Hartree-Fock Many-Body Perturbation Theory for Nuclear Ground-States

0.8 MB·

by  Alexander Tichai

Hartree-FockMany-BodyPerturbationTheoryforNuclearGround-States
Alexander Tichai,1,∗ Joachim Langhammer,1 Sven Binder,2,3 and Robert Roth1,†
1Institut für Kernphysik, Technische Universität Darmstadt, 64289 Darmstadt, Germany
2DepartmentofPhysicsandAstronomy, UniversityofTennessee, Knoxville, TN37996, USA
3Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA
Weinvestigatetheorder-by-orderconvergencebehaviorofmany-bodyperturbationtheory(MBPT)asasim-
pleandefficienttooltoapproximatetheground-stateenergyofclosed-shellnuclei.Toaddresstheconvergence
propertiesdirectly,weexploreperturbativecorrectionsupto30thorderandhighlighttheroleofthepartitioning
forconvergence.TheuseofasimpleHartree-Focksolutiontoconstructtheunperturbedbasisleadstoaconver-
gentMBPTseriesforsoftinteractions,incontrastto,e.g.,aharmonicoscillatorbasis. Forlargermodelspaces
andheaviernuclei,whereadirecthigh-orderMBPTcalculationinnotfeasible,weperformthird-ordercalcula-
tionandcomparetoadvancedabinitiocoupled-clustercalculationsforthesameinteractionsandmodelspaces.
Wedemonstratethatthird-orderMBPTprovidesground-stateenergiesfornucleiupintotinisotopicchainthat
6 areinexcellentagreementwiththebestavailablecoupled-clusterresultsatafractionofthecomputationalcost.
1
0
2 Introduction. The solution of the Schrödinger equation ground-state energies of selected closed-shell medium-mass
for atomicnuclei using realisticnuclear interactions isat the andheavynucleiatthird-orderMBPT,andcomparetorecent
n
heart of ab initio nuclear structure theory. In practice this coupledcluster(CC)calculations[6].
a
J problemissolvedbyconstructingapproximatemethodsfora The Nuclear Hamiltonian. For all following investiga-
4 truncated,i.e.,finite-dimensionalHilbertspace. However,for tions we start from the chiral nucleon-nucleon (NN) interac-
1 thecalculationofground-stateenergiesofheavynucleisignif- tion at next-to-next-to-next-to leading order (N3LO) by En-
icantalgorithmicandcomputationaleffortsareneeded. There temandMachleidt[25]combinedwiththethree-nucleon(3N)

] exist a plethora of different ab initio methods, e.g., coupled interaction at next-to-next-to leading order N2LO in its local
h
cluster(CC)theory[1–6],in-mediumsimilarityrenormaliza- form [26] with three-body cutoff Λ = 400MeV/c. Addi-
t 3N
- tiongroup(IM-SRG) [7–11],orself-consistentGreen’sfunc- tionally, we use the similarity renormalization group (SRG)
l
c tionmethods[12–14]. However,itisdesirabletohaveanal- tosoftentheHamiltonianthroughacontinuousunitarytrans-
u
ternative, light-weight framework available. A conceptually formation controlled by a flow parameter [27–31]. In prin-
n
simple method to solve for the eigenenergies of a physical ciplethistransformationinducesbeyond-3Noperatorsupthe
[
system is many-body perturbation theory (MBPT) [15–17]. massnumberoftheconsiderednucleus, which, however, we

1 A perturbative treatment is the standard approach for many have to neglect. To avoid the complication of dealing with
v problemsfromdifferentfieldsoftheoreticalphysics. Thead- explicit 3N interactions, we make use of the normal-ordered
3
vantage of MBPT compared to other ab initio approaches is two-body approximation (NO2B) of the 3N interaction that
0
its simplicity, which also allows for straightforward gener- has been found to be very accurate for medium-mass nuclei,
7
3 alizations to excited states and open-shell nuclei [18] with- see Refs. [32, 33]. For the matrix-element preparation we
0 out the need of sophisticated equation-of-motion techniques. adopt the procedure we introduced in Ref. [6], i.e., in par-
. The reason why MBPT usually is not considered as ab ini- ticular,weusethelargeSRGmodel-spaceandexploittheit-
1
0 tio technique are convergence issues of the underlying per- erative scheme where necessary. Thus, the matrix elements
6 turbation series. Several studies of high-order MBPT based andthetreatmentofthechiralNN+3Ninteractionareidenti-
1 on Slater determinants constructed from harmonic oscillator caltoRef.[6]andwecancompareddirectlytotheCCresults
: (HO) single-particle states (HO-MBPT) have shown that the presentedthere.
v
i perturbationseriesisdivergentinalmosteverycase[18,19]. Many-Body Perturbation Theory. The essence of
X In such cases one heavily relies on the use of resummation Rayleigh-Schrödinger perturbation theory is the definition
r techniques, e.g., Padé approximants, that enable a robust ex- of an additive splitting, referred to as partitioning, of a
a
tractionofobservablesalthoughtheperturbativeexpansiondi- given Hamiltonian H into an unperturbed part H and a
0
verges[19–21]. perturbationW. Introducinganauxiliaryparameterλyieldsa
InthisLetter, weformulateMBPTbasedonHartree-Fock one-parameterfamilyofoperators,
(HF)single-particlestates(HF-MBPT),and,forthefirsttime,
investigate the convergence behavior of the perturbation se- H = H +λW, (1)
λ 0
ries up to 30th order. We compare the ground-state energies
of 4He and 16O to results from exact diagonalizations in the wheretheperturbationisdefinedbyW = H−H . Asansatz
0
configurationinteraction(CI)approachusingthesamemodel for the solution of the eigenvalue problem of H we take a
space [22–24]. Based on the rapidly converging perturba- powerseriesexpansionintermsofanauxiliaryparameterλ,
tionseriesresultingfromtheuseofHFbasisstates,westudy wheretheexpansioncoefficientsaregivenbytheenergycor-
rectionsandstatecorrections,respectively. Wechoose H to
0
betheHFHamiltonianarisingfromaninitialNN+3Ninterac-
∗[email protected] tion. WehaveshowninRefs.[18,19]thathigh-orderMBPT
†[email protected] corrections are accessible by means of a recursive scheme,
2
allowing for detailed investigations of the convergence char- state with singly-excited determinants [17] and by orthogo-
acteristics of the perturbation series. In general we cannot nality the zero-body part is only present in the expectation
expect that a perturbation series is convergent [34–36], but valueoftheperturbation.Inprinciple,thederivationofenergy
onecanexploitresummation-theorytechniquestoextractin- corrections beyond third order is straight forward. However,
formation on the observables of interest. There are different consideringadiagrammaticapproachintermsofHugenholtz
schemesandtransformationsthatcanbeusedtoextract,e.g., diagrams,thenumberofcontributingdiagramsatagivenper-
the ground-state energy froma divergent expansion [37–39]. turbation order p increases rapidly [41] such that it becomes
Padé approximants have proven to be particularly useful in challengingtogobeyondthird-orderinpractice.Additionally,
thetreatmentofhigh-orderHO-MBPT[18,19].Additionally, terms from higher-order corrections involve expressions that
theyarewell-knowntomathematiciansespeciallyinthefield are notoriously hard to compute, because their effective im-
of convergence acceleration [21, 36, 37]. However, the cal- plementation,e.g.,bymeansofBLAS-enabledmatrixopera-
culation of energy corrections up to sufficiently high orders tions,isnotobvious. Thecomputationalpowerneededtoper-
is only feasible for light nuclei due to increasing computa- formthird-orderMBPTcalculationsuptotheheavy-massre-
tional requirements. When proceeding to the medium-mass gioncaninprinciplebeprovidedbyasinglecomputingnode
regiononemustchooseadifferentstrategy. Dependingonthe within1−3%ofthecomputingtimeneededforstate-of-the-
rateofconvergence,onemightexpectlow-orderpartialsums artCCcalculations.
of the perturbation series to be a reasonable approximation Convergence Characteristics of Hartree-Fock Many-Body
to the exact ground-state energy. Having only low-order in- Perturbation Theory. We start with comparing perturbation
formation available, resummation methods are less effective, series from HO- and HF-MBPT, and focus on their conver-
because one can only construct a small number of approxi- gence characteristics and sensitivity to the SRG flow param-
mantsthatyieldvalidapproximationsonlyifthetransformed eter. In Fig. 1 we present a direct comparison of the order-
sequence converges sufficiently fast [18]. However, one al- by-order behavior for the two partitionings up to 30th order
ternativeistoexploitthefreedominthepartitioning,i.e.,the for 16O. For these high-order calculations we use an N -
max
choice of the unperturbed basis, to improve the convergence truncationofthemany-bodymodelspacesimilartheno-core
oftheperturbationseries. Forlow-ordercalculationsthecor- shell model (NCSM) [23]. The left column of Fig. 1 shows
rectionscanbeexpressedintermsoftheparticle-holeformal- thehigh-orderpartialsumsandtherightcolumntheindivid-
ism. NotethattheHartree-Fockenergycorrespondstothethe ual energy corrections for each order. Panel (a) shows the
first-orderpartialsum, partialsumsfromHO-MBPTforasequenceofmodelspaces
with fixed SRG flow parameter α = 0.08fm4. The partial
E = E(0)+E(1) . (2)
HF sumsaredivergentforeverymodelspace. Thedivergenceis
Therefore, the first contribution to the correlation energy ap- also apparent from panel (c) which reveals the exponentially
pearsinsecond-orderHF-MBPT.Thesecond-andthird-order increasingenergycorrections.Incontrast,panel(b)showsthe
partialsumfortheground-stateenergywithrespecttoHFba- partialsumsarisingfromHF-MBPTthatareconvergentforall
sisforatwo-bodyoperatoraregivenby[40] modelspaces. Furthermore, theconvergedvaluesagreewith
directCIresults. Asseeninpanel(d),theenergycorrections
1(cid:88)<(cid:15)F (cid:88)>(cid:15)F (cid:104)ab|W|ij(cid:105)(cid:104)ij|W|ab(cid:105) areexponentiallysuppressedforhigherorders,givingriseto
E(2) =
4 ((cid:15) +(cid:15) −(cid:15) −(cid:15) ) arobustconvergence.
ab ij a b i j
In Fig. 2 we show the high-order partial sums and energy
1 (cid:88)<(cid:15)F (cid:88)>(cid:15)F (cid:104)ab|W|ij(cid:105)(cid:104)ij|W|cd(cid:105)(cid:104)cd|W|ab(cid:105) corrections in HF-MBPT for different SRG flow parameters.
E(3) =
8 ((cid:15) +(cid:15) −(cid:15) −(cid:15) )((cid:15) +(cid:15) −(cid:15) −(cid:15) ) Panels(a),(b)and(c)showtheconvergentperturbationseries
abcd ij a b c d a b i j for 4He,16O and 24O, respectively. The calculations are per-
1(cid:88)<(cid:15)F (cid:88)>(cid:15)F (cid:104)ab|W|ij(cid:105)(cid:104)ij|W|kl(cid:105)(cid:104)kl|W|ab(cid:105) formedforfixedN = 6for4He,16OandN = 4for24O
+ max max
8 ((cid:15) +(cid:15) −(cid:15) −(cid:15) )((cid:15) +(cid:15) −(cid:15) −(cid:15)) and the flow-parameter dependence of the absolute energies
a b i j a b k l
ab ijkl results from the varying degree of convergence with respect
(cid:88)<(cid:15)F (cid:88)>(cid:15)F (cid:104)ab|W|ij(cid:105)(cid:104)cj|W|kb(cid:105)(cid:104)ik|W|ac(cid:105) tothemany-bodymodelspace.
+ . (3)
((cid:15) +(cid:15) −(cid:15) −(cid:15) )((cid:15) +(cid:15) −(cid:15) −(cid:15) ) A more interesting flow-parameter dependence can be ob-
a b i j a c i k
abc ijk
servedfortheindividualenergycorrectionsinpanels(d),(e),
In the third-order energy correction the first, second and and (f). There is a clear dependence of the convergence rate
thirdtermarecalledparticle-particle(pp),hole-hole(hh)and on the flow parameter for the oxygen isotopes. For 16O the
particle-hole(ph)correction, respectively. The(cid:15) correspond seriesconvergesexponentiallyinallthreecasesandthelarger
i
totheHFsingle-particleenergiesandallmatrixelementsare the flow parameter, i.e. the softer the Hamiltonian, the more
taken to be antisymmetrized. Summation indices a,b,c,.. rapid the convergence—as might be naively expected. For
correspond to particle indices, i.e., above the Fermi level (cid:15) , 24O the behavior is slightly more complicated. For the soft-
F
whereasi, j,k,...correspondtoholeindicesrangingfromlow- est interaction with α = 0.08fm4 there is still a clear expo-
estoccupiedsingle-particlestatesuptotheFermilevel. The nential convergence. However, for the harder interactions,
zero-andone-bodypartsofthenormal-orderedHamiltonian i.e., α = 0.02,0.04fm4, we observe no systematic decrease
onlyenterinthefirst-orderenergycorrection. Brillouin’sthe- ofthehigh-orderperturbativecontributionsanymore,theyre-
oremstatesthatthereisnomixingoftheHartree-Fockground mainapproximatelyconstantandcauseasmall-amplitudeos-
3
40 partialsums energycorrections
(p)E[MeV]sum −−−−−1186200000 (a) (c) 11110000−351||0[MeV]E(p) (p)E[MeV]sum −−−−−222211642086 4He 11110000−−331||[MeVE(p)
−140 10−3 −−28 (a) (d) 10−5 ]
60 103 30
(p)E[MeV]sum −−−−−11184200000 2(b)6 10 14 18 22 26 30 2(d)6 10 14 18 22 26 301111100000−−−−7531||0[MeV]E(p) (p)E[MeV]sum −−−−−−1118642000000 (b) (e) 16O 1111100000−−−3531||[MeV]E(p)
perturbationorderp perturbationorderp 24O 103
-120
V] 10 |E
FNNImNGa+x.31=N. P-2faur(lt(cid:108)li)ai,nlts4eurm(a(cid:72)cst)iooafnn1dw6Oi6thi(n(cid:70)αH)=.OT0.bh0ae8sficmso(r4arae)nsapdnodtnrudHninFcgabtieaonsniesrpg(abyr)acmfooerrrtteehcres- (p)E[Mesum -140 1100−−31|[MeV(p)
]
tions for each order are displayed in (c) and (d), respectively. All -160
calculationsareperformedatoscillatorfrequency(cid:126)Ω=24MeV. (c) (f) 10−5
2 6 10 14 18 22 26 30 2 6 10 14 18 22 26 30
perturbationorderp perturbationorderp
cillatorybehaviorofthepartialsums. However,eveninthese FIG.2. PartialsumsforvaryingflowparametersinHF-MBPTfor
cases we can easily extract a robust estimate for the asymp- 4He(a),16O(b)and24O(c). Thecorrespondingenergycorrections
toticvalue. Inthecaseof4Hethesuppressionisindependent aregivenin(d),(e)and(f),respectively.Themodelspaceforthefirst
ofαandweobservethesamerapidconvergenceforallinter- andsecondpanelaretruncatedatN = 6. Thetruncationforthe
max
actions. thirdpanelisgivenbyNmax =4. Theflowparametersforthediffer-
The numerical values of the partial sums for selected or- entdatasetsareα=0.02fm4((cid:108)),0.04fm4((cid:72))and0.08fm4((cid:70)).All
ders of HF-MPBT for the three nuclei and the different flow calculationsuseaNN+3N-fullinteractionwithoscillatorfrequency
(cid:126)Ω=24MeV.
parametersaresummarizedinTab.Itogetherwiththeresults
ofdirectCIcalculationsforthesameHamiltoniansandmodel
spaces. Thehigher-orderpartialsumsareingoodagreement
culationsfortheselargespaces,however,thecoupled-cluster
withtheCIresults—inmostcasesthedeviationoftheground-
frameworkhasproventoprovideaccurateresultsforground-
stateenergyismuchsmallerthan0.1%.
state energies of closed-shell nuclei [1–4]. We compare the
Basedonourdetailedanalysisofhigh-orderHF-MBPTand HF-MBPTresultstorecentCCcalculationsattheCCSDand
duetotheexponentialsuppressionoftheenergycorrections, theCR-CC(2,3)level[5,6,42]. StartingfromaHFreference
we can take low-order partial sums as a reasonable approxi- statethisapproachprovidesacompleteinclusionofsinglyand
mationtotheconvergedresults. Thismotivatestheinvestiga- doublyexcitedclustersontopofthereferencestateand,inthe
tionofthird-orderpartialsumsforselectedmedium-massand caseofCR-CC(2,3)anapproximatenon-iterativeinclusionof
heavyclosed-shellnucleiinthefollowing. triplyexcitedclusters[43–46].
ExplicitSummationforHeavyNuclei. Forheaviernuclei InFigs.3and4theground-stateenergiespernucleon(a)as
and larger model spaces we cannot compute the high-order wellasthecorrelationenergyE = E−E pernucleon(b)
corr HF
perturbationseriesexplicitlyand,thus,wecannotinvestigate from HF-MBPT and CR-CC(2,3) are depicted for an initial
the convergence characteristics explicitly. We can, however, chiralNN+3NandaninitialchiralNNinteraction. TheSRG-
evaluate the perturbative contributions up to third order very induced three-nucleon contribution are taken into account in
efficiently. To demonstrate the validity of a low-order per- both cases, leading to the NN+3N-full and NN+3N-induced
turbative approximation, we need to compare our results to interactions,respectively.
established ab initio techniques, in our case, coupled-cluster Thesefiguresshowaremarkableresult: Thebindingener-
calculationswithsophisticatedtriplescorrections. gies in third-order HF-MBPT and CR-CC(2,3) are in excel-
Weconsiderasequenceofclosed-shellnucleirangingfrom lent agreement with each other. The relative differences are
4He to 132Sn and perform calculations in second and third- in most cases much smaller than 1%. The same observation
orderHF-MBPTinalargemodelspacetruncatedwithrespect holdsforthecorrelationenergy,i.e.,thecorrectionstotheHF
to the single-particle principal quantum number e = 12. energy. Thethird-orderenergycorrectionscontributeapprox-
max
WerestrictourselvestoSRG-evolvedHamiltonianswithflow imately 0.2 MeV to the overall binding energy per nucleon
parameterα=0.08fm4,whichwasusedextensivelyinprevi- andare,therefore,nonnegligibleeventhoughthethird-order
ouscalculationsandshowedfavorableorder-by-orderconver- energy corrections in HF-MBPT are one order of magnitude
gence in our high-order studies. We cannot perform CI cal- smallerthanthesecond-ordercorrection.
4
6
−
NN+3N-full
7
V] −
Me 8
−
[
A 9
/ −
E
10
− (a)
1
−
V]
e
M 1.5
[ −
A
/
corr −2
E
(b)
2.5
−
4He 16O 24O 36Ca 40Ca 48Ca 52Ca 54Ca 48Ni 56Ni 60Ni 66Ni 68Ni 78Ni 88Sr 90Zr 100Sn116Sn118Sn120Sn132Sn
FIG.3. Panel(a)showstheground-stateenergiespernucleonfromthird-orderHF-M◦BPT((cid:108))incomparisontoCR-CC(2,3)((cid:72))resultsfor
selectedclosed-shellnuclei.Panel(b)showsthecorrelationenergypernucleon,E(2)( )aswellasE(2)+E(3)((cid:108))forHF-MBPT.Additionally,
0 0 0
the correlation energy per nuclei for CCSD ((cid:52)) and CR-CC(2,3) ((cid:72)) are shown. All calculations were performed with the NN+3N-full
interactionwithα=0.08fm4,(cid:126)Ω=24MeVinane =12truncatedmodelspace.Experimentalvaluesareindicatedbyblackbars.
max
6
− NN+3N-induced
V] 7
e −
M
[ 8
A −
/
E
9
− (a)
0.8
−
V] 1
e −
M
1.2
[ −
A
/ 1.4
orr −
Ec 1.6
−
1.8 (b)
−
4He 16O 24O 36Ca 40Ca 48Ca 52Ca 54Ca 48Ni 56Ni 60Ni 66Ni 68Ni 78Ni 88Sr 90Zr 100Sn116Sn118Sn120Sn132Sn
FIG.4.BindingenergyandcorrelationenergyfortheNN+3N-inducedinteraction.AllotherparametersasinFig3.
The third-order energy contribution (3) consists of three ual third-order contributions on the input Hamiltonian show
termscorrespondingtothreeHugenholtzdiagrams. Figure5 thatapartialinclusionofselectedthird-ordertermsmaylead
disentanglestheirindividualcontributionstotheoverallthird- towrongestimates.
order energy correction. The contribution of the pp and hh
Conclusions. We have discussed Rayleigh-Schrödinger
corrections are almost constant over the entire mass range, MBPT as an efficient approach to compute ground-state en-
whereastheenergycorrectionarisingfromthe phtermscales
ergies for closed-shell nuclei up to the heavy-mass region.
with increasing mass number in the case of a NN+3N-full
The use of a HF basis has enabled us to overcome conver-
interaction. For the tin isotopes the third-order energy cor-
genceproblemsthatgenerallyariseinHO-MBPT.Investigat-
rectioncontributes3%oftheoverallbindingenergyinthird- ing 16O in different model spaces showed convergent partial
order HF-MBPT and is not negligible. In particular we see
sums when using HF-MBPT and the limit of the perturba-
thatmostofthethird-orderenergycorrectionarisesfromthe
tion series coincides with the results from explicit CI calcu-
ph diagram. In the case of a NN+3N-induced interaction
lations. Additionally, we found systematic dependencies of
we see that all three terms are suppressed with increasing
the convergence rate on the SRG parameter in the case of
massnumber. Thesesystematicdependenciesoftheindivid- 16Oand24O. Thus,inHF-MBPTwecanimprovetheconver-
5
titioning, or equivalently, defining a starting point of the re-
TABLE I. Ground-state energies for 4He, 16O and 24O in units of
cursive calculation is the most important part. We have seen
[MeV]obtainedinHF-MBPTfordifferentordersupto p = 30and fromtheradicallydifferentbehavioroftheperturbationseries
in CI calculations with NN+3N-full interactions for different flow
in HF-MBPT and HO-MBPT that the partial sums are very
paramtersα. Themodelspacesaretruncatedby N = 6for4He
and 16O and N = 4 in the case of 24O. The HmOaxfrequency is sensitive to the partitioning. When using HF-MBPT we can
max
(cid:126)Ω=24MeV. improvetheorder-by-orderconvergencebyusingsofterinter-
actions corresponding, e.g., to larger SRG flow parameters.
α [fm4]
Even for HF basis sets harder interactions can spoil conver-
0.02 0.04 0.08 gence. The ‘softness’ of the interaction has been character-
E(s2u)m -19.204 -20.269 -23.588 ized in terms of Weinberg eigenvalues, which are connected
E(3) -20.334 -23.224 -26.589 tothespectrumoftwo-bodyGreen’sfunctions[47–49]. Sim-
sum
4He E(s1u0m) -20.507 -24.444 -26.947 ilarexpressionsalsoappearintheformulasforthefirst-order
E(20) -20.526 -24.462 -26.964 statecorrection. Thoughthegeneralconnectionseemsobvi-
sum
E(30) -20.537 -24.469 -26.971 ous,oneshouldbecarefulwithconclusionsabouttheconver-
sum
gence of MBPT for a finite nucleus based on the softness of
CI -20.539 -24.483 -26.994
the interaction. Our work has shown that the partitioning is
E(2) -85.620 -107.241 -120.699
sum key for convergence. Our observation that the convergence
E(3) -89.315 -110.861 -123.863
sum of HF-MBPT deteriorates for harder interactions could sim-
16O E(s1u0m) -83.780 -107.199 -122.561 plybeexplainedbythefactthattheunperturbedHFsolution
E(20) -84.180 -107.341 -122.577
sum becomesamuchworseapproximationforthegroundstatein
E(s3u0m) -84.018 -107.331 -122.577 thesecases.
CI -84.043 -107.330 -122.577 ThesuperiorconvergencepropertiesofHF-MBPThasmo-
E(2) -125.460 -124.459 -149.053 tivatedtheuseoflow-orderapproximationstoinvestigatenu-
sum
E(3) -122.880 -126.670 -151.059 cleiinthemedium-massregion. Wehavevalidatedtheselow-
sum
24O E(s1u0m) -119.705 -121.233 -147.446 order approximations to the most sophisticated CC calcula-
E(20) -119.335 -121.314 -147.508 tionsandfoundexcellentagreementofthird-orderHF-MBPT
sum
E(30) -119.483 -120.948 -147.489 and CR-CC(2,3) at the level of better than 1%. The consis-
sum
tency of high-order partial summations with exact CI diago-
CI -119.131 -120.947 -147.488
nalizationsaswellastheagreementoflow-ordersummations
withcoupled-clusterresultsmayqualifyHF-MBPTasanab
0.2 initio approach. However, the strong dependence of conver-
V] NN+3N-full
e genceonthepartitioningshouldbeareasonforcaution. The
[M 0 HF partitioning seems to be robust for sufficiently soft inter-
A
/ 0.2 actions,butthereisnoformalguaranteeforconvergence.
(3) − Thegreatadvantageoflow-orderHF-MBPTisitssimplic-
E
0.4 ity: Computationally, the third-order calculations are much
−
V] 0.2 NN+3N-induced cheaper than CC or IM-SRG calculations. It is, therefore,
Me ideal for survey calculations over a large range of medium-
0
[ mass nuclei, e.g., to explore the ground-state systematics for
A
(3)/ −0.2 new interactions. Formally, the underlying equations and al-
E gorithms are trivial compared to CC or IM-SRG. As a result
0.4
− 4He 16O 24O 40Ca 48Ca 56Ni 78Ni 90Zr 100Sn132Sn ofthisformalsimplicity, extensionstothedescriptionexited
states and open-shell nuclei are straight-forward. We have
demonstratedthisalreadyforlightnucleiusinghigh-orderde-
FIG.5. Individualcontributionsofthediagramsappearingatthird-
generate HO-MBPT [18]. Alternative multi-configurational
order perturbation theory. Show are the contributions per nucleon
formulationsforopen-shellnucleiareunderinvestigation.
from the pp-diagram((cid:4) ) , the hh-diagram ((cid:70)) and the ph-diagram
Acknowledgements. This work is supported by the DFG
( ).Theoverallcontributionofthethird-ordercorrectionisdepicted
(cid:72)
in((cid:108)).ThefirstpanelcorrespondstotheNN+3N-fullinteractionand through grant SFB 1245, the Helmholtz International Cen-
thesecondpaneltoaNN+3N-inducedinteractionwithα=0.08fm4, terforFAIR,andtheBMBFthroughcontract05P15RDFN1.
(cid:126)Ω=24MeV,ande =12. S.Bindergratefullyacknowledgesthefinancialsupportofthe
max
Alexander-von-Humboldt Foundation (Feodor-Lynen schol-
arship). Numerical calculations have been performed at the
gencebehavioroftheperturbationseriesbyfurtherevolving computing center of the TU Darmstadt (lichtenberg), at the
theHamiltonian,whereasthedivergentHO-MBPTseriesare LOEWE-CSCFrankfurt,andattheNationalEnergyResearch
unaffected. Scientific Computing Center supported by the Office of Sci-
We can identify a hierarchy of elements influencing the ence of the U.S. Department of Energy under Contract No.
convergencepropertiesofperturbationseries. Definingapar- DE-AC02-05CH11231.
6
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