal-ibrated to few-body nuclei can suc-cessfully predict the energy and

Figure 3. Carbon-12 density calculatedwith GFMC using the ADLB library [2].

Figure 2. UNEDF project scope and various active interconnections.

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density distribution of larger nuclei.Unlike quantum chemistry, where theCoulomb interaction is firmly estab-lished, the input nuclear Hamiltonianmust be tested and improved asneeded even as the nuclear DFT itselfis developed. The GFMC/ADLBcombination predicts the 12C experi-mental energy and radius to within1%, which is a compelling validation[2]. The next targets are excited statesin 12C, including the famous Hoylestate.

Configuration interaction ab ini-tio methods use basis expansions andrequire the (partial) diagonalization ofextremely large matrices. These areknown by various names according todetails of the implementation, includ-ing No-Core Shell Model (NCSM)and No-Core Full Configuration(NCFC) methods (see accompanyingarticle by Barrett, Navratil, and Vary).Improvements by applied mathemati-cians working with nuclear theoristson eigensolvers, data structures, andnew combinatorial algorithms haveresulted in substantial performancegains for CI codes such as MFDn,allowing efficient scaling to largenumbers of processors [3].

An example of what this scalingenables is the recent prediction of the 14Fmass and excitation spectrum in advanceof the experimental measurements,which meant solving a Hamiltonian

matrix of dimension 2 billion using30,000 cores [4] (Figure 4). The pre-dictions and measurement agreewithin the combined experimentaland theoretical uncertainties (notshown). Another recent applicationinvolving many UNEDF members isan explanation of the anomalouslylong lifetime of 14C by identifying thecritical role of the three-nucleon(NNN) force in its beta decay [5]. Wecan look forward to many such con-frontations of theory and experimen-tal data from rare isotope facilities.

CI codes used within the UNEDFproject apply both fully microscopicand phenomenological shell modelapproaches, and use different algo-rithms. But common computationalissues are leading UNEDF scientists todevelop the Leadership Class Configu-ration Interaction (LCCI) frameworkto unify, preserve, and disseminatevaluable CI codes and resources. Thiswill provide a user-friendly environ-ment for researchers to download andrun state-of-the-art CI codes neededfor theoretical predictions as well asexperimental analysis.

To make ab initio methods moreconvergent, it can be advantageousto “soften” the initial Hamiltonian bydecoupling low-momentum andhigh-momentum parts. Renormaliza-tion group (RG) methods achievethis decoupling by evolving interac-tions in small steps. The resultingtwo-nucleon (NN) potentials are veryconvergent but the initial NNN inter-action is also changed. Until recentlya consistent NNN evolution had notbeen achieved. But a UNEDF collab-oration successfully combined Simi-larity RG (SRG) evolution methodswith NCSM technology. Figure 5illustrates the improved convergencewith the evolved three-body force[7]. These SRG interactions are now

being applied to larger nuclei and inreaction calculations.

The interactions used in the 14Ccalculations and the SRG evolutionsare systematically constructed (out-side of UNEDF) using chiral effectivefield theory (EFT). UNEDF membershave demonstrated that the coupled-cluster (CC) ab initio method can beused to accurately calculate closed-shell medium-mass nuclei such as40,48C with chiral EFT two-body inter-actions (or RG-softened versions) aswell as proton halo nuclei like 19F [8].The CC formalism has been extended toinclude NNN forces and their inclusionin calculations of the heavier nuclei willbreak new barriers. A recent develop-ment is the first in-medium SRG diago-nalization of closed-shell nuclei such as40C with accuracy comparable to CCresults [9].

Microscopic Inputs to a Nuclear EDF

As implied by the diagram inFigure 2, multiple inputs and con-straints are being used to build a uni-versal nuclear EDF. The foundation is

Figure 5. Improved convergence inhelium-4 using Similarity Renorm-alization Group (SRG) interactions,including an evolved NNN force [7].

Figure 4. Theoretical (“ab-initio”)spectrum from NCFC compared toexperimental results for 14F [6].

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provided by EDF’s of the Skyrmetype, whose phenomenological suc-cesses have been extensively bench-marked as part of the UNEDF project.Ab initio approaches can provide“control-data” to constrain more gen-eral functionals as well as to directlymotivate novel density and momen-tum dependences.

Ab initio calculations can be usedto test candidate EDF’s even for sys-tems not available in the laboratory.Current functionals are least con-strained in their isovector depen-dence, so a system of neutron dropswould be ideal, but the neutronswould not be self-bound. However,DFT says that the functional shouldwork with a theoretical externalpotential, which can be used to “trap”neutrons and adjust their density pro-files. UNEDF calculations applyingGFMC (up to 16 neutrons) and Auxil-iary Field Diffusion Monte Carlo(AFDMC, up to 54 neutrons) methodsto generate such control-data arebeing used to probe deficiencies ofconventional EDF parameterizationsand to improve them. In Figure 6 sucha comparison for radii is shown forthe original SLy4 functional and an

adjusted version [10]. The neutrondrop results are validated by alsocalculating with NCFC and with otherHamiltonians.

A parallel effort uses trapped neu-tron systems as well, but tests ab initioDFT methods against full calculationsbased on the same Hamiltonian. Ahybrid approach uses chiral EFT for thelong-distance (pion range) NN andNNN interactions that can be translated

into novel density dependences througha revitalized incarnation of the densitymatrix expansion. These new function-als are being optimized and testedusing the extensively developed EDFinfrastructure [11].

Enhancing and Extending DFT Infrastructure

Much effort in the UNEDF projecthas been devoted to developing andimproving the algorithmic and com-putational infrastructure needed tooptimize candidate EDF’s and toapply symmetry restoration requiredfor accurate calculations of self-bound nuclei. Performance optimiza-tion of DFT-solver codes has enabledlarge-scale mass table calculations on9,060 processors for 840,000 configu-rations in 9,000 nuclei in a 12-hourrun. The MADNESS (MultiresolutionADaptive NumErical Scientific Simu-lation) framework is an example ofapplying state-of-the-art appliedmathematics technology to create aDFT solver with an adaptive pseudo-spectral method [12].

Figure 7. Greatly improved converg-ence is seen with the novel optimi-zationalgorithm POUNDerS [13].

Figure 6. GFMC calculations of the radii of N neutrons in harmonic oscillator(HO) and Woods-Saxon (WS) traps, which are used to adjust the SLy4 Skyrmefunctional [10].

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Another example is the optimiza-tion algorithm POUNDerS, developedby a UNEDF team of applied mathe-maticians, which yields dramaticcomputational savings over alterna-tive optimization methods, as seen inFigure 7. Using the UNEDF Experi-mental Database and the optimizedsolver HFBTHO, the Skyrme SLy4functional was re-optimized using thederivative-free POUNDerS algorithm.The resulting parameterizationUNEDF0 sets a solid baseline ofnuclear ground-state properties tocompare with future functionals [13].New hybrid functionals with micro-scopic input from chiral EFT [11] arebeing optimized using this approach.

A New Era for Reaction Theory One of the principal aims of the

UNEDF project is to calculate reliablereaction cross-sections for astrophys-ics, nuclear energy, and national secu-rity, for which extensions of standardphenomenology is insufficient. Theinterplay of structure and reactions isessential for a successful descriptionof exotic nuclei as well. Such inter-play is characteristic of the ab initiono-core shell model/resonating-group

method (NCSM/RGM), which treatsbound and scattering states within aunified framework using fundamen-tal interactions between all nucleons.A quantitative proof-of-principle cal-culation of this approach is shown inFigure 8 [14]. A wide range of appli-cations is now possible including3H(d,n)4He fusion, the 7Be(p,γ)8Breaction important for solar neutrinophysics, and many more to come.

Neutron reactions on heaviernuclei are being modeled using DFTresults to predict not just bound states,but also scattering states for nucleons.Microscopic calculations of reactioncross-sections for nucleon-nucleusscattering have been performed bycoupling the elastic channel to all par-ticle-hole excitations in the target,and also with one-nucleon pickupchannels. Target excitations were des-cribed in a random-phase (QRPA)framework using a Skyrme functionaland the resulting transition potentialswere used in large coupled-channelcalculations. As illustrated in Figure 9,the calculated reaction cross-sectionsagree very well with experimentaldata, and also with predictions of glo-bal optical potentials where data arenot available. For the first time, theobserved absorption in the reactioncross-section can be accounted for byexplicit channel coupling [15].

Another important capability forreactions is the calculation of leveldensities. A new proton-neutron algo-rithm for the parallel JMoments codewas recently designed and imple-mented, which improved its scalabil-ity to tens of thousands of cores, andincreased its overall performance by afactor of more than 10,000. Thisdevelopment opens the door to calcu-lating accurate nuclear level densitiesand reaction rates for a large class ofnuclei [16].

Cold Atoms as a Testing Ground UNEDF theorists have made

important contributions to the studyof strongly coupled superfluid sys-tems such as ultracold Fermi atoms,which show many similarities to thecold nuclear matter found in the crustof neutron stars. Cold atoms makeexcellent laboratories for testing andimproving the computational methodsto be used for nuclei; indeed, GFMCand AFDMC calculations fromUNEDF scientists have set the stan-dard for numerical results of the uni-tary gas at zero temperature.

Cold atom systems allow predictionsof superfluid DFT that are testableagainst experiment. An example isshown in Figure 10, where a nuclearDFT code adapted to the antisymmetricsuperfluid local density approximation(ASLDA) [17] is applied to stronglyinteracting spin-imbalanced atomicgases in extremely elongated traps. Fam-ilies of Larkin-Ovchinnikov (LO) stateswith prominent transversal oscillationsof the pairing potential are predicted, asindicated by the radial alignment of

Figure 9. Calculated reaction cross-sections for protons on 90Zr as afunction of incident energy comparedto data. The solid line is the fullcalculation while the dashed curve isfrom the global optical potential [15].

Figure 8. Calculated [14] NCSM/RGMcross-section for 17 MeV neutrons on4He compared to experiment.

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nodes, coexisting with a superfluid statehaving a smooth pairing potential [18].

A recent major UNEDF achieve-ment is the full implementation of thetime-dependent superfluid local den-sity approximation (TDSLDA) on a3D spatial lattice [16]. Unlike manypast approaches, matrix operations arenot needed and the size of the basisset it can handle is 2–3 orders of mag-nitude larger than previous methods,with an implementation that can use97% of the Jaguar supercomputer atOak Ridge National Lab. These codesare being used to simulate the unitarygas (e.g., vortex formation) [16] and aheavy nucleus under the action of var-ious external fields. While stillexploratory, these first-time simula-tions of this kind for fermion super-fluids serve as proof of principle foran eventual treatment of fission.

Quality Control in UNEDF Integral to the UNEDF project is

the verification of methods and codes,the estimation of uncertainties, and

assessment. Methods to verify andvalidate include the cross-checking ofdifferent theoretical methods andcodes (e.g., GFMC vs. AFDMC vs.NCFC), the use of multiple DFT solv-ers with benchmarking, and the con-frontation of ab initio functionals withab initio structure using the sameHamiltonian. Uncertainty quantifica-tion follows using tools for correla-tion analysis to estimate errors andsignificance. A new way to estimatetheory error bars is to use multipleRG-evolved Hamiltonians and exam-ine the cutoff dependence of calcu-lated observables.

The UNEDF assessment compo-nent has required the developmentand application of statistical tools.Particularly important for potentialexperiments is the analysis of exper-imental data significance. For exam-ple, the sensitivity of two optimizedfunctionals to particular data isshown in Figure 11 [13]. Statisticaltools are used to deliver uncertaintyquantification and error analysis fortheoretical studies as well as for theassessment of new experimentaldata. Such technologies are virtu-ally unknown in the low-energynuclear theory community at present,but are essential as new theories andcomputational tools are applied toentirely new nuclear systems and toconditions that are not accessibleto experiment.

Outlook These brief highlights represent

only part of the development withinthe UNEDF project and new resultsare appearing steadily. Please visitunedf.org to find further summaries,updates, and references. WhileUNEDF is in the final year of its 5-year term, its impact will be ongo-ing. Indeed, UNEDF has created

infrastructure and interconnectionsthat are only just beginning to befully exploited. An important spin-off for the future has been the train-ing of young scientists in the newdevelopments of low-energy nucleartheory.

The worldwide impact of theUNEDF collaboration, which isunique in the field of low-energynuclear physics in its scope, size, andstructure, was evident at the 2010INPC meeting in Vancouver, whereUNEDF highlights were quoted inseveral plenary talks. While based andfunded in the United States, UNEDFcollaborates closely with foreignefforts and individual scientists shar-ing similar fundamental science goals.Such collaborations include joint soft-ware developments and benchmark-ing, and representatives of internat-ional collaborating projects attend theUNEDF annual meetings. TheseUNEDF activities have had positiveleveraging effects on our foreign part-ners, including the joint UNEDF-JUSTIPEN and UNEDF-FIDIPROefforts and a new initiative involvinglow-energy nuclear theory in France,FUSTIPEN.

Figure 11. The sensitivity of candidateEDF fits to particular data can beassessed. See Ref. [13] for details.

Figure 10. Pairing potential ofpolarized Fermi gas in an extremelyelongated trap with different aspectratios [18] calculated using nuclearcodes developed for fission.

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High performance computing pro-vides answers to questions that neitherexperiment nor analytic theory canaddress; hence, it becomes a third legsupporting the field of nuclear phys-ics. A series of meetings on extremescale computing in 2009 identifiedseveral major research thrusts thathave a strong foundation in UNEDFachievements [20]. These includepathways to a quantitative micro-scopic description of fission, comput-ing properties of nuclei that determinethe r-process nucleosynthesis path instars, computing properties of nucleiused in double-beta decay experimentsand neutrino-nucleus cross-sectionsfor modeling supernova explosions,and computing the triple-alpha processthat produces 12C, the nucleus at thecore of organic chemistry and thus lifeforms. The future prospects are bright!

Acknowledgments This work was supported by the

UNEDF SciDAC Collaboration underDOE Grant DE-FC02-07ER41457.

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20. www.er.doe.gov/ascr/Misc/Grand-Challenges.html

RICHARD FURNSTAHL

(FOR THE UNEDF COUNCIL)Department of Physics,

Ohio State University

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The UNEDF Project

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