Pressure and size effects in endohedrally confined hydrogen clusters

THE JOURNAL OF CHEMICAL PHYSICS 128, 064316 共2008兲

Pressure and size effects in endohedrally confined hydrogen clusters Jacques Soullarda兲 Departamento de Estado Sólido, Instituto de Física, Universidad Nacional Autonoma de México, A.P. 20-364, México D.F., Mexico

Ruben Santamariab兲 Departamento de Física Teórica, Instituto de Física, Universidad Nacional Autonoma de México, A.P. 20-364, México D.F., Mexico

Julius Jellinekc兲 Chemistry Division, Argonne National Laboratory, Argonne, Illinois 60439, USA

共Received 30 November 2006; accepted 30 November 2007; published online 13 February 2008兲 Density functional theory is used to carry out a systematic study of zero-temperature structural and energy properties of endohedrally confined hydrogen clusters as a function of pressure and the cluster size. At low pressures, the most stable structural forms of 共H2兲n possess rotational symmetry that changes from C4 through C5 to C6 as the cluster grows in size from n = 8 through n = 12 to n = 15. The equilibrium configurational energy of the clusters increases with an increase of the pressure. The rate of this increase, however, as gauged on the per atom basis is different for different clusters sizes. As a consequence, the size dependencies of the configurational energies per atom at different fixed values of pressure are nonmonotonic functions. At high pressures, the molecular 共H2兲n clusters gradually become atomic or dominantly atomic. The pressure-induced changes in the HOMO-LUMO gap of the clusters indicate a finite-size analog of the pressure-driven metallization of the bulk hydrogen. The ionization potentials of the clusters decrease with the increase of pressure on them. © 2008 American Institute of Physics. 关DOI: 10.1063/1.2827487兴 I. INTRODUCTION

The quest for the metallic state of hydrogen induced by high pressure1 and its far-reaching implications in the fields of astrophysics and materials science have motivated experimental and theoretical works on novel techniques and tools for investigation of hydrogen under pressure. On the experimental side, shockwave,2 pulsed-laser,3 and diamond anvil cell techniques4 played an important role. Using the diamond anvil cell methodology metallization is predicted to occur at 450 GPa. Shockwave measurements revealed the metallic state in high-temperature fluid phase of hydrogen. Density functional theory 共DFT兲 computations with nonlocal corrections gave an estimate of the metallization pressure, dependent on the crystal structure, at about 400 GPa or less.5,6 A new state has been proposed using the GuinzburgLandau formalism for hydrogen at low temperatures and high pressures in a strong magnetic field.7 This new state resembles that of a quantum liquid. It has been shown to exhibit a maximum in the melting curve of the TP phase diagram.8 Such a maximum was obtained with firstprinciples calculations, but it is yet to be demonstrated experimentally. The physics behind this maximum is the different degree of softening of the repulsive proton interactions in the solid and the liquid phases, which is attributed to the attractive many-body effects. a兲

Electronic mail: [email protected]. Electronic mail: [email protected]. c兲 Present address: Chemical Sciences and Engineering Division, Argonne National Laboratory, Argonne, Illinois 60439, USA. Electronic mail: [email protected]. b兲

0021-9606/2008/128共6兲/064316/7/$23.00

Structural forms of bulk solid hydrogen have been explored computationally using the DFT framework.5,9,10 The most recent study of Pickard and Needs10 represents a particularly extensive search for the possible conformations of solid hydrogen under different pressures. These authors identified new energetically favorable candidate lattice arrangements. They also illustrated the importance of the inclusion of the zero-point energy 共enthalpy兲 in evaluation of the relative stability of the different structural forms. The effect of the quantum nature of hydrogen on the low-temperature conformations and dynamics of solid hydrogen at different pressures has been discussed by Natoli et al.,11 Biermann et al.,12 and Kitamura et al.13 Computations have also been used to explore the structures and energetics of free hydrogen clusters14–18 as well as the effects of pressure in these systems.19,20 In the latter case, the clusters were endohedrally confined in fullerenelike cages built of hydrogen atoms, and the pressure was mimicked by reducing the radius of the cages. The computations were performed within the Kohn-Sham 共KS兲 formalism of DFT with a nonlocal exchange-correlation potential. The properties characterized included geometric and electronic structure of the clusters as a function of the pressure exerted on them. Analysis of the pressure-induced changes in the equilibrium energies led to a zero-temperature equation of state for clusters with n = 13 and n = 15 molecules, which was found to be consistent with the data measured on a macroscopic hydrogen crystal.21,22 Hydrogen clusters have been studied in other contexts as well. Endohedral hydrogen clusters encapsulated in carbon

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Soullard, Santamaria, and Jellinek

fullerenes have been examined computationally within a study that targeted the hydrogen storage problem.23 The computations suggested formation of encaged hydrogen clusters with octahedral and icosahedral symmetries. Another type of hydrogen clusters, an H+3 ionic core encapsulated in an 共H2兲n environment, plays an important role in the interstellar space.24 Laboratory experiments on beams of these clusters crossing beams of He or photons probed the cluster fragmentation channels and provided information on the liquid-to-gas phase transition in these finite-size systems.25,26 Deuterium clusters placed in intense laser fields have been shown to undergo multiple ionization and consequent Coulomb explosion.27–31 The kinetic energy of the flying apart nuclei is high enough to cause their fusion when they collide. The fusion yield is a function of the initial cluster size. Finally, we mention that spectroscopic studies of molecules embedded in clusters of parahydrogen molecules indicated that at low temperatures 共⬃0.15 K兲 these clusters exhibit superfluid characteristics.32,33 The superfluidity has been also investigated for different sizes of hydrogen clusters at low temperatures14–17 共cf. also superfluidity in bosonic He clusters34–40兲. In our earlier work we examined the effect of pressure on a H2 molecule and, in a limited way, 共H2兲15 cluster19 as well as 共H2兲13.20 The goal of this paper is to extend our earlier analyses41 to a broader range of cluster sizes and pressures. The emphasis is on the size evolution of the properties and the pressure effects. The methodology is outlined in the next section. The results are presented and discussed in Sec. III. A brief summary is given in Sec. IV.

minimization of the energy is performed with a maximum Cartesian step of 0.0015 bohr until the forces become less than 0.0015 hartree/ bohr. The distances of all the atoms of the cage-cluster system from the center of the cage are then reduced by a fixed factor, and new minimizations of the total energy over the positions of the encapsulated atoms are performed. This procedure is repeated on a grid of values of the cage radius that is decreased until the encaged hydrogens begin to form bonds with the cage. For each fixed value of the cage radius RH60, the energy ¯E of the “dressed” 共i.e., interacting with the cage兲 optiHm mized encapsulated m-atom hydrogen cluster is computed as ¯E = E Hm Hm@H60 − EH60 ,

where EHm@H60 is the minimized energy of the cage-cluster system and EH60 is the energy of the frozen cage. The radii RH60 of the cage and RHm of the cluster are obtained using the following formula: k

R Hk =

1 兺 Ri , k i

The computational methodology used in this study is similar to that utilized in Refs. 19 and 20. Hydrogen 共H2兲n clusters of sizes n = 8 – 13 and 15 are encapsulated in a 60atom hydrogen fullerene cage. The systems are treated within the gradient-corrected DFT with the Becke exchange42 and Lee-Yang-Parr correlation43 functionals. The hydrogen atoms are described by a double zeta valence plus polarization Gaussian basis set as implemented in the 44 NWCHEM package. This basis set has been found to yield only small superposition errors.45 The self-consistent cycles are iterated until a convergence level of 10−5 a.u. in electron density and energy is achieved. The confining 60-atom fullerene cage is treated as rigid, and its radius is used as a parameter to simulate different pressure conditions experienced by the enclosed H2 molecules. The initial radius of the cage is chosen to be large enough so as to effectively exert only small pressure. The molecules, initially with the interatomic separations close to that of an equilibrated gas phase H2, are placed within the cage randomly at distances sufficiently large from it to avoid cage-molecule bond formation. The total energy of the cagecluster system is then minimized over the positions of the encapsulated atoms using the quasi-Newton method.46 The

共2兲

where Ri is the distance of the ith hydrogen atom from the center of the cage. The volume ¯VHm of the dressed cluster is computed as ¯V = 4 ␲¯R3 , Hm H 3 m

共3兲

where ¯RHm is the dressed radius, ¯R = R + 共R − R 兲/2. Hm Hm H60 Hm

II. METHODOLOGY

共1兲

共4兲

The pressure ¯PHm exerted on the encapsulated cluster is defined as the rate of change of ¯EHm with ¯VHm, ¯P = − ⳵¯E /⳵¯V , Hm Hm Hm

共5兲

caused by the change of the cage radius.

III. RESULTS AND DISCUSSION A. Geometrical structures

At every pressure, the encapsulated hydrogen assembles into clusters. The equilibrium structures of the clusters depend on their size and the pressure exerted on them. Two different pressure ranges can be identified. The first, which represents pressures up to 100 GPa, leads to the formation of 共H2兲n molecular clusters. The second, which corresponds to pressures larger than 100 GPa, results in gradual dissociation of the H2 molecules and formation of atomic or mixed clusters. The preferred packing of 共H2兲n, n = 8 – 13 and 15, at pressures below 100 GPa is shown in Fig. 1. The displayed structures are configurations that are optimized for the shown relative orientations of the molecules; other orientations produce structures with the same packing and comparable energies. Overall, these structures can be described as antiprisms,


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Endohedrally confined hydrogen clusters

FIG. 1. 共Color online兲 Preferred structures of pressurized 共H2兲n clusters 共the confining H60 cage is not shown兲. The main molecular planes and the axes of highest rotational symmetry are shown. There is a perspective effect where the central axis appears slightly tilted. Darker shading is used to indicate molecules that are closer to the viewer.

or capped antiprisms, with an exact or approximate axis of rotational symmetry that changes from C4 共for n = 8 – 11兲 to C5 共for n = 12– 13兲 and C6 共for n = 15兲. The growth pattern in the size range n = 8 – 11 is based on the square antiprism structure of 共H2兲8, to which first two capping H2 molecules get added sequentially over opposite faces of the antiprism along its C4 axis, and then an extra H2 molecule is placed in its center. 共H2兲12 forms a pentagonal antiprism based conformation with a capping molecule along its C5 axis and a central molecule. The preferred structure of 共H2兲13 is obtained from that of 共H2兲12 by adding the second capping molecule along the C5 axis on the opposite side of the cluster; the overall packing of the resulting structure is that of an icosahedron. 关This structure was also obtained for the free 共H2兲13 Ref. 18兴. The configuration of 共H2兲15 is similar to that of 共H2兲13 with the fivefold molecular rings replaced by sixfold molecular rings. Further details on the preferred structure of 共H2兲15 can be found in Ref. 19. At higher pressures 关the typical range for the cluster sizes considered here is 100– 200 GPa; the corresponding value for the H2 crystal is 350 GPa 共Ref. 47兲兴, the interatomic separation in the H2 molecules begins to noticeably increase and the molecules begin to dissociate 共pressureinduced dissociation of hydrogen was first discussed in 1935 by Wigner and Huntington1兲. For a given cluster size, the fraction of the dissociated molecules depends on the pressure. Of course, the atomization process leads to rearrangements in the preferred structure. Figure 2 depicts the preferred conformations of the 24atom hydrogen cluster at different pressures. It also shows the corresponding distributions of the interatomic distances. At 97 GPa the cluster is comprised of 12H2 molecules and its structure is essentially the same as that shown in Fig. 1 for a lower pressure. The histogram of the interatomic distances indicates a trimodal distribution. The first branch of the distribution corresponds to the interatomic distances in the 12 molecules. The almost continuum of the second and the third branches, which is separated by a large gap from the first one, represents the larger distances between pairs of atoms that belong to different molecules. At 198 GPa the distribution of the interatomic distances

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is still trimodal, but four molecules increased a bit their bond lengths, whereas one 共the central兲 decreased it somewhat. As a consequence and because of the overall reduction of the volume of the cluster, the gap between the first and the continuum of the second and the third branches shrunk. Further increase of the pressure leads to further increase of the intramolecular bond lengths, which takes place in a larger number of molecules, and a gradual filling of the gap between the first two branches of the distribution. At 503 GPa only the central molecule of the cluster has a bond length that does not exceed the equilibrium interatomic distance of a free H2 共about 0.75 Å兲 and its bond length gets, in fact, reduced somewhat. Using the distance of 0.75 Å as the maximum allowed interatomic separation for identification of a pair of H atoms as a molecule, we arrive at values of 0.33, 0.75, and 0.92 for the fraction of the dissociated molecules at pressures of 198, 301, and 503 GPa, respectively. As the pressure on the cluster increases, its volume decreases. The reason for the pressure-induced atomization of the clusters can be understood using the following energy-based considerations. Under pressure-free or low-pressure conditions, the clusters are comprised of H2 molecules, which are separated by relatively large distances. The binding energy of the clusters can be approximated as the sum of the binding energies of their molecules 共strong interactions兲 and the energies of intermolecular bonding 共weak interactions兲. As the pressure on the clusters is increased, their molecules move closer to each other and the weak intermolecular interactions increasingly become stronger interactions between pairs of H atoms that belong to different molecules. An atom of a molecule can form such pairs with atoms of several surrounding molecules. Further increase of pressure leads to an increase of the total number of the newly formed pairs as well as of the strengths of their interactions as the intrapair distances become shorter. This process of atomic pair formation with intrapair distances larger than that in the equilibrium H2 is accompanied by gradual increase in the bond lengths of the original H2 molecules and their eventual atomization. Although the binding energy of the individual newly formed atomic pairs is, in general, weaker than that of bona fide H2 molecules, at some pressure the total binding energy of the completely or partially atomized cluster wins over the binding energy of the corresponding purely molecular clusters at that pressure. The effect of the cluster size on the pressure-induced atomization is illustrated in Fig. 3. The figure presents the preferred structures and distributions of the radial distances of the atoms from the center of the confining cage for 16-, 20-, and 24-atom hydrogen clusters, all at a pressure of about 200 GPa. One notices that all three clusters are partially atomized. The fraction of the atomized molecules is, respectively, 0.25, 0.8 and 0.33. The most peripheral molecules are atomized in all three clusters. Dependent on the cluster size and structure, the atoms that do not form molecules and those that do may share the same or close radial distances. In the case of the 20-atom cluster, each of the two bars that correspond to atoms that form molecules represents a pair of H atoms, which actually belong to different H2 molecules.


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FIG. 2. Preferred structures and histograms of the distributions of the interatomic distances in a 24-atom hydrogen cluster at different pressures 共the atoms of the confining cage are not shown兲. The histograms are obtained with a box size of 0.05 Å. Two atoms are depicted as a molecule when the distance between them does not exceed 0.75 Å 共this somewhat arbitrarily chosen value is close to the equilibrium bond length of free H2兲. Darker shading is used to indicate atoms and molecules that are closer to the viewer.

The nonmonotonic variation of the degree of atomization with the cluster size, as evaluated at 共nearly兲 the same pressure, is a consequence of an interplay between a number of factors. These factors include, in addition to the size, the initial 共i.e., pressure-free or low-pressure兲 structural form and also the more subtle effects of the size and the structure originating from the fact that the value of the pressure, as defined by Eq. 共5兲, depends on them. Figure 3 suggests that the atomization efficiency of the pressure for clusters with initially similar structures 关e.g., 共H2兲8 and 共H2兲10, which do not have a central molecule, cf. Fig. 1兴 increases with the cluster size. But a change in the type of the preferred packing changes this trend. This is illustrated by the case of the larger 24-atom cluster, which when in a fully molecular form has a central molecule 共cf. Fig. 1兲 and which is atomized at ⬃200 GPa to a lesser degree than the 20-atom cluster. The presence of the central molecule reduces the atomization efficiency of the pressure. In fact, we did not observe the atomization of the central molecule in any cluster we have studied even at the highest pressure 共about 500 GPa兲 considered.

B. Energetics and electronic properties

Figure 4 displays the equilibrium cluster energies per atom as a function of pressure 共upper panel兲 and cluster size 共lower panel兲. Naturally, the cluster energies increase with the pressure. Overall, the effect of the pressure, as gauged per atom, is only weakly size dependent for the range of sizes considered. One notices, though, that the rate of the pressure-induced energy change is nonuniform. This nonuniformity is primarily a consequence of the size-dependent structural details. For instance, the 22-atom cluster is the last in the sequence of those whose original preferred configuration is based on two essentially parallel rings with four H2 molecules in each ring 共cf. Fig. 1兲. In addition, it is the first to incorporate a central molecule. It is natural then that its sensitivity to pressure is higher. A complementary picture of the size dependence of the cluster equilibrium energies per atom at different fixed values of pressure, displayed in the lower panel of Fig. 4, shows that this dependence is not monotonic. The nonmonotonicity is a consequence of structural changes. At each fixed pres-


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TABLE I. The HOMO-LUMO gap and the total and exchange-correlation 共XC兲 energy of the equilibrium configuration of the pressurized 24- and 26-atom hydrogen clusters considered with and without the confining 60atom hydrogen cage, as well as of the cage alone, all considered as a function of the cage radius Rc. The corresponding pressure P on the dressed cluster is listed in brackets. Extra data for the cage at Rc = 2.6 Å are added for completeness. See the text for details.

FIG. 3. Preferred structures and radial distance distributions of the atoms from the center of the confining cage 共not shown兲 for 16-, 20-, and 24-atom hydrogen clusters at pressures close to 200 GPa. In the histograms, lighter shading represents atoms that do not form molecules and darker shading atoms that form molecules 共see the text for details兲. The histograms are obtained with a box size of 0.05 Å.

sure, the changes in the equilibrium structures of the clusters arise as a consequence of the change in the number of the hydrogen molecules or atoms in them. The fraction of the atomized molecules increases with the increase of the pressure. Larger fraction of atomized molecules results in a higher structural flexibility and, consequently, more extensive changes in the equilibrium structures. That is the reason why the degree of nonmonotonicity in the size dependence of the equilibrium energies, as analyzed at fixed pressures,

FIG. 4. Equilibrium energies of the clusters per atom as a function of pressure 共upper panel兲 and the cluster size 共lower panel兲.

Cluster size

Rc 关P兴共Å兲关GPa兴

HOMO-LUMO gap 共eV兲

24

3.300 关97兴 2.970关198兴 2.780关301兴 2.538关503兴

Cluster-cage system 1.045 −46.0408 0.740 −45.6066 0.337 −45.0349 0.267 −43.7358

26

3.400 关89兴 3.030关203兴 2.800关341兴

24

3.300 2.970 2.780 2.538

26

3.400 3.030 2.800

7.712 5.676 3.766

−14.7031 −14.3816 −14.0604

−9.6300 −9.8693 −10.0417

3.400 3.300 3.030 2.970 2.800 2.780 2.600 2.538

Cage 1.668 1.750 1.989 2.049 2.231 2.256 2.479 2.566

−32.9409 −33.0353 −33.1769 −33.1773 −33.0918 −33.0722 −32.7699 −32.6049

−18.4899 −18.7887 −19.7219 −19.9576 −20.6914 −20.7832 −21.7002 −22.0515

1.189 1.031 0.882

Total energy 共a.u.兲

XC energy 共a.u.兲

−28.1790 −29.7231 −30.8175 −32.5185

−47.1311 −46.7120 −46.0020

−28.6111 −30.3089 −31.6625

Compressed cluster without cage 7.372 −13.5254 5.772 −13.2347 1.173 −12.8906 0.914 −12.3955

−8.9005 −9.0751 −9.1217 −9.1249

increases with the value of the pressure. At 100 GPa all clusters retain their fully molecular nature, and the degree of nonmonotonicity of the corresponding graph is only moderate. As the value of the pressure increases, the fraction of the atomized molecules increases as well and so does the degree of nonmonotonicity of the corresponding graph. Data that illustrate the electronic and energy effects of the decrease in the radius of the encapsulating cage, which are used to mimic the increase of the pressure on the clusters, are given in Table I. They are listed for three types of systems. The first is an equilibrated pressurized cluster together with its cage; the second is the same cluster, but without the cage; and the third is the cage without the cluster. The clusters considered are those with 24 and 26 atoms. For each fixed value of the cage radius, the corresponding pressure on the dressed cluster as defined by Eq. 共5兲 is also indicated. First, one notices that for the cluster-cage systems and the clusters without the cage, the HOMO-LUMO 共HOMO denotes highest occupied molecular orbital; LUMO denotes lowest unoccupied molecular orbital兲 gap decreases with the decrease of the cage radius. The HOMO-LUMO gap of the cage without the cluster exhibits an opposite trend. Although the cage dominates in terms of the number of atoms, it is the


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much faster rate of change in the HOMO-LUMO gap of the cluster with the cage radius that underlies the resulting trend in the radius dependence of the HOMO-LUMO gap of the combined cluster-cage system. The substantial reduction in the HOMO-LUMO gap of the clusters as the pressure on them is increased can be viewed as the finite-size analog 共cf. Refs. 48–50 and references therein兲 of the pressure-induced metallization of bulk hydrogen discussed in earlier studies.1,2,4–6 A systematic analysis of the size and pressure dependence of the HOMO-LUMO gap in hydrogen clusters will be given in our future publications. The table also presents the radius dependence of the total energies and their exchange-correlation parts. Except for the case of the cage without the cluster, the total energy increases as the cage radius decreases. For the cage, it first decreases as the cage approaches its equilibrium configuration with the radius just below 3 Å and then increases. The exchange-correlation part of the total energy becomes more negative for all cases as the radius decreases. For the clusters under pressure this means that the role of the exchangecorrelation interaction is to counterbalance the pressureinduced increase in the repulsion between the constituents of the clusters. It is worth to mention the pressure effects on the ionization potentials 共IPs兲. As verified in terms of the adiabatic IPs for the case of m = 24 共7.84, 6.54, and 5.84 eV at pressures of 97, 301, and 503 GPa, respectively兲 and vertical IPs for the case of m = 26 共8.36, 7.18, and 6.71 eV at pressures of 20, 320, and 520 GPa, respectively兲, the ionization potentials of the clusters decrease as the pressure on them increases. Overall, this trend is consistent with the pressure-induced increase in the total energies of the clusters. Our final remark is on zero-point energy. This energy should, in general, be included when one is interested in evaluating the relative stability of different structural forms.10 We did not include this energy in our analysis as we are concerned only with what are initially the most stable low-pressure conformations of the clusters and with the changes in these conformations as a function of pressure. IV. SUMMARY

In this paper we examined the size dependence of the pressure-induced changes in the zero-temperature structural features, energetics, and electronic properties of Hm hydrogen clusters with up to 30 atoms. We have shown that under low pressures, the clusters are built of hydrogen molecules and form structures whose symmetry changes from C4 through C5 to C6 as the cluster size changes from m = 16 through m = 24 to m = 30. As the pressure on the clusters increases, their constituent H2 molecules begin to undergo a gradual atomization process. The degree of this atomization 共extent of elongation of the intramolecular bonds and the number of atomizing molecules兲 increases with the increase of the pressure. The equilibrium energy of the clusters, as gauged per atom, is also an increasing function of pressure. The rate of change of the energy with pressure is, in general, size dependent; clusters of different sizes resist pressureinduced changes in their structure to a different degree. The

energy of the equilibrium configurations considered as a function of the cluster size at different fixed values of the pressure exhibits a nonmonotonic trend. The reason for this nonmonotonicity is the structural changes as a function of the cluster size and pressure. At higher pressures the degree of nonmonotonicity is higher, which is caused by an increase in the fraction of the atomized molecules. The pressureinduced changes in the electronic properties are reflected in the increase of the role of the exchange-correlation energy in the cluster cohesion and a substantial reduction in the value of the HOMO-LUMO gap. The latter is indicative of a finitesize analog of the pressure-induced metallization of the bulk hydrogen. ACKNOWLEDGMENTS

The authors thank the IFUNAM and DGSCA computer staff for access to the ultra-f and Kan-Balam clusters and Dr. A. Cordero-Borboa for valuable discussions. J.J. was supported by the Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences, and Biosciences, U.S. Department of Energy under Contract No. DE-AC-0206CH11357. E. P. Wigner and H. B. Huntington, J. Chem. Phys. 3, 764 共1935兲. S. J. Weir, A. C. Mitchell, and W. J. Nellis, Phys. Rev. Lett. 76, 1860 共1996兲. 3 G. W. Collins, L. B. Da Silva, P. Celliers, D. M. Gold, M. E. Foord, R. J. Wallace, A. Ng, S. V. Weber, K. S. Budil, and R. Cauble, Science 281, 1178 共1998兲. 4 P. Loubeyre, F. Ocelli, and R. LeToullec, Nature 共London兲 416, 613 共2002兲. 5 K. A. Johnson and N. W. Ashcroft, Nature 共London兲 403, 632 共2000兲. 6 M. Städele and R. M. Martin, Phys. Rev. Lett. 84, 6070 共2000兲. 7 E. Babaev, A. Sudbo, and N. W. Ashcroft, Nature 共London兲 431, 666 共2004兲. 8 S. A. Bonev, E. Schwegler, T. Ogitsu, and G. Galli, Nature 共London兲 431, 669 共2004兲. 9 J. Kohanoff, S. Scandolo, G. L. Chiarotti, and E. Tosatti, Phys. Rev. Lett. 78, 2783 共1997兲. 10 C. J. Pickard and R. J. Needs, Nat. Phys. 3, 473 共2007兲. 11 V. Natoli, R. M. Martin, and D. Ceperley, Phys. Rev. Lett. 74, 1601 共1995兲. 12 S. Biermann, D. Hohl, and D. Marx, J. Low Temp. Phys. 110, 97 共1998兲; Solid State Commun. 108, 337 共1998兲. 13 H. Kitamura, S. Tsuneyuki, T. Ogitsu, and T. Miyake, Nature 共London兲 404, 262 共2000兲; S. Tsuneyuki, H. Kitamura, T. Ogitsu, and T. Miyake, J. Low Temp. Phys. 122, 291 共2001兲. 14 Ph. Sindzingre, D. M. Ceperley, and M. L. Klein, Phys. Rev. Lett. 67, 1871 共1991兲. 15 D. Scharf, M. L. Klein, and G. J. Martyna, J. Chem. Phys. 97, 3590 共1992兲. 16 M. C. Gordillo and D. M. Ceperley, Phys. Rev. B 65, 174527 共2002兲. 17 S. A. Khairallah, M. B. Sevryuk, D. M. Ceperley, and J. P. Toennies, Phys. Rev. Lett. 98, 183401 共2007兲. 18 J. I. Martínez, M. Isla, and J. A. Alonso, Eur. Phys. J. D 43, 61 共2007兲. 19 J. Soullard, R. Santamaria, and S. A. Cruz, Chem. Phys. Lett. 391, 187 共2004兲. 20 R. Santamaria and J. Soullard, Chem. Phys. Lett. 414, 483 共2005兲. 21 R. J. Hemley, H. K. Mao, L. W. Finger, A. P. Jephcoat, R. M. Hazen, and C. S. Zha, Phys. Rev. B 42, 6458 共1990兲. 22 P. Loubeyre, R. LeToullec, D. Hausermann, M. Hanfland, R. J. Hemley, H. K. Mao, and L. W. Finger, Nature 共London兲 383, 702 共1996兲. 23 R. E. Barrajas-Barraza and R. A. Guirardo-López, Phys. Rev. B 66, 155426 共2002兲. 24 M. Farizon, H. Chermette, and B. Farizon-Mazuy, J. Chem. Phys. 96, 1325 共1991兲. 25 F. Gobet, B. Farizon, M. Farizon, M. J. Gaillard, J. B. Buchet, M. Carré, and T. D. Mark, Phys. Rev. Lett. 87, 203401 共2001兲. 1 2


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