Table Of ContentAb initio coupled-cluster study of 16O
M. W loch,1 D. J. Dean,2 J. R. Gour,1 M. Hjorth-Jensen,3 K. Kowalski,1 T. Papenbrock,2,4 and P. Piecuch1
1Department of Chemistry, Michigan State University, East Lansing, MI 48824, USA
2Physics Division, Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, TN 37831, USA
3Department of Physics and Center of Mathematics for Applications, University of Oslo, N-0316 Oslo, Norway
4Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, USA
(Dated: February 9, 2008)
Wereportconvergedresultsforthegroundandexcitedstatesandmatterdensityof16Ousingre-
alistictwo-bodynucleon-nucleoninteractionsandcoupled-clustermethodsandformalismdeveloped
inquantumchemistry. Mostofthebindingisobtainedwiththecoupled-clustersinglesanddoubles
approach. Additional binding due to three-body clusters (triples) is minimal. The coupled-cluster
method with singles and doubles provides a good description of the matter density, charge radius,
charge form factor, and excited states of a 1-particle-1-hole nature, but it cannot describe the first
5 excited 0+ state. Incorporation of triples has no effect on thelatter finding.
0
0 PACSnumbers:
2
n
a Oneofthemostimportantproblemsinnuclearphysics particle states has been possible thanks to the develop-
J istounderstandhownuclearpropertiesarisefromtheun- ment of general-purpose coupled-cluster computer pro-
6 derlying nucleon-nucleon interactions. Recent progress grams for nuclear structure, using diagram factorization
2 using Monte Carlo [1] and diagonalization[2] techniques techniques adopted by quantum chemists. We pay par-
produced converged results for nuclei with up to A=12 ticular attention to three aspects of the calculations: (i)
1
active particles, yielding a much-improved understand- the convergence of the ground-state energy with respect
v
7 ing of nuclear forces in light systems. One also must to the size of the model space and the role of higher–
6 explore alternative methods that would not suffer from than–two-bodyclustersinsuchstudies,(ii)the abilityof
0 the exponential growth of the configuration space, en- coupled-cluster methods to describe excited states, and
1
ablingaccurateabinitiocalculationsformedium-sizenu- (iii)the performanceofcoupled-clustermethodsinstud-
0
clei. Coupled-cluster theory [3] is a promising candidate ies of nuclear radii, matter density, charge form factor,
5
0 for such developments since it provides an accurate de- and occupation numbers. We have not yet included the
/ scription of many-particle correlations at relatively low three-nucleon interaction that should eventually be con-
h
cost, as has been demonstrated in numerous chemistry sidered[1, 2]. However,our calculationsrepresenta dra-
t
- applications[4,5]. Recently,MihailaandHeisenbergper- matic step forward in nuclear many-body computations
l
c formed coupled-cluster calculations for the binding en- due to the enormous oscillator space we probe through
u
ergyandthe electronscatteringformfactor of16O using application of computationally efficient coupled-cluster
n
: bare interactions [6]. In previous work [7], we took an- methods. They teach us about the nucleon correlations
v
other route and used quantum chemical coupled-cluster and the magnitude of the (missing) three-body forces.
i
X methods and the renormalized Hamiltonian to compute
r ground and excited states of 4He and ground-state ener- We use two variants of effective-field-theory-inspired
a gies of 16O in a small model space consisting of 4 major Hamiltonians, Idaho-A and N3LO [9]. The Idaho-A po-
oscillator shells, demonstrating promising results when tential was derived with up to chiral-order three dia-
compared with exact shell-model diagonalization. grams while N3LO includes chiral-order four diagrams,
and charge-symmetry and charge-independence break-
In this Letter we report, for the first time, converged ing terms. We also include the Coulomb interaction
coupled-cluster calculations for ground- and excited- with the N3LO calculations. Since very slow conver-
state energies and other properties of 16O using mod- gence with the number of single-particlebasis states was
ern nucleon-nucleon interactions derived from effective- obtained using bare interactions [6], we renormalize the
field theory [8]. Our ground-state calculations involv- bareHamiltonianusingano-coreG-matrixapproach[10]
ing one- and two-body components of the cluster op- whichobtainsastarting-energydependenceω˜ inthetwo-
erator are performed in up to 8 major oscillator shells bodymatrixelementsG(ω˜). WeusetheBethe-Brandow-
(480 uncoupled single-particle basis states), while the Petschek [11] theorem to alleviate much of the starting-
correctionsdue to three-body clusters and computations energydependence(see[10]fordetails). Thedependence
of excited states and nuclear properties involve up to 7 uponthestartingenergyisweakfor16O,particularlyfor
major oscillator shells (336 single-particle states). The the matrix elements below the Fermi surface [12]. The
significant progress in going from model calculations us- effective Hamiltonian for coupled-cluster calculations is
ing 80single-particlestates[7]to large-scalecalculations H′ =t+G(ω˜), where t is the kinetic energy. We correct
involving 16 correlated nucleons and almost 500 single- H′ for center-of-mass contaminations using the expres-
2
sion H = H′ +β H . We choose β such that ofH¯ formabiorthonormalset,hΦ|L(µ)R(ν)|Φi=δ . If
c.m. c.m. c.m. µν
the expectation value of the center-of-mass Hamiltonian the only purpose of the calculation is to obtain excita-
H is 0.0MeV. We note that intrinsic excitationener- tionenergies,thelefteigenstateshΦ|L(µ) arenotneeded.
c.m.
gies are virtually independent of β while the unphys- However, for properties other than energy, both right
c.m.
ical, center-of-mass contaminated states show a sharp, and left eigenstates of H¯ are important. In particular,
nearly linear dependence of excitation energies on β . wecalculatethe one-bodyreduceddensitymatrixρ in
c.m. αβ
This allows us to separate intrinsic and center-of-mass quantum state |Ψ i as follows:
µ
contaminated states.
ραβ =hΦ|L(µ) (cid:2)exp(−T)a†αaβexp(T)(cid:3) R(µ)|Φi. (1)
Once the one- and two-body matrix elements of the
In the CCSD ground-state (µ = 0) case, we have
center-of-mass-corrected effective Hamiltonian are con-
T = T + T , R(0) = 1, and L(0) = 1 + Λ + Λ ,
structed, we solve the A-body problem using quantum 1 2 1 2
where the one- and two-body deexcitation operators Λ
chemicalcoupled-clustertechniques. Inthe ground-state 1
and Λ are determined by solving the CCSD left eigen-
calculations, we use the CCSD (“Coupled-Cluster Sin- 2
valueproblem,obtainedbyright-projectingtheequation
gles and Doubles”) approach [13], to describe correla-
hΦ | (1 + Λ)H¯ = E hΦ|(1 + Λ), with E representing
tion effects due to one- and two-body clusters, and the 0 0
the CCSD energy and Λ = Λ +Λ on the singly and
CR-CCSD(T) (“Completely Renormalized CCSD(T)”) 1 2
doubly excited determinants. Thus far, we have focused
method[14],to correctthe CCSDenergiesforthe effects
on the CCSD and EOMCCSD methods which use inex-
of three-body clusters (“Triples”). In the excited-state
pensive computational steps that scale as n2n4, where
andpropertycalculations,weusetheequation-of-motion o u
n (n ) is the number of occupied (unoccupied) single-
(EOM) CCSD method [15] (equivalent to the linear re- o u
particle states. While the full inclusion of triply excited
sponse CCSD approach [16]). We also correct the ener-
clusters is possible, the resulting methods are expensive
gies of excited states obtained with EOMCCSD for the
and scale as n3n5. Thus, we estimate the effects of T
effects of triples using the CR-EOMCCSD(T) approach o u 3
and R on ground- and excited-state energies by adding
[14]. Thedetailsoftheabovemethodscanbefoundelse- 3
thecorrectionstotheCCSD/EOMCCSDenergies,which
where[13,14,15]. Here,weonlymentionthattheCCSD
only require n3n4 noniterative steps. These corrections,
methodisobtainedbytruncatingthemany-bodyexpan- o u
due to T and R , define the CR-CCSD(T) and CR-
sion for the cluster operator T in the exponential ansatz 3 3
EOMCCSD(T) approaches [5, 14]. In this study, we use
exploited in coupled-cluster theory, |Ψ i=exp(T)|Φi,
0
variant“c”ofthe CR-CCSD(T)andCR-EOMCCSD(T)
where | Ψ i is the correlated ground-state wave func-
0
approaches [7].
tion and | Φi is the reference determinant. The trun-
We turn to a discussion of our 16O results. We
cated cluster operator used in the CCSD calculations
has the form T = T1 + T2, where T1 = Pi,atiaa†aai ctohoomseinitmhiezeostchilelatCoCr SeDneregnyer¯hgyω. forFoorurthbeasNis s=tate7s
and T2 = 14Pij,abtiajba†aa†bajai are the singly and dou- and N = 8 oscillator shell runs, h¯ω = 11 MeV,
bly excitedclustersandi,j,...(a,b,...)labelthe single-
and the results are nearly independent of h¯ω [10].
particle states occupied (unoccupied) in |Φi. We deter-
Shown in Fig. 1 are our CCSD/EOMCCSD and CR-
mine the singly and doubly excited cluster amplitudes
CCSD(T)/CR-EOMCCSD(T)ground-andexcited-state
ti and tij by solving the nonlinear system of algebraic
a ab energies as a function of N. The symbols in Fig. 1 rep-
equations, hΦa|H¯|Φi = 0, hΦab|H¯|Φi = 0, where H¯ =
i ij resent our calculations while the lines represent a fit of
exp(−T)Hexp(T)and|Φaiand|Φabiarethesinglyand
i ij the form E(N) = E∞+aexp(−bN), where the extrap-
doublyexciteddeterminants,respectively,relativeto|Φi.
olated energy E and a and b are parameters for the
We calculate the ground-state energy E as hΦ|H¯ |Φi. ∞
0 fit. We also show in Fig. 1 our calculations for the first
We diagonalize the similarity-transformed Hamiltonian excited3− stateandthepositionofthelowestcalculated
H¯ in the relatively small space of singly and doubly ex- 0+ excited state. We now discuss these results.
cited determinants |Φaii and |Φaijbi to obtain the excited- Triples correction to the CCSD ground-state energy.
state wave functions |Ψ i and energies E . The right
µ µ The small model space calculation [7] indicated that the
eigenstates of H¯, R(µ)|Φi, where R(µ) = R +R +R
0 1 2 triplescorrectionstotheground-stateCCSDenergiesare
is a sum of the relevant reference (R ), one-body (R ),
0 1 small. We extended these calculationsfrom4 to 8major
and two-body (R ) components define the excited-state
2 oscillator shells for CCSD calculations and to 7 major
“ket” wave functions |Ψ i = R(µ)exp(T)|Φi, whereas
µ oscillator shells for CR-CCSD(T) calculations, as shown
the left eigenstates hΦ|L(µ) define the “bra” wave func-
inFig. 1. We find that the extrapolatedCCSD energyis
tions hΨ˜ | = hΦ|L(µ)exp(−T). Here, each n-body com-
µ −119.4MeVforIdaho-A.FortheN =7Idaho-Acalcula-
ponent of R(µ) with n > 0 is a particle-hole excitation
tion,thedifferencebetweentheCCSDandCR-CCSD(T)
operator similar to T , whereas L(µ) is a hole-particle
n result is 0.6 MeV, while the extrapolated values differ
deexcitation operator, so that L1 = Pi,aliaa†iaa and by only 1.1 MeV; our extrapolated CR-CCSD(T) en-
L2 = 14Pij,abliajba†ia†jabaa. Therightandlefteigenstates ergy is −120.5 MeV. The Coulomb interaction adds to
3
the binding 11.2 MeV, so that our estimated Idaho-A theory and experiment resides in the Hamiltonian, not
groundstate energy is −109.3MeV (compared to an ex- in the correlation effects which EOMCCSD and CR-
perimental value of −128 MeV). Our N = 7 (N = 8) EOMCCSD(T) describe very well if three-body forces
N3LO CCSD and N = 7 CR-CCSD(T) energies, which play no role and if the state has a 1p-1h nature.
include the Coulomb interaction, are −112.4 (−111.2) Calculation of the first excited 0+ state. This state
and −112.8 MeV, respectively. Thus, the two-body in- (experimentally at 6.05 MeV), believed to have a 4p-4h
teractions underbind 16O by approximately 1 MeV per character, cannot be described by EOMCCSD or CR-
particle, pointing to the need for three-body forces. For EOMCCSD(T). This is confirmed by our calculations as
theIdaho-AandN3LOinteractionsandthe16Onucleus, we see large differences between the EOMCCSD or CR-
we conclude that connectedT clustersare indeed small, EOMCCSD(T) results and experiment (see Fig.1). One
3
contributing less than 1% to the ground-state energy. would need to include 4p-4h operators (T and R ) to
4 4
This is an important finding, since it implies that es- improve coupled-cluster results.
sentially all correlations in a closed-shell nucleus result-
ing from two-body nucleon-nucleon interactions can be −100
captured by the relatively inexpensive CCSD approach. E(0+)−Eg.s.=19.8 MeV CCSD
Another important finding is a rapid convergence of the −105 CR−CCSD(T)
CCSDandCR-CCSD(T)energieswiththenumberofos-
−110 E(3−)−E =12.0 MeV
cillator shells owing to the use of the renormalized form V) g.s.
othfethNe H=am8ilatonndiaNn. =For7eCxaCmSDpl/eI,dtahheod-Aiffeerneenrcgeiebsetiwse0e.n5 E (Me−115
MeV (see Fig. 1). −120
E =−120.5 MeV
Calculations of the first excited 3− state. The first g.s.
−125
excited 3− state in 16O is thought to be principally a
one-particle-one-hole (1p-1h) state [17]. The experience −130
4 6 8 infty
of quantum chemistry is that the EOMCCSD and CR- Number of oscillator shells, N
EOMCCSD(T) methods describe such states well, pro-
FIG. 1: The energies of the ground-state (g.s.) and first-
vided that the three-body interactions in the Hamilto- excited 3− and 0+ states as functions of thenumberof oscil-
niancanbeignored. ThelargestR1 amplitudesobtained latorshellsN obtainedwithcoupled-clustermethodsandthe
in the EOMCCSD calculations indicate that the domi- Idaho-Ainteraction.
nant 1p-1h excitations are from the 0p orbital to the
1/2
0d orbital. The 2p-2h excitations in the EOMCCSD AlthoughweconcentratedonthelowestenergyJ =3−
5/2
wave function, defined as R + R T + R (T + T2/2) andJ =0+excitedstatesandtheroleofthree-bodyclus-
2 1 1 0 2 1
(R =0 in this case), are much smaller than the R am- tersonthese,wealsoperformedpreliminarycalculations
0 1
plitudes, and the CR-EOMCCSD(T) calculation hardly for other negative parity states. The quartet of negative
changes the total energy of the state, which indicates parity states starting with the J =3− state, and includ-
that this state has indeed a 1p-1h nature. Our ex- ingtheJ =1−,2− and0− states,areallbelievedtohave
trapolated Idaho-A results indicate that the 3− state a similar 1p-1h character [17]. The EOMCCSD calcula-
lies at −108.2 and −108.4 MeV in the EOMCCSD and tion with 5 major oscillator shells and Idaho-A confirms
CR-EOMCCSD(T) calculations, respectively. The CR- the existence of this quartet, giving excitation energies
EOMCCSD(T) method yields an excitation energy of of 13.57, 15.37, 17.07, and 17.15 MeV for the J = 3−,
12.0 MeV for this state which experimentally lies at 1−, 2−, and 0− states, respectively. While these states
6.12 MeV. N3LO yields similar results. Based on the are all a few MeV above the experimental values, their
1p-1h structure of the state, we conclude that Idaho-A ordering predicted by EOMCCSD is correct.
and N3LO do not yield an excitation energy for the 3− Calculation of the one-body density. We use Eq. (1),
statewhichiscommensuratewithexperiment. Thesere- where µ = 0, to calculate the ground-state density for
sults agree with recent no-core shell-model calculations 16O.Weshowtheresultingradialdensity,ρ(r),inFig.2.
with similar two-body Hamiltonians [18]. The 3− state The root-mean-square (rms) radius is found through an
is expected to be built on 1p-1h excitations which de- integration r2 = R r4ρ(r)dr/R r2ρ(r)dr. To obtain a
rms
pend on the single-particle splittings. These splittings charge radius, we correct this value for the finite size of
will be affected by three-body forces not included in our the nucleons, which experimentally are r2 = 0.743 fm2
p
Hamiltonian, thus affecting the energy of the 3− state. and r2 = 0.115 fm2, and for the 0s center-of-mass mo-
n
Whether other mechanisms than three-body forces can tion, for which we use hΨ | R | Ψ i = 62.2071 fm2. Our
0 0 Ah¯ω
provideanadditionalbindingof6MeVneedsfurtherre- rms charge radii for 16O for 5, 6, and 7 oscillator shells
search. Our results are converged at the coupled-cluster are 2.45 fm, 2.50 fm, and 2.51 fm, respectively when the
level employing the Idaho-A and N3LO two-body in- Idaho-A potential is used (N3LO gives similar values).
teractions, so it is likely that the discrepancy between The experimental charge radius is 2.73±0.025 fm. We
4
1 lowest J =3− state, which is, quite likely, due to an in-
5 shells, CM corrected
10-2 67 sshheellllss, CM corrected adequatedescriptionoftherelevantnuclearforcesbythe
Hamiltonian. We were unable to accurately describe the
2F(q)|10-4 lowest J =0+ excited state due to connected 4p-4h cor-
|
relations missing in coupled-cluster approximations em-
10-6
ployedinthisstudy. TheCCSDmethodprovidesreason-
10-80 1 2 3 4 ableresultsforthenuclearmatterdensity,chargeradius,
q (fm-1) and charge form factor. The use of the renormalized
0.30
5 shells, r =2.45 fm Hamiltonian guarantees fast convergence of the results
rms
6 shells, rrms=2.50 fm with the number of oscillator shells. All of this makes
-3m)0.20 7 shells, rrms=2.51 fm low-costcoupled-clustermethodsapromisingalternative
ρ(r) (f0.10 16O matter density to traditional shell-model diagonalization techniques.
V = Idaho-A
Research supported by the U.S. Department of En-
0.00 ergy(OakRidgeNationalLaboratory,UniversityofTen-
0 1 2 3 4 5
r (fm) nessee, MichiganState University), the National Science
Foundation (Michigan State University), and the Re-
FIG. 2: Top panel: The charge form factor computed from
search Council of Norway (University of Oslo).
theCCSDdensitymatrix. Bottom panel: thematterdensity
in 16O.The results obtained with theIdaho-A interaction.
also calculated the occupation probability for the natu-
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