The structure of a weakly bound ionic trimer: Calculations ...digital.csic.es/bitstream/10261/76657/1/Gianturco.pdf · Serrano 123, 28006 Madrid, Spain G. B. Bendazzoli and S. Evangelistid)

The structure of a weakly bound ionic trimer: Calculations for the 4He2H−complexF. A. Gianturco, F. Paesani, I. Baccarelli, G. Delgado-Barrio, T. Gonzalez-Lezana et al. Citation: J. Chem. Phys. 114, 5520 (2001); doi: 10.1063/1.1352034 View online: http://dx.doi.org/10.1063/1.1352034 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v114/i13 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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JOURNAL OF CHEMICAL PHYSICS VOLUME 114, NUMBER 13 1 APRIL 2001

The structure of a weakly bound ionic trimer:Calculations for the 4He2HÀ complex

F. A. Gianturco,a) F. Paesani, and I. Baccarellib)

Department of Chemistry, The University of Rome, Citta´ Universitaria, 00185 Rome, Italy

G. Delgado-Barrio, T. Gonzalez-Lezana,c) S. Miret-Artes, and P. VillarrealInstituto de Ma´tematicas y Fısica Fundamental, Consejo Superior de Investigaciones Cientificas,Serrano 123, 28006 Madrid, Spain

G. B. Bendazzoli and S. Evangelistid)

Department of Physical and Inorganic Chemistry, University of Bologna, V.le Risorgimento 4,40136 Bologna, Italy

~Received 27 October 2000; accepted 9 January 2001!

The weakly bound diatomic systems4He2 and 4HeH2 have been found able to support only onebound state forJ50, although the latter also supports an additional bound state forJ51. In thepresent paper we, therefore, study the structure of the bound states which might exist for the weaklybound triatomic4He2H

2, in its J50 state, by describing the full potential as a simple addition oftwo-body ~2B! interactions. We carry out bound state calculations using both Jacobi coordinateswithin a discrete variable representation~DVR! and pair coordinates with a distributed Gaussianfunction ~DGF! expansion. The system is shown to possess two bound states with respect to itslower dissociation threshold and two further ‘‘ghost’’ states before the complete break-up threshold.The spatial structures of such states and of the floppy complex are analyzed in detail, as is thepossibility of detecting Efimov-type states in such a weakly bound aggregate. Finally, the inclusionof three-body~3B! forces in the description of the full interaction and its effect on the number ofpossible bound states is also discussed. ©2001 American Institute of Physics.@DOI: 10.1063/1.1352034#

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I. INTRODUCTION

The experimental use of supersonic jets has beensuccessful over the last two decades in generating beamcold, and nearly isolated, molecules,1–3 thereby providing agreat deal of information on the energetics and structuproperties of very weakly bound molecular species. The aity to cool molecules below 1 K indeed facilitates the exploration of ultracold molecular physics and ultracold collisionas has been shown by recent research.4–7 As an example ofit, small bosonic helium clusters,4HeN , have been the subject of many experimental and theoretical studies andexistence of weakly bound4He dimers has been suggestover the years by independent experiments.8,9 The more re-cent transmission grating observations10 also provide apromising technique to study4HeN clusters withN up toabout 20–30 atoms, thereby indicating in a quantitative wthe relative stabilities of these very weakly molecular scies.

The interest on the properties of such ‘‘molecules’’

a!Author to whom correspondence should be addressed. Fax:139-06-49913305. Electronic mail: [email protected]

b!Present address: Laboratoire de Dynamique des Ions, Atomes etecules, Universit Pierre et Marie Curie, 4 Place Jussieu, 75252 Parisdex, France.

c!Present address: Physical and Theoretical Chemistry Laboratory, SParks Road, Oxford OX1 3QZ, UK.

d!Present address: Laboratoire de Physique Quantique, Universite´ Paul Sa-batier, 118 Route de Narbonne, 31062 Toulouse Cedex, France.

5520021-9606/2001/114(13)/5520/11/$18.00

Downloaded 23 May 2013 to 161.111.22.69. This article is copyrighted as indicated in the abstract.

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both experimental and theoretical in nature, since the pobility of proving the existence of polyatomic aggregates wseveral atoms hinges on our capability of also showing,computational modeling, the specific features of theseusual bound species and the expected values of the binenergies for their ground and excited states. An illuminatreview of such methods11 and some of the results for specificases12,13 have been already published in recent yearswe, therefore, refer any interested reader directly to thwithout giving here any further detail.

Among the variety of studied systems the clusters of4Hepossess a rather rich and intriguing set of properties relato their very weak interaction forces and to their weaksponse to external fields induced by additional atomic amolecular ‘‘impurities.’’ The latter species, in fact, could badded to the initial helium cluster by taking advantage ofease with which these aggregates can pick up one or mosuch impurities.14 In the present study we specifically intento analyze in some detail the structural consequences ofing to the simplest4He cluster, the helium dimer, one of thionic impurities which has been already shown to interwith one 4He atom through a very weak potential, i.e., tH2 impurity.15

The problem here, in fact, is to analyze the overall stial shape of this special triatomic ion and to establishnumber and nature of the bound states of this system in oto relate the present findings with those already obtaineddiscussed by us for another impurity that shows simila

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5521J. Chem. Phys., Vol. 114, No. 13, 1 April 2001 The 4He2H2 complex

weak interactions with4He and 4He2 : the Li atomicspecies.16,17

The present study is organized as follows: The next SII briefly describes the interaction potentials which we eploy in this work while Sec. III reviews the computationmethods used to obtain the bound states and the spatiatures of the triatomic ion. Section IV reports the results aanalyzes them in some detail, while in Sec. V we discussconsequences of scaling the strength of the potential-ensurface in the search for Efimov-type states. Since the nber of bound trimer states can change with the potentiaSec. VI we report some preliminaryab initio calculations ofthe full interaction when the 3B effects are also includeOur conclusions are finally summarized in Sec. VII.

II. THE POTENTIAL-ENERGY SURFACE „PES…

The study of the4HeH2 interaction has been recentrevived because of the corresponding interest in the isoetronic 4He2 dimer and, as mentioned before, its bound stfeatures. The negative ionic character of4HeH2, however,leads to a very different two-body~2B! interaction with re-spect to the neutral dimer and, therefore, to a very differpotential-energy curve. Since the H atom has a sizeable etron affinity ~0.7542 eV! while the He2 system is unstablethe two-particle system will preferably have He1H2 as itslower dissociating threshold for the ground electronic stat15

Hence, the charge-induced interaction at large distancesbehave as proportional toR24 and, therefore, will go to zeromore slowly than the dispersion interaction in4He2 .

The recent full-configuration-interaction ~FCI!calculations15 on the 2B potential have produced a very acurate description of it, suggesting a rather small well deof the potential~;4.0 cm21!, a large value of the distancwhere the minimum is located~about 13 bohr! and a verylong tail for the vibrational ground-state wave function. Tcalculations also showed the presence of an excited rtional state withJ51 in addition to the single bound stafound for J50.

Another recent calculation on the same system18 adoptedinstead a model potential approach and assumed the4HeH2 interaction to consist of the pair interactions betwethe extra electron and the two atomic partners. The authof Ref. 18, therefore, solved the one-electron diatomic mlecular ion, within the Born–Oppenheimer~BO! approxima-tion, by using spheroidal coordinates to obtain a twdimensional eigenvalue equation at each internucseparationR and by then solving the ensuing equation wthe use of two-dimensional B-Spline functions.18 Because ofthe marked differences between the two methods, the cputed data do not agree quantitatively with each otherthough the qualitative conclusions are very similar. We sumarize in Table I the results given by the two methodsthe properties of this dimer. In the present work we haused for the4He–H2 interaction the FCI calculations via thanalytical fit employed by Casalegnoet al.19 in recent MonteCarlo ~MC! calculations for (He)NH2 clusters. For the4He2

interaction potential we have adopted the one proposedAziz and Slaman20 as in our previous calculations.16,17,21,22


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We report in Fig. 1 the relative shapes and strength vues of the 2B potentials considered here: The HeH2 interac-tion is clearly shallower and more repulsive than the corsponding HeHe potential, as already seen by previwork.15,18,23However, the different shape of the long-rantail of the 2B interaction makes the HeH2 potential capableof supporting a more strongly bound state which is mo‘‘compact’’ in space, albeit still very diffuse, than the corrsponding bound state of4He2 .21

The simplest construction of a three-body~3B! interac-tion potential for the trimer is to treat the system as a sum2B interactions between partners

V~R1 ,R2 ,R3!5@VHeH2~R1!1VHeH2~R2!

1VHeHe~R3!#. ~1!

Due to the relative weakness of each 2B potential this mnot be an unrealistic ansatz. It is the one which has been uin all the previous calculations for this trimer.19,23

TABLE I. Computed properties of the4HeH2 interaction potentials fromRefs. 15 and 18.

2B property Ref. 15 Ref. 18

Rmin /bohr 12.88 11.5De/cm21 3.22 4.99e0

J50/cm21 20.4000 20.952e0

J51/cm21 20.0447 20.393^R&J50/bohr 22.54 17.9^R&J51/bohr ¯ 20.2

FIG. 1. Representation of the atom–atom potentials employed in the prework. Distances in Å and potential values in cm21. See text for the refer-ences associated to each potential curve.


alx.

5522 J. Chem. Phys., Vol. 114, No. 13, 1 April 2001 Gianturco et al.

FIG. 2. Two-dimensional energy levels for the potentiof Eq. ~1!, using Jacobi coordinates for the compleThe energy values are given in cm21.

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In Fig. 2 we present the potential-energy maps obtaifrom our potential via Eq.~1! for the range of available(R,r ,u) values as given by writing 2B interactions withinJacobi coordinate representation. One clearly sees therexistence of symmetric,D2h , minima corresponding to theH2 partner located between 7 and 8 Å away from the mid-point of the He2 diatomic. However, as we shall further dicuss below, the overall weakness of the full interaction athe high delocalization of the bound atomic partners makevery difficult to analyze the spatial structure of this vefloppy triatomic within the conventional language of rigmolecular structures.

III. CALCULATION OF THE BOUND STATES

One could approach the construction of the relevant,tal Hamiltonian within a Born–Oppenheimer~BO! picture asbeing given through the Jacobi coordinates described bewhereby the H2 partner is viewed as weakly bound to aequally weakly bound4He2 molecule. One can then emplo


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the PES of Fig. 2 over a broad range of coordinates anrather dense grid of radial and angular values, hence retically map the physical space within which one expectspresent triatomic molecule to be located.

As is well known, Light and co-workers24,25 have pio-neered the use of the so-called discrete variable represetions ~DVR’s! in quantum-mechanical problems, a powerfdevice which becomes applicable whenever the followconditions are satisfied:~i! There exists a basis se$wn(x);n51,N% onto which one can expand the wave funtion of the system, and~ii ! a ~Gaussian! quadrature rule con-sisting of a set of quadrature points$xn ;n51,N% andweights $wn ;n51,N% can be used to compute matrix elments of the relevant operators within the above baWhenever such conditions are satisfied, there exists anmorphism between the finite basis representation~FBR! into$wn% and a discrete representation of coordinate eigenfutions based on the quadrature points. The quadrature pothus become the eigenvalues of the FBR of the coordinoperator so that the corresponding eigenvector ma


na

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specifies the chosen transformation between the coordieigen- functions and the original basis functions.26,27

We have, therefore, carried out calculations of the boustates for the4He2H

2 system by using spherical oscillatorsrepresent the radial basis and Legendre polynomials forangular basis.24,28 In order to reach numerical convergencwe employed 100 functions along the He–He distance~withr going from 0 to 80 Å!, also 110 functions alongR, whichwas varied between 0 and 90 Å, and 24 angular valuestweenu50° and 90°. The details of the numerical implemetation have been given before25 and will not be repeated heras we have followed the same procedure.

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An alternative approach to the problem consists in usatom–atom pair coordinates$R1 ,R2 ,R3%, with the Hamil-tonian written down in terms of 2B interaction potentials

H5T1(i 51

3

V~Ri !, ~2!

the volume element beingR1R2R3dR1dR2dR3 . The corre-sponding kinetic-energy operator for a system of two idencal particles and a third different partner can be written a16

TS52\2

2m H 1

R12

]

]R1R1

2 ]

]R11

1

R22

]

]R2R2

2 ]

]R21

2

R32

]

]R3R3

2 ]

]R31

R122R2

21R32

R1R3

]2

]R1]R31

R222R1

21R32

R2R3

]2

]R2]R3J

2\2

mHH 2

1

R32

]

]R3R3

2 ]

]R31

R121R2

22R32

2R1R2

]2

]R1]R22

R122R2

21R32

2R1R3

]2

]R1]R32

R222R1

21R32

2R2R3

]2

]R2]R3J , ~3!

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wherem5mH•mHe/(mH1mHe). The inclusion of the scalefactor (R1R2R3)1/2 in order to get the standard normalizatioof the total wave functionC(R1 ,R2 ,R3) leads to the finalHamiltonian expression which we have given in detail elwhere for the case of zero total angular momentum.17 As inour previous work, we have expanded the wave functionterms of a set of distributed Gaussian functions28 ~DGF’s!which take into account the proper symmetry of tsystem.16,17

After careful numerical tests, we have finally employa basis of 22 Gaussian functions which depend on eatom–atom pair coordinate and are centered as follo3,3.7,...,12.1,13,14,...,17,20,23, and 26 Å. They are given

wp~Ri !5S 2Ap

p D 1/4

e2Ap~Ri2Rp!2, ~4!

whereRp is the center of thepth Gaussian, the correspondinwidth being controlled by

Ap54b

~Rp112Rp21!2 ~5!

with a selected optimalb value of 1.05.The total wavefunction for the nuclear part of the t

atomic system is expanded over a basis function setf j as

Ck~R1 ,R2 ,R3!5(j

aj~k!f j~R1 ,R2 ,R3!, ~6!

where eachf j function is a Hartree-type product of threewGaussian functions

f j~R1 ,R2 ,R3!5Nlmn21/2 (

PPS2

P@w l~R1!wm~R2!#wn~R3!.

~7!

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Here, as in Eq.~6!, j stands for a collective indexj 5( l<m,n) and R3 represents the He–He distance within ttrimer. In turn, theN normalization factors become

Nlmn52snn~sll smm1slm2 !, ~8!

with the si j ’s being the overlap integrals

si j 5^w i uw j&. ~9!

In order to satisfy the triangular constraint for the mlecular geometries,uR12R2u<R3<(R11R2), then one ac-cepts in Eq.~6! only those three-function products for whicthe centers of the corresponding DGF’s satisfy the requments: Rn<Rl1Rm and Rm<Rl1Rn . Under such condi-tions, the selected 22 DGF’s on each coordinate generabasis of Hartree products for expansion~6! that amounts fora total of 3060f j ’s as defined in Eq.~7!. In this way, thedeviation from the triangular requirement, as we definedRef. 22, is estimated to be lower than 1%.

The advantage of using pair coordinates consists onfact that a variety of spatial indicators, which help usbetter describe the diffuse and floppy nature of trimers,be readily estimated. In particular, it is possible to tacklestudy of the spatial positioning of the partner atoms wrespect to the center-of-mass~c.m.! or to the geometricalcenter~GC! of the trimer molecule. Thus, the distances of tthree atoms to the c.m., in terms of the interparticle coornates, are

r H–CM5mHe

MA2R1

212R222R3

2, ~10!

r He1–CM51

MAmHmHem0~mHe

21R122m0

21R221mH

21R32!,

~11!


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r He2–CM51

MAmHmHem0~mHe

21R222m0

21R121mH

21R32!,

~12!

where M52mHe1mH , m05mHe1mH , and the He atomshave been labeled 1 and 2. Similarly, the distances ofthree atoms to the GC can be written as

r H–GC5 13A2R1

212R222R3

2, ~13!

r He1–GC5 13A2R1

22R2212R3

2, ~14!

r He2–GC5 13A2R1

212R2212R3

2. ~15!

The corresponding expectation values, therefore, reqthe evaluation of integrals of the form

^Ckur X–CuCk&5(j j 8

aj j 8~k!^f j ur X–Cuf j 8&

5(j j 8

aj j 8~k!(

PP8Nj j 8

21/2^P@w lwm#wnur X–C

3uP8@w l 8wm8#wn8&, ~16!

wherer X–C is the corresponding distance between the atX and the center C~CM or GC! of the system,aj j 8

(k)

5aj(k)aj 8

(k) and Nj j 85NlmnNl 8m8n8 . The integrals betweenthe DGFs can be accurately computed by using

^w lwmwnur X–Cuw l 8wm8wn8&'r ~R1† ,R2

† ,R3†! sll 8smm8snn8 ,

~17!

r (R1† ,R2

† ,R3†) is here the distance evaluated at the cen

R1† , R2

† and R3† of the Gaussian functions obtained as t

product of the two Gaussianw-functions associated to eaccoordinate.

In a similar way, as discussed earlier by us,22 the expec-tation values for the cosine of the angle associated to eacatom,^cosuX&, and higher moments,^cosn uX&, can be easilyestimated. Another significant quantity is the root-measquare~rms! value of the area ‘‘covered’’ by the trimerA^S2&, whereS2 can be expressed as

S25 18~R1

2R221R1

2R321R2

2R32!2 1

16~R141R2

41R34!. ~18!

Finally, an additional indicator of the contribution of eactriangular structure,j, to the correspondingkth bound state isprovided by a pseudo-weight defined as22

Pj~k!5aj

~k!^Ckuw j&, ~19!

this piece of information is typical of the DGF variationapproach and will be used extensively below to draw phycal information on the shapes of the floppy triatomic coplex.

IV. COMPUTATIONAL RESULTS

As mentioned before, several calculations have beenried out recently on the structure of the bound states sported by the4He2H

2 triatomic complex19,23 and, therefore,it also becomes of interest to compare the earlier findiwith the present results. Recent variational and diffusMonte Carlo calculations~VMC and DMC!19 have been


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based on the same potential form as that used by us in~1!, that is, the same 2B potential to describe4HeH2 but aslightly different 4He2 potential.29 The potential used herefor the 2B He–He interaction,20 however, differs very littlefrom the latter and, therefore, it is expected to yield a sstantially similar PES for the triatomic complex.

We report in Table II a summary of the eigenvalufound for the present trimer in itsJ50 state using the vari-ous methods discussed above. Since the dissociation thold of the4He2 system is;20.001 cm21, while that for the4HeH2 system is;20.4 cm21, it follows that only the firsttwo energy levels are bound with respect to both asymptowhile those fork52 and 3 are bound only with respect to thhigher energy threshold of4He2 dissociation but are ghosstates with respect to4He release

4He2H2→4HeH214He. ~20!

We clearly see in Table II how the differences represeing the 4HeH2 potential cause very marked energy diffeences within the binding energies of the triatomic systeBoth the number and positions of the bound states fowith hyperspherical coordinates within the adiabaapproximation23 differ markedly from ours.

On the other hand, the present calculations whichtwo entirely different approaches, the DVR treatment andDGF expansion, are seen to be in very good agreementeach other: In fact, the present bound states supported btriatomic complex agree within 1% of each other. This indcates convergence of the two methods to the same seresults. The earlier Monte Carlo calculations19 could only

TABLE II. Calculated4He2H2 bound states, in cm21, using the methods of

this work. For comparison, earlier results are also presented~columns 1–3!.

k Hypersphericala VMCb DMCb DVRc DGFc

0 21.1885 21.0565 21.0912 21.0989 21.09021 20.5421 20.8563 20.84782 20.0037 20.3239 20.32613 20.0911 20.0821

aFrom Ref. 18.bFrom Ref. 19.cPresent calculations.

TABLE III. Average magnitudes of the triatomic complex when using tDGF expansion of the present work. Energy in cm21, distances in Å. Inparentheses, fluctuation values of the corresponding quantities. In therow, the rms of the area,A^S2& is expressed in Å2.

k5E5

0a

21.09120

21.09021

20.84782

20.3261

r HeH2 11.11 10.92~3.37! 11.10~3.68! 13.97~5.77!r HeHe 7.86 9.63~4.90! 14.77~6.70! 21.71~7.50!r He–CM 5.25 4.95~2.45! 7.46~3.31! 10.90~3.77!r H2–CM 6.88 8.39~3.10! 6.62~3.49! 7.92~3.66!cosuHe ¯ 0.40~0.47! 0.64~0.44! 0.73~0.33!cosuH2 ¯ 0.51~0.54! 20.05~0.70! 20.22~0.55!rms(S) ¯ 43 53 91

aFrom Ref. 19.


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produce the ground-state energy but their results alsoout to be very close to ours, in particular their DMresults.19

In Table III we list several properties of the bound staof the trimer obtained by using the DGF method. The flowing comments are in order:

~i! We report results for three bound levels in spite of tfact that only the first two are lower than both dissciation thresholds. The reason to include the th~ghost! level is its proximity in energy to the HeH2

dimer;~ii ! the floppiness of the molecular structure is clearly v

ible from the large fluctuation values displayedradial and angular quantities: The triatomic structuis obviously not given by a unique, rigid moleculastructure. The rms of the area also gives an indicaton the size of the triatomic cluster. The correspondareas found for the different bound states are oforder of those reported for the ground state ofLiHe2 system;16,17

~iii ! for k50, the bond distances are not very differebetween the three particles, although the H2 is foundto be at a larger distance from any He atom than eitof the helium partners. This is in qualitative agrement with the results of Ref. 19 which are reportedthe first column of the table. This situation is just thopposite to thek51, 2 cases, where the He–He ditance becomes the largest;

~iv! due to the large fluctuations in the cosine of boangles, very high moments become necessary toscribe the corresponding distributions, showing agthe strongly floppy character of this cluster. Oshould also notice that the states withk51, 2 chieflycorrespond to a bending excitation of theuH2 angle.In particular, thek52 state presents a marked linegeometry in which the hydrogen is placed in betwethe helium atoms.

FIG. 3. Computed radial distributions along the internal coordinates oftriatomic complex obtained from the variational DGF expansion discusin the main text. The distributions refer to the ground state of the comp


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n

The ground-state (k50) pair distributions are given inFig. 3. One clearly sees there the agreement with whathave mentioned above and found by the VMC–DMC calclations of Ref. 19: The distribution of H2 with respect toeither 4He atoms peaks much further out than the p4He–4He distribution. Our current calculations that usedDMC approach to obtain radial observables for clusters wa larger number of He atoms indeed appear to confirmpresent findings.30

The use of the pseudo-weights defined in Eq.~19! fur-ther allows us to obtain a more appealing pictorial viewthe possible geometries which contribute to the descripof the present system. A summary of the leading trianguconfigurations is reported in Fig. 4, while their relative importance is listed in Table IV for the ground,k50 level.Such configurations were selected taking into accountcenters of the Gaussian functions and allowing a toleragiven by the grid step. We see that the scalene configurat~markedS in Table IV! provide by far the largest contribution to the description of the system,~;60%!. However, thetwo isosceles structures@markedI (1) and I (2) in Fig. 4 andTable IV# also provide sizeable contributions, accounting;25%, while the equilateral structure is almost negligib

ed

x.

FIG. 4. Families of triangular configurations obtained from the dominHartree-type basis functions, Eq.~7!, employed for the ground state of thcomplex. Their corresponding pseudoweights are given in Table IV.

TABLE IV. Computed sum of pseudo-weights, Eq.~21!, for the differentconfigurations collected in Fig. 5, corresponding to the ground trimer stThe S runs over all the triangle structures belonging to a given configution. The highest contributing structure to each family of configurationsalso listed in terms of pair coordinates,Ri , defining the Gaussian centers, awell as the corresponding atom distances from the GC,r X–GC. All distancesin Å.

S j8Pj(0) ~%! R1 R2 R3 r H2–GC r He1–GC r He2–GC

C(1) 4.77 7.9 9.3 17.0 1.0 8.3 8.7C(2) 6.14 8.6 13.0 5.1 7.1 1.8 5.9I (1) 13.42 7.9 8.6 4.4 5.3 3.1 3.7I (2) 12.46 8.6 10.0 7.9 5.6 4.4 5.3E 1.09 7.9 8.6 7.9 4.8 4.4 4.8S 62.13 8.6 10.0 4.4 6.0 3.1 4.3


e

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Finally, it is interesting to note that the collinear structurroughly weight 10% and, therefore, also play a role.

To carry out this analysis even further, we presentTable IV, for thek50 state, the values of the atom–atodistances and of the partner distances from the GC showthe dominant configuration which occurs within each famof triangles analyzed in the table and shown in Fig. 4.looking at ther H2–GC distance in the most important triangular configurations~S, I (1), and I (2)), one immediately re-alizes the agreement with the maximum of the radial disbution reported in Ref. 19 which was foundr H2–GC;5.82 Å. Furthermore, we see that the distancesH2 from the GC, which correspond to keeping it closerGC, (C(1)) or further away, (C(2)) than the two helium at-oms, indeed belong to configurations where the overall shof the complex tends to be quasilinear. They have a vsmall but finite probability, in agreement with the conclusireported by Ref. 19. Moreover, the average of the distandefined in Eqs.~13!–~15!, ^r X–GC&, become 6.30 Å for H2

and 5.89 Å for the He atom, the fluctuations being 2.33 a2.40 Å, respectively. Thus, these results suggest thatdominant configurations indeed correspond to the heliumoms being closer to the GC than the H2 impurity, as ob-tained by the Monte Carlo calculations19 and indicated byour radial distributions of Fig. 3.

V. POTENTIAL SCALING AND EFIMOV STATES

As discussed in our previous work,16,17,21,22one of theadditional interests for the weakly bound three-body agggates relates to the earlier studies in nuclear physics caout by Efimov.31 In this context, a special effect was sugested to occur whenever each dimer involved in theatomic species has no bound states but does exhibit aresonance. The ensuing Efimov effect amounts to a situawhere the 3B system can now support infinitely many boustates which can accumulate at the dissociation threshThe corresponding total wave function for the systethereby acquires a very extended spatial distribution overfairly compact range of action of the full 3B interactiogiven by a sum of 2B potentials. In the case of an identiparticle system~with, therefore, only one 2B potential involved! one can gauge the features of the full interactionintroducing a strength parameterl. In contrast with the standard, expected behavior of common bound states,Efimov-type states gradually disappear with increasingl.

The purpose of this section is to analyze the feasibilitythe existence of Efimov-type states in this trimer. The usprocedure is to varyl and then to estimate the numbeNE(l), of such diffuse states through the followinexpression:31

NE~l!51

pln

ua0~l!ur 0~l!

, ~21!

wherea0 is the scattering length at zero energy fors-wavescattering, andr 0 the effective range of thel-scaled 2B in-teraction. Only whenNE is found to be close to or greatethan unity the 3B system would be expected to showpresence of Efimov states. This prescription works fa


s

n

by

y

i-

f

pery

es

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i-ro

ondld.

e

l

y

e

fal

ey

well in the case of boson helium trimers,21,22,32as well as inthe6Li4He2 system.16 However, the isotopic counterpart sytem, 7Li4He2 , shows the presence of an Efimov-type stafor l values close to one in spite of a value ofNE;0.4 foundfor the 7Li4He dimer.16

One way of extractinga0 andr 0 is to fit the behavior ofthe phase-shift to the expression33

k coth0~k!52a0211 1

2 r 0k210~k4!, ~22!

whereh0(k) is thes-wave phase-shift, estimated throughNumerov propagation in the continuum, at the collisiwave-vectork. In the present situation for the4HeH2 poten-tial, however, the simple definition of above does not apsincer 0 cannot be defined for a potential falling off to zeas r 24 like in this case.34 By using the modified effectiverange theory, however, the scattering length has still physmeaning,35,36 and can be obtained by reaching numericconvergence for thes-wave phaseshift calculations ask goesto zero. In the present case, whenl51.0, a value of10.303Å was found fora0 .

The calculated dependence ofa0 on a range ofl valuesfor the 4HeH2 dimer is reported in Fig. 5, where the resuclearly show that the criticality needed for the presenceEfimov-type states occurs with a modified 2B potential tstrength of which has become unrealistically small~l;0.34!.It, therefore, follows that the present triatomic complex donot seem to show Efimov-type diffuse states unless its oall strength is unrealistically reduced.

Another feature of the diffuse bound states of thesystems, when analyzed in terms of 2B potential details,volves the location of halo states, i.e., of the presence‘‘local’’ 3B bound states when neither of the 2B potentiasupports, when alone, either bound states or zero-enresonances. In the present case we know that the hedimer potential already does not support anymore a bostate forl,1 but very close to it. On the other hand, a vestrong variation of the 2B potential of the ionic dimer with

FIG. 5. Effects of potential scaling on the4HeH2 properties: Behavior of itsJ50 bound state and thes-wave scattering length as a function of thelvalues.


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the trimer is necessary for its bound state to disappear~e.g.,see Fig. 5!. Through the onset of the dimer bound stawhich in turn coincides with the critical behavior of the scatering length, we clearly see again that one needs to dramcally reduce the 2B potential strength in order to be ablelocate halo states in the trimer system. Given the unrealcally reduced interaction, it is very unlikely that the ‘‘exactinteraction within the4He2H

2 complex could give rise tosuch a special type of bound state.

As was mentioned earlier,16 some caution about the reliability of the prediction of the number of Efimov-type 3bound states from 2B properties, as given by Eq.~21!, maybe in order here. In Fig. 6 we show the energy dependencall the relevant bound states by plotting the energy valuEb , as a function of thel parameter. The global prescriptiofor scaling the strength of the 3B interaction is given as

V~R1 ,R2 ,R3!⇒lV~R1 ,R2 ,R3!

5lFVHe2~R3!1(

i 51

2

VHeH2~Ri !G . ~23!

We clearly see in Fig. 6 that, at least in the limited ptential region wherel is varied at most by 20%, the firsexcited state below the4HeH214He dissociation channellabeled asE3B

(1) in the figure, stays well below the dissocition threshold and never tends to disappear into the ctinuum of the first dissociation. The next excited state,beled here asE3B

(2) is clearly above, but close to, the firthreshold for dissociation and only crosses it in the vicinof l51.1.

We also carried out a sort of two-dimensional analysisthe potential strength dependence of the bound states otrimer complex, this time by varying separately the4HeH2

and HeHe diatomic potentials. The corresponding potenscaling, therefore, becomes

FIG. 6. Dependence of the complex bound state energies on the gstrength of the PES as defined by Eq.~23!. See main text for meaning osymbols. All energies are in cm21.


,

ti-oti-

ofs,

-

n--

fthe

al

V~R1 ,R2 ,R3!

⇒@l1VHe2~R3!1l2VHeH2~R1!1l2VHeH2~R2!#, ~24!

and the results are reported in Fig. 7. The six panels shonly the variation of theE3B

(2) in comparison with the firstdissociative threshold, i.e., theEHeH2 binding energy. In theupper three panels the strength of the4HeH2 2B potential iskept fixed in each panel while the strength of the4He2 2Bpotential is varied. The complementary changes are showthe lower three panels of Fig. 7.

One clearly sees that the strengthening of the helidimer potential changes rather little the overall binding eergy of the 3B excited state which varies instead rather maedly as the HeH2 interaction changes its strength~lower pan-els!. On the other hand, the increase of the weakerpotential strength by only 2% or 3% shifts very clearly tE3B

(2) , as indicated in the upper three panels. One alsoserves there that the dependence of the binding energy ostrength of the He2 potential is really very weak, as alreadsurmised by the results of the lower panel. In all exampshown there is no appearance of Efimov-type features for3B excited state under consideration, as already expectethe previous scattering length calculations.

al

FIG. 7. Two-dimensional dependence of the complex second excited-energy,E3B

(2) , in comparison with the location of the lower dissociatiothreshold,EHe2H2

(0) . The upper three panels show their dependence on

variation of the strength parameter of the4He2 interaction at fixed values ofthe 4HeH2 interaction, as given by Eq.~24!. The lower three panels showthe results for the potential changes of the4HeH2 interaction at fixed4He2

potential strength values.


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We can, therefore, say from the above results, thatpossible consequence of strengthening the 2B forces incalculation of the PES of the ionic trimer would be thatmaking the two, already existing, bound states of the trimeven more strongly bound and to induce the eventual appance of an additional third bound state. It deserves, howesome consideration the further possibility of directly peforming ab initio calculations of the full PES involving theadditional 3B contributions in order to assess the reliabiof the results obtained through the use of the simple pairwinteraction of Eq.~2!.

VI. THE FULL TRIATOMIC INTERACTION

In order to have a better understanding of the full Pfeatures, preliminaryab initio calculations have been peformed using the basis set of Ref. 37, uncontracting the futions used there and adding a set of diffuse functio@(2s,2p,2d) on He, (6s,6p,6d) on H#. All calculations wereperformed using theMOLPRO chain, at coupled-cluster singldouble triple~CCSD-T! level, which means that the effect otriple excitations is taken into account in a perturbatway.38 In a previous study of HeH2 it was noticed that,among the different perturbative treatments of triples avable in MOLPRO, CCSD-T gives the results that were thclosest to FCI.15

These full calculations have been carried out for selecgeometries and started with theD2h symmetry. The anglewas then moved away from the T-shape geometry but folimited range of He–He and He–H2 distances. Figures8~a!–8~c! show a comparison between the selected poobtained from the full interaction and the same values frthe surface as computed by using the simpler sum of Eq.~1!.The three subfigures refer to three different values ofHe–He distance and each of them shows four different pels at different values of the Jacobi angle and over the raof H2 distances taken from the midpoints of the4He2

dimers. The solid lines show the behavior of the potenwhen obtained as a sum of 2B interactions, while the dolines refer to the full~2B13B! calculations. The followinginformation could be gathered from the comparison:

~i! In the region of quasi-collinear geometries~small uvalues! the effects coming from the full interactioappear to be rather small as the dashed lines folfairly closely the solid lines for all He–He distance

~ii ! the onset of the repulsive wall and the location of twell are affected by 3B forces mostly in the regiontriangular structures close to the isosceles geomeIn fact, the~2B13B! calculations appear, in all panewith u560° and 90°, to yield interaction potentiawhich have deeper wells and softer repulsive walls

Although we are here talking about fairly small effecthey are rather marked at the geometries close toT-shaped complex when one considers the weakness owhole interaction. Thus, by carrying out our initial study van overall PES obtained as a sum of 2B forces, we shoulaware of the fact that we are possibly constructing intertion forces which are less attractive and more repulsive t


ehe

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full calculations would indicate. From the model analysisthe previous section on the effects ofl scaling, however, wecan already say that no more than one additional third swould become bound for a set of a more strongly interactthree particles. One should also note here that the expatory calculations reported in this section do not intendfully analyze the effects of 3B forces but simply to point othe likely corrections to the binding energy values whithey are inducing in the present system.

VII. PRESENT CONCLUSIONS

In this work we have shown the structure of the boustates of a very floppy trimeric complex in its ground eletronic state, the4He2H

2 complex, by using both Jacobi coordinates within a DVR representation and pair coordinawith a DGF expansion. Assuming a pairwise form of toverall PES, we have presently looked at theJ50 situationand found very good agreement between the two typescalculations: two bound states with respect to the lower dsociation threshold were found to exist. The lower boustate was also found to be in very good accord with otDMC calculations, thus providing an independent checkthe present results, as seen in our Table II. Therefore,remarkable convergence of results when using three, entdifferent, computational methods, shows that the results frthese methods have become highly reliable.

The use of a DGF expansion allows us to easily gsome further information on the spatial properties of tpresent complex since it gives us specific values for thedial size of the bound states, indication on the most preferthree-particle configurations which can describe the compand possible comparisons with the results obtained fromDMC calculations of Ref. 19. The examination of our davis a vis the suggestions made from DMC calculations, fthe ground state, indicates once more good agreementtween the two sets of results~e.g., see Table III!. The mostlikely configuration turns out to be the scalene triangushape followed by isosceles arrangements, with the H2 im-purity outside the two4He atoms and further away. Thprobability of quasilinear structures, finally, is small butnite from both types of computations.

We have also looked into the likelihood for the presesystem to exhibit excited states with Efimov-type behaviordiscussed by us in the case of other, very weakly bouthree particle complexes.16,17,21,22For this purpose we haveexamined the critical behavior of thes-wave scatteringlength, at vanishing collision energies, for the4HeH2 poten-tial as its strength is varied. We have found that the corspondinga0 value appears to show criticality only for aunrealistically weakened 2B potential. The same potentiaalso shown to be able to possibly havehalo states only whenits strength is reduced by strong factors, hence makingabove effects unlikely to occur for any realistic choice of tHeH2 interaction. We have also reduced the overall strenof the global pairwise potential and found that none of tbound states disappear through the threshold for the lodissociation channel. We can conclude, therefore, that4He2H

2 is very unlikely to show the presence of Efimotype states.


rdi-r;


Dow

FIG. 8. Computed full interaction at selected geometries, in Jacobi coonates, for the4He2–H2 complex.~a! For the He–He distance of 5.0 boh~b! For an He–He distance of 6.0a0 and~c! For an He–He distance of 7.0bohr. The solid lines labeled ‘‘2B sum’’ refer to the potential from Eq.~1!while the dashed lines labeled ‘‘3B’’ report the fullab initio CCSD-Tcalculations.

nloaded 23 May 2013 to 161.111.22.69. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

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On the other hand, one of the excited bound stateseen to cross the threshold for the lower dissociation chaat moderate strengthening values,l;1.1, suggesting thasuch a state could become an actual bound level if theforces were to be properly included. The contribution of suforces on the full interaction has been analyzed at a prelnary level by carrying out additional CCSD-T calculatiofor a set of selected geometries of the three-particle compA comparison of such computations with the interaction otained instead as a sum of 2B terms showed that, especfor the nuclear configurations near to the T-shape geetries, the 3B forces are not negligible and cause the corinteraction to be both more attractive in the well region awith a ‘‘softer’’ repulsive wall. Thus, we expect that thpresent pairwise PES is likely to be qualitatively correalthough room for improvement still exists at the quantitatlevel.

ACKNOWLEDGMENTS

The financial support of The Italian Nat. Research Cocil ~CNR! and of The Italian Ministry for Universities anResearch~MURST! is gratefully acknowledged, togethewith the DGICYT ~Spain! under contract PB95-0071. Walso acknowledge the support of the Italy–Spain Collabotive Integrated Action No. HI1999-0157.

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The structure of a weakly bound ionic trimer: Calculations ...digital.csic.es/bitstream/10261/76657/1/Gianturco.pdf · Serrano 123, 28006 Madrid, Spain G. B. Bendazzoli and S. Evangelistid)