Page 1

arXiv:1005.1599v1 [nucl-th] 10 May 2010

Systematics of binding energies and radii based on realistic two-nucleon

plus phenomenological three-nucleon interactions

A. G¨ unther,1, ∗R. Roth,1, †H. Hergert,2and S. Reinhardt1

1Institut f¨ ur Kernphysik, Technische Universit¨ at Darmstadt, 64289 Darmstadt, Germany

2National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, MI 48824, USA

(Dated: May 11, 2010)

We investigate the influence of phenomenological three-nucleon interactions on the systematics of ground-

state energies and charge radii throughout the whole nuclear mass range from4He to208Pb. The three-nucleon

interactions supplement unitarily transformed two-body interactions constructed within the Unitary Correlation

Operator Method or the Similarity Renormalization Group approach. To be able to address heavy nuclei as

well, we treat the many-body problem in Hartree-Fock plus many-body perturbation theory, which is sufficient

to assess the systematics of energies and radii, and limit ourselves to regularized three-body contact interactions.

We show that even with such a simplistic three-nucleon interaction a simultaneous reproduction of the experi-

mental ground-state energies and charge radii can be achieved, which is not possible with unitarily transformed

two-body interactions alone.

PACS numbers: 21.30.Fe,21.45.Ff,21.60.Jz

I.INTRODUCTION

Nuclearstructuretheoryisapproachinganeraofsystematic

many-body calculations using nuclear Hamiltonians based on

Quantum Chromodynamics (QCD). An important step along

these lines is the formulation of nuclear interactions within

chiral effective field theory [1–3], leading to a consistent hier-

archy of two-, three- and many-nucleon interactions starting

from the relevant degrees of freedom and symmetries for the

low-energy nuclear structure regime. The use of these two-,

three- and many-nucleoninteractions in nuclear structure cal-

culations is a formidable task.

Inadditiontofew-bodycalculationsthemostpromisingnu-

clear structure calculations using the chiral two- plus three-

nucleon interaction consistently have been performed in the

no-core shell model (NCSM) for mid p-shell nuclei [4]. An

immense numerical effort is needed to compute and man-

age the three-body matrix elements in these calculations,

which limits the range of applicability of these calculations

at present. Recently, the use of consistent two- plus three-

nucleon interactions resulting from a Similarity Renormaliza-

tion Group evolution of the chiral two- plus three-nucleon in-

teraction was demonstrated also in the context of the NCSM

[5]. This approach, a unitary transformation of the chiral

Hamiltonianaimingat a pre-diagonalizationthat improvesthe

convergence properties of NCSM substantially, holds great

potential also for the use in other many-body schemes and

will play a significant role in the future. However, the com-

putational effort for including those two- plus three-nucleon

interactions into many-body calculations, be it exact or ap-

proximate, is still the limiting factor for many applications.

In this paper we follow a more pragmatic route to explore

the impact of three-body forces in connection with unitarily

transformed two-nucleon interactions. We start from the Ar-

∗Electronic address: anneke.guenther@physik.tu-darmstadt.de

†Electronic address: robert.roth@physik.tu-darmstadt.de

gonne V18 high-precision two-nucleon potential [6], which

is still widely used although it does not have the same sys-

tematic link to QCD like the chiral effective field theory in-

teractions and is consideredphenomenologicalin this respect.

We then use the Similarity Renormalization Group [7–11] as

well as the Unitary Correlation Operator Method [10, 12–14]

to construct a transformedtwo-nucleoninteraction, which has

a much better convergence behavior and allow us to use sim-

plified many-body schemes. At this level neither genuine nor

induced three-nucleon interactions are included. From vari-

ous applications of these unitarily transformed two-nucleon

interactions we know that there are characteristic deviations

of basic nuclear observables from the experimental systemat-

ics that might be connected to three-body interactions. For

example, unitarily transformed two-body interactions which

yield a realistic systematics for binding energies tend to un-

derestimate the charge radii [10, 15]. Here we study to what

extend these systematic deviations can be cured by including

a three-bodyinteraction. Note that we are not aiming at a pre-

cision description of individual nuclei but rather the complete

systematics from light nuclei,4He, to heavy nuclei,208Pb.

To facilitate calculations for the full mass range from4He

to208Pb we have to simplify the approach compared to the

consistent. The first simplification consists in the use of a

phenomenologicalthree-bodyinteraction,whichallowsforan

efficient computation of matrix elements but violates the con-

sistency discussed above. The second simplification consists

in the use of Hartree-Fock plus many-body perturbation the-

ory for the approximate solution of the many-body problem.

Despite ofthese simplifications, we will obtainvaluableinfor-

mation on the interplay between realistic two-body and phe-

nomenological three-body interactions and on how well the

experimental systematics of ground-state energies and charge

radii can be reproduced. Furthermore, these studies prepare

thegroundforcalculationswithconsistentlytransformedtwo-

plus three-nucleon interactions.

After a brief reminder of the basic concepts of the Unitary

CorrelationOperatorMethodandthe Similarity Renormaliza-

tion Groupwe introducethe phenomenologicalthree-bodyin-

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2

teraction and calculate the matrix elements in the harmonic

oscillator basis in the second section. In the third section, we

discuss the inclusion of the three-body interaction in Hartree-

Fock and many-body perturbation theory and discuss the sys-

tematics of ground-state energies and charge rms-radii across

the whole nuclear mass range and its dependence on the two-

and three-nucleon interaction.

II.FORMALISM

A. Unitary Correlation Operator Method and Similarity

Renormalization Group

The Unitary Correlation Operator Method and the Similar-

ity RenormalizationGroupprovidetwo conceptuallydifferent

but physically related approaches for the construction of soft

phase-shift equivalent interactions.

The Similarity Renormalization Group (SRG) [7–11] aims

at the pre-diagonalizationof the Hamiltonian for a given basis

by means of a unitary transformation implemented through

the renormalization-groupflow equation:

dHα

dα

= [ηα,Hα] ,

(1)

where α is the flow parameter and Hαthe evolved Hamilto-

nian, with H0 = H being the initial or ‘bare’ Hamiltonian.

The anti-hermitian generator ηα defines the specifics of the

flow evolution, e.g. the representation with respect to which

the Hamiltonian should become diagonal or block-diagonal.

Various choicesfor this generatorhavebeen investigated[11],

we restrict ourselves to the simple generator [7, 9]

ηα= [Tint,Hα] (2)

with Tint= T − Tcmbeing the intrinsic kinetic energy, which

leads to a pre-diagonalizationof the Hamiltonian with respect

to the eigenbasis of the kinetic energyor momentumoperator.

Once the generator is fixed, the Hamiltonian and all operators

of interest can be evolved easily using a matrix representation

of the flow equation (1).

In A-bodyspace the evolutiongenerates up to A-bodyoper-

atorseveniftheinitialHamiltoniancontainsonlyuptotwo-or

three-body operators. For reasons of practicability one has to

truncate the evolution at some low particle number—typically

this is done by solving the evolution equation in a matrix rep-

resentation in two- or three-body space. For the moment we

restrict ourselves to transformations in two-body space, i.e.,

we will discard any induced three-body interactions.

The aim of the Unitary Correlation Operator Method

(UCOM) [10, 12–14, 16] is to explicitly treat short-range cor-

relations inducedby the nuclearinteractionvia a static unitary

transformation. This transformationcan either be used to cor-

relate the many-body states or to similarity transform opera-

tors of interest, e.g. the Hamiltonian

˜ H = C†HC ,

(3)

using the correlation operator C. The dominant short-range

correlations are induced by the strong short-range repulsion

and the tensor part of the nuclear interaction. Therefore the

correlation operator is written as a product of two unitary op-

erators, Crfor the central correlations and CΩfor the tensor

correlations. We choose an explicit form of the correlation

operators:

C = CΩCr= exp

?

− i

?

j<k

gΩ,jk

?

exp

?

− i

?

j<k

gr,jk

?

(4)

with the following ansatz for hermitian generators grand gΩ:

gr=1

gΩ=3

2[qrs(r) + s(r)qr] ,

2[(σ1· r)(σ2· qΩ) + (σ1· qΩ)(σ2· r)] ,

(5)

where qr=1

The strengths and radial dependencies of the two transforma-

tions are governed by the correlation functions s(r) and ϑ(r)

for the central and tensor correlations, respectively. One can

obtain these functions via an energy minimization in the two-

body system [16]. Recently, we have also employed the SRG

as tool for the determination of the UCOM correlation func-

tions s(r) and ϑ(r) as discussed in Refs. [9, 10]. Here, we will

use these SRG-optimized UCOM correlation functions only.

Though the SRG- and UCOM-transformations have a dif-

ferent formal background, they address the same physics of

short-range correlations. A first connection becomes clear at

the level of the generators [8]—the SRG generator (2) in two-

body space at α = 0 reveals the same operator structures that

appear in the UCOM-generators (5). At the level of matrix

elements, both the SRG- and UCOM-transformations lead to

a suppression of the off-diagonal momentum-space matrix el-

ements and an enhancement of the low-momentum matrix el-

ements as discussed in detail in Ref. [10].

In the following, we employ both transformations to gener-

ate one-parameterfamilies of phase-shiftequivalenttwo-body

interactions starting from a specific initial NN-interaction,

the Argonne V18 (AV18) in our case.

transformation the flow parameter α directly spans this fam-

ily of two-body interactions. We will study two versions of

the SRG-transformation, one where the flow equations are

solved for all partial waves and one where only the partial-

waves containing relative S-waves, i.e. the1S0and the cou-

pled3S1−3D1partial waves, are transformed. The latter is

motivated by the fact that short-range correlations affect the

S-wave channels most, because for all higher orbital angular

momenta the relative wave functions are suppressed by the

centrifugal barrier at short distances. We use the label ‘SRG’

for the fully transformed interactions and ‘S-SRG’ for the S-

wave-only transformations. For the UCOM-transformation

we use correlation functions determined from SRG-evolved

two-body wave functions as discussed in Refs. [9, 10], thus

the flow parameter α also spans a family of different UCOM-

transformed interactions. Note that the standard formulation

of UCOM only uses different transformationsfor the different

(S,T)-channels. We thus use the SRG-evolved wave func-

tions for the lowest partial waves for each (S,T)-channel to

define the correlation functions, leading to a transformed in-

teraction labelled ‘UCOM(SRG)’. Analogously to the S-SRG

2(r

r·q+q·r

r), qΩ= q−r

r·qr, and q =1

2[p1−p2].

For the SRG-

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00.040.080.12

0.16

0.2

α [fm4]

-30

-29

-28

-27

-26

-25

-24

.

E(4He) [MeV]

FIG. 1:

the flow parameter α obtained from a converged no-core shell

model calculation using the UCOM(SRG)-transformed (•), the S-

UCOM(SRG)-transformed (?), the SRG-transformed (?), or the S-

SRG-transformed (?) AV18 potential. The horizontal lines indicate

the experimental binding energy (

energy for the bare AV18 two-body interaction (

(color online) Binding energy of

4He as function of

) and the exact ground state

) [17].

transformation,we canalso use an S-wave-onlyUCOM trans-

formation, denoted ‘S-UCOM(SRG)’, which acts only in the

1S0and the coupled3S1−3D1partial waves.

So far, we have assumed that both transformationsare eval-

uated in two-body space, leading to a transformed interaction

containing two-body terms only. A consistent first-principles

treatment requires the transformation to be performed in A-

body space, leading to a hierarchy of induced interactions up

to the A-body level, as mentioned earlier. The most advanced

attempts along these lines use the full SRG-evolution at the

three-body level to construct a consistently transformed two-

plus three-nucleon interaction [5]. The use of those two- plus

three-body interactions in many-body calculations is very de-

manding and presently limited to rather small model spaces.

Therefore, we follow a more pragmatic path in this work.

We evaluate the unitary transformations at the two-body level

and mimic the three-body contributions (genuine plus in-

duced) through a simple phenomenological three-body inter-

action. By using a simplified three-nucleon (3N) interaction,

e.g., a regularized contact or a Gaussian interaction, the cal-

culation of the three-body matrix elements becomes formally

and computationally much less demanding. This allows us to

study the impact of 3N interactions on various nuclear struc-

ture observables for nuclei and model spaces beyond the do-

main accessible with realistic 3N interactions. Furthermore,

we can developand benchmarkapproximatetreatments of the

three-body contributions and establish the technical frame-

worktoinclude3Ninteractionintodifferentmany-bodymeth-

ods.

The parameters of the phenomenological 3N interactions

will be adjusted depending on the flow parameter α of

the transformed two-nucleon (NN) interaction. For a wide

range of α parameters the transformed two-body interaction

alone produces an overbinding compared to the experimen-

tal ground state energy. This is illustrated in Fig. 1 for the

ground-state energy of4He as function of α obtained in con-

vergedno-coreshellmodelcalculationsfortheUCOM(SRG)-

, the S-UCOM(SRG)-, the SRG-, and the S-SRG-transformed

AV18 interaction. Thus the additional phenomenological in-

teraction, which mimics the net effect of the genuine and the

induced 3N interaction, has to be repulsive in order to lead

to a4He binding energy consistent with experiment. Note

that the phenomenological three-body forces that are used in

connection with the bare AV18 interaction are generally at-

tractive. Thus the induced 3N interaction resulting from the

unitary transformation of the NN interaction alone has to be

repulsiveand sufficiently strong to create an over-all repulsive

three-body contribution.

B.Three-Body Contact Interaction

The simplest choice for a phenomenological3N interaction

is a spin-isopin-independentcontact interaction

V3N= C3Nδ(3)(x1− x2) δ(3)(x1− x3)(6)

with variable strength C3N. Despite its simplicity it allows us

to study the impact of a 3N interaction on bulk observables

like ground-stateenergies or chargeradii. Obviouslythis sim-

plistic choice offers substantial computational advantages.

For evaluating the matrix elements of a realistic 3N in-

teraction for the use in configuration-space Hartree-Fock or

no-core shell model type calculations one typically adopts

a two-step procedure: First the matrix elements are eval-

uated in a Jacobi-coordinate basis for the relative motion

in the three-nucleon system. Then, through a sequence of

Talmi-Moshinski transformations and angular momentum re-

couplings, the matrix elements are transformed into the m-

scheme to perform the many-body calculation. Both steps

are non-trivial and computationally demanding, thus limiting

the model-space sizes for which those matrix elements can be

handled.

In contrast, the matrix elements of the contact interac-

tion can be directly evaluated in the m-scheme in a straight-

forward manner. We first consider the matrix elements of the

3N contact interaction with respect to the spatial part of three-

particle product states in the harmonic oscillator basis

?n1l1ml1,n2l2ml2,n3l3ml3|V3N|n4l4ml4,n5l5ml5,n6l6ml6? . (7)

The spin and isospin quantum numbers and the antisym-

metrization will be included subsequently. We can insert a

unitoperatorinpositionrepresentationusingcartesiancoordi-

nates and directly evaluate the Kronecker-deltas. This leaves

us with a single integration over a single-particle coordinate,

which can be rewritten in spherical coordinates. Introducing

the position representation of the harmonic oscillator single-

particle states, φnlml(x) = Rnl(x)Ylml(Ω), with radial wave

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4

functions Rnl(x) and spherical harmonics Ylml(Ω), we obtain:

?n1l1ml1,n2l2ml2,n3l3ml3|V3N|n4l4ml4,n5l5ml5,n6l6ml6?

= C3N

dxx2Rn1l1(x)Rn2l2(x)Rn3l3(x)

?

× Rn4l4(x)Rn5l5(x)Rn6l6(x)

dΩ Y∗

×

?

l1ml1(Ω)Y∗

× Yl4ml4(Ω)Yl5ml5(Ω)Yl6ml6(Ω) .

l2ml2(Ω)Y∗

l3ml3(Ω)

(8)

The integraloverthe six radial wave functionsRnl(x) has to be

calculatednumericallywhiletheintegraloverthesixspherical

harmonics Ylml(Ω) can be evaluated analytically. The product

of three spherical harmonics can be reduced to one spherical

harmonic and the integral over the remaining two spherical

harmonics can be solved analytically, leading to

?

dΩY∗

l1ml1(Ω)Y∗

1

16π2ˆl1ˆl2ˆl3ˆl4ˆl5ˆl6

l2ml2(Ω)Y∗

l3ml3(Ω)Yl4ml4(Ω)Yl5ml5(Ω)Yl6ml6(Ω)

1

2L2+ 1

=

?

L1L2L3

ML1ML2ML3

?L1 l3

ML1

c

L3

ML3

c

× c

× c

× c

?l1l2

?l1

?l4

0 0

???L1

0

?

c

0 0

?

?

???L2

ML1ml3

?L3

0

?

c

?l4l5

???

???

0 0

L2

ML2

L2

ML2

???L3

?

?

0

?

c

?L3 l6

0 0

???L2

0

?

l2

ml1ml2

???

???

L1

?L1

ML3ml6

l3

l5

ml4ml5

l6

(9)

withˆl =

coefficients.

We precompute and store those angular integrals as well as

theradialintegralsin(8). Theinclusionofthespinandisospin

quantum numbers, the coupling of the single-particle orbital

angular momenta and the spins, and the antisymmetrization

are then done on the flight during the many-body calculation.

This makes calculations in large model spaces feasible.

For applications beyond the mean-field level a regulariza-

tion of the contact interaction is inevitable. However, the

regularization should preserve the simplicity of the matrix-

element calculation,which rules out momentum-spacecutoffs

and such. Hence, we introduce an energy cut-off parameter

e3N, which defines an upper bound for total oscillator energy

ofthethree-particlestate, (2n1+l1)+(2n2+l2)+(2n3+l3) ≤ e3N.

The implementation of this cutoff is trivial and it preserves all

computational advantages of the contact interaction.

√2l + 1 and c

?

l1

l2

ml1ml2

???L

ML

?

being Clebsch-Gordan

III.MANY-BODY CALCULATIONS

We adopt the 3N contact interaction together unitarily

transformed NN interactions for the study of the systematics

of nuclear ground-state energies and charge radii throughout

the whole mass range from4He to208Pb using Hartree-Fock

and many-bodyperturbation theory.

A.Hartree-Fock Approximation

WehaveemployedtheHartree-Fock(HF)approximationas

a first indicator for the gross systematics of binding energies

andchargeradiiobtainedwith unitarilytransformedtwo-body

interactions in Refs. [10, 15] already. In order to assess the

impact of 3N contact interactions we extend our HF frame-

work in a first step.

All calculations are based on the translationally invariant

Hamiltonian

Hint= Tint+ VNN+ V3N= H(2)

int+ V3N

(10)

with VNNbeing the UCOM- or SRG-transformed NN inter-

action and Tint = T − Tcmthe intrinsic kinetic energy. This

Hamiltonian includes all charge dependent and electromag-

netic terms of the transformed AV18 potential as well as the

phenomenologicalthree-body force.

The HF equations are formulated in a harmonic oscilla-

tor basis representation, i.e., the single-particle states are ex-

panded in the harmonic oscillator states:

|νljmmt? =

?

n

C(νljmt)

n

|nljmmt? ,

(11)

where |nljmmt? denotes the harmonic oscillator eigenstates

with radial quantum number n, orbital angular momentum l,

total angular momentum j with projection m, and isospin pro-

jection quantum number mt. Since we only consider closed-

shell nuclei in the following, the expansion coefficients are

independent of m. The HF equations can now be written as

?

¯ n

h(ljmt)

n¯ n

C(νljmt)

¯ n

= ε(νljmt)C(νljmt)

n

(12)

with the single-particle energies ε(νljmt). The matrix elements

of the single-particle HF Hamiltonian

h(ljmt)

n¯ n

=

?

l′j′m′

t

?

n′¯ n′

?nljmt,n′l′j′m′

t|H(2)

int|¯ nljmt, ¯ n′l′j′m′

t? ̺(l′j′m′

t)

¯ n′n′

+1

2

?

l′j′m′

l′′j′′m′′

t

t

?

n′n′′

¯ n′¯ n′′

?nljmt,n′l′j′m′

t,n′′l′′j′′m′′

t|×

× V3N|¯ nljmt, ¯ n′l′j′m′

t, ¯ n′′l′′j′′m′′

t? ̺(l′j′m′

t)

¯ n′n′

̺(l′′j′′m′′

¯ n′′n′′

t)

(13)

are obtained by contractions of the antisymmetrized matrix

elements of the two-bodypart of the Hamiltonian H(2)

three-body interaction V3Nwith the one-body density matrix

given by

intand the

̺(ljmt)

¯ nn

=

?

ν

O(νljmt)C(νljmt)∗

¯ n

C(νljmt)

n

(14)

with O(νljmt)being the numberof occupiedmagneticsublevels

which is 2j + 1 for closed-shell nuclei.

In the following the HF approach is applied to selected

closed-shell nuclei from4He to208Pb. The HF equations are

solved iteratively until full self-consistency is reached. The

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5

-14

-12

-10

-8

-6

-4

.

E/A[MeV]

2

3

4

5

.

rch[fm]

4He

16O

24O

34Si

40Ca

48Ca

48Ni

56Ni

60Ni

78Ni

88Sr

90Zr

100Sn

114Sn

132Sn

146Gd

208Pb

FIG. 2: (color online) Ground-state energies per nucleon and charge

radii of selected closed-shell nuclei resulting from HF calculations

based on pure two-body interactions for emax = 10: UCOM(SRG)

with α = 0.16fm4(•), S-UCOM(SRG) with α = 0.16fm4(?), SRG

with α = 0.10fm4(?), S-SRG with α = 0.10fm4(?). The bars

indicate the experimental values [18, 19].

model space is truncated at a given major oscillator quantum

number e = 2n+l ≤ emax, where emax= 10 is sufficient to ob-

tain convergedground-stateenergies and radii at the HF level.

The oscillator parameteris chosen for each nucleus separately

such that the experimental charge radius is reproduced by a

shell-model Slater determinant built from harmonic oscillator

single-particle states.

As a first illustration of the behavior of unitarily trans-

formed two-body interactions Fig. 2 summarizes the ground-

state energies per nucleon and the charge radii obtained at

the HF level for nuclei up to208Pb. We adopt four differ-

ent two-body interactions—UCOM(SRG), S-UCOM(SRG),

SRG, and S-SRG—with flow parameters relevant for the later

calculations including the 3N contact interaction. We observe

that the general trend of the binding energies and charge radii

is similar for the UCOM(SRG), the S-UCOM(SRG), and the

S-SRG interactions. All three interactions produce binding

energies that are within 2 MeV per nucleon of the experimen-

tal values for the whole mass range. By includingcorrelations

beyond HF, e.g., through many-body perturbation theory, all

interactions would lead to an overbindingcompared to exper-

iment. At the same time the charge radii are underestimated

for all but the lightest isotopes. Those systematic deviations

can be remedied by a repulsive 3N interaction, as it will be

included in the next step.

The SRG-transformed interaction exhibits a vastly differ-

ent behavior.The binding energies per nucleon increase

rapidly with mass number, leading to an completely unphysi-

cal overbinding already at the HF level for intermediate and

heavy nuclei. At the same time the charge radii are even

smaller than the ones obtained with the other transformed

interactions. Those strong systematic deviations have to be

compensated by the 3N interaction that is generated from the

-8

-6

-4

.

E/A[MeV]

2

3

4

5

.

rch[fm]

4He

16O

24O

34Si

40Ca

48Ca

48Ni

56Ni

60Ni

78Ni

88Sr

90Zr

100Sn

114Sn

132Sn

146Gd

208Pb

FIG. 3: (color online) Ground-state energies per nucleon and charge

radii of selected closed-shell nuclei resulting from HF calculations

for the pure two-body interaction S-UCOM(SRG) for emax= 10 and

different flow parameters: α = 0.04fm4(•), α = 0.12fm4(?), α =

0.16fm4(?). The bars indicate the experimental values [18, 19].

initial NN potential in the course of the SRG-evolution. Be-

cause of the mere size of the three-body corrections needed

one cannot expect a simple phenomenological interaction to

beadequatetocapturethemainphysicscontainedinthethree-

body contributions. Therefore, we will not consider the fully

SRG-transformed interactions in the following.

Before including the 3N contact interaction explicitly, we

analyze the dependence of the HF results obtained with the

transformed two-body interactions on the flow parameter α.

In Fig. 3 the binding energies and charge radii for the S-

UCOM(SRG) interactions with α = 0.04fm4, 0.12fm4, and

0.16fm4are shown. For the smallest flow parameter α =

0.04fm4the ground-state energies reproduce the systemat-

ics of the experimental values up to a constant shift. The

missing binding energy can be explained by beyond-HF cor-

relations that can be recovered, e.g., by perturbation theory.

This flow parameter would be used for calculations based on

the pure NN interaction, as they are discussed in detail in

Refs. [9, 10, 15].

When increasing the flow parameter entering into the con-

struction of the S-UCOM(SRG) interaction to α = 0.12fm4

or 0.16fm4the ground-state energy at the HF level decreases

substantially. For most nuclei the binding energy per nucleon

morethandoubleswhengoingfromα = 0.04fm4to 0.16fm4.

For heavier nuclei the increase is larger, thus leading to a tilt

of the ground-state energy systematics with respect to the ex-

perimental behavior. Unlike the energies, the charge radii ex-

hibit a very weak α-dependence as shown in the lower panel

of Fig. 3. For all flow parameters considered here, the radii

are somewhat underestimated. The situation is very similar

for the UCOM(SRG) and the S-SRG interactions.

This general phenomenology of ground-state energies and

charge radii obtained from unitarily transformed interactions

at larger flow parameters illustrates that the purely repulsive