thermal effects on small para-hydrogen clusters

Thermal Effects on SmallPara-Hydrogen Clusters

JESÚS NAVARRO, RAFAEL GUARDIOLAIFIC, CSIC-Universidad de Valencia, Apartado 22085, 46071 Valencia, Spain

Received 10 December 2009; accepted 29 December 2009Published online 30 March 2010 in Wiley Online Library (wileyonlinelibrary.com).DOI 10.1002/qua.22575

ABSTRACT: A brief review of different quantum Monte Carlo simulations of small(p-H2)N clusters is presented. The clusters are viewed as a set of N structureless p-H2

molecules, interacting via an isotropic pairwise potential. Properties as superfluidity,magic numbers, radial structure, excitation spectra, and abundance production of (p-H2)N

clusters are discussed and, whenever possible, a comparison with 4HeN droplets ispresented. All together, the simulations indicate that temperature has a paradoxical effectof the properties of (p-H2)N clusters, as they are solid-like at high T and liquid-like at lowT, due to quantum delocalization at the lowest temperature. © 2010 Wiley Periodicals, Inc.Int J Quantum Chem 111: 463–471, 2011

Key words: atomic and molecular clusters; phase transitions in clusters; structure ofclusters; molecular hydrogen and isotopes

1. Introduction

U nder normal conditions of pressure and tem-perature, hydrogen is a gas formed by H2

molecules. The molecule exists in two isomericforms, para-hydrogen (p-H2) and ortho-hydrogen(o-H2), which differ in the coupling of their nuclearspins: antiparallel (J = 0) and parallel (J = 1), respec-tively. The energy difference between the lowestlevels of the isomers is 170.5 K, the p-H2 being the iso-mer with lower energy. At room temperature thereis a statistical mixture of the two isomers, and equi-librium hydrogen is 75% ortho and 25% para. At

Correspondence to: J. Navarro; e-mail: [email protected] grant sponsor: MCyT (Spain).Contract grant number: FIS2007-60133.

low temperatures, near the boiling point (20.3 K),hydrogen is nearly all para-hydrogen.

The p-H2 molecule and the 4He atom have incommon that both are spinless bosons subject toweak van der Waals interaction. This analogy ledGinzburg and Sobyanin [1] to suggest that bulk para-hydrogen could also exhibit superfluidity. For anideal nondegenerate Bose gas, the temperature forBose-Einstein condensation is given by 3.31�

2n2/3/m,where n is the number density and m is the massof the boson. For liquid 4He at saturation one hasn � 0.022 Å−3, and a transition temperature of 3 Kis predicted, close to the observed value of 2.17 K.Although the p-H2 molecule mass is smaller thanthe 4He atom mass by a factor of two, the attrac-tion between two p-H2 molecules is greater thanthe He-He one by a factor of four, and the stablelow-temperature bulk phase of hydrogen is actually

International Journal of Quantum Chemistry, Vol 111, 463–471 (2011)© 2010 Wiley Periodicals, Inc.

NAVARRO AND GUARDIOLA

an hcp solid. Near the triple point (0.72 MPa, 13.8K), the density number of liquid para-hydrogen isn � 0.023 Å−3, and the predicted transition temper-ature is around 6 K. To produce superfluid p-H2,it is, therefore, necessary to supercool the liquidbelow its normal freezing temperature or to producenegative pressure. Attempts to produce a super-fluid by supercooling the normal liquid below thetriplet point have been insofar unsuccessful [2]. Apossible way to circumvent this problem is to con-sider finite clusters or two-dimensional geometriesto reduce the effective attraction between molecules.Indeed, the lowering of the melting point comparedwith the bulk is a well known and rather general phe-nomenon in clusters (see e.g., Ref. [3]). Path integralMonte Carlo (PIMC) simulations by Sindzingre et al.[4] predicted that (p-H2)N clusters with N = 13 and18 molecules are superfluid below about 2 K. Thisprediction motivated two experiments [5–7] and amuch larger number of theoretical studies [8–29] onsmall (p-H2)N clusters.

It is worth mentioning that superfluidity hasnot been the sole motivation for studying hydro-gen or deuterium clusters, and we mention here afew of them. Many investigations have focused onsingly charged hydrogen clusters, which are activespecies in the stratosphere and in interstellar clouds.Laser irradiation and Coulomb explosion of deu-terium clusters have been studied in connection withnuclear fusion reactions. The study of hydrogenclusters encapsulated in fullerenes and carbon nan-otubes has been motivated as a possible way to buildefficient hydrogen containers for energy transport(see [30] for details and references on these issues).

In this article we briefly review the results givenby quantum Monte Carlo simulations of small(p-H2)N clusters. These include path integral MonteCarlo (PIMC), which allows calculations at T >

0, diffusion Monte Carlo (DMC), which is a T =0 method, and path integral ground state (PIGS),which is an adaptation of PIMC for calculating (T =0) ground states. The clusters are viewed as a setof N structureless p-H2 molecules, interacting viaan isotropic pairwise potential. Among the differentinteraction potentials two are of particular interestbecause they combine ab initio properties with prop-erties of the gas (or solid) as well as experimentalinformation from collisions. One is due to Silvera andGoldman [31], the other to Buck et al. [32], hereafterreferred to as SG and BHKOS, respectively, the mostnoticeable difference between them is that the latterproduces slightly more binding than the former.

There is an apparent contradiction between thesuperfluidity of (p-H2)N clusters, which indicates aliquid-like character, and their structured radial dis-tribution functions as well as the existence of magicnumber stabilities, which point toward a solid-likerigidity. We shall discuss here properties as super-fluidity, magic numbers, radial structure, excitationspectra and abundance production of (p-H2)N clus-ters. Whenever possible a comparison with 4HeN

droplets will be presented. All together the theo-retical studies indicate that small (p-H2)N clustersbecome more solid-like at high T and more liquid-like at low T due to quantum delocalization at thelowest temperature.

2. Superfluidity

Evidence of superfluidity of (p-H2)N clusters wasgiven in Ref. 5, based on an infrared spectroscopicexperiment similar to the one performed by thesame group to show superfluidity in 4HeN droplets[33]. Clusters of p-H2 molecules were assembledaround a carbonyl sulfide (OCS) chromophore mol-ecule inside helium droplets in a molecular beam.The central OCS molecule serves as a spectroscopyprobe for measuring the effective moments of iner-tia, by studying the decoupling of its rotation andthe surrounding medium. It was found that (p-H2)N

clusters with N = 14, 15, and 16 molecules are super-fluid at sufficiently low temperatures. The sameexperiment failed to detect superfluid behavior ofortho-deuterium clusters, which is a clear indicationof the importance of the zero-point motion.

PIMC calculations predict a sizeable global super-fluid response for (p-H2)N clusters for sizes at least upto N � 30 at T ≤ 1.5 K, whereas clusters of larger sizehave a significantly depressed superfluid response.The superfluid fractions obtained by Mezzacapo andBoninsegni [18] at T = 1 K, and by Khairallahet al. [25] at T = 1, 1.5, 3, and 4.5 K are plottedin Figure 1. The results obtained by Khairallah etal. at T = 0.5 K have been omitted in Figure 1for the sake of clarity. The values exceeding 100%are attributed to the ambiguity in the definition ofthe moments of inertia in small clusters [25]. It canbe seen that clusters with N ≤ 22 are essentiallysuperfluid for temperatures T ≤ 1 K. The super-fluid fraction is only somewhat reduced in clusterswith N = 13 and 19, which are classical magic sizeswith highly symmetric icosahedral structures. Inter-estingly, in the size range 22 ≤ N ≤ 30, superfluiditystrongly depends on N and T. A few clusters behave

464 INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY DOI 10.1002/qua VOL. 111, NO. 2

THERMAL EFFECTS ON SMALL PARA-HYDROGEN CLUSTERS

FIGURE 1. Superfluid fraction of (p-H2)N clusters as afunction of N , calculated by PIMC methods at severaltemperatures. Labels (a) and (b) correspond to Ref.[18, 25], respectively. [Color figure can be viewed in theonline issue, which is available at wileyonlinelibrary.com.]

in a way consistent with coexistence of nonsuper-fluid and superfluid phases. Along the simulationprocess, at a given value of T, these clusters jumpfrom ordered solid-like configurations to delocal-ized liquid-like configurations, melting at low T as aconsequence of the zero-point motion and freezingat higher temperature. This intriguing behavior hasbeen named as “quantum melting” by Mezzacapoand Boninsegni. We shall come back to that pointlater on.

The two mentioned PIMC calculations are in over-all agreement about the N and T dependence of thesuperfluid fraction. However, there is a discrepancyconcerning the localization of the superfluid density.In the next section, we shall see that the radial densityprofile is characterized by the presence of spher-ical geometrical shells. Khairallah et al. [25] havefound that superfluidity is localized at the surfacebeyond the outer maximum in the radial density pro-file, mainly arising from exchange cycles involvingloosely bounded surface molecules. Consequently,superfluidity of (p-H2)N would lie in the surfaceregion, a behavior which is at variance with 4Hedroplets, where superfluidity is greatest near the cen-ter. Khairallah et al. conclude that “with increasingcluster size, the strong many-body intermolecularinteractions lead to a rigid solid-like inner corethereby pushing the delocalization-induced super-fluidity toward the surface, where it is favored bythe reduced coordination and weak inward interac-tion with the small central core.” Mezzacapo and

Boninsegni [28] have arrived to a different con-clusion, namely that in the low temperature limitsmall (p-H2)N clusters are uniformly superfluid. Forthese authors, superfluidity “crucially depends onthe onset of long exchange cycles involving allmolecules, not just those on the surface,” and theirresults do not indicate the existence of clusters witha rigid core and a superfluid outer shell. These twoso different interpretations call for further theoreticalresearch.

3. Magic Numbers

Small neutral (p-H2)N clusters have been pro-duced in a cryogenic free jet expansion and studiedby Raman spectroscopy [6, 7]. The Q(0) Raman lineof the H2 monomer is shifted as the number N ofmolecules in the cluster changes, thus providing amethod to identify the cluster mass. The first sevenresolved peaks next to the monomer line have beenassigned to clusters with N = 2, . . . , 8 molecules.Although in that experiment the resolution was notenough to resolve larger sizes, broad maxima wereobserved at N ≈ 13 and 33, and perhaps 55, whichhave been interpreted as a propensity for geomet-ric shell structures. Indeed, classical static [34–36]and molecular dynamics [36, 37] results, based ona generic Lennard-Jones interaction potential, indi-cate that the expected structures for such clustersare the so-called Mackay icosahedra [38], exhibitingmagic sizes (13, 33, 55…) related to the packing ofmolecules in closed icosahedral arrangements.Adis-cussion of these classical geometrical patterns can befound in [3, 39] and the references therein.

The calculated total energies E(N) of (p-H2)N

clusters decrease with the cluster size N in a seem-ingly monotonic behavior for both SG and BHKOSinteractions. It would be tempting to fit these ener-gies with a mass-formula, which works so well for4HeN droplets [40]. However, a closer examinationof the energies indicates local deviations, whichare magnified by looking at the separation energydefined as

δ(N) = E(N − 1) − E(N). (1)

It corresponds to the energy required to remove a p-H2 molecule from the (p-H2)N cluster, and it gives asensitive measure of its stability. Whereas in 4HeN

droplets δ(N) varies smoothly with N and tendsto the binding energy per particle in the bulk asN → ∞, the separation energy of (p-H2)N clusters

VOL. 111, NO. 2 DOI 10.1002/qua INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 465


FIGURE 2. Separation energies of (p-H2)N clusters asa function of N , calculated with several MC methods.See text for details. [Color figure can be viewed in theonline issue, which is available at wileyonlinelibrary.com.]

displays local maxima for some particular values ofN. The corresponding clusters are more stable thantheir neighbors, and this is an indication of theirmagic character.

The separation energies δ(N) as a function of N,calculated with the BHKOS interaction are plottedin Figure 2. PIMC calculations [29] at T = 1.5 Kgive magic numbers N = 13, 26, 29, 32, 34, 37, and39. PIGS calculations [21, 27] yield the same num-bers except 32 and 37, although there are noticeabledifferences with the PIMC values of δ(N). DMC cal-culations [19, 22, 26] result in a smoother separationenergy, with magic numbers limited to N = 13 and36. However, many of the PIMC magic numbers dis-appear when lowering the temperature, as shown inthe insert of Figure 2, where it can be seen that PIMCat T = 0.25 K and DMC results are in agreementwithin statistical errors. Nevertheless, we have noexplanation for the disagreement with PIGS, whichis a T = 0 method as DMC.

The precise magic numbers depend on the usedinteraction. For the SG one, PIMC calculations [25]yield magic numbers at N = 13, 19, 23, 26, 29,32, and 37 in the temperature range T = 0.5−3 K;PIGS [21, 27] gives 13, 21, 27, 34, and 39; DMC cal-culations [19, 22, 26] only yield N = 13. It appearsthat N = 13 is a robust magic number, independentof the interaction and the temperature. The experi-ments based on Raman spectroscopy [6] observedclear broad peaks at N � 13 and 33. It is worthmentioning that for N > 8 the relation betweenthe Raman shift and the number of molecules in theclusters is an extrapolation from small values of N,

and the position of the peak at N = 33 may beimprecise.

4. Structure

Relevant information about the structure of theclusters is provided by the one-body density distri-bution function, defined as the expectation value ofthe operator

N∑i=1

δ[r − (ri − R)], (2)

where ri stands for the ith-particle coordinates, andR = ∑

i ri/N for the center-of-mass coordinate.The number densities n(r) are plotted in Figure 3,normalized as

4π

∫ ∞

0dr r2 n(r) = N, (3)

obtained in a DMC calculation [19] with the BHKOSinteraction. As anticipated earlier, p-H2 moleculesare arranged in quite sharply defined sphericalshells, with radii growing slowly but steadily withthe number of molecules. It should be mentionedthat this shell structure does not appear in the den-sities obtained with the optimized VMC trial func-tions, so that they are developed along the DMCevolution. Near the cluster center-of-mass, the den-sity profile presents a very large peak around N = 13and 50, and no particle is present for sizes in between.The existence of geometrical shells is a strong indi-cation that small (p-H2)N clusters cannot be directlyidentified with simple liquid-like systems. Actually,it is very instructive to compare these density distri-butions with those obtained for 4HeN droplets. It canbe seen in Figure 3 that the latter are rather constantin the bulk of the cluster, as expected for a liquiddrop, presenting only slight oscillations that die outat the surface.

Alonso and Martínez [30] have pointed out anintriguing similarity between the growth of (p-H2)N

clusters and that of metallic clusters based on thesimple SAPS cluster model [41], in which the totalionic pseudopotential acting on the electrons isspherically averaged around the cluster centre. Theminimization of the total cluster energy leads to welldefined concentric shells of atoms. For instance, CsN

clusters with N < 7 have a hollow shell and in thesize range 7 ≤ N ≤ 18 the clusters have a centralatom surrounded by a spherical shell whose radius



FIGURE 3. One-body density profiles n(r ) of (p-H2)N clusters as a function of the distance to the center-of-mass of thecluster, calculated with DMC [19] using the BHKOS pairwise interaction. [Color figure can be viewed in the online issue,which is available at wileyonlinelibrary.com.]

increases as N grows. When 19 ≤ N ≤ 39, the clusteris formed by two shells, with no atom at the cen-ter, and beyond this size the structure consists oftwo shells surrounding a central atom. The similaritywith the DMC structural features of (p-H2)N clustersis striking, although the two systems are physicallydifferent.

PIMC and DMC one-body densities have beenpresented in Ref. [42], using a Lennard-Jones inter-action, with parameters ε = 34.2 K and σ = 2.96 Å,to allow for a comparison with classical calculations.On closer examination, some additional maxima areobserved in the PIMC density profiles at T=1.5 K,which are not found in the (T = 0) DMC groundstate. Such maxima are magnified by looking at thedensity distributions defined as

P(r) = 4π

Nn(r)r2, (4)

but the molecules near the cluster center are not visi-ble due to the factor r2. The T = 1.5 K PIMC densitiesindicate for N = 18 to 31 the presence of an addi-tional shell of molecules interposed between the twoshells obtained by DMC and for N ≥ 32 altogetherfour shells instead of two. Interestingly, the extrashells lie within the size range in which the finite tem-perature PIMC calculations predict magic numberswhich are not found in the DMC calculations.

As a typical example, the radial distribution func-tion P(r) for the cluster size N = 40 are plotted in

Figure 4, which are calculated with PIMC at two val-ues of T and with DMC. It can be seen that PIMCP(r) at T = 1.54 K has three peaks at around 1,3.7, and 6.5 Å. However, at T = 0.385 K there isonly one peak at 6.5 Å and a shoulder at 3.7 Å,and the density profile practically coincides withthe DMC ground state one, the small visible dif-ferences are likely due to a dependence of DMCcalculations on the trial function. The transition tem-perature at which PIMC and DMC densities nearly

FIGURE 4. Radial density distributionP(r ) = 4πn(r )r 2/N of the N = 40 cluster, calculated withDMC and PIMC. Adapted from Ref. [42]. [Color figurecan be viewed in the online issue, which is available atwileyonlinelibrary.com.]



coincide is estimated to be around 0.75−1 K [42],close to the temperature where the large clustersalso become superfluid [25], although the precisevalue could depend on the specific interaction. PIMCcalculations for distinguishable bosons were alsopresented in Ref. [42], and it was shown that bosonsymmetrization plays no role into the density dis-tribution at such high temperature. Classical den-sities have also been calculated and compared toPIMC ones at T = 1.54 K [42]. Both profiles havesome similarity, apart from a global outer shift of thePIMC density due to the zero-point motion energy.In conclusion, these results indicate that (p-H2)N

clusters are essentially classical clusters at high Tor, in other words, small (p-H2)N clusters are morequantum delocalized at T = 0 than at higher T.

5. Excitation Spectra

Besides its own interest, the determination of theexcitation spectrum of (p-H2)N clusters is interestingbecause it permits a comparison with the spectrumof 4HeN droplets and it also gives access to the knowl-edge of the partition function, as we shall see inthe next section. These spectra have been calculatedwith DMC in Ref. [40] for (p-H2)N and in Ref. [43]for 4HeN droplets; it turns out that they are verydifferent. An important feature of DMC is that theimportance sampling function allows one to controlnot only the required bosonic symmetry but also theangular momentum quantum number of the cluster.The ground state of a bosonic system is completelysymmetric and has total angular momentum equalto zero. A simple way to describe excited states withangular momentum L �= 0 is to include such a depen-dence into the importance sampling function in thefollowing way

N∑i=1

|ri − R|ν(L) YLL(ω̄i) �GS(r1, . . . , rN), (5)

where �GS is the trial function used for the groundstate, ω̄i stands for the spherical angles of the vectorri − R which defines the position of the i-th particlewith respect to the center-of-mass coordinate R, andYLM is a spherical harmonic. In this expression, wehave taken ν(L) = L for any L �= 0, except ν(1) = 3to avoid an identically vanishing function.

In the time evolution process, all componentswith angular momentum different from the speci-fied value L vanish exponentially, and thus one getsan estimate of the energy of the lowest state with

FIGURE 5. Reduced excitation spectra of (p-H2)N

clusters and 4He droplets, calculated with DMC. Thedotted lines correspond to the chemical potentials.Adapted from references Ref. [40, 43]. [Color figure canbe viewed in the online issue, which is available atwileyonlinelibrary.com.]

angular momentum L. However, the importancesampling function (5) has nodal surfaces and wehave resorted to the fixed-node approximation,where the random walk is restricted so as not cross-ing the nodal surfaces.An upper bound to the energyof the lowest state with angular momentum L is thusobtained, contrarily to the estimate of the groundstate energy which is exact within the statisticalerrors.

To make a sensible comparison between (p-H2)N

clusters and 4HeN droplets, the respective reducedspectra are plotted in Figure 5, which are obtainedby dividing the excitation energies with the corre-sponding Lennard-Jones energy parameter, 34.2 Kand 10.2 K, respectively. Each panel includes theseparation energy δ(N), which is the stability limitfor excitations. A cluster of size N with an excita-tion energy greater than δ(N) will likely evaporateone molecule. Whereas 4HeN droplets can sustain asmall number of excited states and their spectrumis relatively simple, (p-H2)N clusters display a rathercomplex spectrum and their level density is muchdepending on the specific size.

The excitation levels for the lowest states, withL = 2 to 6 are plotted in Figure 6. Notice the typi-cal energy values: a few kelvin for 4HeN droplets, oneorder of magnitude larger for (p-H2)N clusters. More-over, the 4HeN spectrum presents a rather smoothbehavior, which as a matter of fact can be nicelyfitted assuming the liquid drop model, includingsurface corrections for these small sizes. These exci-tations correspond to ripplons at the surface of theliquid droplet. In contrast, no general trend can be



FIGURE 6. Low-lying excitation energies of (p-H2)N

clusters and 4He droplets, calculated with DMC. Adaptedfrom Refs. [40, 43]. [Color figure can be viewed in theonline issue, which is available at wileyonlinelibrary.com.]

identified for (p-H2)N clusters, apart from the factthat the lowest state for all sizes correspond to L = 2,as in 4HeN droplets, with a few exceptions where thestate with L = 3 is either the lowest one or degen-erate with L = 2. The excitation energies cannot befitted with a simple formula in terms of L and N.

The magic character of the N = 13 cluster isreflected in some characteristics of its spectrum. Thestates with L = 1 − 5 show a prominent peak, whichbecomes a minimum for L = 6 and a discontinuityfrom L = 7 to 9. Excited states with higher values ofL are not supported by this specific size. Incidentally,L = 6 is the first non-null multipole of the classicalMackay N = 13 cluster.

6. Cluster Abundance

The knowledge of the excitation energies allowsus to estimate the partition function and thus theeffect of temperature on different magnitudes. Wecan calculate the ground state energy of the clusterat a given temperature, or the dependence on T of theseparation energy. We shall concentrate on the chem-ical equilibrium process because thermal effects aremagnified.

In an usual experiment, clusters are produced ina supersonic gas expansion through a nozzle. Vir-tually all theories of the kinetics of homogeneousnucleation assume that cluster growth is dominatedby a three-body reaction

(pH2)N−1 + pH2 + X � (pH2)N + X, (6)

where X is a spectator particle required to fulfill theenergy and momentum conservation. The relatedchemical equilibrium constant is independent ofthe spectator particle, and is given by the ratio ofpartition functions

KN = N

N−11, (7)

whose behavior as a function of the cluster size andtemperature is given by

N

N−1

=(

NN − 1

)3/2

eδ(N)/kT 1 + ∑L(2L + 1)e−EL(N)/kT

1 + ∑L(2L + 1)e−EL(N−1)/kT

·(8)

Neither the separation energy δ(N) nor the excita-tion energies EL(N) are smooth functions of N,as shown in Figures 5 and 6. One can expect thatthe aforementioned ratio should display such a non-monotonic behavior as jumps at particular values ofN, which would be an indication of an enhanced pro-duction for these specific sizes, related to their magiccharacter. The ratio of partition functions N/N−1

is displayed in Figure 7 as a function of N for severalvalues of temperature. As these ratios span severalorders of magnitude as T varies, we have arbitrarilygiven the unit value to the case N = 40 at any T.

Let us first consider the results for 4HeN droplets,which present a regularly spaced spectrum, verywell fitted in terms of a liquid drop formula. The

FIGURE 7. The ratio of the partition functionsN/N−1 as a function of N for several values oftemperature (p-H2)N clusters and 4He droplets. In bothcases the scale has been arbitrarily fixed at N = 40.Adapted from Refs. [40, 43]. [Color figure can be viewedin the online issue, which is available atwileyonlinelibrary.com.]



reaction constant has thus a smooth behavior as afunction of N, with some sudden jumps at N = 8,14, and 25, which correspond precisely to clustersizes which can accommodate one more additionalstable excitation with L = 2, 3, and 4, respectively.This is in nice agreement with the temperature-dependent production abundances of such clustersexperimentally observed in ultrasonic expansion ofpressurized gas [44]. That agreement is an indirectconfirmation of the excitation levels pattern. Themagic character of these sizes is not a consequenceof a locally enhanced binding energy, but of the exis-tence of stability thresholds which are manifested inthe production mechanism.

This interpretation is no longer applicable for(p-H2)N clusters because, as they can accommodatemore excited states than 4HeN droplets, the stabilitythreshold varies rather smoothly with N. The pres-ence of peaks in the reaction constant KN are relatedto the nonmonotonic behavior of both the separa-tion energy δ(N) and the excitation energies EL(N).In Figure 7, clearly pronounced peaks are observedat any temperature value at N = 13, 31 and 36,and less pronounced peaks at N = 26. Other peaksalso appear, at N = 19 and N = 29, but they onlyappear at specific values of T. These results providean additional support to the previous assertion thatthe apparent disagreement between DMC and PIMCmagic numbers is a temperature effect. The separa-tion energies calculated with PIMC at T = 0.25 K andwith DMC are in essential agreement, and so are themagic numbers. Figure 7 shows the effect of T on thecluster abundance calculated with DMC inputs: newmagic numbers appear as T is increased.

7. Conclusions

A brief review of different quantum Monte Carlosimulations of small (p-H2)N clusters has been pre-sented. Properties as superfluidity, magic numbers,radial structure, excitation spectra, and abundanceproduction have been discussed and, in some cases,compared with the corresponding 4HeN results.

Small para-hydrogen clusters exhibit a clear geo-metrical order, with the molecules distributed inspherical shells, reminiscent of the structure of clas-sical clusters and metallic clusters. There is an openquestion regarding the behavior of p-H2 moleculesinside the shells. They could move as in a liquid, theycould be tied to some fixed positions as in a Mackayicosahedron, and we should not exclude a more com-plex structure as liquid-like within an external layer

and solid-like internally. In connection to this latterpossibility, it is worth reminding the two so differentinterpretations about the localization of the super-fluid density, either at the surface of the cluster of inits bulk volume.

The spectrum of (p-H2)N is very rich and con-tains bound excited states up to angular momentumL = 13 in the size region considered with N ≤ 40,their binding threshold increasing as L increases. Nosimple pattern can be guessed to fit the spectrumas a function of the angular momentum of the stateand the size number. In particular, and contrarily towhat happens in the analogous system of heliumdroplets, the liquid drop model formula is excluded.Perhaps, an appropriate description could be foundconsidering the interplay between solid and liquidstructures.

Both experiment and theory clearly establish themagic number N = 13, which is interpreted as thecompletion of a first shell around a central molecule.Other magic numbers are suggested, but their pre-cise values are still uncertain. From the experimentalpoint of view, there is evidence about N = 33 and 55,identified as broad peaks. Further theoretical studyis required to explain the variety of magic num-bers other than N = 13, changing with temperature,interaction, and computational method.

All together, Monte Carlo simulations indicatethat temperature has an important effect on clustersproperties, as would be expected. However, it is arather paradoxical effect since the clusters becomemore solid-like at high T and more liquid-like atlow T, an intriguing behavior which has been namedas quantum melting. In other words, small (p-H2)N

clusters are more quantum delocalized at T = 0 thanat higher T.

ACKNOWLEDGMENTS

Stimulating conversations with M. Barranco andJ.P. Toennies are gratefully acknowledged.

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thermal effects on small para-hydrogen clusters

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