Attachment Energetics of Quantum Dopants in a Weakly Interacting

Jan 25, 2010 - ... further confirmation, in the realm of nanoscopic-size systems like the helium droplets, of the well-known nonmixing and nonsolvatin...
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J. Phys. Chem. A 2010, 114, 3221–3228

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Attachment Energetics of Quantum Dopants in a Weakly Interacting Quantum Solvent: 1H, 2 H and 3H in Small 4He Clusters† E. Coccia and F. A. Gianturco* Department of Chemistry and CNISM, UniVersity of Rome La Sapienza, Piazzale A. Moro 5, 00185 Rome, Italy ReceiVed: September 30, 2009; ReVised Manuscript ReceiVed: January 7, 2010

Small 4He clusters doped with a single atomic impurity, 1H, 2H, and 3H, have been studied via a quantum Monte Carlo approach with the intent of establishing their binding behavior in nanoscopic clusters. Our calculations find that the only trimer xH (He)2, which exhibits a bound state, is that with the tritium dopant (x ) 3), in agreement with previous calculations using hyperspherical coordinates in the adiabatic approximation. The lightest dopant 1H is seen not to stabilize the small helium clusters, while 2H and 3H are weakly bound to this solvent: our computed exchange energies and probability distribution functions reveal the “heliophobic“ nature of hydrogen, leading thus to a further confirmation, in the realm of nanoscopic-size systems like the helium droplets, of the well-known nonmixing and nonsolvating features of hydrogen in macroscopic liquid bulk helium. 1. Introduction The elementary effects of the interaction between the two simplest atoms, H and He, is of fundamental interest in the field of molecular physics and of chemistry in general. A great number of studies have been dedicated to shedding new light on the mixing properties of quantum fluids, their interactions at very low temperatures and the role of statistics in the macroscopic properties of such systems. Early calculations on binary mixtures of atomic hydrogen, deuterium, and tritium found that the boson systems 1H-4He, 2H-4He and 3H-4He completely phase separate at zero temperature and that 1H, 2H, and 3H do not penetrate the surface of liquid 4He since their chemical potentials µ are positive.1-5 It is thus well-known that only 3He gets solvated into bulk 4He up to a concentration of 6.6%:6 the lighter impurities as atomic hydrogen or deuterium are only weakly adsorbed onto the 4He surface and do not penetrate the bulk.6 In fact, ref 1 describes a series of variational calculations that show that bosonic mixtures of spin aligned atomic hydrogen and tritium interacting in the b3Σ+u state and 4He prefer to remain completely separated at 0 K, while the case of a Fermi-Bose mixture is instead totally different since one can obtain a full range of occurrences, from complete separation to complete mixing.1 Guyer and Miller2 found strongly bound physisorbed states for spin-aligned 3H and 2H, and a weakly bound physisorbed state for 1H, hence confirming the nonmixing nature of the two-component fluid. The above studies were enriched by additional variational calculations of the Feynman-Lekner form3 regarding the interaction of a single 1H, 2H, and 3H with the free surface of liquid 4He, which revealed the very small binding energy for such surface states, e.g., 1.83 K for tritium, 1.39 K for deuterium, and 0.63 K for hydrogen, i.e., less than the 4.64 K of the bound state for 3He, as expected. A theoretical analysis in ref 4 has shown the different behavior of the atomic and molecular isotopes of hydrogen in liquid 4He. In Figure 1 of that work one can see the rapid †

Part of the “Benoît Soep Festschrift”. * Corresponding author. E-mail: [email protected]. Fax: +39 0.06.49913305.

decrease of µ for all the three atomic impurities as the helium density decreases; the possible binding between the hydrogen and the solvent becomes energetically favorable under conditions of low solvent density because the repulsive contribution of the H-He interaction becomes drastically reduced with dilution. The authors tried to modify the ε term of the Lennard-Jones potential for the H-He interaction (from 6.6 to 8.1 K; see also Table 1) to verify the possibility of solvating the tritium in bulk helium; their conclusion was that under particular conditions of helium density the tritium could be considered as a 3He atom and thus penetrate into the liquid.4 The situation is completely different when looking instead at the diatomic hydrogen molecules (1 H2, 1 H2H, and 3 H2 studied there4), which present negative chemical potentials for all the densities values. Similar results were obtained by using the Jastrow-Feenberg theory in ref 5: the chemical potentials of the three species maintain positive values under any condition of helium density. Marin et al.6 published diffusion Monte Carlo calculations regarding a single hydrogen atom (1H, 2H, or 3H) or a hydrogen molecule (1H2 or 2H2) in bulk superfluid 4He. They say once again that all the three atomic species do not dissolve in the bulk, the chemical potential decreasing when going from the hydrogen to the tritium. Only one experimental result7 exists in relation to the solvation or the mixing of atomic hydrogen in bulk 4He: an atomic deuterium gas has been studied at temperatures just above 1 K, coming from the dissociation of solid 2H2. The resulting 2H atoms were observed by a pulsed magnetic resonance on the β-δ hyperfine transition.7 From the analysis of the temperature dependence of the lifetimes it is possible, through a simple “penetration” model, to extract information about the chemical potential of deuterium; µ was seen to get the value of 13.6(6) K which, when compared with the theoretical estimates of 15.51 K4 and 14.2(5) K,6 makes one fairly confident about the quality and the reliability of such variational approaches. From the “few-body” standpoint, two attempts to find bound states for complexes formed by H and He atoms have been published in the last ten years:8,9 the first employed the adiabatic hyperspherical method10,11 to solve the Schro¨dinger equation

10.1021/jp909403t  2010 American Chemical Society Published on Web 01/25/2010

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Figure 1. Potential curves for the He-He pair (solid lines, from ref 12, labeled as HFDID) and the H-He pair (dashed lines, from ref 23).

TABLE 1: Equilibrium Well Depth (De) and Bond Distance (Re)a reference

De, cm-1

Re, Å

23 14 13 22 24 4 4

5.237 4.697 4.951 4.535 4.726 4.586 5.629

3.519 3.588 3.524 3.600 3.704 3.600 3.600

a Ref 23 is the H-He potential model used in the present work. The H-He potential from ref 14 was employed in ref 9, that from ref 13 in ref 8, and that from ref 22 in ref 6. The potential in ref 24 was obtained by quantum Monte Carlo calculations. The last two rows refer to a Lennard-Jones potential with two different ε values, used in the calculations reported in ref 4.

for the H-He-He, He-He-H-, and He-H-H systems;8 the second presents an accurate approach to calculate the eigenstates of floppy rare gas hydrides by using Pekeris coordinates and a symmetry-adapted Lanczos recursion.9 For the He-He pair the potential from Aziz et al.12 was used in both works; two different potential models were instead employed for the H-He interaction,13,14 whose differences in well depth and equilibrium distances are reported in Table 1. In ref 8 a full analysis of the effects from the isotopic substitution of the impurity was carried out, leading to the result that only the heavier hydrogenic species gives rise to a bound state of -0.00473 cm-1; calculations in ref 9 confirm that the lighter 1H(He)2 is not bound. It therefore becomes interesting to further investigate the above research line so that, by monitoring the number of He atoms in the complex, one could better understand the solvation properties of such atomic dopants in finite-size solvent aggregates as opposed to the macroscopic bulk behavior. Furthermore, to be able to reliably assess the values and signs of the binding energies in ultraweak interacting nanoscopic aggregates is already an interesting result for the present methods. We have studied a set of small (4He)N clusters,15,16 with N up to 10, doped with the three isotopic variants of hydrogen, focusing our work on the possible solvation mechanisms of such light impurities in nanoscopic systems and thus underline differences and similarities with respect to the bulk regime. To do this, we have carried out variational and diffusion Monte

Carlo (VMC and DMC) as described below:17 the present study is meant to help to understand better the evolution of van der Waals systems from the more elementary ”few-atom world“ to the bulk environment of a large system. The work is organized as follows: in section 2 we shall describe the potential interaction we have used; the quantum Monte Carlo (QMC) methods will be briefly reviewed in section 3; in section 4 we shall report our results and, finally, in section 5 we present our conclusions. For the sake of clarity, in the following 4He will become He, 1H will simply be H, while 2H and 3H will become D and T, respectively. 2. Potential Interaction For such very weakly bound many-body systems the overall potential is usually approximated by a two-body picture; i.e., disregarding the effects of three-body (3B) terms, one writes N

VTot(R) )

∑ i

N(N-1)/2

VH-He(Ri) +

∑

VHe-He(Rjk) + O(3B)

j Ψ and give birth in regions where ΨT < Ψ. We have employed a DMC version27-32 in which the number of walkers is changed during the simulation. Each walker is characterized by a cumulative weight; at the jth iteration the weight wij for the ith walker is given by

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j

wij )

∏

(10)

bik

k)1

where bik is defined as

{[

bik ) exp ER -

] }

EL(Ri) + EL(Ri′) τeff 2

(11)

The reference energy ER is updated during the random walk according to the following formula33

ER ) ER +

R N(τj-1) ln τ N(τj)

(12)

where R is a control parameter (to be chosen small) and N(τj-1) and N(τj) represent the population of walkers in two successive steps of the random walk. The quantity present in the numerator can be substituted by the starting population N(0) or by the desired population N. A larger global stability has been obtained if the choice is the one indicated by eq 12: updating ER is a fundamental tool to minimize the fluctuations in the ensemble, and τeff in eq 12 is the effective time step corrected by the acceptance ratio.26 The average over the ensemble at the jth iteration is given by N

∑

ˆ 〉j ) 1 〈O wO Wj i)1 ij i,j

(13)

where Wj ) ΣNi wij. The final estimate of the observable O is obtained by the usual definition of Nb blocks of M steps, to minimize the correlation of the data

∫ Ψ0(R) OˆΨT(R) dR 1 N -1 = ∑ ∫ Ψ0(R) ΨT(R) dR Nb k)0 b

ˆ〉 ) 〈O

[

(k+1)×M

∑

1 ˆ 〉j 〈O M j)k×M+1

]

(14) 4. Results We have performed VMC calculations for all the clusters studied here to start with an optimized many-body trial wave function ΨT. The Powell method34 to find the absolute minimum in the parameter space has therefore been coupled to standard VMC simulations; a set of 10 parameters (5 for the impurity-helium interaction and 5 for the He-He pair) is modified during the overall optimizing procedure, which stops when convergence is reached. The cost function we have chosen to be minimized is the variance of the local energy EL.35,25 The DMC parameters used in our calculations are listed in the following: 1. For the tritium-doped systems, we have employed 2000 walkers (4000 for T(He)2 and τ2 ) 300 hartrees-1), 40 blocks of 60 000 (N ) 2, 3, and 4), 30 000 (N ) 5), 18 000 (N ) 6 and 7), 15 000 (N ) 8), 12 000 (N ) 9), and 9000 (N ) 10) time steps each block; the time step τ varies with N according to the formula τN ) τ2/N1/2; 2. The calculations for the deuterium are characterized by 2000 walkers and 40 blocks of 60 000 time steps for N ) 3 and 4, with τ ) 300 and 150 hartrees-1, respectively; 30 000 (N ) 5), 18 000 (N ) 6 and 7), and 15 000 (N )

8) time steps with τ ) 50 hartrees-1; 12 000 (N ) 9) and 9000 (N ) 10) time steps with τ ) 20 hartrees-1; 3. For the hydrogen case, 2000 walkers are propagated along 40 blocks of 3000 time steps and τ ) 120 hartrees-1 for H(He)3, 60 000 time steps and τ ) 150 hartrees-1 for N ) 4, 30 000 time steps and τ ) 134 hartrees-1 (N ) 5), 18 000 time steps and τ ) 80 hartrees-1 (N ) 6), 18 000 time steps and τ ) 70 hartrees-1 (N ) 7), 15 000 time steps and τ ) 25 hartrees-1 (N ) 8), 12 000 time steps and τ ) 20 hartrees-1 (N ) 9) and, finally, 1000 time steps and τ ) 15 hartrees-1 for N ) 10. We have encountered some difficulties when dealing with the hydrogen atom as a dopant, especially for N ) 3 and 10; the convergence to the DMC ground state turns out to be very slow because of the large amplitude of ΨT (and the consequent need to explore a huge portion of configurational space), and for this reason we expect that our convergence there would provide only a semiquantitative answer for the energies, although it would not change the present conclusions. A better balance between the computational effort and the general quality of calculations has been reached for all the other systems thanks to a correct choice of τ. We have also carried out calculations for pure He clusters up to N ) 10, the corresponding energies being fundamental for estimating the binding and the exchange energies, as we shall explain below. The mass for He we have used is 4.002 60 amu; 1.007 94 amu for H, 2.014 10 amu for D, and 3.016 05 amu in the case of T. Total van der Waals energies for the doped and pure clusters are reported in Table 2. It is worth noting that our code is able to reproduce the very weakly bound state for He2; furthermore, we also find that only the trimer complex with T is bound, whereas the other two isotopes do not form any bound state with He2. This finding is in agreement with the results of refs 8 and 9, the main difference being the energy value, equal to -0.018 98(20) cm-1 (τ ) 300 hartrees-1, with the smallest error, Figure 2) in the present work and -0.004 73 cm-1 in ref 8. This can be due in principle to the H-He potentials23,13 employed in the two calculations, although the possibility of not having fully converged calculations with the adiabatic hyperspherical method should be taken into account. Recent results with the potential defined in ref 23 were obtained by using the DGF method36 and by the Faddeev equations,37 and both gave a bound state value of -0.0141 cm-1, i.e., closer to our present result. In the case of very small binding energies, it is well-known that the “exact” DMC ground state energy should be obtained by extrapolating it at τ ) 0, after a series of calculations carried out at different time step values. We report in Figure 2 the DMC energy for T(He)2 as a function of τ; we have fitted the set of data (at τ ) 10, 50, 100, 200, 300, and 400 hartrees-1) by a linear function, according to the predicted behavior of the ˆ for the algorithms used by us.38,39 The expectation value of H extrapolation procedure gives us an energy value equal to -0.019 52 cm-1, by taking into account all the points; if we neglect the first point at τ ) 10 hartrees-1, the extrapolated result is -0.019 04 cm-1. Given the very good agreement between the extrapolated results and that for τ ) 300 hartrees-1 (the first extrapolation produces a trimer that is more bound by about 3% while -0.019 04 cm-1 is within the error bar of the τ ) 300 simulation) and the numerical instability at small τ (τ ) 10 hartrees-1, with the largest error), we can say that the extrapolation procedure represents a necessary step in the DMC analysis. In this particular situation, however, in which we

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Figure 2. DMC energy for T(He)2 as a function of the time step for different values of τ: 10, 50, 100, 200, 300, and 400 hartrees-1. The solid line corresponds to a linear fitting for the extrapolation at τ ) 0.

Figure 3. Total energies (left panel) and evaporation energies (right panel) for the three different isotopes of hydrogen in interaction with helium clusters, as a function of the number of the solvents atoms. See text for details.

choose T(He)2 to be a representative example for all the other clusters of this work, our choice to carry out calculations by employing only one single time step without extrapolation, as discussed before, turns out to be an excellent compromise between computational effort and general accuracy of the conclusions, with no loss of physical information. This is certainly even more so when we shall deal with clusters with N > 2, where larger binding energies are involved. H(He)N clusters are seen to produce binding energies very close to the ones we obtain for the pure systems: it is therefore difficult to decide whether or not the stabilization occurs. This effect is mainly due to the lighter mass of H with respect to D and T since the zero-point energy (ZPE) is dominated in such very floppy systems by the weakness of the interaction and by the mass of the atoms at play. In the present case, given the same pair potential for the dopant-helium interaction, the marked change in the mass when one goes from H to T (mT ∼ 3mH and ∼ 1.5mD) causes the appearance of a bound state only for the heaviest impurity. In the left panel of Figure 3 the total energies are reported as a function of N. As expected, there is

no evidence of “magic” numbers40 or of particular structuring effects since the energies E are dominated by the helium-helium interactions. The mass effect obviously makes the T(He)N more bound with respect the other isotopes, but a similar behavior (essentially depending on N2) by increasing N characterizes the energetic features of all the present atomic aggregates. To better assess the cluster stabilization effects induced by the impurity (Imp), it is useful to evaluate a quantity like the evaporation energy, defined as the amount of energy needed to extract one He atom from the cluster under study with a given impurity

∆E ) E(Imp(He)N-1) - E(Imp(He)N)

(15)

the right panel of Figure 3 presents the evolution of ∆E for the three isotopes as a function of N. It is interesting to note that the tritium gets the largest evaporation energies within the size range examined here and that an oscillatory behavior characterizes the systems from N ) 8 to 10, with a local minimum at N

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Figure 4. Binding (left panel) and exchange energies (right panel) for the H(He)N systems as a function of N. See text for details.

) 9. However, the energy window is so small (∼2 cm-1) that no significant information on the stabilization during the growth process can be inferred by the present data. D and H have the same behavior of their ∆E values up to N ) 8; then the helium atoms seem much more bound in the presence of deuterium, whereas the evaporation energies for H(He)N resemble those of the pure clusters (see Table 2). A further quantity of interest could be had by evaluating the binding energy Ebind, defined as the gain in energy relative to the attachment of the impurity to the helium cluster

Ebind ) E(HeN) - E(Imp(He)N)

(16)

For T and D Ebind is always positive, indicating that their addition leads to a general stabilization of the overall cluster (Figure 4, left panel); the slope for tritium is more steep than the one for deuterium, a fact that can be considered as a further demonstration of the crucial role played by the mass effect in producing bound states in very floppy systems. For H the Ebind values oscillate around 0 and assume a negative value for N ) 10. In the other cases a maximum is visible at N ) 8 (Ebind ∼ 0.1 cm-1). In other words, our DMC calculations strongly suggest that a nanoscopic helium cluster is not able to bind an H atom and therefore that the (N + 1) system is energetically equivalent to the pure N cluster. To gain a deeper insight on the energetics of these van der Waals complexes, we further define another quantity: the exchange energy Eexc, given by the amount of energy absorbed by the system when one helium atom is replaced by the dopant:

Eexc ) E(HeN+1) - E(Imp(He)N)

(17)

At variance with the Ebind defined by eq 16, the exchange energy estimate involves knowledge of the energies E of systems with the same number of components; this analysis can be useful to better understand the intrinsic importance of the hydrogen (deuterium or tritium) atom in an helium environment. Eexc is negative for all three species (Figure 4, right panel); D and H show a similar slope up to N ) 8, after which the exchange energy of deuterium becomes constant. The substitution of a solvent atom with H (D or T) is therefore not an energetically favorable process. This conclusion fits well with several results

for the bulk regime,4-6 where it was shown that the isotopic variants of hydrogen cannot penetrate the helium as a liquid bulk. Our simulations show now that this also occurs in nanoscopic aggregates of He atoms. We have performed a geometrical analysis only on the Tand D-doped clusters because for H(He)N we would find the same radial distributions and static structure factors found for pure helium, with the H impurity pushed away from the clusters. We have used the DMC “mixed estimator”.17 By testing the application of a forward walking algorithm on T(He)2,41,42 we have obtained no essential difference between the T-He distance distribution obtained by the mixed estimator and that extracted by the more correct, unbiased one. In Figure 5 we report the average distances for the T(D)-He and He-He pairs as a function of the number of helium atoms. T(He)2 (Figure 5, left panel) is characterized by 〈RT-He〉 ∼ 14 Å and 〈RHe-He〉 ∼ 17 Å, as expected for a very diffuse atomic trimer; even more delocalized is D(He)3 (Figure 5, right panel) in which the average D-He distance is ∼80 Å, the helium pairs being instead at a distance equal to 10 Å. The addition, in both cases, of He atoms makes the clusters more compact and the He-He distance nearly independent of N: the solvent seems to be more bound and localized in the presence of T (with 〈RHe-He〉 ∼ 7 Å) while the interaction with D induces a more “diffuse” spatial arrangement for the solvent, where the average distance for the latter oscillates around 8.5 Å. The impurity-He distance is always larger than the corresponding He-He one, for each specific N value. Apart from the clear cases with N ) 2 and 3, one can see from the two panels of Figure 5 that the T(D)-He radial distance assumes values around 8(10) Å; this finding reflects the weaker H-He interaction and the lighter mass with respect to the helium network and it is more marked for the deuterium dopant. A kind of progressive “compacting” effect seems to start occurring for T as N increases, reminiscent of the behavior we found earlier in calculations for the H2 molecule in He clusters,19 although this suggestion needs to be confirmed by further calculations that shall have to extend to larger clusters (in ref 19 N went up to 100). From ref 6 we know that the radial distribution functions for both D and T (in the bulk regime) exhibit the main peak at 4.10 Å (Figure 1 in ref 6), the largest clusters examined here (N ) 10) show instead mean peaks at 11.30 Å for D and 7.07

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Figure 5. On the left: evolution of the average distances for the T-He and He-He pairs by increasing N. On the right: the same but for the D dopant.

Figure 6. On the left: average distances of T and He atoms from the geometric center (GC) of the system. On the right: the same but for D (the inset shows the size evolution of the cluster from N ) 4 to 10).

Å for T; the difference can be explained in terms of surface states (present in the bulk case but absent in the nanoscopic regime) that bind the impurity by a physisorption process and therefore produce more dense aggregates.2,3 A very important feature here is also given by the possibility of determining the dopant’s location with respect to the solvent; using He density functional theory calculations, Ancilotto et al.43 derived a simple model that relates the gain in energy due to helium-dopant interactions with the cost in energy necessary to carve out a cavity for the solute inside the helium droplet. This feature was expressed by a dimensionless parameter λ

λ ) 2-1/6σ-1FDeRe

(18)

where σ is the surface tension of liquid helium (0.179 cm-1 Å-2), F is the number density of liquid helium (0.022 Å-3), and Re and De are the equilibrium bond distance (in Å) and the well depth (in cm-1) of the dopant-helium pair potential, respectively. Ancilotto predicted that dopants with values of λ greater than 1.9 should reside inside the droplet, whereas others would reside on the surface; the H dopant has λ ) 1.8 and

therefore should be expected to be located outside the small droplet, although this simple analysis does not take into account the mass of the impurity. We therefore decided to carry out DMC calculations to obtain a realistic picture for the dopant and solvent radial distributions with respect to the geometric center (GC) of the cluster, thereby following the evolution of the T(D) position with increasing N. In Figure 6 we describe the location of T(D) and of the He atoms with respect the GC as a function of the growth process. Apart from the N ) 3 case for the deuterium, one clearly sees that the He atoms maintain the same geometric arrangement (∼5 Å), independent of N and of the nature of the dopant. The two impurities show instead an oscillatory behavior: the radial window goes from 5 to 7 Å for T (Figure 6, left panel) and from 7 to 12 Å for D (same figure, right panel). Both isotopes therefore reside outside the cluster, confirming by more realistic quantum calculations the empirical prediction from the Ancilotto’s parameter λ of a nonsolvating hydrogen atom in interaction with a helium aggregate of nanoscopic size. Finally, Figure 7 compares the partial static structure factors S(k) for the solvent component, calculated within the DMC framework, for He10, D(He)10, and T(He)10. No substantial

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Coccia and Gianturco Acknowledgment. The financial support of the University of Rome Research Committee, of the CASPUR Supercomputing Center and of the PRIN 2006 Research Program is gratefully acknowledged. References and Notes

Figure 7. Partial static structure factors for the helium component in He10, D(He)10, and T(He)10.

difference is seen to occur among the curves of the three systems: the impurities do not alter the spatial features of helium, which tends to behave as a pure, unperturbed system of bound atoms that acquires no new structuring features. 5. Conclusions Our QMC calculations on nanoscopic, finite-size He clusters doped with a single H, D, or T atom confirm the heliophobic character of hydrogen, already established by early variational estimates1-6 in the bulk regime and by the experimental findings about deuterium.7 By using a realistically stronger H-He potential, we have obtained for T(He)2 a van der Waals energy about 4.5 times larger in absolute value than the result reported in ref 8. Total and binding energies (as defined in eq 16) indicate that the H attachment to the solvent aggregate is not an energetically favorable process, while the exchange energies (eq 17) show negative values for D and T (other than H, obviously), reinforcing the idea of the fundamental role played by the He atoms in the stabilization of weakly bound nanoscopic clusters and of the marginal effect that the presence of a single hydrogen atom has on the spatial features of the solvating droplet. From the structural point of view, the small clusters present geometric features related to the energy landscape described above: the H atom is pushed away, i.e., does not bind. The H(He)N systems essentially correspond to pure HeN clusters where the D and T dopants get bound to them but reside outside the solvent environment, the T being closer than D and producing a more compact helium aggregate. It could be of some interest to carry out further DMC calculations for larger N values to verify the “progressive solvation“ for tritium and the possibility that both D and T dopants reach the bulk limit in terms of distance from the He surface. Such calculations could also be extended to H to understand how many helium atoms are necessary to bind the impurity to the surface of that cluster: in the future, some of the above work will be tackled in our laboratory. The present calculations, however, already tell in quantitative terms the likely behavior of hydrogen atoms in nanoscopic He aggregates and unequivocally confirm, using realistic quantum calculations, the heliophobic nature of its interaction with small clusters of bosonic helium.

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