Table Of ContentQuantum Monte Carlo study of large spin-polarized tritium clusters
I. Beˇsli´c
Faculty of Science, University of Split, HR-21000 Split, Croatia and
ICAM/I2CAM, 4415 Chem Annex, UC Davis, One Shield Avenue, Davis, CA 95616
L. Vranjeˇs Marki´c
Faculty of Science, University of Split, HR-21000 Split, Croatia and
Institut fu¨r Theoretische Physik, Johannes Kepler Universita¨t, A-4040 Linz, Austria
J. Boronat
0
Departament de F´ısica i Enginyeria Nuclear, Campus Nord B4-B5,
1
Universitat Polit`ecnica de Catalunya, E-08034 Barcelona, Spain
0
2 (Dated: January 8, 2010)
n This work expands recent investigations in the field of spin-polarized tritium (T↓) clusters. We
a reporttheresultsforthegroundstateenergyandstructuralpropertiesoflargeT↓clustersconsisting
J of up to 320 atoms. All calculations have been performed with variational and diffusion Monte
8 Carlo methods, using an accurate ab initio interatomic potential. Our results for N ≤ 40 are
in good agreement with results obtained by other groups. Using a liquid-drop expression for the
] energyperparticle,weestimatetheliquidequilibriumdensity,whichisingoodagreementwithour
r
e recently obtained results for bulk T↓. In addition, the calculations of the energy for large clusters
h have allowed for an estimation of the surface tension. From the mean-square radius of the drop,
t determined using unbiased estimators, we determine the dependence of the radii on the size of the
o
cluster and extract theunit radius of the T↓ liquid.
.
t
a
PACSnumbers: 67.65.+z,02.70.Ss
m
-
d INTRODUCTION mentmethod,wasreportedbySalcietal[11]. Becauseof
n
evidentresemblanceofbosonicT↓and4HeatomsBlume
o
c The extreme quantum nature of electron spin- et al. [1] also compared general properties of both types
[ of clusters. They showed that common attributes of T↓
polarized hydrogen (H↓) and its isotopes, spin-polarized
clusters are weakerbinding and greaterinterparticle dis-
1 deuterium (D↓) and spin-polarized tritium (T↓), pro-
v moted renewed theoretical interest [1–6] in their con- tances between atoms, in comparison with 4He clusters
4 having the same number of atoms. In the same work,
densedphases. Theyarecharacterizedbythesmallmass
2 resultsofcoupled-channelscatteringcalculationsfor two
oftheiratomsandtheirweaklyattractiveinteratomicpo-
2
1 tential, which is very accurately determined in ab initio T↓ atoms are reported, indicating the possibility for for-
mation of a tritium condensate using its broadFeshbach
. calculations. Already in the seventies, several theoret-
1 resonance.
ical predictions of the spin-polarized hydrogen systems
0
0 appeared[7,8]. In1976,StwallyandNosanow[9]under- Mixed clusters consisting of spin-polarized hydrogen-
1 linedH↓asafirstcandidateforachievingaBose-Einstein tritium and deuterium-tritium atoms have also been in-
: condensate(BEC)state. Theirtheoreticalpredictionwas vestigated. [4, 5] It has been shown that three T↓ atoms
v
i experimentallyconfirmedin1998byFriedetal.[10],who areneededtobindoneD↓atominastablesystem,mak-
X
succeeded to overcome demanding experimental obsta- ing thus clusters (T↓)ND↓ stable for all N ≥ 3. On
r cles and reported the formation of a BEC state with H↓ the contrary, it has been shown that even 60 T↓ are not
a
atoms. enough to bind one H↓ in a stable system. Namely, the
Furtherinvestigationrelatedtopossiblecandidatesfor ground-stateenergyofthecluster(T↓)60H↓iswithinthe
BECstateinhydrogensystemswasdonebyBlumeetal.. errorbarequaltothegroundstateenergyof(T↓)60,lead-
Intheirwork[1],spin-polarizedtritiumclustersaswellas ing to the conclusion that clusters (T↓)NH↓ for N ≤ 60
optically pumped tritium condensate were theoretically areeffectivelyunstableorareatthethresholdofbinding.
investigatedforthefirsttime. Blumeetal. reportedtheir Due to the more complicated calculations in the case of
DMC results for the ground state energy and structural severalfermionicD↓atoms,sofaronlythestabilitylimits
propertiesofT↓clustersconsistingofupto40atoms. In of small mixed spin-polarizeddeuterium-tritium clusters
addition, it was also shown that the smallest T↓ cluster having up to 5 D↓ atoms have been examined. [4]
is a trimer, i.e. (T↓) ; negative ground-state energy was Despite the lack of experimental verification, bulk
3
not obtained for the dimer (T↓) , which means that T↓ properties of all spin-polarized hydrogen isotopes have
2
trimer is an example of Borromean or halo state. The been theoretically predicted. Bulk properties of the D↓
same conclusion for (T↓) , obtained with the finite ele- system are conditioned by the number of occupied nu-
3
2
clear spin states. [12–14] If only one nuclear spin state METHOD
is occupied (D↓ ), the system is in a gas state at zero
1
pressure, while in the case of two (D↓2) or three (D↓3) The starting point of the DMC method is the
equally occupied nuclear spin states, the systemremains Schr¨odinger equation written in imaginary time,
liquid at zero pressure and zero temperature. Exten-
∂Ψ(R,t)
sive investigations of the H↓ and T↓ bulk systems have −¯h =(H −E )Ψ(R,t) , (1)
r
been carriedout recently with the diffusion Monte Carlo ∂t
(DMC)method[2,6],whichprovidesexactresultswithin where E is a constant acting as a reference energy and
r
errorbars for bosonic systems. Using the DMC method R≡(r ,...,r ) collectively denotes particle positions.
1 N
for bulk H↓ and T↓, the energy per particle, structural The N-particle Hamiltonian, H, is given as
properties, as well as densities and the pressure of the
gas (liquid)-solid transition have been predicted. An ac- ¯h2 N N
H =− ∇2+ V(r ) , (2)
curatecalculationoftheground-stateenergyperparticle 2m i ij
i=1 i<j
inbulkH↓hasallowedthe confirmationofits gasnature X X
inthe limitofzerotemperatureandupto170bar,point where V(r) is the interaction potential. The inter-
at which H↓ solidifies. A similar investigation of bulk atomic interaction between tritium atoms is described
T↓ has revealedthat the system is a liquid up to 9 bars, with the spin-independent central triplet pair potential
where it crystallizes. [6] b3Σ+, which was determined in an essentially exact way
u
by Kolos and Wolniewicz [15]. As in our recent DMC
calculations of bulk H↓ and T↓,[2, 6] we have used the
recent extention of Kolos and Wolniewicz data to larger
interparticle distances by Jamieson et al. (JDW). [16]
Inthis work,weexpandpreviouslyreportedstudies of
The potential is finally constructed using a cubic spline
pure T↓ clusters.[1, 4, 5] We report the ground-state en-
interpolationofJDW data,whichis smoothly connected
ergy of clusters having up to 320 atoms, as well as their
tothe long-rangebehaviorofthe T↓-T↓potentialascal-
structural properties, obtained with the DMC method.
culatedbyYanetal..[17]TheJDWpotentialusedinthe
From the density profiles, we estimate the thickness of
presentworkhasacorediameterσ =3.67˚A andamini-
the clusters’ surface. Justification for carrying out de-
mumof−6.49Katadistance4.14˚A.Wehavepreviously
manding calculations for large clusters lies in the fact
verified that the addition of mass-dependent adiabatic
thatthepresentresultsforclusterscanbeusedtoextrap-
corrections(as calculated by Kolos and Rychlewski [18])
olate precise equilibrium T↓ bulk properties. A goal of
totheJDWpotentialdoesnotchangetheenergyofbulk
our investigationis to examine the validity of the liquid-
spin-polarized tritium. [6] It is worth mentioning that
drop formulas when they are applied to T↓ clusters, as
withintheBorn-Oppenheimerapproximationithasbeen
it was done in the past for 3He and 4He clusters.[24, 25]
explicitlyshownthatinthespin-alignedelectronicstate,
Inheliumclusters,liquid-dropformulasweresuccessfully
tritium nuclei behave as effective bosons. [19]
applied and the results for the equilibrium energy per
DMC solves stochastically the Schr¨odinger equation
particle and the unit liquid radius were in good agree-
(1) by multiplying Ψ(R,t) with the ψ(R), a trial wave
ment with experimental studies. [24, 25] We have used
function used for importance sampling, and rewriting
the energy per particle of the T↓ clusters to extrapo-
Eq. (1) in terms of the mixed distribution Φ(R,t) =
late the equilibrium energy per particle in bulk T↓. We
Ψ(R,t)ψ(R). Within the Monte Carlo framework,
comparetheestimatedresultwiththeenergyperparticle
Φ(R,t) is represented by a set of walkers. In the limit
calculatedinarecentDMCstudyofabulkT↓.[6]Inad-
t → ∞ only the lowest energy eigenfunction, not or-
dition, the surface tension of liquid T↓ is estimated and
thogonalto ψ(R), survivesand then the sampling of the
compared with known results for 3He and 4He liquids.
ground state is effectively achieved. Apart from statisti-
Furthermore,we extractthe unit radius of the liquid us-
caluncertainties,theenergyofaN-bodybosonicsystem
ing the average distance of the particles to the centre of
is exactly calculated.
mass of the cluster.
InthepresentsimulationsJastrowtrialwavefunctions
have been used,
N
ψ (R)= f(r ), (3)
J ij
In Sec. II, we report briefly the DMC method and
i<j
discussthetrialwavefunctionsusedforimportancesam- Y
plingoftheclusters. Sec. IIIreportstheresultsobtained with a two-body correlationfunction f(r),
by the DMC simulations. Finally, Sec. IV comprises a
b 5
summary of the work and an account of the main con- f(r )=exp − −srn , (4)
clusions. ij " (cid:18)rij(cid:19) ij#
3
where n = 1 or n = 2, depending on the size of the (T↓) . These differencesaremainly due to the factthat
N
cluster, and b and s are variational parameters. Previ- there are small differences in the potential of interaction
ous experience in work with small pure and mixed spin- employed in two simulations. Namely, Blume et al. [1]
polarized tritium clusters [4] has shown that the best haveincludedintheHamiltonianthedampedthree-body
choice for the two-body correlation function is obtained Axilrod-Teller potential term [22] which causes a slight
with n=1. We have used this type of function for clus- raise of the ground-state energy.
ters having N ≤ 60 atoms, but for larger clusters the InRef. 1,weakerbindingandgreaterspreadintheT↓
variational energies obtained with this type of function clusters were emphasized as the main difference between
worsensignificantly. Then,forlargerclusters,itisbetter small(T↓) and(4He) clusters,forthesameN andup
N N
toconsiderthemodelwithn=2,whichissimilartof(r) to 40. Similar comparisoncan be done for largeclusters.
whichhasbeenusedinrecentinvestigationsofvorticesin Namely, the energy per particle of the largest investi-
large 4He clusters. [20] Eq. (4) with n= 2 defines much gated 4He cluster with the DMC method, (4He) , is
112
bettertheconfinementofspaceinwhichlargenumberof -3.780(3) K [23]. This can be compared with the energy
T↓ atoms is settled and definitely provides better VMC per particle of the largest T↓ cluster, (T↓) , which is -
320
energies for clusters having N ≥80 atoms. 2.286(8)K(TableI). Itisclearfromtheseresultsthatin
The optimization of the trial wave functions has been the(4He) cluster,whichconsistsofalmostthreetimes
112
done for all clusters by means of the variational Monte smallernumber ofatomsthan(T↓) cluster,bindingis
320
Carlo method. For clusters with N ≤ 60 the best varia- significantly stronger. From that, we conclude that the
tionalparametersvaryfromb=3.574˚Atob=3.605˚A and binding in large T↓ clusters is very weak, as it was al-
from s=0.0328 ˚A−1 to s=0.0073 ˚A−1 for increasing N. ready concluded for small clusters [1].
In case of clusters having N ≥ 80 atoms, the parame- The energy per particle of quantum liquid clusters
ter b assumes values from 3.574 ˚A to 3.605 ˚A, while at as a function of N is well reproduced by a liquid-drop
the same time s varies from 000162 ˚A−2 to 0.0000145 model, [24, 25]
˚A−2. It is worthnoticing thatinbothtypes oftwo-body
E(N)/N =E +xE +x2E , (5)
correlation functions (n =1,2) the parameter b remains v s c
practically constant, while s always decreases with N. whereE ,E andE arerespectivelythevolume,surface
v s c
For severalclusters we have verified that 1000walkers and curvature terms, and the variable x is defined as
are enough for excluding the bias coming from the size x=N−1/3.
of the population ensemble used in a simulation. Thus, In Fig. 2, we have plotted results for the energy per
we have decided to employ this number of walkers in all particle from Table I, as well as a line on the top of
the remaining DMC calculations. The same conclusion data, which represents the best fit with expression (5).
emergedalsofrompreviousexperienceinpureandmixed We have included in our fit all the investigated clusters
T↓ clusters. and the best set of parameters obtained with the above
In order to eliminate bias coming from the time-step mentioned fit is: E = -3.66(3) K, E = 10.2(2) K and
v s
value usedin simulations,allcalculationshavebeen per- E = -6.1(4) K.
c
formed with several ∆t time-steps which assume values TheparameterE representstheenergyperparticleof
v
withintheinterval5×10−4−1.3×10−3K−1. Fromtheob- bulk liquid T↓ at the equilibrium density. This extrapo-
tainedresults,wehaveextrapolatedtheresultto∆t→0. lated result is in a very good agreement with our recent
In accordance with the DMC method used in this work, results obtained in calculations of bulk T↓ [6]. With the
whichisaccuratetosecondorderinthetimestep[21],the DMC method we obtained ρ = 0.007466(7) ˚A−3 as the
0
extrapolationis made witha quadraticfunction. Second equilibrium density of liquid T↓ and e = −3.656(4) K
0
order DMC enables the use of greater time-steps than as the energy per particle at that density.
the linear DMC method. It is also important to emphasize that the parameters
E , E and E have been obtained without including in
v s c
the fit the bulk energy per particle at equilibrium. Fur-
RESULTS
thermore, the parameters in Eq. (5) remain practically
the same when the energy per particle at the equilib-
Our DMC results for the ground-stateenergyper par- rium density is included in the fit. Thus, the demanding
ticle and radii of the investigated clusters are given in DMC calculations have been worthwhile because the re-
Table I. For clusters consisting of up to 10 T↓ atoms sult of our liquid-drop model (5) does not depend on
we have already shown [4] good agreement with results the knowledge of the equilibrium energy per particle of
obtained by Blume et al. [1]. Here, we extend this com- the bulk. Also, we have determined that (T↓) is the
280
parison for clusters consisting of up to 40 T↓ atoms. In ’smallest’ cluster needed to be included in the fit in or-
Fig. 1, comparisonof our results and the ones by Blume der to extrapolate the parameter E properly. Namely,
v
et al. is shown. As in the case of small clusters, we using just DMC results for clusters havingN =20−280
report slightly lower ground-state energy for all clusters atoms in the fit we get E = -3.69(3) K, which is within
v
4
the error bars the same as the result obtained including and
the largest cluster (N=320, E =-3.66(3) K). However,
v d
the extrapolation of the equilibrium energy per particle g (x)=cx+ . (10)
2
x
with fits including just results for clusters smaller than
(T↓)280 always produces lower energy than the one cal- The parameters extracted from the fits are: a =
culatedforthe bulk. Forexample,the extrapolatedbulk 1.85(12)˚A, b = 2.34(3)˚A, c = 2.55(1)˚A and d =
energy with results for clusters having up to 120 atoms 3.73(12)˚A.Using these interpolationparametersand the
is -3.85(2) K, which is around 5% lower than e0. Sim- definition of unit radiigiven in (8), in the limit N →∞,
ilar conclusions about this fit emerged for 4He clusters we extract an equilibrium radius r = 3.02(4) ˚A using
0
[25], where it was emphasized that an accurate extrapo- the function (9) and r = 3.29(1) ˚A using the function
0
lation of the bulk equilibrium energy from finite cluster (10). Since in both cases the quality of the fit is very
calculations should include relatively large clusters. good,we cannotstate which of the two extractedresults
Wehavealsotriedtofitallthe obtaineddataforE/N for equilibrium unit radius should be considered as the
with a linear function in N−1/3, as in Ref. 25, but that better estimation. We can thus only conclude that the
kind of fit has not been so precise as the one performed equilibriumunit radiusassumesavalue withinthe inter-
including a quadratic dependence on N−1/3. With the val from 3.02(4) ˚A to 3.29(1) ˚A.
linear fit Ev= -3.26(3) K, which is around 11% higher Therefore, the estimation of the equilibrium unit ra-
than the e0 obtained for bulk, showing the necessity of dius from the results obtained in clusters calculations
including the second-order term. is very sensitive to the choice of interpolating function.
The second parameter Es extracted from Eq. (5) is Because of that, we decided to include the unit radius
related to the surface tension of liquid T↓ through 3.18(1) ˚A , derived from the bulk T↓, in the estimation
of the surface tension.
E
t= s , (6) It is useful to compare the value r with σ = 3.67 ˚A.
4πr2 0
0 If we consider r as the radius of the sphere that one T↓
0
atom occupies at equilibrium density in a liquid, σ has
where r is the unit radius ofthe liquid. The unit radius
0
tobe smallerthan2r ,asitis inourcase. Onthe other
of the liquid can be determined in two ways, using the 0
hand,thevalueofr ofliquidT↓isalsogreaterthanthe
result of the equilibrium density of the bulk liquid 0
unit radius of liquid 4He (r =2.1799 ˚A) [25]. This also
0
4πr3ρ =1 , (7) explains the greater spread in T↓ clusters because it is
3 0 0 clear that T↓ atoms occupy more space than 4He atoms
at equilibrium density.
or from the expression
From r , it is possible to calculate the liquid surface
0
5 1/2 tensiont using expression(6); we haveobtainedt=0.08
r0(N)= hr2(N)i N−1/3 , (8) K˚A−2. There is no experimental result for the surface
3
(cid:20) (cid:21) tension of liquid T↓, and this prediction is, to the best
wherehr2(N)iisthemean-squareradiusofaclusterwith of our knowledge, the first estimation of t for liquid T↓.
N atoms. Contrary to liquid T↓, the surface tensions of 4He and
Using the result for the equilibrium density of the T↓ 3He liquids have been experimentally investigated and
liquid ρ = 0.007466(7) ˚A−3 [6] and Eq. (7), we obtain themeasuredvaluesarerespectively0.27K˚A−2 and0.11
r =3.180(1) ˚A . K˚A−2. [24] In the case of liquid T↓ our estimated value
0
The second method for obtaining the unit radius, of the surface tension is even smaller than the surface
Eq. (8), was previously used in the study of 4He clus- tension of the 3He liquid, although bulk T↓ is a bosonic
ters. [24, 25] In Ref. 24, it is emphasized that only those system. Explanationforsuchasmallvalueofthesurface
4He clusters with more than ten atoms have a radius tension lies in the fact that the interaction between T↓
whichincreasesapproximatelyasN1/3. Sinceinourcal- atoms is described with a very shallow potential.
culations we obtain unbiased mean-square radii of clus- In addition to the ground-state energy, we have also
ters with pure estimators, we tried to interpolate our studied the structure of T↓ clusters. Exact estimators
data for clusters radii, R (N) = hr2(N)i, with sev- of DMC method have been employed to calculate values
cm
eral polynomial functions of the variable x = N1/3. As such as the pair distribution function P(r), as well as
inthecaseoftheinterpolationofthpeenergyperparticle, the distribution of particles with respect to the centre
wehaveincludedallclustershavingN =20−320atoms. of mass of the cluster ρ(r). Possible bias in our results
In Fig. 3, clusters’ radii obtained from the calculations coming from the type of trial wave function used in the
are plotted, as well as two lines on top of data which simulations is resolved with the use of pure estimators
represent two interpolations, using functions which ensure unbiased results. [26]
In Fig. 4, the density distributions of T↓ clusters hav-
g (x)=a+bx, (9) ing 40,80,120,180,240and320atoms areplotted. The
1
5
densityprofilesshowthattheclustersizegrowswhenthe caseofT↓clusters,weobservethatthesurfacethickness
number of atoms in the cluster increases, and that the is almost a linear function of N, up to N= 320 atoms.
central densities of the largest clusters are very similar We havetriedto predict the surface thickness ofclusters
to the bulk equilibrium density ρ =0.007466(7)˚A−3. having more than 320 atoms by fitting our data with
0
The increase of cluster size with increasing number of the function used in Ref. 27 to predict the width of a
atomscanalsobeseenfromthepairdistributionfunction free surface. However, with the present results we have
P(r) shown in Fig. 5 for the same clusters. P(r) is not been able to determine the asymptotic value of the
normalizedsuchthat P(r)r2dr =1. Asignificantdecay surface thickness. Saturation should be probably seen
of the peak height for the largest clusters, as well as the withresultsforclustershavingmorethanN=320atoms,
R
growing probability for larger interparticle distances in but the DMC calculations are already difficult with 320
large clusters, is a clear evidence of the size spreading atoms.
tendency.
The surface thickness of clusters s can be estimated
t
from the density profiles as a difference of radiiat which CONCLUSIONS
the central density ρ =ρ(r =0) has decreased from 90%
c
to 10% of its value. From the plotted density profiles in General characteristics of large spin-polarized tritium
Fig. 4 it is obvious that the density error bars are large T↓ clusters have been investigated using the DMC ap-
for small distances. In order to determine the central proach. The ground-state energies of clusters consisting
density as precisely as possible we have tried to fit the of up to 40 T↓ atoms have been compared with previ-
density profile with the function used in Ref. 27 ously published results. For clusters having more than
40 atoms the ground-state energies, as well as the struc-
ρ
0
ρ(r)= (1+eβ(r−r0))δ , (11) ture description, are determined for the first time. This
prediction relies on the use of a very precise potential of
where ρ , β, r and δ are fitting parameters and r is the interactionbetweenT↓-T↓atoms. Thepresentresultsfor
0 0
distancetothecentreofmassofthecluster. Wefindthat theground-stateclusters’energyarealsousedtoextract
for T↓ clusters Eq. (11) can be employed to model den- the energy per particle of liquid T↓ at equilibrium den-
sity profiles of small clusters, while for greater clusters sity using a liquid-drop model. The extrapolation using
thesamemodelreproducespoorlythecalculateddensity a liquid-drop formula gives a value Ev =−3.66(3) K for
profiles at small distances. Since the small distances are the energy per particle in the equilibrium bulk system,
important for our calculation, we have decided to fit the which is in very good agreement with the result from a
calculated density profiles to a constant function for dis- recentDMCcalculationofthebulk,e0 =−3.656(4)K[6].
tances up to some value r . We have varied the value of The radii of clusters are calculated with pure estima-
1
r from 2 ˚A to 4 ˚A, increasing it with the growing size tors and those results are used to estimate the interval
1
ofthecluster. Wehaveconsideredtheconstantobtained in which the unit radius of the liquid is expected. The
with the fit as a central density value and used it in fur- result for the unit radius from the bulk calculation lies
ther estimation of the clusters’ surface thickness. The in the estimated interval. The latter value of the bulk
results for the surface thickness are reported in Table I. unit radius has been employed to estimate the surface
Wecancompareourresultswiththe surfacethicknessof tensiont of bulk T↓, t=0.08K˚A−2. In addition, the sur-
4He clusters [24, 28]. Using the VMC method Pandhari- face thickness of clusters has been estimated from the
pande et al. [24] showed that in 4He clusters the surface clusters’ density profiles.
thickness is ∼7˚A for clusters N ≥ 112. In the case of As it is already shown for clusters consisting of up to
T↓ clusters, for N ≥ 100, the surface thickness is signif- 40T↓atoms [1], itis concludedthatlargespin-polarized
icantly greater than in 4He (Table I). This is expected tritiumclustersarelessboundandaremoredilutedthan
due to the evidently greaterinterparticledistances inT↓ 4He clusters with the same number of atoms, i.e., inter-
clusters, which is a direct consequence of the shallow at- particle distances are significantly greater in the corre-
tractive part of T↓-T↓ interaction potential. With the sponding T↓ clusters.
density functional approach, Stringari et al. [28] calcu- J. B. acknowledges support from DGI (Spain) Grant
latedthesurfacethicknessofseveral4He clustersandwe No. FIS2005-04181and Generalitatde Catalunya Grant
cancomparethoseresultswithourresultsforT↓clusters No. 2008SGR-04403. I.B and L.V.M. acknowledge
having20,40and240atoms. Thereportedsurfacethick- support from MSES (Croatia) under Grant No. 177-
nessforclusters4He ,4He and4He arerespectively 1770508-0493. I. B. also acknowledges support from
20 40 240
8.8˚A,9.0˚Aand9.3˚A.Acomparisonwith(T↓) reveals L’Or´ealADRIA d.o.o. andCroatiancommissionfor Un-
N
largersurfacethicknessof4He forN=20,40andsmaller esco, as well as from U.S. National Science Foundation
N
surfacethicknessforN=240. Also,itcanbenoticedthat I2CAM International Materials Institute Award, Grant
the surfacethicknessreportedbyStringariet al. isnota DMR-0645461. We also acknowledge the support of the
linear function of the number of atoms. Contrary,in the CentralComputingServicesattheJohannesKeplerUni-
6
versity in Linz, where part of the computations was per- [26] J. Casulleras and J. Boronat, Phys. Rev. B 52, 3654
formed. In addition, the resources of the Isabella cluster (1995).
atZagrebUniversityComputingCentre(Srce)andCroa- [27] J.M.Mar´ın, J.Boronat andJ.Casulleras, Phys.Rev.B
71, 144518 (2005).
tianNationalGridInfrastructure(CRONGI)wereused.
[28] S. Stringari and J. Treiner, J. Chem. Phys. 87, 5021
(1987).
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FIG. 1: Comparison of calculated ground-state ener-
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7
FIG. 1: I. Beˇsli´c et al.
FIG. 2: I. Beˇsli´c et al.
FIG. 3: I. Beˇsli´c et al.
8
FIG. 4: I. Beˇsli´c et al.
FIG. 5: I. Beˇsli´c et al.
