Ground-state path integral Monte Carlo simulations of positive ions in $^4$He clusters: bubbles or snowballs?

0.32 MB·English

by  Stefano Paolini

Ground-state path integral Monte Carlo simulations of positive ions in 4He clusters:
bubbles or snowballs?
Stefano Paolini,1,2,∗ Francesco Ancilotto,1,2,† and Flavio Toigo1,2,‡
1 – Dipartimento di Fisica ”G. Galilei”, Universita’ di Padova, via Marzolo 8, I-35131 Padova, Italy
2 CNR-INFM–DEMOCRITOS National Simulation Center, Trieste, Italy
(Dated: February 6, 2008)
7
0 The local order around alkali (Li+ and Na+) and alkaline-earth (Be+, Mg+ and Ca+) ions in
0 4He clusters has been studied using ground-state path integral Monte Carlo calculations. We ap-
2 ply a criterion based on multipole dynamical correlations to discriminate between solid-like versus
n liquid-like behavior of the 4He shells coating the ions. As it was earlier suggested by experimental
a measurementsin bulk4He,ourfindingsindicatethat Be+ producesasolid-like (“snowball”) struc-
J ture,similarlytoalkaliionsandincontrasttothemoreliquid-like4Hestructureembeddingheavier
alkaline-earth ions.
0
1
PACSnumbers: 36.40.-c,36.40.Mr,67.40.Yv,67.70.Jg,02.70.Ss

]
r
e I. INTRODUCTION itive ions have been studied both in bulk 4He,11,14 and
h 4He clusters,14 with the aim of computing the effective
t
o masses of charged impurities moving in liquid 4He and
In past decades, the implantation of ions in liquid he-
. of establishing, at the same time, the surrounding local
at lium has often been used to probe the bulk properties structure of the solvent (for a calculation of the effective
m of the superfluid state.1 The attention has also focused
masses within the hydrodynamical model see Ref. 15).
on the details of the structure of the helium solvent
- Thedistinctionbetweensolid-andliquid-likebehavior
d aroundtheions.2,3,4 Asaconsequenceofelectrostriction,
of 4He around atomic/molecular impurities is a subtle
n astrongincreaseintheheliumdensitywithrespecttoits
o issuewhichrequiresacarefulexaminationofthedynam-
bulk value is expected in the surroundings of the impu-
c ical evolution of the system. Within reptation quantum
rity site.2,3 Within a crude picture, ions in liquid helium
[ Monte Carlo (RQMC),16,17 the use of suitable (imagi-
can be divided in two categories, on the basis of the dif-
nary) time-correlation functions has been exploited to
1 ferentlocalstructuretheycreateinthesolvent. Strongly
v proposeadynamicalcriterionfordiscriminatingbetween
attractiveionstendtoformasolid-like structure,theso-
5 freezing versus quantum melting in small para-hydrogen
called snowball, characterized by a very inhomogeneous
1 clusters.13 In the present paper, we resort to the same
helium density in the surroundings of the impurity. Al-
2 criterion to investigate the structural order in 4He clus-
1 kali ions are believed to belong to this category. In con-
ters doped with alkali and alkaline-earth positive ions,
0 trast, singly charged alkaline-earth cations, due to their
simulated by a more standard ground-state path inte-
7 larger radii and weaker interaction with the He atoms,
0 are expected to produce a cavity surrounded by com- gral (PIGS) Monte Carlo algorithm.18 We take Li+ and
/ Na+ as representativesof alkalimetals, while among the
at hperelisusmed,(bluebssblestirnuctthuerefdol,loawnidngm).o3st likely not solidified, alkaline-earths,weconsiderBe+,Mg+andCa+. Lithium
m and sodium -doped 4He clusters have been chosen be-
- Severalinvestigationsofthe ion-heliumstructure have cause the strong interaction of these ions with the 4He
d beenbasedonmeasurementsofionsmobilitiesinbulkhe- solvent makes them prototypes of snowball structures,
n lium.5,6 In particular, the experimental results of Ref. 6 whose expected solid-like signature can provide a useful
o haveshownthatthe temperature dependence ofthe mo- benchmark in the study of the other ions, where a clear-
:c bility of Be+ is much more similar to that of alkali ions cutdistinctionbetweenbubblesandsnowballsmightnot
v (K+,Rb+ andCs+)5 andHe+,6 thantothatoftheother be easy to be resolved. As a paradigmatic case of liq-
Xi alkaline-earth ones (Mg+,Ca+,Ba+,Sr+).6 It was thus uidhelium-impuritystructures,wealsosimulatedasmall
suggested that Be+ forms a snowball, similarly to He+ 4Heclusterseededwithacarbonmonoxidemolecule. We
r
a and alkali ions, and at variance with heavier alkaline- considered ion impurities in clusters rather than in bulk
earthions,where“bubble”statesarebelievedtodevelop helium in order to avoid the possibility of finding ficti-
instead.6 The recent discovery that positive ions can be tious structures imposedby the symmetry ofthe period-
captured in helium nanodroplets7 promises an extension ically repeated simulation box. Moreover, the local 4He
to charged particles of experimental techniques which structure around the ion in large enough clusters should
proved extraordinarily fruitful for neutral species.8,9,10 be very similar to that developed in bulk 4He.14
Quantum many-body simulation methods allow for a We show that the criterion proposed in Ref. 13 is ap-
microscopicdescriptionofsuchsystems,11,12aswellasfor plicable alsoto largersystems than those investigatedin
the investigation of solid-like behavior in quantum clus- that paper, and that it is able to discriminate between a
ters.13 Using quantum Monte Carlo (QMC) simulations, crystalline or a liquid behaviorof the impurity-He struc-
withintheshadowwavefunctionapproach,differentpos- ture. In qualitative agreement with experimental data,6
2
oursimulationsshowinparticularthatthe4Hestructure system, and O is diagonal in the coordinate represen-
around Be+ is much more similar to that found around tation. By sampling the probability P(X) via a gener-
b
alkali ions, where a snowball structure is found, than to alized Metropolis algorithm, and accumulating the val-
the one around heavier alkaline-earth ions, where 4He ues assumed by the operator O over the sampled paths,
exhibits a more liquid-like behavior. the desired expectation value can be accessed within a
b
Ourpaper is organizedasfollows. In Sec.II we briefly known statistical error and without any mixed estimate
reviewourtheoreticalframework. SectionIII Acontains nor population-controlbias.
our results for very small 4He clusters (seeded with Li+ For the relatively small CO@He , it is sufficient to
15
andCO),thathavebeenconsideredfor testing ourcom- usetheprimitiveapproximation26fortheimaginary-time
putational strategy. In Sec. III B we present simulations propagator G, and then to sample the probability P(X)
for several ions solvated in larger 4He clusters, and we by the RQMC algorithm.16,17 In the case of ionic im-
discuss the implications of these results on the main is- purities, we resorted to the pair approximation for G,26
sue we are addressing in this paper (i.e. the local struc- employingthebisection-multilevelsamplingprocedure.26
ture of the 4He solvent around Be+ in comparison with
The dependence ofhOi onβ andǫ needs to be checked,
β
thefindingsforalkaliionsandotheralkaline-earthones).
inordertoextrapolatetheresulttotheexactvalue. Our
Finally, Section IV contains our conclusions. projection times β vbaried between 0.1K−1 and 0.7K−1,
while we used time steps ǫ ranging between 0.0025 K−1
and 0.0125 K−1 for clusters seeded with ions, and equal
II. THEORY to0.001K−1 forCO@He . Thesevaluesofβ andǫhave
15
been empirically found to ensure the stability of our re-
Our clusters are described by a realistic Hamiltonian, sults.
H,inwhichthe N 4He atomsandthe ionic impurityare We used a Jastrow-type trial wave function:
treated as point particles interacting via pair potentials.
b
When the CO molecule is used as dopant, we neglect its N N
internaldegreesof freedomand we treatit as a rigidlin- ΨT =exp − u1(ri)− u2(rij) , (2)
ear rotor with only translational and rotational degrees h Xi=1 Xi<j i
of freedom. The interactions are derived from accurate
quantum chemistry calculations, either by an interpola- where ri and rij indicate, respectively, the distances be-
tion of tabulated values, as for Li+-He and Ca+-He,19,20 tween the i-th 4He atom and the ion, and that between
orbyparametrizedanalyticalexpression,asforHe-He,21 the i-th and the j-th 4He atoms. The radial functions
He-CO,22 and for the remaining He-ions pairs.23,24 The u1 and u2 have been optimized independently for each
size ofthe ion-dopedclusters varies up to a maximumof one of the simulated systems, by minimizing the cluster
704He atoms,whereas,inthe caseofCO,weconsidered variationalenergywithrespecttoasmallnumber(10-12)
a smaller cluster, with only fifteen particles (CO@He15), of adjustable parameters. For CO@He15, the one-body
approximativelycorrespondingtoonecompletesolvation termu1 alsodependsontheanglebetweenthemolecular
shell.25 axisandthevectorpositionofthe4Heatomwithrespect
Wesimulatethesystemusingtheground-statepathin- to the molecule center of mass (details can be found in
tegral Monte Carlo scheme,18 where the imaginary-time Ref. 25).
evolution operator, e−βH, is exploited to improve sys- Within the path integral scheme, O can be ei-
tematically a suitable trial function Ψ , by projecting it ther a static operator (such as the Hamiltonian
b T b
out onto the state |Ψ i = e−βH|Ψ i. The latter con- itself) or a time-dependent one, such as those
β T
neededtocalculateimaginary-timecorrelationfunctions,
verges to the exact ground state, Ψ , for β → ∞. By
b 0
Trotter slicing the imaginary-time evolutioninto m time Aexp(−τH)B. Their evaluation requires an extended
steps of length ǫ = β/m, and by choosing an appro- tbime rangbe, b2β + τ, but it is nevertheless straightfor-
priate definite positive short-time approximation for the ward: cAB(τ) = hΨβ|A(τ0)B(τ0 + τ)|Ψβi is estimated
imaginary-time propagator, G(R′,R;ǫ) ≈ hR′|e−ǫH|Ri as hn1i iA(Ri)B(Ri+kb)i, wbhere k = τ/ǫ, h·i indicates
between the configurations R (a vector which describes a statisPtical average over the sampled paths, and the ni
the coordinates of all 4He atoms in the cluster, in abddi- values of i span the range β/ǫ < i < 2m− 2β/ǫ−k.
tiontothoseoftheimpurity)andR′,expectationvalues The results that will be shown in the following Sections
have been obtained by varying τ within 1 K−1. Notice
of quantum operators, O, on the state |Ψ i, can be ex-
β thattime-translationinvarianceimpliesthatthesecorre-
pressed in a path-integral form:
b lations are independent of τ0.
As it was proposedin Ref. 13, imaginary-time correla-
hΨ |O|Ψ i tionscanbeexploitedtounveileitherasolid-oraliquid-
hOi = β β ≈ dXO(R )P(X), (1)
β hΨ |Ψ i Z m likebehaviorinquantumclusters,bymonitoringtheper-
βb β
b sistenceofgeometricalsignaturesofasolid-likestructure.
where P(X) = Ψ (R ) 2m G(R ,R ;ǫ)Ψ (R ), The criterion of Ref. 13 is based on the definition of the
T 0 i=1 i i−1 T 2m
X = {R0,···,R2m} is aQpath of configurations of the rotationally invariant time-correlation functions
3
0.3 0 1
-100
0.8
-200
0.6
0.2 )
-300 τ(
L
°-3ρ(r) [A] -400 µ [K] C 0.4
-500 0.2
0.1
-600
-700
0 -800
2 3 4 0 2 4 6 8 10 12
r [°A] N
FIG.1: Leftpanel: Radialdensitydistributionof4Heatoms τ)
(
aroundtheLi+ ioninLi+@HeN forN =8(solidline),N =9 D
(long dashed line), and N = 10 (short dashed line). Right
panel: 4HechemicalpotentialµN =EN−EN−1asafunction
oftheclustersize,N,inLi+@HeN,whosetotalenergyisEN.
0 0.2 0.4 0.6 0.8 1
τ [K-1]
hQ∗ (τ )Q (τ +τ)i
c (τ)= M LM 0 LM 0 , (3)
L P hQ∗ (τ )Q (τ )i FIG. 2: Upper panel: Multipole time-correlation functions
PM LM 0 LM 0 cL(τ) of Li+@He8 (L = 4: diamonds, L = 5: filled circles)
of the multipole moments, and CO@He15 (L = 5: triangles). Lower panel: Diffusion
coefficient of the 4He atoms in Li+@He8, as calculated from
the original configurations (empty circles) and from configu-
Q = 4π/(2L+1) drρ(r)rLY∗ (θ,φ), (4) rations rotated back as described in the text (filled circles).
LM Z LM
p
ofthe clustermassdistributionρ(r), aroundits centerof
III. RESULTS AND DISCUSSION
mass. InEq. 3, Q (τ ) is the multipole momentof the
LM 0
systemat(imaginary)timeτ andQ (τ +τ)indicates
0 LM 0
A. Testing the approach on small clusters
the multipole calculated after the configuration at time
τ +τ has been rotated back in order to minimize the
0
particles diffusion between the two time instants. As a test case, we first discuss the results for a Li+
For a rigid system c (τ) is independent of time, ion in small 4He clusters (Li+@He ). The left panel of
L N
whereas it decays to zero at large τ if the system un- Fig. 1 displays the radial distribution functions of 4He
dergoes random fluctuations. A large value of c (τ) at atoms around the impurity ion, for cluster sizes in the
L
large times is thus a distinctive mark of the persistence rangeN =8−10. ForN ≥9the magnitudeofthe max-
of the shape (“rigidity”) of the system. This criterion imum of these functions stays nearly constant while the
has been originally applied to small clusters of para- tail of the distribution extends to larger distances from
hydrogen molecules (both pure an doped with a CO thedopant. Correspondingly,thechemicalpotentialrises
molecule), for cluster sizes around twelve molecules, at steeply from N = 8 to N = 9. These findings indicate
which a first solvation shell is completed.13 As a pre- thatforN =8afirstsolvationshellhasbeencompleted.
liminary step in the study of the correlation functions We notice that an even more marked jump in the chem-
c (τ), we evaluate the static multipole correlation func- ical potential occurs at N = 6, thus suggesting a great
L
tion, q = hQ∗ Q i, which provides a rotation- stability of Li+@He . In particular some of the mul-
L M LM LM 6
ally invariaPnt characterization of a rigid body.13 This tipole imaginary-time correlation functions of Li+@He6
quantityisnotsuitablefordiscriminatingthecaseswhere show an extremely slow decay in imaginary time, which
a given multipole vanishes on average but q is different is indicative of the rigidity of the system (see below).
L
from zero because of fluctuations, from those where a However,sinceweaimatrevealingtheexistenceofsolid-
non-vanishingq valueisduetotheaverageshapeofthe like signatures within the first shell of largerclusters, we
L
system. However, non-vanishing q values allow to se- will not discuss any longer the case of Li+@He , focus-
L 6
lectasetofL’samongwhichto searchforslow-decaying ing instead on the solvent structure in a cluster of eight
time-correlations c (τ). 4He atoms, i.e. one representing a fully developed first
L
4
0.3
+
Li @He
+ 70
Na @He
0.25 + 70
Be @He
+ 70
Mg @He
0.2 + 70
3] Ca @He70
-A
°) [ 0.15
r
(
ρ 0.1
0.05
0
1 2 3 4 5 6 7 8 9 10
°
r [A]
FIG.4: Radial density distribution function of theHeatoms
in Li+@He70 (solid black line), Na+@He70 (dashed line),
Be+@He70 (red), Mg+@He70 (green), Ca+@He70 (blue).
FIG. 3: He density distribution for Li+@He8. The dis- twoparallelsquares,rotatedbyπ/4withrespecttoeach
other. Indeed, the lowest non-vanishing q values for a
played isosurface contains about 50% of the total number of L
He atoms. The Li+ ion (not shown) is located at the center polyhedron with such a structure correspond to L = 4
of the cluster. and L = 5. The results reported in Figs. 2 and 3 seem
toindicatethatLi+@He behavesmuchlikearigidbody
8
rather than as an ion embedded within a liquid shell.
solvation shell.
The upper panel of Fig. 2 shows the multipole time-
correlation functions of Li+@He for L = 4,5 and, for
8
comparison, of CO@He15 for L = 5. All the multi- B. Multiple-shell ion-doped 4He clusters
poles correlations of CO@He decay rapidly, indicating
15
a liquid-like behavior of the first He shell around the
We come now to the main concern of this paper, i.e.
molecule, and we arbitrarily chose to show the L = 5
todiscriminatebetweensolid-likeversusliquid-likestruc-
curve as representative of the general behavior of the
tures around helium-solvated ions. To this aim we have
c (τ) correlations for any L. On the other hand, in
L
Li+@He , all the multipole time-correlations with L<4 tried to apply the criterion of Ref. 13 to the more gen-
8
eralcaseofclustersconsistingofmorethanonesolvation
decay rapidly, whereas for L = 4 and, more evidently,
shell.
for L = 5, after a short transient in which a very steep
drop occurs, the correlations decays much more slowly, Insteadofextending the integralin Eq.4 to the entire
indicating the long persistence of anaveragegeometrical cluster, one may calculate multipole moments QLM of a
shapeofthecluster. InthelowerpanelofFig.2wecom- region within a suitably chosen distance from the center
parethediffusioncoefficientD(τ)=hN1 Ni=1[ri(τ+β)− of mass. Thus, the criterion described in the previous
ri(β)]2iofthe4Heatoms,ascalculatedfPromtheoriginal Section allows to identify a solid-like behavior of that
sub-region of the cluster. In Fig. 4 we show the radial
configurations of each quantum path generated during
solvent density profiles for 70 4He atoms clusters doped
the simulation, with that obtained after rotating back-
with different ions. By taking the first minima of the
ward the configurations at time τ in order to minimize
curves as cut-off distances in the integral which defines
the particles diffusion with respect to the configuration
Q , we accessed the imaginary-time auto-correlation
attimeβ. Thehighdegreeofrigidityoftheclusterstruc- LM
functions of the multipoles of the first shell of the cor-
ture near the impurity is confirmed by the considerable
responding clusters. Table I reports the integral of the
effectiveness of the diffusion minimization procedure.
Inordertovisualizetheintrinsicshapeofthe4Heclus- 4Hedensitywithinthefirstsolvationshellforthevarious
ions that we have considered.
ter, we have calculated the helium density distribution
ρ(r) = h Ni=1δ(r − ri)i around the ion, by applying
the mentiPoned backward-rotation procedure. The con-
tribution to ρ(r) from each quantum path sampled by TABLEI: Integralofthe4Hedensitydistributionwithinthe
the PIGS simulation is then further rotated to minimize firstsolvationshellof4He70clustersdopedwithdifferentions.
the diffusion of the particles centroids of different paths.
Li+ Na+ Be+ Mg+ Ca+
Fig 3 shows the density distribution for Li+@He . The
8
solventaccumulates in eight regions roughly arrangedin 8.24 12.02 14.55 19.02 23.18
5
standing of the behavior of the time-correlations c (τ).
1 L
Inaverysimplifiedandidealpicture,thediffusionofthe
4He atoms of a solid first solvation shell during the sim-
0.8
ulationcanbethoughtasthe combinationoftwokindof
motions: one approximatively corresponding to a coher-
0.6
τ) entrotationofthewholeshell,andtheotherduetohigh-
(
C4 frequencysmall-amplitudefluctuationsaboutthesmooth
0.4
trajectoryoftheformer. Whentheclusterconfigurations
along the path are rotated backward in order to mini-
0.2
mize the particles diffusion, the neat rotation is totally
suppressed, whereas the fast fluctuations are not. Thus
0
the only survivingmotions canbe seenas the incoherent
1 fluctuations of4He atomsinthe firstshellaboutaverage
positions. Morerealistically,othermotions,notremoved
0.8 by the backwardrotationprocedure,enter the decompo-
sitionofthe4Heatomsdiffusioninthefirstshell. Ifthese
0.6 motionsaresufficientlyslow,i.e. iftheirenergiesaresuf-
)
τ
(5 ficiently low, the average shape of the first shell is not
C 0.4 disruptedtooquickly. Correspondingly,sinceimaginary-
time correlation functions can be written as a sum of
0.2 decaying exponentials whose decaying constants are the
excitation energies of the system, the c (τ) correlations,
L
0 characteristic of the system geometry, exhibit a slow de-
0 0.2 0.4 0.6 0.8 1 cay.
τ [K-1] Within the same qualitative picture, the atoms in the
outer regions undergo random motions, none of which is
reducible to a neat rotation. The slowest modes of the
FIG. 5: Comparison of the multipole time-correlation func-
tions cL(τ) for L=4(upperpanel) and L=5 (lower panel), whole cluster are thus fast enough to prevent the persis-
as calculated for thewhole Li+@He70 (emptycircles), within tenceofanyglobalshape(andtoproducearapiddecayof
the first solvent shell of the same cluster (filled circles), and the corresponding time-correlations cL(τ)), but anyway
for CO@He15 (triangles). theyaremuchslowerthanthevibrationsoftheatomsin
the first shell, thus explaining the lower decaying rate of
thec (τ)correlationfunctionsatsmallτ withrespectto
L
1. Alkali ions doped 4He clusters those of the first shell.
Fig. 6 depicts the 4He density distribution within the
In Fig. 5 we compare the multipole time-correlation radius of the first shell, where we found an integrated
functions c (τ) for L = 4 (upper panel) and L = 5 density of ∼ 8.24 4He atoms. The solvent concentrates
L
(lowerpanel), ascalculatedforthe firstsolvationshellin mainly in eight regions arranged in two parallel squares
Li+@He70,forthesamewholecluster,andforCO@He15. rotated by π/4 with respect to each other. At the oppo-
The time correlations of the L = 4 and L = 5 multi- site poles of this structure, along an axis perpendicular
polesofthefirstshell,albeitdifferinginmagnitude,both to the planes of the squares, two small lower-density ac-
show a very slow decay (almost a saturation in the case cumulations, probably due to incursions of 4He density
of L=4), whereas the same quantities, when considered fromthe secondshell,arefound(they disappearwellbe-
forthewholecluster,decayrapidlytozero,verysimilarly forethe othereightoneswhenweincreasethe isodensity
to what happens in CO@He15. Also the L = 6 correla- value chosen to draw the picture of Fig. 6). As we ex-
tion function (not reported) exhibits a slow decay, just plainedindiscussingLi+@He ,foratworotated-squares
8
slightly morerapidthanfor theL=4and L=5 curves. structure the lowest non-vanishing q values correspond
L
Onthe contrary,the decayofallthe otherc (τ) correla- to L = 4 and L = 5. We note that the addiction of two
L
tions, calculated within the first shell, is extremely fast. morevertexesonalineperpendiculartothesquarespro-
We interpret this result as an indication that the 4He duces apolyhedronhavingthe lowestnon-vanishingq ’s
L
structureinthefirstsolvationshellmaintainsanaverage for L=5 and L=6. The behavior of the c (τ) correla-
L
shape,behavinglikeasolidcagewhichseparatestheLi+ tion functions can thus be seen as the result of the con-
ion from the liquid environment of the outer shells. The comitant signals corresponding to these two structures.
signature of the rigidity of the first shell is lost among Westressatthispointtheneedofconsideringdynamical
the random fluctuations in the more external regions, correlations c (τ) of the multipoles of the cluster mass
L
and thus it disappears in the multipole correlations of density, rather than their static magnitudes q , in order
L
the entire system. to reveal a solid-like behavior of the solvent. In fact, by
At least qualitatively, we can attempt a better under- calculatingthequantitiesq oftheHedensityofthefirst
L
6
FIG.6: Hedensitydistributionwithinthefirstsolvationshell
of Li+@He70. The displayed isosurface contains 60% of the
numberof Heatoms in thefirst shell. TheLi+ ion is located
at thecenter of thepicture, but it is not displayed.
solvation shell of Li+@He , we obtain value at L = 4
70
which is larger but of the same order of magnitude of FIG. 7: Upper panel: multipole time-correlation functions
the value at L=3, whose corresponding imaginary-time cL(τ) for L=6, as calculated for the whole Na+@He70 clus-
correlation function, instead, decays extremely rapidly. ter (empty circles) and within the first solvent shell (filled
circles). Lower panel: He density distribution in the first
Evenmore evidentis the existence ofa solidfirstshell
inclustersdopedwithaNacation. Oursimulationsindi- shell of Na+@He70. The isodensity surface contains the 60%
of the density of the first shell. The Na+ ion resides at the
catethatthe time correlationsofthe firstfivemultipoles
centerof thestructureand it is not displayed.
within the first shell decay rapidly, whereas the L = 6
shows an extremely slow decay. These results are com-
patible with an icosahedralstructure, whose lowest non-
vanishing multipoles correspond to L = 6 and L = 10. a time step ǫ = 0.0125 K−1. By integrating the 4He
For the first shell of Na+@He70, we show in Fig. 7 the density within the first shell of Na+@He70, we obtainan
slow-decaying L = 6 multipole time-correlation function estimate oftwelve4He atoms,in agreementwithshadow
(upper panel) and the 4He density distribution (lower wave function and path-integral Monte Carlo (PIMC)
panel). The latter clearly shows an icosahedral order, in results,11,27 but at variance with other existing PIMC
agreementwithpreviousQMCcalculations,basedonthe calculations.12 Our result is stable, within the statisti-
shadow wave functions technique.11 cal error, when reducing the time step by a factor 2.5,
Interestingly enough, despite the stronger interaction thus showing a negligible time-step dependence in this
potential characterizing the Li+-He pair, we find a more ǫ range. For larger time steps, however, the reliability
pronounced solid-like behavior of the first shell of 4He of the simulations is not guaranteed. In Ref. 12, where
arounda Na+ ionthaninthe caseofLi+. This is due to a time step of 1/20 K−1 is used, sixteen 4He atoms are
a subtle balance between the depth and the position of found,insteadoftwelve,inthefirstshellofaNa+@He
100
the minimum of the He-ion interaction potential, which clusteratthetemperatureof1K. Theabovediscrepancy
determines the averagenumber of 4He atoms in the first may originate from the choice of the He-Na+ interaction
solvation shell, and, hence, the average distances among potential12 or, more likely, it is a time-step effect. In
them. Forinstance,thenearestneighborHeatomsinthe fact, for strongly attractive potentials, such as the Li+-
fi[email protected]˚A,i.e. He and the Na+-He ones, we noted that the use of a
they find each other in the repulsive regionof the He-He large time step (like the one used in Ref.12) may lead to
interaction potential. In contrast, the nearest neighbor overestimate the 4He density in the first shell. In par-
distance in the first shell of Na+@He70 is slightly less ticular, by simulating Na+@He70 with the same rather
than 2.8 ˚A, that is within the well of the He-He pair large time step ǫ = 0.05K−1, we reproduced the result
potential and closer to its minimum (∼3.0 ˚A). of Ref. 12. We also observed that the 4He density in
The results for Na+@He have been obtained using the firstsolvationshellis muchmoredelocalized,lacking
70
7
1
0.8
0.6
)
τ
(
7
C
0.4
0.2
0 0.2 0.4 0.6 0.8 1
τ [K-1]
FIG.8: Multipoletime-correlationfunctionscL(τ)forL=7,
ascalculated forthewholeBe+@He70 cluster(emptycircles)
and within the first solvation shell (filled circles).
FIG.9: HedensitydistributioninthefirstshellofBe+@He70.
the icosahedral structure previously found, and that the
Theisodensity surfacecontains the60% of thedensityofthe
lowesttwelvemultipolestime-correlationfunctionsdecay first shell. The Be+ ion resides at thecenterof thestructure
now much more rapidly. and it is not displayed.
2. Alkaline-earth ions doped 4He clusters shape assumed by the solvent around the Be+ ion. The
average shape of the solvent in the first shell is shown
We now turn our attention to the case of 4He clus- in Fig. 9. Despite being less localized than for alkali ion
ters doped with alkaline-earth ions. Due to their larger doped clusters, the 4He density in the vicinity of Be+
radiiandtheirrelativelyweakerinteractionswithhelium appears anyway significantly structured.
atoms,these ions are expected to give rise to bubble-like Fig. 10 compares the diffusion coefficient in imaginary
structures of the 4He liquid around the impurity. This time, D(τ), as calculated after the backward rotation,
expectationisconfirmedbymobilitymeasurementsatfi- for the 4He atom in the whole cluster and within the
nite temperature for several alkaline-earth ions but not first shell, for Li+@He70 and Be+@He70. For both sys-
for Be+, for which results of similar measurements sug- tems, with increasing the (imaginary) time τ, the coeffi-
gest instead a snowball structure of the defect.6 For this cientD(τ)ofthewholeclusterincreasesmuchmorethan
reason,inthe following,wewillfocus particularlyonthe withinthefirstsolvationshell,duetotheliquidcharacter
solvent structure around Be+. The rigidity of the defect of the external regions. We interpret the small diffusion
structures of Be+ and Mg+ can hardly be distinguished rateoftheHeatomsinthefirstshell,and,moreover,the
on the basis of static correlation functions among 4He close similarities in D(τ) for Be+@He70 and Li+@He70
atoms,14 thus spurring us to consider the multipole mo- (which we consider a prototype of a cluster with a first
ments dynamical correlations criterion.13 shell with solid-like order) as an indication of a nearly
In order to detect even weak signatures of a possible rigid-body behavior of the first shell of Be+@He70. This
solid-like order around the alkaline-earth ions, the clus- conclusion will be strengthened by the comparison with
ters configurations, used for calculating both static and the results for Mg+@He70, which will be discussed next.
dynamicalquantities,havebeenrotatedbackinorderto For Mg+@He the calculation of q , within the first
70 L
minimize theparticlesdiffusionwithinthefirst solvation solvation shell, gives a faint signal for L = 8. How-
shell, and not in the whole clusters. According to our ever,the correspondingimaginary-timecorrelationfunc-
results,all the multipoles correlationsof the firstshellof tion c (τ), which is reported in Fig. 11, exhibits an ex-
L
Be+@He up to L = 5 decay rapidly (even faster than tremely fast decay, despite being the slowest decaying
70
in the whole cluster), whereas for L = 6 and, more ev- one among those we have analyzed (up to L=12). Our
idently, for L = 7 the correlation decay is much slower findingssuggestthattheHeatomssurroundingMg+ un-
thanthatofthewholecluster. Alsointhiscasethemag- dergorandomfluctuations which disruptthe localstruc-
nitude of the multipoles q providea guide in the search ture of the solvent much more quickly than in the other
L
of the slow decaying time-correlations. Fig. 8 displays cases examined so far, thus preventing the formation of
theL=7multipolecorrelationinthefirstsolvationshell anylonglypersistingsolid-likeorder. ForMg+@He the
70
andinthewholecluster. Inspiteofthefasterdecaythan rapid decaying of the c (τ) correlations of the first shell
L
for Li+@He and Na+@He , the behavior of c (τ), for is very similar to that of the corresponding ones of the
70 70 L
L=7, indicates an extremely slow disappearance of the entirecluster,thusindicatingaliquid-likebehaviorofthe
8
+
Li @He
70
)
τ
(
D
+
Be @He
70
)
τ
(
D
0 0.2 0.4 0.6 0.8
τ [K-1]
FIG.10: Upperpanel: diffusioncoefficientD(τ)asafunction FIG. 12: Upper panel: He density distribution in the first
ocofn(fiimguargaintiaornys),ftoimrtehτeHcaelcautloamtesdinfrtohmewthheolbeaLcki+w@arHde-r7o0tcaltuesd- [email protected] athmeoudnetnssittoy6c0o%ntoafinthede
ter (empty circles), and for the He atoms in its first solva- numberof Heatoms in thefirst shell. Lower panel: diffusion
tion shell (filled circles). Lower panel: same quantities for coefficient D(τ) as a function of (imaginary) time τ for the
Be+@He70, displayed with thesame symbols. He atoms in Mg+@He70, as calculated form the backward-
rotated configurations, for the whole cluster (empty circles)
and for thefirst solvation shell (filled circles).
whole system.
As it is shown in the upper panel of Fig. 12, the 4He
densitydistributioninthefirstshellisconsiderablymore The lower panel of Fig. 12 compares the imaginary-time
diffuse than what found for the other ions. By increas- diffusionofthe4Heatomsintheentireclusterandinthe
ing the isodensity value chosenforthe graph,it becomes firstsolvationshell,afterthebackwardrotation. Thetwo
possibletoidentifyregionsofhigherdensity,nevertheless curvesarenowcloserandtheyevenmergewithintheer-
thesolventismuchmoredelocalizedthanforBe+@He70. rorbars,thusprovingthatthe rotationaldiffusionofthe
particles in the first shell cannot be reduced to a neat
rotation. The liquid-like behavior of the solvent is even
1 moreevidentinCa+@He ,forwhichthefirstshellmul-
70
tipole time-correlation functions decay very rapidly, the
0.8 density distribution in the first solvation shell is largely
delocalized and the diffusion coefficients of 4He atoms of
) 0.6 the inner and of the outer shells are much less distin-
τ
(8 guishable.
C
0.4
0.2
IV. CONCLUSIONS
0 0.2 0.4 0.6 0.8 1
UsingPIGSsimulations,wehavestudiedthesolvation
τ [K-1] of alkali and alkaline-earth ions in 4He clusters. In or-
der to discriminate between solid-like versus liquid-like
FIG.11: Multipoletime-correlationfunctioncL(τ)forL=8, behavior of the solvent in the vicinity of the impurity,
as calculated within thefirst solvation shell of Mg+@He70. we employeda criterionbasedon dynamicalcorrelations
9
of the multipole moments of the 4He density in the first these two categories, whose features are expected to be
shell, previously proposed for quantum clusters consist- different both theoretically3,4 and experimentally,5,6 the
ing of just one completed shell. We also used other indi- multipolecorrelationssignaturesareclearlydifferentand
catorstoclarifythestructureof4Hearoundtheimpurity. unambiguously distinguishable.
Thedynamicalcorrelationscriterion,whoseapplicabil-
ity to clusters with more than one shell was not granted ForBe+,oursimulationsindicateaborder-linebehav-
a priori, proved suitable to address situations in which ior, which, however, has much more similarities with al-
regions with different rigidity coexist within the system, kali ions than with heavier alkaline-earth ones. This re-
withoutrequiringanypriorknowledgeofthedefectstruc- sultssuggeststhatBe+ formsasnowballuponsolvation,
ture. Inparticularwefindbubble(liquid-like)structures possibly elucidating the “anomalous” mobility of Be+
for heavier alkaline-earth ions (Mg+ and Ca+), whereas in liquid helium with respect to Mg+, Ca+ and other
alkali-ions are caged by a solid-like, snowball structure, alkaline-earth ions.6
further embedded in an outer liquid environment. For
∗ Electronic address: [email protected] 16 S.Baroni andS.Moroni, inQuantumMonteCarlo Meth-
† Electronic address: [email protected] ods in Physics and Chemistry, edited by P. Nightingale
‡ Electronic address: fl[email protected] and C.J. Umrigar. NATO ASI Series, Series C, Math-
1 A. L. Fetter, in The Physics of Liquid Helium, edited by ematical and Physical Sciences, Vol. 525, (Kluwer Aca-
K. H.Bennemann and J. B. Ketterson (Wiley,New York, demic Publishers, Boston, 1999), p. 313, also available at
1976). cond-mat/9808213.
2 K. R.Atkins,Phys. Rev. 116, 1339 (1959). 17 S.BaroniandS.Moroni,Phys.Rev.Lett.82,4745(1999).
3 M. W. Cole and R. A. Bachman, Phys. Rev. B 15, 1388 18 A. Sarsa, K. E. Schmidt, W. R. Magro, J. Chem. Phys.
(1977). 113, 1366 (2000).
4 M. W.Cole and F. Toigo, Phys. Rev.B 17, 2054 (1978). 19 M. T.Elford, I.Røeggen, and H.R.Skullerud,J. Phys.B
5 W.I.Glaberson andW.W.Johnson,J.LowTemp.Phys. 32, 1873 (1999).
20, 313 (1975). 20 E. Czuchaj, F. Rebentrost, H. Stoll, H. Preuss, Chem.
6 M. Foerste, H. Guenther, O. Riediger, J. Wiebe, and G. Phys. 207, 51 (1996).
zu Pulitz, Z. Phys. B: Condens. Matter 104, 317 (1997). 21 The He-He interactions are calculated by the potentials
7 P. Claas, S. -O. Mende, and F. Stienkemeier, Rev. Sci. given in A. Aziz, A. R. Janzen, M. R. Moldover, Phys.
Instrum.74, 4071 (2003). Rev. Letts. 74, 1586 (1995) for ion-doped clusters and in
8 J. P. Toennies and A. F. Vilesov, Angew. Chem. Int. Ed. T. Korona, H.L. Williams, R.Bukowski,B. Jeziorski and
43, 2622 (2004). K. Szalewicz, J. Chem. Phys. 106, 5109 (1997) for the
9 F.Stienkemeierand K.K.Lehmann,J.Phys.B 39, R127 CO-doped one.
(2006) 22 T. G. A. Heijmen, R. Moszynski, P. E. S. Wormer, and
10 M. Barranco, R. Guardiola, S. Hern´andez, R. Mayol, and A. vander Avoird,J. Chem. Phys. 107, 9921 (1997).
M. Pi. J. Low Temp. Phys. 142, 1 (2006). 23 R. Alrichs, H. J. Bohm, S. Brode, K. T. Tang, and J. P.
11 M. Buzzacchi, D. E. Galli, and L. Reatto, Phys. Rev. B Toennies, J. Chem. Phys. 88, 6290 (1988).
64, 094512 (2001). 24 D.Bellert and W.H.Beckenridge, Chem. Rev.(Washing-
12 A. Nakayama and K. Yamashita, J. Chem. Phys. 112, ton, D.C.) 102, 1595 (2002).
10966 (2000). 25 P.Cazzato,S.Paolini,S.Moroni,andS.Baroni,J.Chem.
13 S.BaroniandS.Moroni, ChemPhysChem 6,1884 (2005). Phys. 120, 9071 (2004)
14 M. Rossi, M. Verona, D. E. Galli, and L. Reatto, Phys. 26 D. M. Ceperley, Rev.Mod. Phys. 67, 279 (1995).
Rev.B 69, 212510 (2004). 27 M. Rossi, PhD. Thesis (2006)
15 K. K. Lehmann,Phys. Rev.Lett. 88, 145301 (2002).