Variational calculation of 4He tetramer ground and excited states using a realistic pair potential

0.83 MB·English

by  E. Hiyama

Variational calculation of 4He tetramer ground and excited states
using a realistic pair potential
E. Hiyama∗
RIKEN Nishina Center, RIKEN, Wako 351-0198, Japan
M. Kamimura†
Department of Physics, Kyushu University, Fukuoka 812-8581, Japan,
RIKEN Nishina Center, RIKEN, Wako 351-0198, Japan
(Dated: January 12, 2012)
2
1 Wecalculated the4Hetrimerandtetramergroundandexcitedstateswith theLM2M2 potential
0 using our Gaussian expansion method (GEM) for ab initio variational calculations of few-body
2
systems. Themethod hasextensivelybeenusedfor avarietyofthree-,four- andfive-bodysystems

n in nuclear physics and exotic atomic/molecular physics. The trimer (tetramer) wave function is
a expanded in terms of symmetric three-(four-)body Gaussian basis functions, ranging from very
J compact to very diffuse, without assuming any pair correlation function. Calculated results of the

1 trimer ground and excited states are in excellent agreement with the literature. Binding energies
1 of the tetramer ground and excited states are obtained to be 558.98 mK and 127.33 mK (0.93 mK
below thetrimer ground state), respectively. Wefound that precisely the same shape of the short-

h] range correlation (rij <∼ 4˚A) in the dimer appear in the ground and excited states of trimer and
tetramer. The overlap function between the trimer excited state and the dimer and that between
p
the tetramer excited state and the trimer ground state are almost proportional to the dimer wave
-
m functionintheasymptoticregion(upto∼1000˚A).Alsothepaircorrelationfunctionsoftrimerand
tetramerexcitedstatesarealmost proportionaltothesquareddimerwavefunction. Wethencome
o to propose a model which predicts the binding energy of the first excited state of 4HeN (N ≥ 3)
at measuredfromthe4HeN−1 groundstatetobenearly 2(NN−1)B2 usingthedimerbindingenergyB2.
.
s
c
i I. INTRODUCTION tialshaveshownthatthe4Hetrimerpossessestwobound
s
y states with binding energies of nearly 126.4 mK and 2.3
h mK, ii) it is already rather well established that, if the
In early 1970’s, Efimov pointed out a possibility of
p 4He trimer excited state exist, it should be Efimov na-
having an infinite number of three-body bound states
[ ture, and iii) it is suggested that the 4He trimer ground
even when none exists in the separate two-body subsys-

2 tems [1–3]. This occurs when the two-body scattering state may be considered as an Efimov state since the
v ground- and excited-state binding energies move along
length is much larger than the range of the two-body
0 the same universal scaling curve under any small defor-
interaction. As a candidate of such three-body states,
7 mation of the two-body potential (for details, see, e.g.,
Efimov discussed about the famous Hoyle state [4] (the
43 second0+ state at7.65MeVinthe 12C nucleus)taking a Sec.III of Ref. [13]). Experimentally, the 4He trimer
. modelofthreeαparticles(clustersofthree4Henuclei)as ground state has been observed in Ref. [14] to have the
1 well as about the three-nucleon bound state (3H nuclei). 4He-4He bond length of 11+−45 ˚A in agreement with the-
1
oretical predictions, whereas a reliable experimental evi-
1 In nuclearsystems,the Borromeanstates, weaklybound
dence for the 4He trimer excited state is still missing.
1 three-bodystatesthoughhavingnoboundtwo-bodysub-
: systems, are familiar but not classified as Efimov states. Only very recently, experimental evidences of Efi-
v
i In atomic systems, triatomic 4He (trimer) have been mov trimer states have been reported in the work us-
X expected to have bound states of Efimov type since the ing the ultracold gases of cesium atoms [15, 16], pota-
r realistic 4He-4He interactions [5? –8] give a large 4He- sium atoms [17], lithium-7 atoms [18, 19], and lithium-6
a
4He scattering length ( 115˚A), much greater than the atoms[20–24],inwhichthetwo-bodyinteractionbetween
potential range ( 10˚A≃), and a very small 4He dimer those alkali atoms was manipulated so as to tune the
binding energy (∼1.3 mK). (Experimentally, Ref. [10] scattering length to values significantly greater than the
evaluated a scatte≃ring length of 104+8 ˚A and a binding potential range. These experiments have been access-
−18
energy of 1.1+0.3 mK). ing the study of a wide variety of interesting physical
−0.2 systems in the atomic and nuclear fields. Recently, the
As is mentioned in recent reviews about the 4He
study extends to the Efimov physics and universality of
trimer [11, 12] (further references therein), i) a lot of
four-atomic systems (tetramers).
three-body calculations using the realistic pair poten-
Though the interactions between 4He atoms can not
be manipulated, the study of 4He trimer using the re-
alistic pair potentials has been providing fundamental
∗Electronicaddress: [email protected] information to the Efimov physics. Now it is one of the
†Electronicaddress: [email protected] challengingsubjectstopreciselyinvestigatethestructure
2
of 4He tetramer using the realistic 4He-4He potential. ground and excited states showing that the calculated
So far there exist in the literature a large number of results agree excellently with the literature. In Sec. III,
4He trimer calculations [25–39] giving well convergedre- thefour-bodycalculationofthe4Hetetramergroundand
sults with the realistic 4He-4He interactions. However, excitedstatesispresented. SummaryisgiveninSec. IV.
calculationsofthetetramerremainlimited[25,28–31];in
those papers, the binding energy of the tetramer ground
stateagreeswellwitheachother,whilethatoftheloosely II. 4He TRIMER
boundexcitedstatedifferssignificantlyfromoneanother.
Thus the main purpose of the present paper is to per- The 4He trimer bound states have extensively been
form accurate calculations of the 4He tetramer ground studied in many theoretical work using realistic poten-
and excited states using a realistic 4He-4He interaction, tials. Monte-Carlo, hyperspherical, variational and Fad-
the LM2M2 potential [6]. We employ the Gaussian ex- deev techniques were used to calculate accurately the
pansion method (GEM) for ab initio variational calcu- binding energies of the ground and excited states [25–
lations of few-body systems [40–43]. The method has 39] (see also recent reviews [11, 12]). Nevertheless, in
been proposed and developed by the present authors this section, we explain our Gaussian expansion method
and collaboratorsand applied to various types of three-, (GEM) and present the calculated result for the 4He
four-andfive-bodysystemsinnuclearphysicsandexotic trimer in order to demonstrate high accuracy of our cal-
atomic/molecular physics (cf. review papers [43–45]). culation before we report our investigation of the 4He
Advantage of using the GEM for the 4He tetramer tetramer in the next section.
calculation in the presence of the strong short-range re-
pulsive potential is as follows: Some 30000 symmetrized
four-body Gaussian basis functions, ranging from very A. Three-body wave function
compact to very diffuse, are constructed on the full 18
setsofJacobicoordinateswithoutassuminganypaircor- WetakeallthethreesetsofJacobicoordinates(Fig.1),
relation function. They forms a nearly complete set in x =r r and y =r 1(r +r ) and cyclically for
1 2− 3 1 1− 2 2 3
thefinitecoordinatespaceconcerned,sothatonecande- (x ,y ) and (x ,y ), r being the position vector of ith
2 2 3 3 i
scribe accurately both the short-rangestructure and the particle. Hamiltonian of the system is expressed as
long-range asymptotic behavior (up to 1000 ˚A) of the
∼ ¯h2 ¯h2 3
four-body wavefunction, which makes it possible to find H = 2 2 + V(r ), (2.1)
new facets of 4He clusters. −2µx∇x− 2µy∇y 1=Xi<j ij
Wethusfindthatpreciselythesameshapeoftheshort-
range correlation (rij < 4 ˚A) in dimer appears in the where µx = 12m and µy = 32m, m being mass of a 4He
ground and excited sta∼tes of trimer and tetramer. This atom. V(rij) is the two-body 4He-4He potential as a
gives a foundationto an a priori assumptionthat a two- function of the pair distance rij =rj ri.
−
particle correlation function (such as the Jastrow’s) so
as to simulate the short-range part of the dimer wave
functionisincorporatedinthetrimerandtetramerwave
functions from the beginning.
By illustrating the asymptotic behavior of the 4He
trimer and tetramer, we discuss about an interesting
relation between their excited-state wave functions and
the dimer wave function. We then come to propose a
’dimerlike-pair’ model that predicts the binding energy
of the first excited state of the N-cluster system, 4He ,
N
measuredfromthegroundstateof4He tobeapprox- FIG.1: ThreesetsoftheJacobi coordinates for 4Hetrimer.
N−1
imately N B using the dimer binding energy B .
2(N−1) 2 2
We calculate the three-body bound-state wave func-
We explicitly write the asymptotic form of the total
tion, Ψ , which satisfies the Schr¨odinger equation
wave function of 4He trimer (tetramer). The asymptotic 3
normalization coefficient (ANC) [26, 41, 46–48], namely (H E)Ψ =0. (2.2)
3
−
theamplitudeoftailfunctionofthedimer-atom(trimer-
Since we consider the 4He atom as a spinless boson,
atom)relativemotioninthepresentcase,isaquantityto
we expand the wave function of three identical spinless
reflecttheinternalstructureoftrimer(tetramer). There-
bosons in terms of L2-integrable, fully symmetric three-
fore, attention to the ANC might be useful when one
body basis functions:
intends to reproduce the non-universal variation of the
4He trimer (tetramer) states by means of parametrizing αmax
effective models beyond Efimov’s universal theory. Ψ3 = AαΦ(αsym), (2.3)
Thepaperis organizedasfollows: InSec. II,weapply αX=1
theGEMtothethree-bodycalculationofthe4Hetrimer Φ(sym) = Φ (x ,y )+Φ (x ,y )+Φ (x ,y ). (2.4)
α α 1 1 α 2 2 α 3 3
3
Itisofimportancethatthosebasisfunctions Φ(sym);α= B. Gaussian basis functions
α
{
1,...,α , which are nonorthogonal to each other, are
max
}
constructed on the full three sets of Jacobi coordinates; The radial function φ (x) in (2.9) is taken to be a
this makes the function space of {Φ(αsym)} quite wide. Gaussian multiplied by xnlxnlxx (similarly for ψnyly(y) ):
The eigenenergiesE andamplitudes A ofthe ground
and excited states are determined by theαRayleigh-Ritz φnxlx(x)=xlx e−(x/xnx)2, (2.10)
variational principle: ψnyly(y)=yly e−(y/yny)2, (2.11)
Φ(sym) H E Ψ =0, (2.5) wherenormalizationconstantsareomittedforsimplicity.
h α | − | 3i Setting of the ranges by stochastic or random choice
does not seem suitable for describing the strong short-
where α = 1,...,α . Eqs.(2.5) results in a generalized
max rangecorrelationandthelong-rangeasymptoticbehavior
eigenvalue problem:
of the wave function. Any intended choice of the ranges
is necessary. The GEM recommends to set them in a
αmax
geometric progression:
α,α′ E α,α′ Aα′ =0. (2.6)
H − N
αX′=1(cid:2) (cid:3) x =x anx−1 (n =1,...,nmax), (2.12)
nx 1 x x x
y =y any−1 (n =1,...,nmax), (2.13)
The matrix elements are written as ny 1 y y y
with common ratios a > 1 and a > 1. This
x y
Hα,α′ = hΦ(αsym)|H |Φ(αs′ym)i, (2.7) greatly reduces the nonlinear parameters to be opti-
mized. We designate a set of the geometric sequence
Nα,α′ = hΦ(αsym)| 1 |Φ(αs′ym)i. (2.8) iblyar{lynmxfoarx,nxm1,axx,nymxax,}yinstea,dwohfi{chnmxisaxm,oxr1e,caoxn}veannidenstimto-
The lowest-lying two S-wave eigenstates, Ψ(v)(v = 0,1), consider t{heyspati1al dnimystarxi}bution of the basis set. Opti-
3
willbeidentifiedasthetrimerground(v =0)andexcited mizationofthenonlinearrangeparametersisinprinciple
(v =1) states. bytrialanderrorprocedurebutmuchofexperiencesand
WeexpresseachbasisfunctionΦ (x ,y )asaproduct systematicshavebeenaccumulatedinmanystudiesusing
α i i
of a function of x and that of y : the GEM.
i i
The basis functions φ have the following proper-
nl
{ }
ties: i) They range from very compact to very diffuse,
Φ (x ,y )=φ (x )ψ (y ) Y (x )Y (y ) ,
α i i nxlx i nyly i lx i ly i more densely in the inner region than in the outer one.
h iJM
(2.9) While the basis functions with small ranges are respon-
b b
where α specifies a set of quantum numbers sible for describing the short-range structure of the sys-
tem, the basis with longest-range parameters are for the
α= n l ,n l ,JM asymptotic behavior. ii) They, being multiplied by nor-
x x y y
{ } malization constants for φ φ =1, have a relation
nl nl
h | i
commonly for the components i = 1,2,3. J is the total 2ak l+3/2
angular momentum and M is its z-component. In this φ φ = , (2.14)
paper, we consider the trimer bound states with J = 0. h nl| n+kli (cid:18)1+a2k(cid:19)
Then, the totally symmetric three-body wave function
which tells that the overlap with the k-th neighbor is
requires l =l =even.
x y independentofn,decreasinggraduallywithincreasingk.
One of the most important issues of the present vari- We then expect that the coupling among the whole
ational calculation is what type of radial shape we use basis functions take place smoothly and coherently so
for φnxlx(xi) and ψnyly(yi). The basis functions should as to describe properly both the short-range structure
be capable of precisely describing the strong short-range and long-range asymptotic behavior simultaneously. We
correlation(without assuminganycorrelationfunction a notethatasingleGaussiandecaysquicklyasxincreases,
priori) and the long-range asymptotic behavior of very but appropriate superposition of many Gaussians can
loosely bound states. decay even exponentially up to a sufficiently large dis-
The GEM recommends two types of functions which tance. A good example is shown in Fig. 3 of Ref. [43]
are tractable in few-body calculations and work accu- for the 4He dimer wave function (with the HFDHE2 po-
lately. One is the Gaussian function and the other, tential) that is accurate up to 1000 ˚A with the use of
more powerful one, is the complex-range Gaussian func- the nonlinear parameters nma∼x = 60, x = 0.14 ˚A and
1
tion[43]. Inthenextsubsection,weintroducetheformer x =700˚A (thesame-{qualitydimerwavefunctionis
nmax
}
thatwassuccessfullyusedinourpreviousstudy(Sec. 3.1 seen in Fig. 2 below in Sec.II.D using the complex-range
of Ref. [43]) of the 4He trimer ground and excited state Gaussians with the LM2M2 potential).
with the use of the HFDHE2 potential [5]. The latter Alotofsuccessfulexamplesofthethree-andfour-body
function is introduced in Sec.II.C. GEMcalculationsareshowninreviewpapers[43–45]and
4
inpapersoffive-bodycalculations[49,50]. Theexamples Note that, when calculating the matrix elements (2.7)
includes our previous calculation of the ground and ex- and(2.8) using φ(cos)(x) and φ(sin)(x), we explicitly take
cited states of 4He trimer using the HFDHE2 potential; (2.15)andthe rignhlt-mostexprenslsionof (2.16)and (2.17)
the binding energies were in good agreement with those since the computation programming is almost the same
given by a Feddeev-equation calculation [33]. As for the as that for (2.10) though some of real variables are
trimerwavefunction,weshowed,inFigs.3,13and14in changed to complex ones.
Ref.[43], that the strong short-range correlation (x < 4 A great advantage of the real- and complex-range
˚A) and asymptotic behavior (up to x 1000 ˚A) of∼the Gaussianbasisfunctionsisthatthecalculationofmatrix
∼
trimer groundandexcitedstatesare simultaneouslywell elements (2.7) and (2.8) is easily performed. As for the
described. Also, the three-body basis functions (2.9)– overlapandkinetic-energymatrix elements ofthe trimer
(2.13) together with the LM2M2 potential were used re- (tetramer), all the six-(nine-)dimensional integrals give
cently by Naidon, Ueda and one of the present authors analytical expression. In the case of the potential ma-
(E.H.) [51]to study the universalityandthe three-body trix, we have analytical expression except for the one-
parameter of 4He trimers. dimensional numerical integral having the final form
∞
x2me−λx2V(x)x2dx. (2.20)
C. Complex-range Gaussian basis functions Z
0
We explained,inRef.[43],varioustechniquesto perform
Before we proceed to the calculation of the 4He the three- and four-body matrix-element calculations as
tetramer ground and excited states, we improve the easily, accurately and rapidly as possible.
Gaussianshape ofthe basisfunctions so asto havemore It is to be emphasized that the GEM few-body cal-
sophisticated(but stilltractable)radialdependence. We culations need neither introduction of any a priori pair
then test the new basis in the calculation of the trimer correlation function (such as the Jastrow function) nor
states below. separationof the coordinate space by x<r and x>r ,
c c
In Ref. [43], we proposed to improve the Gaussian r being the radiusofa stronglyrepulsivecorepotential.
c
shape by introducing complex range instead of the real Proper short-range correlation and asymptotic behavior
one: of the total wave function are automatically obtained by
φ(ω)(x)=xle−(1+iω)(x/xn)2, (2.15) solving the Schr¨odinger equation (2.2) using the above
nl basis functions for ab initio calculations.
where n = 1,...,nmax and x are given by (2.12). Using
x n
φ(±ω)(x), we construct two kinds of real basis functions:
nl D. Pair interaction and 4He dimer
φ(cos)(x) = xle−(x/xn)2cosω(x/x )2
nl n
= [φ(−ω)(x)+φ(ω)(x)]/2, (2.16) To describe the interaction between the 4He atoms,
nl nl we employ one of the most widely used 4He-4He inter-
φ(sin)(x) = xle−(x/xn)2sinω(x/x )2 actions, the LM2M2 potential by Aziz and Slaman [6].
nl n
= [φ(−ω)(x) φ(ω)(x)]/2i, (2.17) Useismadeof h¯m2 =12.12K˚A2 astheinputmassof4He
nl − nl atom. We can then precisely compare calculated results
whereweusuallytakeω =1. Thethree-bodybasisfunc- for the tetramer ground and excited states with those
tion Φ (x ,y ) in (2.9) is replaced by obtained by Lazauskas and Carbonell [25] who made a
α i i
Faddeev-Yakubovsky (FY) equation calculation taking
Φα(xi,yi)=φ(ncsxionlsx)(x)ψnyly(yi)hYlx(xi)Yly(yi)iJM, athuethsoarmseopfoRteenf.ti[a5l2a]ncdla4imHetmhaatssaasmaobroevpe.reRciesceenvtallyu,ethoef
(2.18)
where α specifies a set b b h¯2 = 12.11928 K˚A2 should be employed. We shall ad-
m
ditionallyshowthe trimerandtetramerbindingenergies
α ‘cos’ or ‘sin’,ω,n l ,n l ,JM . (2.19)
x x y y in the case of using this value.
≡ { }
The new basis φ(csions)(x) apparently extend the func- We calculated the 4He dimer binding energy, say B2,
{ nxlx } and the wave function, Ψ2( Ψ2(x)Y00(x)), using the
tionspacefromtheoldones(2.10)sincetheyhavetheos- ≡
same prescription as described above. We expanded
cillating components; see Sec.2.4 and Sec.2.5 of Ref. [43] Ψ (x) with 100 basis functions of (2.16) abnd (2.17) as
2
for some examples taking this advantage in calculations
ofhighlyvibrationalexcitedstates(with 25nodes)and nmax
x
scattering states. The sin-type basis (2.∼17) particularly Ψ (x)= A(cos)φ(cos)(x)+A(sin)φ(sin)(x) (2.21)
2 n n0 n n0
work when the wave function is extremely suppressed at nX=1 (cid:2) (cid:3)
x 0duetothestronglyrepulsiveshort-rangepotential.
∼ with a parameter set
In the following calculations,we employ the new basis
(2.16) and (2.17) for the x-space instead of (2.10), but nmax =50, x =0.5˚A, x =600.0˚A, ω =1.0 .
{ x 1 nmxax }
keep (2.11) for the y-space. (2.22)
5
0.03 2a xt 1x0=60 K 4He dimer TABLE I: Mean values for 4He trimer ground and excited
states with the use of the LM2M2 potential and h¯2 = 12.12
m
−3/2(Å) 0.02 Kcle˚A2d.istBan3(vc)eiasntdheribGinidsinthgeendeisrtgayn,creijofstaanpdasrtfoicrleinftreormpartthie-
x) center-of-massofthetrimer. Seetextfortheasymptoticnor-
ψ(2 0.01 malization coefficient C3(v)(v=0,1).
trimer ground state
0
present Ref.[25] Ref.[26] Ref.[27]
−10.9 K at x=3 Å
0 5 10 B3(0) (mK) 126.40 126.39 126.4 126.40
x (Å) hTi(mK) 1660.4 1658 1660
100 hVi (mK) −1786.8 −1785 −1787
4He dimer hri2ji (˚A) 10.96 10.95 10.96
10−2 phriji (˚A) 9.616 9.612 9.610
2) hri−j1i (˚A−1) 0.134 0.135
−1/Å hri−j2i (˚A−2) 0.0228 0.0230
x) ( 10−4 hri2Gi (˚A) 6.326 6.49 6.32
ψx (2 pC3(0)(˚A−12) 0.562 0.567
10−6
trimer excited state
present Ref.[25] Ref.[26] Ref.[27]
10−8
0 500 1000 1500
x (Å) B3(1) (mK) 2.2706 2.268 2.265 2.2707
hTi(mK) 122.15 122.1 121.9
hVi (mK) −124.42 −124.5 −124.2
FIG. 2: Short-range structure (upper) and asymptotic be- hr2i (˚A) 104.5 104.3 101.9
havior (lower) of the radial wave function Ψ (x) of the 4He ij
dimer obtained by using the complex-range2Gaussian basis phriji (˚A) 84.51 83.53 83.08
functions (2.21) and (2.22). The open circles stands for the hri−j1i (˚A−1) 0.0265 0.0267
exact asymptotic form. The dotted line (upper) illustrates hr−2i (˚A−2) 0.00216 0.00218
ij
theLM2M2 potential in arbitrary unit. hr2 i (˚A) 60.33 58.8 59.3
iG
pC3(1)(˚A−12) 0.179 0.178
We obtained B = 1.30348 mK, x2 = 70.93 ˚A, and
2
h i
x = 52.00 ˚Awhich are the sampe as those obtained in E. Trimer bound states
thhei literature. Experimentally, x =52 4 ˚A [10] from
which B =1.1+0.3 mK was esthimiated. ±
2 −0.2 We calculatedthewavefunctions ofthe trimerground
state, Ψ(0), and the excited state, Ψ(1), and their bind-
3 3
AsshowninFig.2,boththestrongshort-rangecorrela-
ing energies, B(0) and B(1), respectively, as well as some
tion(x<4˚A) andthe asymptotic behaviorofthe dimer 3 3
are well∼described. In the lower panel, xΨ2(x) precisely meanvalueswiththeΨ3(v) (v =0,1). Someofresultsare
reproducestheexactasymptoticshape0.1498exp( κ x) summarized in Table I together with those obtained in
2
(˚A−1/2)withκ =√mB /¯h=0.0104˚A−1 uptox−1200 the literature. Our results excellently agree with those
2 2
˚A which is large enough for our discussions. ∼ by Refs.[25–27]. The 4He-4He bond length in the trimer
groundstatewasmeasuredas r =11+4 ˚A[10],which
h iji −5
is well explained by the calculations, r =9.61 ˚A.
There are 30 basis functions whose Gaussian ranges ij
h i
x < 4 ˚A, which is sufficiently dense to describe the Those converged results were given by taking the
shnoxrt-range structure of the wave function precisely. An symmetric three-body basis function Φ(sym);α =
α
{
interesting issue is whether the same shape of the short- 1,...,α with α = 4400, in which the shortest-
max max
range correlationin Fig. 2 appear also in the trimer and rangeseti}s(x =0.3˚A, y =0.4˚A)andthe longest-range
1 1
tetramergroundandexcitedstateswithoutassumingany oneis(x =150˚A, y =600˚A).Allthenonlinearpa-
max max
two-body correlationfunction. rameters of the Gaussian basis set are listed in Table II.
6
TABLEII: AllthenonlinearparametersoftheGaussianbasis TABLE III: Convergence of the 4He trimer calculations
functions used for the4He-trimerstates with J =0(lx =ly). with respect to the increasing maximum partial wave (lmax).
Those in column a) are commonly for φ(cos)(x) and φ(sin)(x) The four columns present trimer ground (B(0)) and excited
nxlx nxlx 3
andb)forψnyly(y). Totalnumberofthebasis,αmax,is4400. (B(1)) state energies in comparison with those obtained by
3
theFaddeev-equation calculation of Ref.[25].
a) φ(cos)(x),φ(sin)(x) b) ψ (y)
nxlx nxlx nyly
trimer present Ref.[25]
lx nmxax x1 xnmxax ω ly nmyax y1 ynmyax number
[˚A] [˚A] [˚A] [˚A] of basis l B(0) (mK) B(1) (mK) B(0) (mK) B(1) (mK)
max 3 3 3 3
0 22 0.3 150.0 1.0 0 50 0.4 600.0 2200
0 121.00 2.2397 89.01 2.0093
2 17 0.6 150.0 1.0 2 40 0.8 400.0 1360
2 126.39 2.2705 120.67 2.2298
4 14 0.8 130.0 1.0 4 30 1.0 200.0 840
4 126.40 2.2706 125.48 2.2622
8 126.34 2.2677
12 126.39 2.2680
14 126.39 2.2680
There are neither additional parameter nor assump-
tions. The present calculation is so transparent that it
is possible for the readers to repeat the calculation and
check the results reported here. The parameters for the
F. Short-range correlation and asymptotic
Gaussian ranges are in round numbers but further op-
behavior
timization of them do not improve the binding energies
(B(0) =126.40mKandB(1) =2.2706mK)aslongaswe
3 3
calculate them with five significant figures (cf. another In order to see how the present method describes the
short-range structure of trimer, we calculated the pair
check in Sec.II.H about the accuracy of the calculation).
correlation function (pair distribution function or two-
(0) (1)
ConvergenceofthebindingenergiesB andB with
3 3 body density) P(v)(x) defined by
respect to increasing partial waves l (= l ) is shown in 3
x y
Table III in comparisonwith the Faddeev calculation by
Lazauskas and Carbonell [25]. The case lxmax = 4 is P3(v)(x1)Y00(x1)=hΨ3(v)|Ψ3(v)iy1, (2.23)
sufficient in the present work as long as the accuracy of
five significant digits is required. where the symbol h biy1 means the integration over y1
only. This integration gives an analytical expressionow-
The convergence of the present result is more rapid
ing to the use of the Gaussian basis functions; here, we
than that of the Faddeev solution (the same will be seen
inthetetramercalculationinSec.III).Thereasonisthat explicitly rewrite Ψ3(v) as a function of (x1,y1) by trans-
boththeinteractionandthewavefunctionaretruncated forming the other coordinates (x2,y2) and (x3,y3) into
in the angular-momentum space (l ) in the Faddeev (x ,y ). P(v)(x )isindependentofi(=1,2,3)andisap-
max 1 1 3 i
calculations, but the full interaction is included in the parently normalized as P(v)(x)x2dx = 1. It presents
3
presentcalculation(withnopartial-wavedecomposition)
theprobabilityoffindingRtwoparticlesataninterparticle
thoughthewavefunctionistruncated(l ). Thediffer-
max distance x.
ence of the convergence in the two calculation methods In Fig. 3, short-range structure of P(v)(x)(v = 0,1) is
3
was precisely discussed in the case of the three nucleon illustrated together with P (x)(= Ψ (x)2) for the 4He
bound states (3H and 3He nuclei) in our GEM calcu- 2 | 2 |
dimer. The dashed line is for the trimer ground state
lation [41–43] and in a Faddeev calculation [53]; for an
(v = 0). The solid line for the excited state (v = 1)
illustration of the difference, see Fig. 15 in Ref. [43]. In
and the dotted line for the dimer have been multiplied
thiscontext,itisworthpointingoutthat,inTableI,our
byfactors14.5and6.0,respectively. Itisofinterestthat
result precisely agrees with another Faddeev calculation
precisely the same shape of the short-range correlation
by Ref.[27] with no the partial wave decomposition. (x<4˚A)asseeninthedimerappearsbothinthetrimer
Use of the value h¯2 = 12.11928 K˚A2 [52] results in gro∼und and excited states (the same will be seen in the
m
(0) (1) tetramer ground and excited states in the next section).
B = 126.499 mK and B = 2.27787 mK, while
3 3 This gives a foundation to an a priori assumption that
Ref.[52]gives126.499mKand2.27844mK,respectively.
a two-particle correlation function (such as the Jastrow
Calculation of the binding energy was also made pertur-
batively with B = 12.11928 T + V , where T and function) to simulate the short-range part of the dimer
3 12.12 h i h i h i radial wave function Ψ (x) is incorporated in the three-
V are those obtained with h¯2 =12.12 K˚A2; this gives 2
h i m body wave function from the beginning.
B3(0) = 126.498 mK and B3(1) = 2.27787 mK. The calcu- To investigate the trimer configuration in the asymp-
lations below in Sec.II.F-H take h¯2 =12.12 K˚A2. totic region where one atom is far from the other two,
m
7
0.006 100
4He trimer 4
He trimer
)
−3(Å) 0.004 −1/2(Å 10−2
(v)P(x)3 (v)O(y) 3
0.002 v=0 (dimer) y 10−4 v=1
v=0
(dimer)
v=1
0
0 5 10 10−6
x (Å) 0 500 1000
y (Å)
FIG.3: Short-rangestructureofthepaircorrelationfunction
FIG. 4: Overlap function O(v)(y), multiplied by y, between
P3(v)(x) of the 4He trimer calculated by (2.23). The dashed the 4He trimer wave function3 (v = 0,1) and the dimer one,
line is for the trimer ground state (v = 0), the solid line for
whichisdefinedby(2.24). Opencirclesrepresentthefitofthe
theexcitedstate(v=1)andthedottedlineforthe4Hedimer
b(|yΨf2a(cxt)o|r2s).14T.5heansodli6d.0a,nrdesdpoetctteivdellyin,etsohbaevneobrmeeanlizmeudltaitpltiehde nasoyrmmpaltioztaitciofnunccoteiffioncie(2n.t25C)3(vt)o. TOh3(ve)(syo)lidusliinneg(tvh=e a1s)yimspfotuontidc
peak. The same shape of the short-range correlation (x <∼ 4 to be parallel to the dotted line for the dimer wave function
˚A) appears in thethree states. (yΨ2(y)).
wecalculatetheoverlapfunction (v)(y )[26,41,46–48] TheANC,C(v),isaquantitytoconveytheinteriorstruc-
todescribetheoverlapbetweenthOe3trim1erwavefunction turalinforma3tionofthe trimer to the asymptotic behav-
Ψ(v)(v =0,1) and the dimer one Ψ (x): ior. Itisknown,inthenuclearperipheralreactionswhere
3 2 only the asymptotic tails of the wave functions of react-
(v)(y )Y (y )= Ψ (x ) Ψ(v) . (2.24) ingparticlescontributeto the reactionprocess,the cross
O3 1 00 1 h 2 1 | 3 ix1 section is proportional to the squared ANC which can
In Fig. 4, we plot yb (v)(y) for the ground and excited be measured in some specific systems [47, 48, 54]. The
O3
states. They should asymptotically satisfy ideaofANC mightbe availabletothe calculationof4He
atoms reactionssuchas dimer+dimer trimer+atom.
y (v)(y) y→∞ C(v)exp( κ(v)y), (2.25) Also, attention to the ANC might be→useful when one
O3 −−→ 3 − 3
intends to reproduce the non-universal variations of the
where κ(3v) is the binding wave number given by trimer states by parametrizing effective models.
κ(v) = 2µ (B(v) B )/¯h (κ(0) = 0.117 ˚A−1, κ(1) =
3 q y 3 − 2 3 3
0.0103˚A−1). TheamplitudeC3(v) iscalledtheasymptotic G. ’Dimerlike-pair’ model in asymptotic region
normalzation coefficient (ANC) [26, 41, 46–48] defining
the amplitude of the tail of the radial overlap function.
In Fig. 4, we find that the solid line (v =1) is parallel
The asymptotic functions (2.25) with C3(0)= 0.562˚A−21 to the dotted line (dimer); namely, κ(1)(= 0.0103 ˚A−1)
and C3(1)=0.179˚A−12 (see the open circles) are precisely is verycloseto κ2(=0.0104˚A−1). Thi3s agreementis not
reproduced by the dashed line (v =0) and the solid line accidental, but is understandable from a model, which
(v = 1), respectively, up to y 1000 ˚A, which demon- we refer to as a ’dimerlike-pair’ model, for the asymp-
∼
strates the accuracyofour wavefunctions in the asymp- totic behavior of the trimer excited state (Fig. 5a). The
totic region. The values of C(v) agree with those given model tells that i) particle a, located far from b and c
3
by Barletta and Kievsky [26] using a variationalmethod which are loosely bound (dimer), is little affected by the
with correlated hyperspherical harmonics functions (see interaction between b and c, ii) therefore, the pair a and
Table I). b at a distance x is asymptotically dimerlike, iii) since
The total three-body wave function Ψ(v)(v = 0,1) is x y asymptotically, the amplitude of particle a along
3 ≃
represented asymptotically as y is dimerlike, namely κ(1) κ .
3 ≃ 2
If this model is acceptable, we can predict that, in
Ψ(v) C(v) 3 Ψ (x )e−κ3(v)yiY (y ). (2.26) the asymptotic region, the pair correlation function of
3 −→ 3 Xi=1 2 i yi 00 i the trimer excited state, x2P3(1)(x), should decay ex-
b
8
100
dimerlike pair
4
He trimer
)
1
−
Å
(
)
x
v)( 10−5
(3
P
dimerlike pair 2
x
v=1
v=0
(dimer)
10−10
0 500 1000
x (Å)
FIG. 5: ’Dimerlike-pair’ model for the asymptotic behavior
of the trimer and tetramer excited states (see text).
FIG. 6: Asymptotic behavior of the pair correlation (distri-
bution)functionP(v)(x),multipliedbyx2,ofthe4Hetrimer.
3
The dashed line is for the trimer ground state (v = 0). The
ponentially with the same rate as that in the dimer
solid line fortheexcited state(v=1) and thedottedlinefor
(x2P (x)). This is clearly seen in Fig. 6; the solid and
2 the dimer are found to have almost the same exponentially-
dottedlineshavealmostthesameexponentially-decaying
decaying rate, 2κ(1) ≃2κ .
rateof2κ(1)( 2κ ). ThesameevidenceisseeninFigs.3 3 2
and14of3our≃prev2iouscalculationofthe4Hetrimerusing
the HFDHE2 potential reported in Ref. [43].
Once we accept the dimerlike-pair model (κ(1) κ ), We try to apply the same model to the tetramer
3 ≃ 2
we can estimate B(1), the trimer excited-state binding excited state (Fig. 5b) and predict its binding energy
energy, using B . W3ith the use of the definitions of the B(1). Asymptotically, particle a decays from the trimer
2 4
binding wave numbers: (b+c+d) as exp( κ(1)z) with
− 4
κ = 2µ B /¯h, (2.27)
2 x 2 κ(1) = 2µ (B(1) B(0))/¯h, (2.30)
p 4 q z 4 − 3
κ(1) = 2µ (B(1) B )/¯h, (2.28)
3 q y 3 − 2 where µ = 3m is the reduce atom-trimer mass. Taking
z 4
where µx = 21m and µy = 32m. Taking κ(31) ≃κ2, we can κ(41) ≃κ2, we predict B4(1) as
then predict
µ 2
(1) (0) x (0)
B B + B =B + B =127.27mK (2.31)
B(1) B + µxB = 7B =2.281mK, (2.29) 4 ≃ 3 µz 2 3 3 2
3 ≃ 2 µ 2 4 2
y
when employing the calculated values of B(0) and B
3 2
which is close to 2.2706 mK by the present three-body with LM2M2. In Sec.III, we make a four-body calcula-
calculationusingtheLM2M2potentialforwhichwehave tion of the tetramer with LM2M2 and check the above
the ratio B(1)/B =1.74( 7/4). prediction of B(1).
3 2 ≃ 4
In order to see a deviation of the ratio from 7/4 de-
pending on the realistic potentials in the literature, we
refer to B(1)/B = 1.59 [26] (SAPT2 [8]), 1.65 [39, 52] H. Generalized eigenvalue problem
3 2
(SAPT2007 [9]), 1.74 [26, 27] (TTY [7]), 1.74 (LM2M2),
2.01 [37] (HFDHE2). The dimerlike-pair model provides In this subsection, we discuss about a technical sub-
a reason why the ratio B(1)/B is located around 7/4 in ject on a numerical trouble which arises when solving
3 2
a narrow region of 1.6–2.0. the generalized eigenvalue problem (2.6). This is due to
We note that this model should be considered under the fact that the overlap matrix becomes almost sin-
N
the condition that the 4He atoms are interacting with gular when a very large number of nonorthogonal basis
(sym)
a realistic pair potential and should not be discussed in functions Φ employed. Inthiscase,becauseofthe
α
{ }
any situation where a large deformation of the strength non-negligibleround-offerrorindouble-precisioncompu-
is posed to the potential (cf. a discussion in Sec.III of tation ( 16 decimal digits), we may obtain no solution
Ref. [13] on the Efimov states in 4He trimer). of(2.6)≃orasolutionthatincludessomeunphysicallytoo
9
deep erroneous bound state. In order to overcome this
TABLE IV: Stability of the calculated trimer binding ener-
trouble, we took the following two steps:
gies of thelowest-lying two states against decreasing number
Step i): we first diagonalize the overlap matrix :
N (Nmax) of symmetrized orthonormal three-body basis func-
αmaxNα,α′Cα(N′ ) =νNCα(N), (2.32) tcilbounsiso{nΦbB(Ns3(y0m))=}1o2f6E.4q0.(m2.K33)a.nTdhBis3(1a)s=su2r.e2s70a6ccmurKaciynoTfatbhleecoIn.-
αX′=1
whereα,N =1,...,αmax. TheeigenvaluesνN arepositive N (∆N ) B(0) (mK) B(1) (mK)
definite since α,α′ = α′,α. We then define a new, max max 3 3
N N
symmetrized orthonormal basis set: 3b250 b− 126.3999 2.270606
1 αmax 3240 (−10) 126.3998 2.270605
Φ(sym) = C(N)Φ(sym), (2.33) 3200 (−50) 126.3995 2.270602
N √νN αX=1 α α 3150 (−100) 126.3991 2.270594
b
hΦ(Nsym)|Φ(Nsy′m)i=δN,N′, (2.34) 2950 (−300) 126.3975 2.270533
2750 (−500) 126.3954 2.270484
where N,N′b= 1,...b,α . The generalized eigenvalue
max 2250 (−1000) 126.3657 2.270163
problem(2.6)arethenequivalentlyconvertedintoastan-
dard eigenvalue problem:
αmax
N,N′ EδN,N′ AN′ =0, (2.35) the lowest two states. By checking the stability of the
H −
NX′=1(cid:2)b (cid:3) b energy values againstfurther decreasing Nmax, we verify
where N =1,...,α , and the values of B(0) = 126.40 mK and B(1) = 2.2706 mK
max 3 3 b
in Table I.
HN,N′ =hΦ(Nsym)|H |Φ(Nsy′m)i. (2.36)
Here,we arrabnge{Φ(Nsymb)}in the debcreasingorderofνN: III. 4He TETRAMER
ν >ν > b >ν > >ν . (2.37)
1 2 ··· N ··· αmax Calculation of the 4He tetramer using realistic poten-
When the nonorthogonality among the basis functions tials has been performed in Refs. [25, 28–31]. Although
Φ(sym) is very large, some of ν become extremely binding energy of the ground state obtained in the pa-
α N
{ }
small and therefore the large factor 1/√ν may cause pers agrees well with each other ( 558 mK), that of
N ∼
a serious cancellation in the summation in (2.33). Since the loosely bound excited state differs significantly from
the present calculation is performed by double-precision each other; namely, the binding energy with respect to
computation, such a large cancellation may generate a the trimer ground state (126.4 mK) is given as 1.1 mK
substantial round-off error in (2.33) and hence in the bythe Faddeev-Yakubovski(FY)equationsmethod[25],
matrix elements (2.36). This may give rise to some erro- 6.6mKby MonteCarlomethodscombinedwiththe adi-
neous eigenstates in (2.35) that have unphysically huge abatic hyperspherical approximation [30] and 52 mK re-
binding energies. centlybyusingamethodofthecorrelatedpotentialhar-
Step ii): We therefore omit such members of Φ(sym) monic basis functions [31]. Though the Faddeev result
that have too small ν . The binding energies{ofNsuch} (1.1 mK) seems to the present authors the most accu-
N
unphysicalstatesdecreasesquicklyasthebasissizbeisre- rate, the excited state was not solved as a bound-state
probleminRef.[25]buttheresultwasextrapolatedfrom
duced. Finally, we reach an appropriate size, say N ,
max
of the basis Φ(sym) for which those unphysical states the atom-trimer scattering phase shifts.
{ N } b Thus the purpose of this section is to perform, us-
have disappeared from the low-energy region, and ener-
b ing the same LM2M2 potential as in Ref. [25], accurate
gies of the lowest-lying (deepest) states take physically
bound-statecalculationofthetetramerexcitedstate,not
reasonable values. It is to be emphasized that the bind-
only giving a precise binding energy but also describing
ing energies of so-obtained lowest-lying physical states
the short-range correlation and the asymptotic behavior
are stable against further reduction of N .
max of the wave function properly.
Table IV explicitly demonstrates Step ii). We start
b The GEM has extensively been employed in bound-
with α = 4400 basis functions Φ(sym) whose pa-
max α state calculations of various four-body systems in nu-
{ }
rameters are givenin Table II. When the size of the new
clear and hypernuclear physics (cf. review papers [43–
basis {Φ(Nsym)} is reduced from αmax to Nmax = 3250 45]). Extension from three-body GEM calculations to
according to (2.37), the solution of (2.35) has come to four-body ones in the presence of strong short-range re-
b b
include no unphysical state and give the binging ener- pulsionisafamiliarsubjectinnuclearphysics. Forexam-
gies B(0) = 126.3999 mK and B(1) = 2.270606 mK for ple,thestudyofthree-nucleonboundstates(3Hand3He
3 3
10
nuclei) in Ref. [41] was extended to that of four-nucleon with
ground state (4He nucleus, Jπ = 0+) [55] and the first
excited, very diffuse state (Jπ = 0+) [56]. The study 12
Φ(sym;K) = Φ(K)(x ,y ,z ), (3.4)
of the three-α-particle system (12C nucleus) [43, 57, 58] αK αK i i i
was extended to that of the four-α-particle system (16O Xi=1
18
nucleus) [58] with the strongly repulsive Pauli-blocking
Φ(sym;H) = Φ(H)(x ,y ,z ), (3.5)
projection operator on the α-α motion. Therefore, ex- αH αH i i i
tensionofthe 4He trimercalculationtothetetramerone iX=13
is straightforwardon account of those experiences. inwhichΦ(x ,y ,z )isafunctionofi-thsetofJacobico-
i i i
ordinates. ItisofimportancethatΦ(sym;K) andΦ(sym;H)
αK αH
are constructed on the full 18 sets of Jacobi coordinates;
A. Method
this makes the function space of the basis quite wide.
(K) (H)
The eigenenergies E and amplitudes A (A ) are
αK αH
We take two types of Jacobi coordinate sets, K-type determined by the Rayleigh-Ritz variational principle:
and H-type (Fig. 7). Namely, for K-type, x = r r ,
1 2 1
y1 =r3−12(r1+r2)andz1 =r4−31(r1+r2+r3)andc−ycli- hΦ(αsKym;K)|H −E|Ψ4i=0, (3.6)
cally for x ,y ,z ; i = 2,...,12 by the symmetrization
between t{hei foiuriparticles. For}H-type, x13 = r2 r1, hΦ(αsHym;H)|H −E|Ψ4i=0, (3.7)
y =r r ,z = 1(r +r ) 1(r +r )andcycli−cally
fo1r3 x ,4y−,z3; i1=3 142,...3,18 4. −An2 ex1plici2t illustration of where αK =1,...,αmKax and αH =1,...,αmHax. This set of
{ i i i } equations results in a generalized eigenvalue problem:
the totally 18 sets of the rearrangement Jacobi coordi-
nates of four-body systems is seen inFig. 18 ofRef. [43]. αmc′ax
(c,c′) E (c,c′) A(c′) =0, (3.8)
c′X=K,H αXc′=1hHαc,αc′ − Nαc,αc′i αc′
where c = K,H and α = 1,...,αmax. The matrix ele-
c c
ments are given by
(c,c′) = Φ(c) H Φ(c′) , (3.9)
Hαc,αc′ h αc | | αc′ i
(c,c′) = Φ(c) 1 Φ(c′) . (3.10)
Nαc,αc′ h αc | | αc′ i
Uptohereisthemostgeneralwayofvariationalcalcula-
tions for bound states of identical spinless four particles.
We describe the basis function Φ(K)(Φ(H)) in the form
FIG. 7: Jacobi coordinates, K-type and H-type, for the 4He αK αH
tetramer. Symmetrization of the four particles generates the Φ(K)(x ,y ,z )=φ(csions)(x )φ (y )ϕ (z )
sets i=1,...,12 (K-type)and i=13,...,18 (H-type). αK i i i nxlx i nyly i nzlz i
Y (x )Y (y ) Y (z ) ,
Thetotalfour-bodywavefunctionΨ4istobeobtained ×h(cid:2) lx i ly i (cid:3)Λ lz i iJM
by solving the Scho¨dinger equation b b (i=1b,...,12) (3.11)
(H −E)Ψ4 =0 (3.1) Φ(H)(x ,y ,z )=φ(csions)(x )ψ (y )ϕ (z )
αH i i i nxlx i nyly i nzlz i
with the Hamiltonian
Y (x )Y (y ) Y (z ) ,
H = ¯h2 2 ¯h2 2 ¯h2 2+ 4 V(r ), (3.2) ×h(cid:2) lx bi ly bi (cid:3)(Λi=lz13b,i..i.J,1M8) (3.12)
−2µ ∇x− 2µ ∇y− 2µ ∇z ij
x y z 1=Xi<j where α specifies a set
K
where µx = 12m, µy = 32m and µz = 34m on the K-type αK ≡ {‘cos’ or ‘sin’,ω,nxlx,nyly,nzlz,Λ,JM},(3.13)
coordinates, and µ = µ = 1m and µ = m on the
x y 2 z
H-type ones. which is commonly for the components i=1,...,12; and
Ψ4 is expanded in terms of the symmetrized L2- similarly for αH commonly for i=13,...,18.
integrable K-type and H-type four-body basis functions: Since we consider the case of J = 0 in this paper,
the totally symmetric four-body wave function requires
αmKax αmHax i) lx = even, ly +lz = even and Λ = lz for the K-type
Ψ4 =αXK=1A(αKK)Φ(αsKym;K)+αXH=1A(αHH)Φ(αsHym;H), (3.3) bthaesiHs a-tnydpeii)balxsis=. even, ly =even and Λ = lz =even for